ON CERTAIN CLASS OF UNIVALENT FUNCTIONS WITH CONIC DOMAINS INVOLVING SOKÓ L - NUNOKAWA CLASS

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1 U.P.B. Sci. Bull., Series A, Vol. 80, Iss. 1, 018 ISSN ON CERTAIN CLASS OF UNIVALENT FUNCTIONS WITH CONIC DOMAINS INVOLVING SOKÓ L - NUNOKAWA CLASS S. Sivasubramanian 1, M. Govindaraj, K. Piejko 3 The aim of this investigation is to introduce a new class of analytic univalent functions k MN that are connected to domains bounded by conic sections and obtain certain differential subordination results involving k MN. Certain comparisons of the differential subordination is being analyed with the classical results existing in the literature. Apart from obtaining other results related to the class k MN, we also obtain a containment relation between the class k MN and the class of starlike functions under certain condition. A slight improvement of a recent work of On some class of convex functions, C. R. Math. Acad. Sci. Paris, , , by Sokó l and Nunokawa [19 is also obtained. Keywords: Key Words: Analytic function, Starlike function, Convex function, Conic section. MSC010: Primary 30C45, 33C50; Secondary 30C Introduction and Definitions Let A denote the class of all functions f of the form f = a n n, 1 which are analytic in the open unit disk U = { : C and < 1}. Let S denote the subclass of A consisting of univalent functions. A function f S is said to be convex if and only if R f /f > 1 for U and a function f S is said to be starlike if and only if R f /f > 0 for U. The class of all convex and starlike functions are denoted by K and S respectively. Let P denote the class of analytic functions of the form p = 1 p 1 p such that Rp > 0 in U. In 1999, Kanas and Wiśniowska [7 and [8 introduced the class of k-uniformly convex functions, denoted by k-ucv and the class of k-starlike functions, denoted by k-st respectively. The analytic conditions of these classes are the following see [6,[7,[8 for details. n= 1 Associate Professor, Department of Mathematics, University College of Engineering Tindivanam, Anna University, Tindivanam , Tamil Nadu, India, sivasaisastha@rediffmail.com Research Scholar, Department of Mathematics, University College of Engineering Tindivanam, Anna University, Tindivanam , Tamil Nadu, India, govink7@gmail.com 3 Professor, Department of Mathematics, Resów University of Technology, Al. Powstanców Warsawy 1,, Resów, Poland, piejko@pr.edu.pl 13

2 14 S. Sivasubramanian, M. Govindaraj, K. Piejko For 0 k <, k UCV = { f S : R 1 f f > k f f }, U and { f k ST = f S : R > k f } f f 1, U. 3 Note that for k = 1, we get the well known classes of UCV and S p studied by Goodman [3,[4 and Rønning [15, while for k = 0, we get defined above families of convex K and starlike S functions, respectively. It is easy to see that the conditions and 3 may be rewritten into the form Rp > k p 1 U, 4 where p = 1 f /f or p = f /f is a function from the class P. Also, it is easy to see that pu is the conic domain or Ω k = Ω k = {ω C : Rω > k ω 1 } 5 {ω = u iv : u > k u 1 v }, 6 where 0 k <. Note that Ω k is such that 1 Ω k and Ω k is a curve defined by Ω k = { ω = u iv : u = k u 1 k v }. 7 Elementary computations show that Ω k represents a conic section symmetric about the real axis. It follows that the domain Ω k is bounded by an ellipse for k > 1, by a parabola for k = 1 and by a hyperbola if 0 < k < 1. Finally, for k = 0, Ω k is the right half plane. Recently Sokó l and Nunokawa [19 defined an interesting new class of functions MN defined by { MN = f S : R 1 f f > f } f 1, U. 8 It is clear that MN K. Motivated by the work [19 related works are also done in [,[5,[13,[14,[16,[17,[18 and the conic domain defined by Kanas and Wiśniowska [7, one may construct a new class k MN defined by { k MN = f S : R 1 f f > k f } f 1, U. 9 A function f is subordinate to the function g, written as f g, provided that there is an analytic function w defined on U with w0 = 0 and w < 1 such that f = g[w for U. In particular, if the function g is univalent in U then f g is equivalent to f0 = g0 and fu gu. Denoting by p k the conformal mapping that maps U onto Ω k, we obtain the family of conformal mappings depending on k k [0,. Therefore, for each fixed k, the family k UCV is the family of all f S for which 1 f f p k and k ST consists of all functions f S such that f p k, U. f The idea of subordination was used for defining many of classes of functions studied in geometric function theory. For obtaining the main result, we shall use the methods of differential subordinations. The theory of differential subordinations were introduced by Miller and Mocanu in [11 and [1.

