SECOND HANKEL DETERMINANT FOR A CERTAIN SUBCLASS OF ANALYTIC AND BI-UNIVALENT FUNCTIONS
|
|
- Virgil Rose
- 5 years ago
- Views:
Transcription
1 TWMS J Pure Appl Math V7, N, 06, pp85-99 SECOND HANKEL DETERMINANT FOR A CERTAIN SUBCLASS OF ANALYTIC AND BI-UNIVALENT FUNCTIONS BA FRASIN, K VIJAYA, M KASTHURI Abstract In the present paper, we consider a subclass of the function class Σ of bi-univalent analytic functions in the open unit disk we obtain the functional a a 4 a 3 for the function class Also we give upper bounds for a a 4 a 3 Our result gives corresponding a a 4 a 3 for the subclasses of Σ defined in the literature Keywords: Univalent functions, analytic functions, Bi-univalent functions, coefficient bounds, Hankel determinant AMS Subject Classification: 30C45 Introduction Let A denotes the class of functions of the form f(z = z + a n z n ( which are analytic in the open unit disc = {z : z < } normalized by the conditions f(0 = 0 f (0 = Further, let S denote the class of all functions in A which are univalent in Some of the important well-investigated subclasses of the univalent function class S include (for example the class S (α of starlike functions of order α (0 α < in the class K(α of convex functions of order α (0 α < in It is well known that every function f S has an inverse f, defined by where n= f (f(z = z (z f(f (w = w ( w < r 0 (f; r 0 (f /4 f (w = g(w = w a w + (a a 3 w 3 (5a 3 5a a 3 + a 4 w 4 + ( A function f A is said to be bi-univalent in if both f f are univalent in Let Σ denote the class of bi-univalent functions in given by ( Earlier, Brannan Taha [4] introduced certain subclasses of bi-univalent function class Σ, namely bi-starlike functions of order α denoted by SΣ (α bi-convex function of order α denoted by K Σ(α corresponding to the function classes S (α K(α respectively A function f A is in the class of strongly bi-starlike( strongly bi-convex functions SΣ [α]( K Σ[α] [4, 9] of order α (0 < α if each of the following conditions is satisfied: ( zf arg (z f(z < απ ( wg arg (w g(w Department of Mathematics, Al al-bayt University, Mafraq, Jordan School of Advanced Sciences,VIT University,Vellore, Tamilnadu, India bafrasin@yahoocom, kvijaya@vitacin, kasthurim@vitacin Manuscript received February < απ
2 86 TWMS J PURE APPL MATH, V7, N, 06 ( arg + zf (z f < απ (z ( arg + wg (w g < απ (w where g is the extension of f to For each of the function classes S Σ [α]( K Σ[α] nonsharp estimates on the first two Taylor-Maclaurin coefficients a a 3 were found [4, 9] But the coefficient problem for each of the following Taylor-Maclaurin coefficients: a n (n N \ {, }; N := {,, 3, } is still an open problem (see [3, 4, 3, 6, 9] An analytic function f is subordinate to an analytic function g, written f(z g(z, provided there is an analytic function w defined on with w(0 = 0 w(z < satisfying f(z = g(w(z Ma Minda [4] unified various subclasses of starlike convex functions for which either of the quantity z f (z f(z or + z f (z f (z is subordinate to a more general superordinate function For this purpose, they considered an analytic function ϕ with positive real part in the unit disk, ϕ(0 =, ϕ (0 > 0 ϕ maps onto a region starlike with respect to symmetric with respect to the real axis The class of Ma-Minda starlike functions consists of functions f A satisfying the subordination z f (z f(z ϕ(z Similarly, the class of Ma-Minda convex functions consists of functions f A satisfying the subordination + z f (z f (z ϕ(z A function f is bi-starlike of Ma-Minda type or bi-convex of Ma-Minda type if both f f are respectively Ma-Minda starlike or convex These classes are denoted respectively by SΣ (ϕ K Σ (ϕ (see [] In the sequel, it is assumed that ϕ is an analytic function with positive real part in the unit disk, satisfying ϕ(0 =, ϕ (0 > 0 ϕ( is symmetric with respect to the real axis Such a function has a series expansion of the form ϕ(z = + B z + B z + B 3 z 3 +, (B > 0 (3 We need the following lemma for our investigation Lemma (see [7], p 4 Let P be the class of all analytic functions p(z of the form p(z = + p n z n (4 satisfying R(p(z > 0 (z p(0 = Then n= p n (n =,, 3, This inequality is sharp for each n In particular, equality holds for all n for the function p(z = + z z = + z n In 976, Noonan Thomas [7] defined the qth Hankel determinant of f for q by H q (n = n= a n a n+ a n+q a n+ a n+ a n+q a n+q a n+q a n+q Further, Fekete Szegö [8] considered the Hankel determinant of f A for q = n =, H ( = a a a a 3 They made an early study for the estimates of a 3 µa when
3 BA FRASIN et al: SECOND HANKEL DETERMINANT FOR 87 a = with µ real The well known result due to them states that if f A, then 4µ 3 if µ, a 3 µa + exp( µ µ if 0 µ, 3 4µ if µ 0 Furthermore, Hummel [0, ] obtained sharp estimates for a 3 µa when f is convex functions also Keogh Merkes [] obtained sharp estimates for a 3 µa when f is close-to-convex, starlike convex in Here we consider the Hankel determinant of f A for q = n =, H ( = a a 3 a 3 a 4 To prove our main result, we need the following lemmas Lemma If the function p(z P is given by the series (4, then p = p + x(4 p, (5 4p 3 = p 3 + (4 p p x p (4 p x + (4 p ( x z, (6 for some x, z with x, z p [0, ] Lemma 3 [9] The power series for p(z given in (4 converges in to a function in P if only if the Toeplitz determinants p p p n p p p n D n =, n =,, 3, (7 p n p n+ p n+ p k = p k, are all nonnegative They are strictly positive except for m p(z = ρ k p 0 (e itkz, ρ k > 0, t k real k= t k t j for k j in this case D n > 0 for n < m D n = 0 for n m Many researchers (see [,, 5, 8, 0, ] have introduced investigated several interesting subclasses ( by generalization or in associated with certain linear operators see [5] of the bi-univalent function class Σ they have found non-sharp estimates on the first two Taylor- Maclaurin coefficients a a 3 The object of the present paper is to determine the functional a a 4 a 3 for a subclass of the function class Σ Also we give upper bounds for a a 4 a 3 Our result gives corresponding a a 4 a 3 for the subclasses of Σ defined in the literature coefficient bounds for the function class F Σ (ϕ, α For α 0 we let a function f Σ given by ( is said to be in the class F Σ (ϕ, α, if the following conditions are satisfied: ( zf ( (z ( α + α + zf (z f(z f ϕ(z (8 (z ( wg ( (w ( α + α + wg (w g(w g ϕ(w, (9 (w where g is the inverse of f given by (
4 88 TWMS J PURE APPL MATH, V7, N, 06 Definition Taking α = 0, we let F Σ (ϕ, α SΣ (ϕ if f S Σ (ϕ, then zf (z f(z wg (w g(w where the function g is the inverse of f given by ( ϕ(z (0 ϕ(w, ( Definition Taking α =, we let F Σ (ϕ, α K Σ (ϕ if f K Σ (ϕ, then + zf (z f (z + wg (w g (w where the function g is the inverse of f given by ( ϕ(z ( ϕ(w, (3 Theorem Let f given by ( be in the class F Σ (ϕ, α Then H(, f(α, B, B, B 3 0, C(α, B, B, B 3 0 a a 4 a 3 { } B max, H(, f(α, B 4(+α, B, B 3 > 0, C(α, B, B, B 3 < 0 B 4(+α, f(α, B, B, B 3 0, C(α, B, B, B 3 0 (4 max {H(t 0, H(}, f(α, B, B, B 3 < 0, C(α, B, B, B 3 > 0, where H( = B B B + B 3 ( + α ( + α + B 4 ( + α + B B B ( + α 3 ( + α 3( + α 3 ( + α ( + 3α H ( C(α,B t 0 =,B,B 3 f(α,b,b,b 3 = 4( + α [C(α, B, B, B 3 ] 48( + α 3 ( + α ( + 3αf(α, B, B, B 3, f(α, B, B, B 3 = 4B B B + B 3 ( + α ( + α + 4B 4 ( + α 3B 3 ( + α( + α( + 3α B ( + α ( + α +3B ( + α 3 ( + 3α, C(α, B, B, B 3 = 3B 3 ( + α( + α( + 3α + 8B B B ( + α 3 ( + α +B ( + α ( + α 6B ( + α 3 ( + 3α ] Proof Let f F Σ (ϕ, α g = f Then there are analytic functions u, v :, with u(0 = v(0 = 0, satisfying ( α ( α ( zf ( (z + α + zf (z f(z f = ϕ(u(z (5 (z ( wg ( (w + α + wg (w g(w g = ϕ(v(w (6 (w
5 BA FRASIN et al: SECOND HANKEL DETERMINANT FOR 89 Define the functions p(z q(z by It follows that, u(z := p(z p(z + = p(z := + u(z u(z = + p z + p z + p 3 z 3 + q(z := + v(z v(z = + q z + q z + q 3 z 3 + [ ( ( ] p z + p p z + p 3 p p + p3 z v(z := q(z q(z + = [ ( ( ] q z + q q z + q 3 q q + q3 z 3 + (8 4 Then p(z q(z are analytic in with p(0 = = q(0 Using (7 (8, it is clear that, ϕ(u(z = + B [ ( p z + p p + ] 4 B p z (9 [ ( + p 3 p p + p3 + B ( p p p + B 3p 3 ] z ϕ(v(w = + B q + [ w + [ ( q q ( q 3 q q + q3 4 Equating the coefficients in (5 (6, we get, (7 + ] 4 B q w (0 + B q ( q q + B 3q 3 ] w ( + αa = B p, ( ( + αa 3 ( + 3αa = B ( p p + 4 B p, ( = B 3( + 3αa 4 3( + 5αa a 3 + ( + 7αa 3 ( p 3 p p + p3 + B ( p p p + B 3p (3 ( + αa = B q, (4 ( + αa 3 + (3 + 5αa = B ( q q + 4 B q, (5 = B From ( (4 gives 3( + 3αa 4 + 6( + 5αa a 3 (5 + αa 3 ( q 3 q q + q3 + B ( q q q + B 3q (6 a = B p ( + α = B q ( + α, (7
6 90 TWMS J PURE APPL MATH, V7, N, 06 which implies Now from (, (5 by using (7, we obtain p = q (8 a 3 = B p 4( + α + B (p q 8( + α (9 On the otherh, subtracting (6 from (3 by using (7, (9, we get a 4 = 5B3 p3 6( + α 3 + 5B p (p q 3( + α( + α + B (p 3 q 3 ( + 3α Thus we establish that + p3 (B B + B 3 4( + 3α + (B B p (p + q ( + 3α a a 4 a 3 = B (B B + B 3 ( + α 3 ( + α 4 96( + α 4 p 4 B (p q ( + 3α 64( + α According to Lemma, we have hence by (8, we have further p3 B3 ( + 9α 48( + α 3 (30 + B3 p (p q 64( + α ( + α + B (B B p (p + q + B p (p 3 q 3 4( + 3α 4( + α( + 3α (3 p = p + x(4 p q = q + y(4 q, (3 p q = 4 p (x y (33 p + q = p + 4 p (x + y (34 4p 3 = p 3 + (4 p p x p (4 p x + (4 p ( x z, 4q 3 = q 3 + (4 q q y q (4 q y + (4 q ( y w, for some x, y, z, w with x, y, z, w p, q [0, ] Thus, p 3 q 3 = p3 + p (4 p (x+y p (4 p (x +y + 4 p [ ( x z ( y w ] (35 4 Using (33 - (35 in (3, we get, ( a a 4 a (B B + B 3 ( + α 4 3 = + B (B B ( + α 3 48( + α 3 p 4 ( + 3α + B3 p (4 ( p (x y 8( + α ( + α + (B B ( + α + (4 p 48( + α( + 3α p (x + y + (4 p p 96( + α( + 3α (x + y B (4 p (x y 56( + α B p (4 p 48( + α( + 3α [( x z ( y w] (36 Since p(z P, so p Thus, letting p = t applying triangle inequality on (36, with λ = x µ = y, we obtain a a 4 a 3 C + C (λ + µ + C 3 (λ + µ + C 4 (λ + µ = F (λ, µ, (37
7 where C = C (t = C = C (t = BA FRASIN et al: SECOND HANKEL DETERMINANT FOR 9 t { 48( + α 3 B B + B 3 ( + α t 3 + B 4 t 3 ( + 3α +B B B ( + α 3 t 3 + B ( + α (4 t } 0, B (4 t t { 3B 384( + α ( + α( + 3α ( + 3α +8 B B ( + α ( + α + B ( + α( + α } 0, C 3 = C 3 (t = B (4 t t(t 96( + α( + 3α 0, C 4 = C 4 (t = B (4 t 56( + α 0 Now, we need to maximize function F (λ, µ in the closed square, S = {(λ, µ : 0 λ, 0 µ } Since, coefficients of the function F (λ, µ has dependent variable t, we need to maximize F (λ, µ in the cases t = 0, t = t (0, Firstly, let t = 0 Therefore, from (37, we write F (λ, µ = B 6( + α (λ + µ We can see easily the maximum of function F (λ, µ occurs at λ = µ = max{f (λ, µ : 0 λ, 0 µ } = Secondly, let t = In this case, F (λ, µ is a constant function as follows B 4( + α (38 F (λ, µ = B B + B 3 B ( + α + B 4 + B B B 3( + α 3 (39 ( + 3α 3 Thirdly, let t (0, In this case, if we change λ + µ = ξ λµ = η, then F (λ, µ = C (t + C (tξ + [C 3 (t + C 4 (t]ξ C 3 (tη = G(ξ, η, 0 ξ, 0 η (40 Now, we investigate maximum of G(ξ, η in D = {(ξ, η : 0 ξ, 0 η } From definition of function G(ξ, η, we have G ξ (ξ, η = C (t + [C 3 (t + C 4 (t]ξ = 0, G η(ξ, η = C 3 (t = 0 From this, it is clear that, the function has no critical point in D Thus, F (λ, µ has no critical point in square S Then, the function can not take maximum value in square S Now, we investigate maximum of F (λ, µ on the boundary of the square S 3 Firstly, let λ = 0, 0 µ (similarly, µ = 0, 0 λ In this case, we write Then, F (0, µ = C (t + C (tµ + [C 3 (t + C 4 (t]µ = φ (µ φ (µ = C (t + [C 3 (t + C 4 (t]µ Case (i If C 3 (t + C 4 (t 0, then φ (µ > 0 the function is increasing the maximum occurs at µ = Case (ii Let C 3 (t + C 4 (t < 0 Since C (t + [C 3 (t + C 4 (t] > 0, we have, C (t + [C 3 (t + C 4 (t]µ C (t + [C 3 (t + C 4 (t]
8 9 TWMS J PURE APPL MATH, V7, N, 06 is true for all µ [0, ] So, φ (µ > 0 Therefore, φ (µ is an increasing function maximum occurs at µ =, Thus, max{f (0, µ : 0 µ } = C (t + C (t + C 3 (t + C 4 (t (4 3 Secondly, let λ =, 0 µ (similarly, µ =, 0 λ Then F (, µ = C (t + C (t + C 3 (t + C 4 (t + [C (t + C 4 (t]µ + [C 3 (t + C 4 (t]µ = φ (µ We can show that φ (µ is an increasing function as similar to previous case Therefore, max{f (, µ : 0 µ } = C (t + [C (t + C 3 (t] + 4C 4 (t (4 Also, for every t (0,, we can see easily that Therefore we obtain, C (t + [C (t + C 3 (t] + 4C 4 (t > C (t + C (t + C 3 (t + C 4 (t max{f (λ, µ : 0 λ, 0 µ } = C (t + [C (t + C 3 (t] + 4C 4 (t Since φ ( φ ( for t [0, ], max F (λ, µ = F (, on the boundary of the square S Thus the maximum of F occurs at λ = µ = in the closed square S Let us define H : (0, R as H(t = max F (λ, µ = F (, = C (t + [C (t + C 3 (t] + 4C 4 (t (43 On substituting the value of C (t, C (t, C 3 (t C 4 (t in the above function, we obtain where H(t = 4( + α + f(α, B, B, B 3 t 4 + C(α, B, B, B 3 t 9( + α 3 ( + α, ( + 3α f(α, B, B, B 3 = 4B B B + B 3 ( + α ( + α +4B 4 ( + α 3B 3 ( + α( + α( + 3α B ( + α ( + α + 3B ( + α 3 ( + 3α, C(α, B, B, B 3 = 3B 3 ( + α( + α( + 3α + 8B B B ( + α 3 ( + α +B ( + α ( + α 6B ( + α 3 ( + 3α Now, we investigate the maximum value of H(t in the interval (0, By simple calculation, we obtain H (t = [f(α, B, B, B 3 t 3 + C(α, B, B, B 3 ]t 48( + α 3 ( + α ( + 3α Let us examine the different cases of f(α, B, B, B 3 C(α, B, B, B 3 as follows: Case : Let f(α, B, B, B 3 0 C(α, B, B, B 3 0, then H (t 0, so the function is increasing Thus, maximum point must be on the boundary of t [0, ], that is, t = Thus, max{f (λ, µ : 0 λ, 0 µ } = H( = B B B + B 3 ( + α ( + α + 4( + α + B B B ( + α 3 ( + α 3( + α 3 ( + α ( + 3α (44