3 On certain class of univalent functions with conic domains involving Sokó l - Nunokawa class 15 The purpose of the present paper is to obtain interesting new results for the class k MN by using the method of differential subordination to improve a recent result obtained in [19. Finally, we compare our conclusions with the classical results in the univalent function theory.. Main Results Let us start with the following theorem to prove that the class k MN is non empty. In fact, the theorem will show that there are plenty of functions in the class k MN. Theorem.1. A function is in the class k MN if and only if f = 1 A Proof. If f is given by 10, then R 1 f 1 A f = R 1 A 10 A 1, k k and k f f 1 = k A 1 A. 13 We know that f k MN if and only if k A 1 A 1 A < R A It is suffices to study for = 1. Setting A = r and A = re iθ in 14, then k re iθ 1 re iθ 1 re iθ < R 1 re iθ. 15 Following a computation, From 15 and 16, we get On simplification, we easily get 1 1 re iθ R 1 re iθ = 1 r 1 re iθ. 16 kr 1 re iθ 1 r 1 re iθ r kr. 18 [1 r cos θ r 1 The right-hand side of 18 is seen to have a minimum for θ = π and this minimal value is 1 r. Hence, a necessary and sufficient condition for 18 is kr 1 r or A = r 1 1 k. This completes the proof of Theorem.1. Corollary.1. The function f = belongs to the class k MN if and only if k = 0. 1 The following proposition follows directly from the definition. Proposition 1. If k 1 k, then k 1 MN k MN.

4 16 S. Sivasubramanian, M. Govindaraj, K. Piejko Next, we state a basic lemma and Theorems which are required prove our main results. Lemma.1. [11 Let h be an analytic function on U except for at most one pole on U, and univalent on U, p be an analytic function in U with p0 = h0 and p p0, U. If p is not subordinate to h, then there exist points 0 U, ζ 0 U and m 1 for which p < 0 hu, p 0 = hζ 0, 0 p 0 = mζ 0 h ζ 0. Theorem.. [8 If f S α for some α [1/, 1, then f 1 R > 3 α. 19 f Theorem.3. [8 If R > α for some α [1/, 1, then f R > α Theorem.4. Let k [0,. Also, let p be an analytic function in the unit disk such that p0 = 1. If R p p k p 1 > 0, 1 p then p 1 1 α 1 where α αk, and αk is given by 1 αk = 1 k 4 1 k = h, 8 1 k 1 k. 3 1 k Proof. We may assume that α 1, since we have the condition, R p p p > 0 implies that at least Rp > 1. Suppose now, on the contrary, that p h. Then by Lemma.1 of Miller and Mocanu [11, there exist a 0 U, ζ 0 U, ζ 0 1 and m 1, such that p 0 = α ix, 0 p 0 = my, where y 1 α x, x, y R. 1 α Making use the above relations, we have R p 0 0p 0 p 0 = R α ix my α ix k p 0 1 k α 1 ix = α αmy α x k 1 α x α = rx α 1 α 1 α x α x k 1 α x

5 On certain class of univalent functions with conic domains involving Sokó l - Nunokawa class 17 The function rx is even as regards x. Now, we show that rx attains its maximum at x = 0 when 1 α < 1. Clearly, [ r αα 1 x = x 1 αα x k. 1 α x Equating r x = 0, one can easily see that r x = 0 if and only if x = 0. For α 1, the expression α 1 is nonnegative. Therefore, we have r 0 = 1 [α 1 kα < 0. α1 α Therefore, we have rx has a maximum at x = 0. Hence, rx r0 = α 1 α α k1 α = 0, 4 for α = αk, as given by 3, which contradicts the assumption. This essentially completes the proof of Theorem.4. Applying Theorem.4 we may formulate the following: Theorem.5. If 0 k <, then k MN S α, where S α is the class of starlike functions of order α, α αk and αk is given by 3. Proof. Let f k MN. Then, by 9 R 1 f f > k f f 1, U. 5 Setting p = f, p0 = 1, the above condition can be written as f R p p > k p 1 6 p or R p p k p 1 > 0. 7 p Now applying Theorem.4, we obtain the assertion as stated in Theorem.5. In view of Theorem. and Theorem.3, we compare the classical results concerning the right half plane with these obtained for domains bounded by conic sections. This comparison shows that such a substitution provides a step to the right. For instance classical results give R 1 f f > 0 R whereas, Theorem. and Theorem.4, gives us that f > 1 f f R > 1, That is, R 1 f f > k f f 1 f R f > α R > η, f f f f k MN R > α R > η, f

6 18 S. Sivasubramanian, M. Govindaraj, K. Piejko 1 where α = αk = 1 k 4 1 k η = 1 3 α. 8 1 k 1 k and 1 k Setting k = 1, we obtain from the above the following: R 1 f f > f f 1 f R > 1 [ f 8 f 4 R > That is, f f MN R > 1 [ f R > f The above observation in 8 may stated as below. Corollary.. If f MN then MN S α 0, where α = It is to be remarked here that Kanas [8 has obtained a similar improvement of order of starlikeness for uniformly convex functions from 3 = 0.75 to by applying the method 4 of differential subordination techniques for domains involving conic section. Our corollary. is for the class MN. Although it is a slight improvement, the technique adopted by us is different. Moreover, we have also given one more implication in 8 relating to the class MN, which is a slight improvement of the recent result obtained by Sokó l and Nunokawa [19 as the order of starlikeness is slightly increased. Remark.1. Observe that in the case, k = 0, we recover the classical sharp result that each convex function is starlike at least of order 1/. Furthermore, we know that, R 1 f f > 0 R f > 1 f R > 1, whereas, Theorem.4 and Theorem.3, gives us f R f f > k f 1 R f f > α R > δ, 1 where α = αk = 1 k 8 1 k 4 1 k 1 k and δ = α 1. 1 k 3 Setting k = 1, we obtain from the above the following: f R f f > f 1 R f > 1 [ f 4 R >

7 On certain class of univalent functions with conic domains involving Sokó l - Nunokawa class 19 For k = 0, we get one more classical result as given below f R f f > 0 R f > 1 f R > 1. Setting p = f/andp = f, Theorem.4 reduces to the following corollaries: Corollary.3. Let 0 k <. Let f A is analytic in U. Then f R f f 1 > k f 1 f R > αk. 9 Corollary.4. Let 0 k <. Let f A is analytic in U. Then f R f f > k f 1 R f > αk. 30 For k = 1 and p = f Corollary.5. f R f f 1 > in Theorem.4, we obtain the following corollary f 1 f R > For k = 1, p = f in Theorem.4, we obtain the following corollary Corollary.6. f R f f > f 1 R f > 1. 3 From 9 we obtain R p p > k p 1 k R 1 p U 33 p and hence R p p > k 1 kp k The above inequality is equivalent to the familiar Briot-Bouquet differential subordination of the form, 1 k p p 1 1 kp 1k. 1 In view of Theorem 3.3d, [11, p.109, we obtain 1 p q = k 1 k 0 1 t tdt 1 4 = F 1 1 k, 1, 3; = g k say A simple computation yields g k = 1 4 1k 8 5 k 85 k3 k 1 31 k 1 k 151 k In order to prove the results involving coefficient inequalities, we need the following lemmas.

8 130 S. Sivasubramanian, M. Govindaraj, K. Piejko Lemma.. [1 If p = 1 p 1 p P, then for each k 1, p k, and p p 1 p Lemma.3. [10 If q = 1 c 1 c is an analytic function with positive real part in U, then c vc 1 max{1; v 1 }. 39 In particular, if v is a real number, then 4v if v 0, c vc 1 if 0 v 1, 4v if v 1. When v < 0 or v > 1, the equality in 39 holds true if and only if q = 1 or one of 1 its rotations. If 0 < v < 1, then the equality holds true if and only if q = 1 or one 1 of its rotations. If v = 0, then the equality holds true if and only if 1 q = λ λ 1 0 λ 1 1 or one of its rotations, while for v = 1, equality holds if and only if q is a reciprocal of one of the functions such that the equality holds true in the case when v = 0. Theorem.6. If f k MN, then we have f f where g k is as given by 37. q k = 1 g k U, 40 Theorem.7. Let 0 k < and let f, given by 1, be in the class k MN. Then a 4 31 k and 9 9k a k for k < 17 3, 4 for k k 3. For any complex number µ { } a 3 µa 31 k max 1; 13 9k 11 k 4 31 k 8µ 31 k. 43 Proof. If f k MN, then exist a Schwar function ω is analytic in U, with ω0 = 0 and ω < 1 in U, then f = q k ω. 44 f Define the function ϕ 1 is given by 41 ϕ 1 := 1 ω 1 ω = 1 c 1 c