9 BA FRASIN et al: SECOND HANKEL DETERMINANT FOR 93 Case : If f(α, B, B, B 3 > 0 C(α, B, B, B 3 < 0, t 0 = C(α,,B,B 3 f(α,b,b,b 3 of H(t Since H (t 0 < 0, the maximum value of function H(t occurs at t = t 0 is critical point In this case, Therefore, H(t 0 = 4( + α [C(α, B, B, B 3 ] 48( + α 3 ( + α ( + 3αf(α, B, B, B 3 (45 H(t 0 < B 4( + α max{f (λ, µ : 0 λ, 0 µ } = { max 4( + α, B B B + B 3 ( + α ( + α + B 4 ( + α + B B B ( + α 3 ( + α 3( + α 3 ( + α ( + 3α Case 3: If f(α, B, B, B 3 0 C(α, B, B, B 3 0, H(t is a decreasing function on the interval (0, Thus, max{f (λ, µ : 0 λ, 0 µ } = } (46 B 4( + α (47 Case 4: If f(α, B, B, B 3 < 0 C(α, B, B, B 3 > 0, t 0 is a critical point of H(t Since H (t 0 < 0, the maximum value of H(t occurs at t = t 0 In this case, Therefore, B 4( + α < H(t 0 max{f (λ, µ : 0 λ, 0 µ } = max {H(t 0, B B B + B 3 ( + α ( + α + B 4 ( + α + B B B ( + α 3 ( + α 3( + α 3 ( + α ( + 3α } (48 Thus, from (44,(46,(47 (48, the proof is completed Corollary Let f given by ( be in the class F Σ (ϕ, α B <, B = B Then a a 4 a 3 B B B + B 3 ( + α ( + α + 4( + α + B B B ( + α 3 ( + α 3( + α 3 ( + α (49 ( + 3α In particular, if B = /, B = /4 B 3 =, then a a 4 a 3 4( + α( + 3α + 48( + α 3 ( + 3α + ( + 3α (50
10 94 TWMS J PURE APPL MATH, V7, N, 06 Corollary Let f given by ( be in the class F Σ (ϕ, α B <, B B Then { a a 4 a 3 max 4( + α, B B B + B 3 ( + α ( + α + B 4 ( + α + B B B ( + α 3 ( + α 3( + α 3 ( + α ( + 3α In particular, if B = /, B = / B 3 =, then a a 4 a 3 3( + α( + 3α + 48( + α 3 ( + 3α Corollary 3 Let f given by ( be in the class F Σ (ϕ, α B Then In particular, if B =, then a a 4 a 3 a a 4 a 3 B 4( + α 4( + α Corollary 4 Let f given by ( be in the class F Σ (ϕ, α B, B = B Then { a a 4 a 3 max H(t 0, B B B + B 3 ( + α ( + α + 4( + α + B B B ( + α 3 ( + α } 3( + α 3 ( + α ( + 3α (5 In particular, if B =, B = / B 3 = 4, then a a 4 a 3 The following theorems are results of Theorem Theorem Let f given by ( be in the class S Σ (ϕ If B, B B 3 satisfy the conditions } (5 5 3( + α( + 3α + 3( + α 3 ( + 3α + 3( + 3α (53 4B 3 3B B + 4 B 4B + B 3 8 B 0, 3B B + 8 B 0, If B, B B 3 satisfy the conditions a a 4 a 3 B ( 3 + B 4B + B 3 3 4B 3 3B B + 4 B 4B + B 3 8 B > 0, 3B B + 8 B < 0, { B a a 4 a 3 max 4, B ( 3 + B } 4B + B If B, B B 3 satisfy the conditions 4B 3 3B B + 4 B 4B + B 3 8 B 0, 3B B + 8 B 0,
11 BA FRASIN et al: SECOND HANKEL DETERMINANT FOR 95 4 If B, B B 3 satisfy the conditions a a 4 a 3 B 4 4B 3 3B B + 4 B 4B + B 3 8 B < 0, 3B B + 8 B > 0, { a a 4 a ( 3 3 max + B 4B + B 3, 3 Theorem 3 Let f given by ( be in the class K Σ (ϕ If B, B B 3 satisfy the conditions B 4 B (3B B + 8 B 48[4B 3 3B B + 4 B 4B + B 3 8 B ] 3B 3 6B 4B + B 4B + B 3 4 B 0, 3B B + B 0, If B, B B 3 satisfy the conditions a a 4 a 3 B ( B 4B + B B 3 6B 4B + B 4B + B 3 4 B > 0, 3B B + B < 0, { B a a 4 a 3 max 36, B ( B } 4B + B If B, B B 3 satisfy the conditions 3B 3 6B 4B + B 4B + B 3 4 B 0, 3B B + B 0, 4 If B, B B 3 satisfy the conditions a a 4 a 3 B 36 3B 3 6B 4B + B 4B + B 3 4 B < 0, 3B B + B > 0, { a a 4 a ( 3 3 max + 4 B 4B + B 3, B (3 B + B 88[3 3 6B 4B + B 4B + B 3 4 B ] } }
12 96 TWMS J PURE APPL MATH, V7, N, 06 Corollary 5 By choosing ϕ(z of the form (??, we state the following results for functions f F Σ (ϕ, α, 6( β 4 +4(+α ( β, β [0, β 3(+α 3 (+3α 0 ] a a 4 a 3 } (54 max {H(t 0, 6( β4 +4(+α ( β, β ( β 0,, 3(+α 3 (+3α where β 0 = 3(+α(+α(+3α 9(+α (+α (+3α 6(+α [3(+α 3 (+3α 8(+α (+α ] 6(+α, H (t 0 = ( β ( + α [C(α, β] 48( + α 3 ( + α ( + 3αf(α, β, f(α, β = 4( β { 6( + α ( β 6( + α( + α( + 3α( β +3( + α 3 ( + 3α 8( + α ( + α }, C(α, β = 4( β {( + α( + α( + 3α( β +( + α ( + α ( + α 3 ( + 3α] } Proof Let f F Σ (ϕ, α, with ϕ(z of the form (?? We need to maximize function F (λ, µ, definition by the formula (37, in the closed square S = {(λ, µ : 0 λ, 0 µ } This proof will be completed as proof of Theorem For t = 0, F (λ, µ = ( β 4( + α (λ + µ This function has no critical point in square S, so it has no maximum point Then max{f (λ, µ : 0 λ, 0 µ } = F (, = If t =, F (λ, µ is a constant function: F (λ, µ = C ( According to this, ( β ( + α (55 max{f (λ, µ : 0 λ, 0 µ } = 6( β4 + 4( + α ( β 3( + α 3 (56 ( + 3α 3 Now let t (0, In this case, F (λ, µ will take a maximum value depend on t : max{f (λ, µ : 0 λ, 0 µ } = H(t, where H(t is given in (43 If we write B = B = B 3 = ( β in value of C (t, C (t, C 3 (t, C 4 (t we consider these in H(t, we obtain where H(t = ( β ( + α + f(α, β t4 + 4 C(α, β t 9( + α 3 ( + α ( + 3α, f(α, β = 4( β { 6( + α ( β 6( + α( + α( + 3α( β +3( + α 3 ( + 3α 8( + α ( + α }, C(α, β = 4( β {( + α( + α( + 3α( β +( + α ( + α ( + α 3 ( + 3α] }
13 BA FRASIN et al: SECOND HANKEL DETERMINANT FOR 97 Now, we investigate maximum of H(t in the open interval (0, The derivative of H(t is as follows: H (t = [f(α, βt + C(α, β]t 48( + α 3 ( + α ( + 3α For all values of α [0, ] β [0,, C(α, β > 0 Moreover, for all α [0, ] β [0, β 0 ], f(α, β 0 In here, β 0 = 3(+α(+α(+3α 9(+α (+α (+3α 6(+α [3(+α 3 (+3α 8(+α (+α ], In this 6(+α case, H (t > 0, so H(t is an increasing function in (0, However, this function doesn t take maximum value in (0, Thus, for β [0, β 0 ], max{f (λ, µ : 0 λ, 0 µ } = 6( β4 + 4( + α ( β 3( + α 3 (57 ( + 3α If β ( β 0, ], f(α, β < 0 In this case, t 0 = C(α, β f(α, β is a critical point of H(t We observe that t 0 <, that is, t 0 is interior point of the interval (0, Since H (t 0 < 0, the maximum value of H(t occurs at t = t 0 max{h(t : 0 < t < } = H(t 0 = In this case, Therefore, ( β ( + α [C(α, β] 48( + α 3 ( + α ( + 3αf(α, β ( β ( + α < H(t 0 max{f (λ, µ : 0 λ, 0 µ } { ( β = max ( + α [C(α, β] 48( + α 3 ( + α ( + 3αf(α, β, Thus, from (57 (58, the proof is completed 6( β 4 + 4( + α ( β 3( + α 3 ( + 3α } (58 Corollary 6 Taking α = 0 α = in the Corollary 5, we obtain the results for the classes S Σ (ϕ K Σ(ϕ, which leads to the results obtained in Theorem 3 of [6], respectively Corollary 7 Putting β = 0 in the Corollary 6, we get the boundary estimates for the second Hankel determinant in the classes of bi-starlike bi-convex functions as a a 4 a 3 0/3 a a 4 a 3 /3 The boundary estimates for the second Hankel determinant obtained in the Corollary 7 verifies to the Corollary 4 of [6], respectively
14 98 TWMS J PURE APPL MATH, V7, N, 06 3 Acknowledgment We would like to thank the referee(s for his comments suggestions on the manuscript References [] Ali, RM, Lee, SK, Ravichran, V, Supramaniam, S, (0, Coefficient estimates for bi-univalent Ma-Minda starlike convex functions, ApplMath Lett, 5(3, pp [] Altinkaya, S, Yalcin, S, Coefficient bounds for a subclass of bi-univalent functions, TWMS J Pure Appl Math, 6(, 05, pp80-85 [3] Brannan, DA, Clunie(Eds, JG, (980, Aspects of Contemporary Complex Analysis (Proceeding of the NATO Advanced study Institute held at the University of Durham, Durham: July -0, 979, Academic Press, New York London [4] Brannan, DA, Taha, TS, (986, On some classes of bi-univalent functions, Studia Univ Babes-Bolyai Math, 3(, pp70-77 [5] Deniz, E, (03, Certain subclasses of bi-univalent functions satisfying subor dinate conditions, JClassAnal, (, pp49 60 [6] Deniz, E, Caglar, M, Orhan, H, (05, Second Hankel determinant for bi-starlike bi-convex functions of order β, Appl MathComp, 7, pp [7] Duren, PL, (983, Univalent Functions, in: Grundlehren der Mathematischen Wissenschaften, B 59, Springer-Verlag, New York, Berlin, Heidelberg Tokyo [8] Fekete, M, Szegö,G, Eine Bemerkung ü, (933, ber ungerade schlichte Funktionen, J London Math Soc, 8, pp85 89 [9] Grener, U, Szegö,G,(958, Toeplitz Forms Their Applications, Univ of California Press, Berkeley Los Angeles [0] Hummel, J, (957, The coefficient regions of starlike functions, Pacific J Math, 7, pp [] Hummel, J, (960, Extremal problems in the class of starlike functions, Proc Amer Math Soc,, pp [] Keogh, FR Merkes, EP, (969, A coefficient inequality for certain classes of analytic functions, Proc Amer Math Soc, 0, pp8 [3] Lewin, M, (967, On a coefficient problem for bi-univalent functions, Proc Amer Math Soc, 8, pp63 68 [4] Ma, WC, Minda, D, (99, A unified treatment of some special classes of functions, Proceedings of the Conference on Complex Analysis, Tianjin, pp57-69, Conf ProcLecture Notes Anal Int Press, Cambridge, MA, 994 [5] Murugusundaramoorthy, G, Janani,T, Inclusion results associated with certain subclass of analytic functions involving calculus operator, TWMS J Pure Appl Math, 7(, 06, pp63-75 [6] Netanyahu, E, (969, The minimal distance of the image boundary from the origin the second coefficient of a univalent function in z <, Arch Rational Mech Anal, 3, pp00- [7] Noonan, JW, Thomas, DK, (976, On the second Hankel determinant of areally mean p valent functions, Trans Amer Math Soc, 3(, pp [8] Srivastava, HM, Mishra, AK, Gochhayat, P, (00, Certain subclasses of analytic bi-univalent functions, Appl Math Lett, 3, pp88-9 [9] Taha, TS, (98, Topics in Univalent Function Theory, PhD Thesis, University of London [0] Xu, Q-H, Gui, Y-C, Srivastava, HM, (0, Coefficient estimates for a certain subclass of analytic bi-univalent functions, Appl Math Lett, 5, pp [] Xu, Q-H, Xiao, H-G, Srivastava, HM, (0, A certain general subclass of analytic bi-univalent functions associated coefficient estimate problems, Appl Math Comput, 8, pp46 465
15 BA FRASIN et al: SECOND HANKEL DETERMINANT FOR 99 Dr BA Frasin received his PhD degree in complex analysis (Geometric Function Theory from the National University of Malaysia (UKM, in 00 He is having twenty five years of teaching research experience His research areas include special classes of univalent functions, special functions harmonic functions DrK Vijaya works as a Professor of mathematics at the School of Advanced Sciences, Vellore Institute of Technology, VIT University, Vellore-63 04, Tamilnadu, India She received her PhD degree in Complex Analysis (Geometric Function Theory from VIT University, Vellore, in 007 Her research areas include special functions, harmonic functions M Kasthuri is a Research Associate, pursuing PhD in Mathematics, VIT University, Vellore6304 She got Master of Science degree from Thiruvalluvar University, Vellore, TN,India Her research areas include Bi-univalent, harmonic Bessel functions
arxiv: v1 [math.cv] 30 Sep 2017
SUBCLASSES OF BI-UNIVALENT FUNCTIONS DEFINED BY SĂLĂGEAN TYPE DIFFERENCE OPERATOR G. MURUGUSUNDARAMOORTHY, AND K. VIJAYA 2, arxiv:70.0043v [math.cv] 30 Sep 207 Corresponding Author,2 School of Advanced
More informationAPPLYING RUSCHEWEYH DERIVATIVE ON TWO SUB-CLASSES OF BI-UNIVALENT FUNCTIONS
International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:1 No:06 68 APPLYING RUSCHEWEYH DERIVATIVE ON TWO SUB-CLASSES OF BI-UNIVALENT FUNCTIONS ABDUL RAHMAN S. JUMA 1, FATEH S. AZIZ DEPARTMENT
More informationSecond Hankel Determinant Problem for a Certain Subclass of Univalent Functions
International Journal of Mathematical Analysis Vol. 9, 05, no. 0, 493-498 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.988/ijma.05.55 Second Hankel Determinant Problem for a Certain Subclass of Univalent
More informationOn a subclass of analytic functions involving harmonic means
DOI: 10.1515/auom-2015-0018 An. Şt. Univ. Ovidius Constanţa Vol. 231),2015, 267 275 On a subclass of analytic functions involving harmonic means Andreea-Elena Tudor Dorina Răducanu Abstract In the present
More informationSubclasses of bi-univalent functions related to shell-like curves connected with Fibonacci numbers
Acta Univ. Sapientiae, Mathematica, 10, 1018 70-84 DOI: 10.478/ausm-018-0006 Subclasses of bi-univalent functions related to shell-like curves connected with Fibonacci numbers H. Özlem Güney Dicle University,
More informationHankel determinant for p-valently starlike and convex functions of order α
General Mathematics Vol. 17, No. 4 009, 9 44 Hankel determinant for p-valently starlike and convex functions of order α Toshio Hayami, Shigeyoshi Owa Abstract For p-valently starlike and convex functions
More informationCOEFFICIENTS OF BI-UNIVALENT FUNCTIONS INVOLVING PSEUDO-STARLIKENESS ASSOCIATED WITH CHEBYSHEV POLYNOMIALS
Khayyam J. Math. 5 (2019), no. 1, 140 149 DOI: 10.2204/kjm.2019.8121 COEFFICIENTS OF BI-UNIVALENT FUNCTIONS INVOLVING PSEUDO-STARLIKENESS ASSOCIATED WITH CHEBYSHEV POLYNOMIALS IBRAHIM T. AWOLERE 1, ABIODUN
More informationarxiv: v1 [math.cv] 1 Nov 2017
arxiv:7.0074v [math.cv] Nov 07 The Fekete Szegö Coefficient Inequality For a New Class of m-fold Symmetric Bi-Univalent Functions Satisfying Subordination Condition Arzu Akgül Department of Mathematics,
More informationCoefficient Bounds for a Certain Class of Analytic and Bi-Univalent Functions
Filomat 9:8 (015), 1839 1845 DOI 10.98/FIL1508839S Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Coefficient Bounds for a Certain
More informationCOEFFICIENT BOUNDS FOR A SUBCLASS OF BI-UNIVALENT FUNCTIONS
TWMS J. Pure Appl. Math., V.6, N.2, 205, pp.80-85 COEFFICIENT BOUNDS FOR A SUBCLASS OF BI-UNIVALENT FUNCTIONS ŞAHSENE ALTINKAYA, SIBEL YALÇIN Abstract. An analytic function f defined on the open unit disk
More informationResearch Article Coefficient Estimates for Two New Subclasses of Biunivalent Functions with respect to Symmetric Points
Function Spaces Volume 2015 Article ID 145242 5 pages http://dx.doi.org/10.1155/2015/145242 Research Article Coefficient Estimates for Two New Subclasses of Biunivalent Functions with respect to Symmetric
More informationSecond Hankel determinant for the class of Bazilevic functions
Stud. Univ. Babeş-Bolyai Math. 60(2015), No. 3, 413 420 Second Hankel determinant for the class of Bazilevic functions D. Vamshee Krishna and T. RamReddy Abstract. The objective of this paper is to obtain
More informationCOEFFICIENT ESTIMATES FOR A SUBCLASS OF ANALYTIC BI-UNIVALENT FUNCTIONS
Bull. Korean Math. Soc. 0 (0), No. 0, pp. 1 0 https://doi.org/10.4134/bkms.b170051 pissn: 1015-8634 / eissn: 2234-3016 COEFFICIENT ESTIMATES FOR A SUBCLASS OF ANALYTIC BI-UNIVALENT FUNCTIONS Ebrahim Analouei
More informationInitial Coefficient Bounds for a General Class of Bi-Univalent Functions
Filomat 29:6 (2015), 1259 1267 DOI 10.2298/FIL1506259O Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://.pmf.ni.ac.rs/filomat Initial Coefficient Bounds for
More informationFaber polynomial coefficient bounds for a subclass of bi-univalent functions
Stud. Univ. Babeş-Bolyai Math. 6(06), No., 37 Faber polynomial coefficient bounds for a subclass of bi-univalent functions Şahsene Altınkaya Sibel Yalçın Abstract. In this ork, considering a general subclass
More informationA Certain Subclass of Analytic Functions Defined by Means of Differential Subordination
Filomat 3:4 6), 3743 3757 DOI.98/FIL64743S Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat A Certain Subclass of Analytic Functions
More informationCoefficient Inequalities of Second Hankel Determinants for Some Classes of Bi-Univalent Functions
mathematics Article Coefficient Inequalities of Second Hankel Determinants for Some Classes of Bi-Univalent Functions Rayaprolu Bharavi Sharma Kalikota Rajya Laxmi, * Department of Mathematics, Kakatiya
More informationTHE FEKETE-SZEGÖ COEFFICIENT FUNCTIONAL FOR TRANSFORMS OF ANALYTIC FUNCTIONS. Communicated by Mohammad Sal Moslehian. 1.
Bulletin of the Iranian Mathematical Society Vol. 35 No. (009 ), pp 119-14. THE FEKETE-SZEGÖ COEFFICIENT FUNCTIONAL FOR TRANSFORMS OF ANALYTIC FUNCTIONS R.M. ALI, S.K. LEE, V. RAVICHANDRAN AND S. SUPRAMANIAM
More informationCOEFFICIENT INEQUALITY FOR CERTAIN SUBCLASSES OF ANALYTIC FUNCTIONS
NEW ZEALAND JOURNAL OF MATHEMATICS Volume 42 (2012), 217-228 COEFFICIENT INEQUALITY FOR CERTAIN SUBCLASSES OF ANALYTIC FUNCTIONS D. VAMSHEE KRISHNA 1 and T. RAMREDDY (Received 12 December, 2012) Abstract.
More informationCoefficient Bounds for Certain Subclasses of Analytic Function Involving Hadamard Products
Mathematica Aeterna, Vol. 4, 2014, no. 4, 403-409 Coefficient Bounds for Certain Subclasses of Analytic Function Involving Hadamard Products V.G. Shanthi Department of Mathematics S.D.N.B. Vaishnav College
More informationSecond Hankel determinant obtained with New Integral Operator defined by Polylogarithm Function
Global Journal of Pure and Applied Mathematics. ISSN 0973-768 Volume 3, Number 7 (07), pp. 3467-3475 Research India Publications http://www.ripublication.com/gjpam.htm Second Hankel determinant obtained
More informationCOEFFICIENT INEQUALITY FOR CERTAIN SUBCLASS OF ANALYTIC FUNCTIONS
Armenian Journal of Mathematics Volume 4, Number 2, 2012, 98 105 COEFFICIENT INEQUALITY FOR CERTAIN SUBCLASS OF ANALYTIC FUNCTIONS D Vamshee Krishna* and T Ramreddy** * Department of Mathematics, School
More informationON ESTIMATE FOR SECOND AND THIRD COEFFICIENTS FOR A NEW SUBCLASS OF ANALYTIC AND BI-UNIVALENT FUNCTIONS BY USING DIFFERENTIAL OPERATORS
Palestine Journal of Mathematics Vol. 8(1)(019), 44 51 Palestine Polytechnic University-PPU 019 ON ESTIMATE FOR SECOND AND THIRD COEFFICIENTS FOR A NEW SUBCLASS OF ANALYTIC AND BI-UNIVALENT FUNCTIONS BY
More informationCoefficient bounds for some subclasses of p-valently starlike functions
doi: 0.2478/v0062-02-0032-y ANNALES UNIVERSITATIS MARIAE CURIE-SKŁODOWSKA LUBLIN POLONIA VOL. LXVII, NO. 2, 203 SECTIO A 65 78 C. SELVARAJ, O. S. BABU G. MURUGUSUNDARAMOORTHY Coefficient bounds for some
More informationBounds on Hankel determinant for starlike and convex functions with respect to symmetric points
PURE MATHEMATICS RESEARCH ARTICLE Bounds on Hankel determinant for starlike convex functions with respect to symmetric points Ambuj K. Mishra 1, Jugal K. Prajapat * Sudhana Maharana Received: 07 November
More informationCoefficient Estimates for Bazilevič Ma-Minda Functions in the Space of Sigmoid Function
Malaya J. Mat. 43)206) 505 52 Coefficient Estimates for Bazilevič Ma-Minda Functions in the Space of Sigmoid Function Sunday Oluwafemi Olatunji a, and Emmanuel Jesuyon Dansu b a,b Department of Mathematical
More informationTHIRD HANKEL DETERMINANT FOR THE INVERSE OF RECIPROCAL OF BOUNDED TURNING FUNCTIONS HAS A POSITIVE REAL PART OF ORDER ALPHA
ISSN 074-1871 Уфимский математический журнал. Том 9. т (017). С. 11-11. THIRD HANKEL DETERMINANT FOR THE INVERSE OF RECIPROCAL OF BOUNDED TURNING FUNCTIONS HAS A POSITIVE REAL PART OF ORDER ALPHA B. VENKATESWARLU,
More informationFekete-Szegö Inequality for Certain Classes of Analytic Functions Associated with Srivastava-Attiya Integral Operator
Applied Mathematical Sciences, Vol. 9, 015, no. 68, 3357-3369 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ams.015.543 Fekete-Szegö Inequality for Certain Classes of Analytic Functions Associated
More informationFABER POLYNOMIAL COEFFICIENTS FOR GENERALIZED BI SUBORDINATE FUNCTIONS OF COMPLEX ORDER
Journal of Mathematical Inequalities Volume, Number 3 (08), 645 653 doi:0.753/jmi-08--49 FABER POLYNOMIAL COEFFICIENTS FOR GENERALIZED BI SUBORDINATE FUNCTIONS OF COMPLEX ORDER ERHAN DENIZ, JAY M. JAHANGIRI,
More informationFekete-Szegö Problem for Certain Subclass of Analytic Univalent Function using Quasi-Subordination
Mathematica Aeterna, Vol. 3, 203, no. 3, 93-99 Fekete-Szegö Problem for Certain Subclass of Analytic Univalent Function using Quasi-Subordination B. Srutha Keerthi Department of Applied Mathematics Sri
More informationRosihan M. Ali and V. Ravichandran 1. INTRODUCTION
TAIWANESE JOURNAL OF MATHEMATICS Vol. 14, No. 4, pp. 1479-1490, August 2010 This paper is available online at http://www.tjm.nsysu.edu.tw/ CLASSES OF MEROMORPHIC α-convex FUNCTIONS Rosihan M. Ali and V.
More informationA NEW SUBCLASS OF MEROMORPHIC FUNCTION WITH POSITIVE COEFFICIENTS
Bulletin of Mathematical Analysis and Applications ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 3 Issue 32010), Pages 109-121. A NEW SUBCLASS OF MEROMORPHIC FUNCTION WITH POSITIVE COEFFICIENTS S.
More informationOn the Second Hankel Determinant for the kth-root Transform of Analytic Functions
Filomat 3: 07 7 5 DOI 098/FIL707A Published by Faculty of Sciences and Mathematics University of Niš Serbia Available at: http://wwwpmfniacrs/filomat On the Second Hankel Determinant for the kth-root Transform
More informationESTIMATE FOR INITIAL MACLAURIN COEFFICIENTS OF CERTAIN SUBCLASSES OF BI-UNIVALENT FUNCTIONS
Miskolc Matheatical Notes HU e-issn 787-43 Vol. 7 (07), No., pp. 739 748 DOI: 0.854/MMN.07.565 ESTIMATE FOR INITIAL MACLAURIN COEFFICIENTS OF CERTAIN SUBCLASSES OF BI-UNIVALENT FUNCTIONS BADR S. ALKAHTANI,
More informationSUBORDINATION RESULTS FOR CERTAIN SUBCLASSES OF UNIVALENT MEROMORPHIC FUNCTIONS
SUBORDINATION RESULTS FOR CERTAIN SUBCLASSES OF UNIVALENT MEROMORPHIC FUNCTIONS, P.G. Student, Department of Mathematics,Science College, Salahaddin University, Erbil, Region of Kurdistan, Abstract: In
More informationThe Relationships Between p valent Functions and Univalent Functions
DOI:.55/auom-25-5 An. Şt. Univ. Ovidius Constanţa Vol. 23(),25, 65 72 The Relationships Between p valent Functions and Univalent Functions Murat ÇAĞLAR Abstract In this paper, we obtain some sufficient
More informationON THE FEKETE-SZEGÖ INEQUALITY FOR A CLASS OF ANALYTIC FUNCTIONS DEFINED BY USING GENERALIZED DIFFERENTIAL OPERATOR
Acta Universitatis Apulensis ISSN: 58-539 No. 6/0 pp. 67-78 ON THE FEKETE-SZEGÖ INEQUALITY FOR A CLASS OF ANALYTIC FUNCTIONS DEFINED BY USING GENERALIZED DIFFERENTIAL OPERATOR Salma Faraj Ramadan, Maslina
More informationSOME INCLUSION PROPERTIES OF STARLIKE AND CONVEX FUNCTIONS ASSOCIATED WITH HOHLOV OPERATOR. II
italian journal of pure and applied mathematics n. 37 2017 117 126 117 SOME INCLUSION PROPERTIES OF STARLIKE AND CONVEX FUNCTIONS ASSOCIATED WITH HOHLOV OPERATOR. II M. Kasthuri K. Vijaya 1 K. Uma School
More informationSubordination and Superordination Results for Analytic Functions Associated With Convolution Structure
Int. J. Open Problems Complex Analysis, Vol. 2, No. 2, July 2010 ISSN 2074-2827; Copyright c ICSRS Publication, 2010 www.i-csrs.org Subordination and Superordination Results for Analytic Functions Associated
More informationCoefficient Inequalities of a Subclass of Starlike Functions Involving q - Differential Operator
Advances in Analysis, Vol. 3, No., April 018 https://dx.doi.org/10.606/aan.018.300 73 Coefficient Inequalities of a Subclass of Starlike Functions Involving q - Differential Operator K. R. Karthikeyan
More informationApplied Mathematics Letters
Applied Mathematics Letters 3 (00) 777 78 Contents lists availale at ScienceDirect Applied Mathematics Letters journal homepage: wwwelseviercom/locate/aml Fekete Szegö prolem for starlike convex functions
More informationON SOME LENGTH PROBLEMS FOR ANALYTIC FUNCTIONS
Nunokawa, M. and Sokół, J. Osaka J. Math. 5 (4), 695 77 ON SOME LENGTH PROBLEMS FOR ANALYTIC FUNCTIONS MAMORU NUNOKAWA and JANUSZ SOKÓŁ (Received December 6, ) Let A be the class of functions Abstract
More informationFABER POLYNOMIAL COEFFICIENT ESTIMATES FOR A NEW SUBCLASS OF MEROMORPHIC BI-UNIVALENT FUNCTIONS ADNAN GHAZY ALAMOUSH, MASLINA DARUS
Available online at http://scik.org Adv. Inequal. Appl. 216, 216:3 ISSN: 25-7461 FABER POLYNOMIAL COEFFICIENT ESTIMATES FOR A NEW SUBCLASS OF MEROMORPHIC BI-UNIVALENT FUNCTIONS ADNAN GHAZY ALAMOUSH, MASLINA
More informationarxiv: v1 [math.cv] 19 Jul 2012
arxiv:1207.4529v1 [math.cv] 19 Jul 2012 On the Radius Constants for Classes of Analytic Functions 1 ROSIHAN M. ALI, 2 NAVEEN KUMAR JAIN AND 3 V. RAVICHANDRAN 1,3 School of Mathematical Sciences, Universiti
More informationON CERTAIN CLASS OF UNIVALENT FUNCTIONS WITH CONIC DOMAINS INVOLVING SOKÓ L - NUNOKAWA CLASS
U.P.B. Sci. Bull., Series A, Vol. 80, Iss. 1, 018 ISSN 13-707 ON CERTAIN CLASS OF UNIVALENT FUNCTIONS WITH CONIC DOMAINS INVOLVING SOKÓ L - NUNOKAWA CLASS S. Sivasubramanian 1, M. Govindaraj, K. Piejko
More informationarxiv: v1 [math.cv] 11 Aug 2017
CONSTRUCTION OF TOEPLITZ MATRICES WHOSE ELEMENTS ARE THE COEFFICIENTS OF UNIVALENT FUNCTIONS ASSOCIATED WITH -DERIVATIVE OPERATOR arxiv:1708.03600v1 [math.cv] 11 Aug 017 NANJUNDAN MAGESH 1, ŞAHSENE ALTINKAYA,,
More informationCoecient bounds for certain subclasses of analytic functions of complex order
Hacettepe Journal of Mathematics and Statistics Volume 45 (4) (2016), 1015 1022 Coecient bounds for certain subclasses of analytic functions of complex order Serap Bulut Abstract In this paper, we introduce
More informationFUNCTIONS WITH NEGATIVE COEFFICIENTS
A NEW SUBCLASS OF k-uniformly CONVEX FUNCTIONS WITH NEGATIVE COEFFICIENTS H. M. SRIVASTAVA T. N. SHANMUGAM Department of Mathematics and Statistics University of Victoria British Columbia 1V8W 3P4, Canada
More informationCERTAIN SUBCLASSES OF STARLIKE AND CONVEX FUNCTIONS OF COMPLEX ORDER
Hacettepe Journal of Mathematics and Statistics Volume 34 (005), 9 15 CERTAIN SUBCLASSES OF STARLIKE AND CONVEX FUNCTIONS OF COMPLEX ORDER V Ravichandran, Yasar Polatoglu, Metin Bolcal, and Aru Sen Received
More informationSOME SUBCLASSES OF ANALYTIC FUNCTIONS DEFINED BY GENERALIZED DIFFERENTIAL OPERATOR. Maslina Darus and Imran Faisal
Acta Universitatis Apulensis ISSN: 1582-5329 No. 29/212 pp. 197-215 SOME SUBCLASSES OF ANALYTIC FUNCTIONS DEFINED BY GENERALIZED DIFFERENTIAL OPERATOR Maslina Darus and Imran Faisal Abstract. Let A denote
More informationSufficient conditions for certain subclasses of meromorphic p-valent functions
Bol. Soc. Paran. Mat. (3s.) v. 33 (015): 9 16. c SPM ISSN-175-1188 on line ISSN-0037871 in ress SPM: www.sm.uem.br/bsm doi:10.569/bsm.v33i.1919 Sufficient conditions for certain subclasses of meromorhic
More informationCertain properties of a new subclass of close-to-convex functions
Arab J Math (212) 1:39 317 DOI 1.17/s465-12-29-y RESEARCH ARTICLE Pranay Goswami Serap Bulut Teodor Bulboacă Certain properties of a new subclass of close-to-convex functions Received: 18 November 211
More informationA note on a subclass of analytic functions defined by a generalized Sălăgean and Ruscheweyh operator
General Mathematics Vol. 17, No. 4 (2009), 75 81 A note on a subclass of analytic functions defined by a generalized Sălăgean and Ruscheweyh operator Alina Alb Lupaş, Adriana Cătaş Abstract By means of
More informationarxiv: v1 [math.cv] 28 Mar 2017
On third Hankel determinants for subclasses of analytic functions and close-to-convex harmonic mappings arxiv:703.09485v math.cv] 28 Mar 207 Yong Sun, Zhi-Gang Wang 2 and Antti Rasila 3 School of Science,
More informationDifferential Operator of a Class of Meromorphic Univalent Functions With Negative Coefficients
Differential Operator of a Class of Meromorphic Univalent Functions With Negative Coefficients Waggas Galib Atshan and Ali Hamza Abada Department of Mathematics College of Computer Science and Mathematics
More informationA Class of Univalent Harmonic Mappings
Mathematica Aeterna, Vol. 6, 016, no. 5, 675-680 A Class of Univalent Harmonic Mappings Jinjing Qiao Department of Mathematics, Hebei University, Baoding, Hebei 07100, People s Republic of China Qiannan
More informationCorrespondence should be addressed to Serap Bulut;
The Scientific World Journal Volume 201, Article ID 17109, 6 pages http://dx.doi.org/10.1155/201/17109 Research Article Coefficient Estimates for Initial Taylor-Maclaurin Coefficients for a Subclass of
More informationA general coefficient inequality 1
General Mathematics Vol. 17, No. 3 (2009), 99 104 A general coefficient inequality 1 Sukhwinder Singh, Sushma Gupta and Sukhjit Singh Abstract In the present note, we obtain a general coefficient inequality
More informationAn Investigation on Minimal Surfaces of Multivalent Harmonic Functions 1
General Mathematics Vol. 19, No. 1 (2011), 99 107 An Investigation on Minimal Surfaces of Multivalent Harmonic Functions 1 Hakan Mete Taştan, Yaşar Polato glu Abstract The projection on the base plane
More informationAN EXTENSION OF THE REGION OF VARIABILITY OF A SUBCLASS OF UNIVALENT FUNCTIONS
AN EXTENSION OF THE REGION OF VARIABILITY OF A SUBCLASS OF UNIVALENT FUNCTIONS SUKHWINDER SINGH SUSHMA GUPTA AND SUKHJIT SINGH Department of Applied Sciences Department of Mathematics B.B.S.B. Engineering
More informationOn a New Subclass of Salagean Type Analytic Functions
Int. Journal of Math. Analysis, Vol. 3, 2009, no. 14, 689-700 On a New Subclass of Salagean Type Analytic Functions K. K. Dixit a, V. Kumar b, A. L. Pathak c and P. Dixit b a Department of Mathematics,
More informationOn sandwich theorems for p-valent functions involving a new generalized differential operator
Stud. Univ. Babeş-Bolyai Math. 60(015), No. 3, 395 40 On sandwich theorems for p-valent functions involving a new generalized differential operator T. Al-Hawary, B.A. Frasin and M. Darus Abstract. A new
More informationOn certain subclasses of analytic functions associated with generalized struve functions
BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Number 3(79), 2015, Pages 3 13 ISSN 1024 7696 On certain subclasses of analytic functions associated with generalized struve functions G.
More informationAn ordinary differentail operator and its applications to certain classes of multivalently meromorphic functions
Bulletin of Mathematical analysis and Applications ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 1, Issue 2, (2009), Pages 17-22 An ordinary differentail operator and its applications to certain
More informationSubclass Of K Uniformly Starlike Functions Associated With Wright Generalized Hypergeometric Functions
P a g e 52 Vol.10 Issue 5(Ver 1.0)September 2010 Subclass Of K Uniformly Starlike Functions Associated With Wright Generalized Hypergeometric Functions G.Murugusundaramoorthy 1, T.Rosy 2 And K.Muthunagai
More informationOn Quasi-Hadamard Product of Certain Classes of Analytic Functions
Bulletin of Mathematical analysis Applications ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 1, Issue 2, (2009), Pages 36-46 On Quasi-Hadamard Product of Certain Classes of Analytic Functions Wei-Ping
More informationConvex Functions and Functions with Bounded Turning
Tamsui Oxford Journal of Mathematical Sciences 26(2) (2010) 161-172 Aletheia University Convex Functions Functions with Bounded Turning Nikola Tuneski Faculty of Mechanical Engineering, Karpoš II bb, 1000
More informationInt. J. Open Problems Complex Analysis, Vol. 3, No. 1, March 2011 ISSN ; Copyright c ICSRS Publication,
Int. J. Open Problems Complex Analysis, Vol. 3, No. 1, March 211 ISSN 274-2827; Copyright c ICSRS Publication, 211 www.i-csrs.org Coefficient Estimates for Certain Class of Analytic Functions N. M. Mustafa
More informationUniformly convex functions II
ANNALES POLONICI MATHEMATICI LVIII. (199 Uniformly convex functions II by Wancang Ma and David Minda (Cincinnati, Ohio Abstract. Recently, A. W. Goodman introduced the class UCV of normalized uniformly
More informationA NOTE ON UNIVALENT FUNCTIONS WITH FINITELY MANY COEFFICIENTS. Abstract
A NOTE ON UNIVALENT FUNCTIONS WITH FINITELY MANY COEFFICIENTS M. Darus, R.W. Ibrahim Abstract The main object of this article is to introduce sufficient conditions of univalency for a class of analytic
More informationInclusion and argument properties for certain subclasses of multivalent functions defined by the Dziok-Srivastava operator
Advances in Theoretical Alied Mathematics. ISSN 0973-4554 Volume 11, Number 4 016,. 361 37 Research India Publications htt://www.riublication.com/atam.htm Inclusion argument roerties for certain subclasses
More informationMULTIVALENTLY MEROMORPHIC FUNCTIONS ASSOCIATED WITH CONVOLUTION STRUCTURE. Kaliyapan Vijaya, Gangadharan Murugusundaramoorthy and Perumal Kathiravan
Acta Universitatis Apulensis ISSN: 1582-5329 No. 30/2012 pp. 247-263 MULTIVALENTLY MEROMORPHIC FUNCTIONS ASSOCIATED WITH CONVOLUTION STRUCTURE Kaliyapan Vijaya, Gangadharan Murugusundaramoorthy and Perumal
More informationCertain subclasses of uniformly convex functions and corresponding class of starlike functions
Malaya Journal of Matematik 1(1)(2013) 18 26 Certain subclasses of uniformly convex functions and corresponding class of starlike functions N Magesh, a, and V Prameela b a PG and Research Department of
More informationSubclasses of Analytic Functions with Respect. to Symmetric and Conjugate Points
Theoretical Mathematics & Applications, vol.3, no.3, 2013, 1-11 ISSN: 1792-9687 (print), 1792-9709 (online) Scienpress Ltd, 2013 Subclasses of Analytic Functions with Respect to Symmetric and Conjugate
More informationA SUBORDINATION THEOREM WITH APPLICATIONS TO ANALYTIC FUNCTIONS
Bulletin of Mathematical Analysis and Applications ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 3 Issue 3(2011, Pages 1-8. A SUBORDINATION THEOREM WITH APPLICATIONS TO ANALYTIC FUNCTIONS (COMMUNICATED
More informationOn a subclass of certain p-valent starlike functions with negative coefficients 1
General Mathematics Vol. 18, No. 2 (2010), 95 119 On a subclass of certain p-valent starlike functions with negative coefficients 1 S. M. Khairnar, N. H. More Abstract We introduce the subclass T Ω (n,
More informationOn Starlike and Convex Functions with Respect to 2k-Symmetric Conjugate Points
Tamsui Oxford Journal of Mathematical Sciences 24(3) (28) 277-287 Aletheia University On Starlike and Convex Functions with espect to 2k-Symmetric Conjugate Points Zhi-Gang Wang and Chun-Yi Gao College
More informationThe Fekete-Szegö Theorem for a Certain Class of Analytic Functions (Teorem Fekete-Szegö Bagi Suatu Kelas Fungsi Analisis)
Sains Malaysiana 40(4)(2011): 385 389 The Fekete-Szegö Theorem for a Certain Class of Analytic Functions (Teorem Fekete-Szegö Bagi Suatu Kelas Fungsi Analisis) MA`MOUN HARAYZEH AL-ABBADI & MASLINA DARUS*
More informationRosihan M. Ali, M. Hussain Khan, V. Ravichandran, and K. G. Subramanian. Let A(p, m) be the class of all p-valent analytic functions f(z) = z p +
Bull Korean Math Soc 43 2006, No 1, pp 179 188 A CLASS OF MULTIVALENT FUNCTIONS WITH NEGATIVE COEFFICIENTS DEFINED BY CONVOLUTION Rosihan M Ali, M Hussain Khan, V Ravichandran, and K G Subramanian Abstract
More informationHarmonic Mappings for which Second Dilatation is Janowski Functions
Mathematica Aeterna, Vol. 3, 2013, no. 8, 617-624 Harmonic Mappings for which Second Dilatation is Janowski Functions Emel Yavuz Duman İstanbul Kültür University Department of Mathematics and Computer
More informationA NOVEL SUBCLASS OF UNIVALENT FUNCTIONS INVOLVING OPERATORS OF FRACTIONAL CALCULUS P.N. Kamble 1, M.G. Shrigan 2, H.M.
International Journal of Applied Mathematics Volume 30 No. 6 2017, 501-514 ISSN: 1311-1728 printed version; ISSN: 1314-8060 on-line version doi: http://dx.doi.org/10.12732/ijam.v30i6.4 A NOVEL SUBCLASS
More informationConvolutions of Certain Analytic Functions
Proceedings of the ICM2010 Satellite Conference International Workshop on Harmonic and Quasiconformal Mappings (HQM2010) Editors: D. Minda, S. Ponnusamy, and N. Shanmugalingam J. Analysis Volume 18 (2010),
More informationThe Order of Starlikeness of New p-valent Meromorphic Functions
Int. Journal of Math. Analysis, Vol. 6, 2012, no. 27, 1329-1340 The Order of Starlikeness of New p-valent Meromorphic Functions Aabed Mohammed and Maslina Darus School of Mathematical Sciences Faculty
More informationBulletin of the Transilvania University of Braşov Vol 8(57), No Series III: Mathematics, Informatics, Physics, 1-12
Bulletin of the Transilvania University of Braşov Vol 857), No. 2-2015 Series III: Mathematics, Informatics, Physics, 1-12 DISTORTION BOUNDS FOR A NEW SUBCLASS OF ANALYTIC FUNCTIONS AND THEIR PARTIAL SUMS
More informationOn second-order differential subordinations for a class of analytic functions defined by convolution
Available online at www.isr-publications.co/jnsa J. Nonlinear Sci. Appl., 1 217), 954 963 Research Article Journal Hoepage: www.tjnsa.co - www.isr-publications.co/jnsa On second-order differential subordinations
More informationON STRONG ALPHA-LOGARITHMICALLY CONVEX FUNCTIONS. 1. Introduction and Preliminaries
ON STRONG ALPHA-LOGARITHMICALLY CONVEX FUNCTIONS NIKOLA TUNESKI AND MASLINA DARUS Abstract. We give sharp sufficient conditions that embed the class of strong alpha-logarithmically convex functions into
More informationSubclasses of Analytic Functions. Involving the Hurwitz-Lerch Zeta Function
International Mathematical Forum, Vol. 6, 211, no. 52, 2573-2586 Suclasses of Analytic Functions Involving the Hurwitz-Lerch Zeta Function Shigeyoshi Owa Department of Mathematics Kinki University Higashi-Osaka,
More informationQuasi-Convex Functions with Respect to Symmetric Conjugate Points
International Journal of Algebra, Vol. 6, 2012, no. 3, 117-122 Quasi-Convex Functions with Respect to Symmetric Conjugate Points 1 Chew Lai Wah, 2 Aini Janteng and 3 Suzeini Abdul Halim 1,2 School of Science
More informationSome basic properties of certain subclasses of meromorphically starlike functions
Wang et al. Journal of Inequalities Applications 2014, 2014:29 R E S E A R C H Open Access Some basic properties of certain subclasses of meromorphically starlike functions Zhi-Gang Wang 1*, HM Srivastava
More informationBIII/. Malaysinn Math. Sc. Soc. (Second Series) 26 (2003) Coefficients of the Inverse of Strongly Starlike Functions. ROSlHAN M.
BULLETIN of the MALAYSIAN MATHEMATICAL SCIENCES SOCIETY BIII/. Malaysinn Math. Sc. Soc. (Second Series) 26 (2003) 63-71 Coefficients of the Inverse of Strongly Starlike Functions ROSlHAN M. ALI School
More informationMajorization Properties for Subclass of Analytic p-valent Functions Defined by the Generalized Hypergeometric Function
Tamsui Oxford Journal of Information and Mathematical Sciences 284) 2012) 395-405 Aletheia University Majorization Properties for Subclass of Analytic p-valent Functions Defined by the Generalized Hypergeometric
More informationSOME PROPERTIES OF A SUBCLASS OF ANALYTIC FUNCTIONS DEFINED BY A GENERALIZED SRIVASTAVA-ATTIYA OPERATOR. Nagat. M. Mustafa and Maslina Darus
FACTA UNIVERSITATIS NIŠ) Ser. Math. Inform. Vol. 27 No 3 2012), 309 320 SOME PROPERTIES OF A SUBCLASS OF ANALYTIC FUNCTIONS DEFINED BY A GENERALIZED SRIVASTAVA-ATTIYA OPERATOR Nagat. M. Mustafa and Maslina
More informationSECOND HANKEL DETERMINANT PROBLEM FOR SOME ANALYTIC FUNCTION CLASSES WITH CONNECTED K-FIBONACCI NUMBERS
Ata Universitatis Apulensis ISSN: 15-539 http://www.uab.ro/auajournal/ No. 5/01 pp. 161-17 doi: 10.1711/j.aua.01.5.11 SECOND HANKEL DETERMINANT PROBLEM FOR SOME ANALYTIC FUNCTION CLASSES WITH CONNECTED
More informationConvolution properties for subclasses of meromorphic univalent functions of complex order. Teodor Bulboacă, Mohamed K. Aouf, Rabha M.
Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Filomat 26:1 (2012), 153 163 DOI: 10.2298/FIL1201153B Convolution properties for subclasses of meromorphic
More informationarxiv: v3 [math.cv] 11 Aug 2014
File: SahSha-arXiv_Aug2014.tex, printed: 27-7-2018, 3.19 ON MAXIMAL AREA INTEGRAL PROBLEM FOR ANALYTIC FUNCTIONS IN THE STARLIKE FAMILY S. K. SAHOO AND N. L. SHARMA arxiv:1405.0469v3 [math.cv] 11 Aug 2014
More informationSUBCLASS OF HARMONIC STARLIKE FUNCTIONS ASSOCIATED WITH SALAGEAN DERIVATIVE
LE MATEMATICHE Vol. LXIX (2014) Fasc. II, pp. 147 158 doi: 10.4418/2014.69.2.13 SUBCLASS OF HARMONIC STARLIKE FUNCTIONS ASSOCIATED WITH SALAGEAN DERIVATIVE H. E. DARWISH - A. Y. LASHIN - S. M. SOILEH The
More informationSUBCLASS OF HARMONIC UNIVALENT FUNCTIONS ASSOCIATED WITH THE DERIVATIVE OPERATOR
Hacettepe Journal of Mathematics and Statistics Volume 41(1) (2012), 47 58 SUBCLASS OF HARMONIC UNIVALENT FUNCTIONS ASSOCIATED WITH THE DERIVATIVE OPERATOR A.L. Pathak, S.B. Joshi, Preeti Dwivedi and R.
More informationCertain classes of p-valent analytic functions with negative coefficients and (λ, p)-starlike with respect to certain points
BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Number 379), 2015, Pages 35 49 ISSN 1024 7696 Certain classes of p-valent analytic functions with negative coefficients and λ, p)-starlike
More informationDIFFERENTIAL SUBORDINATION RESULTS FOR NEW CLASSES OF THE FAMILY E(Φ, Ψ)
DIFFERENTIAL SUBORDINATION RESULTS FOR NEW CLASSES OF THE FAMILY E(Φ, Ψ) Received: 05 July, 2008 RABHA W. IBRAHIM AND MASLINA DARUS School of Mathematical Sciences Faculty of Science and Technology Universiti
More informationBOUNDEDNESS, UNIVALENCE AND QUASICONFORMAL EXTENSION OF ROBERTSON FUNCTIONS. Ikkei Hotta and Li-Mei Wang
Indian J. Pure Appl. Math., 42(4): 239-248, August 2011 c Indian National Science Academy BOUNDEDNESS, UNIVALENCE AND QUASICONFORMAL EXTENSION OF ROBERTSON FUNCTIONS Ikkei Hotta and Li-Mei Wang Department
More information