9 On certain class of univalent functions with conic domains involving Sokó l - Nunokawa class 131 Since ω is a Schwar function, we see that Rϕ 1 > 0 and ϕ 1 0 = 1. function p is given by p := f f Define the = 1 a a 3 a 3a 4 a 3 3a 3 a In view of above equations 44, 45 and 46, we have ϕ1 1 p = q k. 47 ϕ 1 1 Since ϕ 1 1 ϕ 1 1 = 1 [ c 1 c c 1 c 3 c3 1 4 c 1c 3. Therefore, ϕ1 1 q k = ϕ k c 1 From 46 and 47, we get and It is easily get a 3 = 1 [ a 3 a = [ 31 k 31 k a = c c k 361 k c k c 1, 49 c c k 31 k 361 k c c c k 361 k c k c Now applying the coefficient estimate c k for k = 1,,..., it is easily to get by Lemma.3, a 31 k c 4 1 = 5 31 k and That is a 3 1 [ c 1 31 k a k 361 k c k c [ 13 9k 31 k 71 k 91 k 1 c k This gives the bound for a 3 is given by 4. where σ = 1 a 3 µa = 1 [ c σc k [ 8µ 1 31 k k k 11 k This completes proof of the Theorem.7 by Lemma.3. From Theorem.7, we get the following corollary.

10 13 S. Sivasubramanian, M. Govindaraj, K. Piejko Corollary.7. Let 0 k < and let f, given by 1, be in the class k MN. Then, for any real number µ 9 9k 181 k 16µ 91 k if µ σ 1, a 3 µa if σ 1 µ σ, k 9 9k 181 k 16µ 91 k if µ σ, where The result is sharp. σ 1 = σ = 91 k 3 91 k k k k 3 91 k 13 9k, 91 k Theorem.8. Let 0 k <. Also, let f k MN. Then fu contains an open disk of radius 31 k 10 6k. 58 Proof. Let ω 0 0 be a complex number such that f ω 0 for U. Then f 1 = Since f 1 is univalent in U, so that ω 0f ω 0 f =. a 1ω0. 59 a ω 0 Now using Theorem.7, we have and hence k. 61 ω 0 ω 0 31 k 10 6k. 6 Since, f S, the inverse of f has a Maclaurin expansion in a disk of radius at least 1 4, say F ω = f 1 ω = ω d ω d 3 ω 3. In an earlier investigation, Libera and Z lotkiewic [9 obtained few earlier coefficients of the inverse of a regular convex function f. Now, we obtain the first two early coefficient estimates of the inverse when f k MN. Theorem.9. Let f k MN and F ω = f 1 ω = ω d ω d 3 ω 3. Then d 4 31 k 63

11 and On certain class of univalent functions with conic domains involving Sokó l - Nunokawa class 133 Proof. As F f =, we have, d k 181 k. 64 d = a, d 3 = a a From 49 and 51, we have d = 31 k c 1 66 and 4 91 k c 1 1 [ c c k k 11 k c 1 31 k c [ 1 = 31 k c 13 9k 91 k 71 k 1 c 61 k Now applying Lemma.3, the proof of Theorem.9 is completed. Concluding remarks and observations Very similar to the interesting classes MN defined by Sokó l and Nunokawa [19, and the class k MN defined in this paper by the authors, one might think of the emerging classes that appear in 9,30, 31and 3. Acknowledgements The authors are very much thankful to the referee for his/her critical reviews that essentially improved the article. The work of first two authors are supported by a grant from Department of Science and Technology, Government of India vide ref: SR/FTP/MS- 0/01 under fast track scheme. R E F E R E N C E S [1 P. Duren, Subordination in complex analysis, 599 of Lecture notes in Mathematics, 9, Springer, Berlin, Germany, [ J. Diok, M. Darus, J. Sokó l and T. Bulboacă, Generaliations of starlike harmonic functions, C. R. Math. Acad. Sci. Paris., , No. 1, [3 A. W. Goodman, On uniformly convex functions, Ann. Polon. Math., [4 A. W. Goodman, On uniformly starlike functions, Ann. Polon. Math., [5 R. Jurasińska and J. Sokó l, Some problems for certain family of starlike functions, Math. Comput. Modelling, [6 S. Kanas and A. Wiśniowska, Conic regions and k-uniform convexity, II, Folia Scient. Univ. Tech. Res. Mat., [7 S. Kanas and A. Wiśniowska, Conic regions and k-uniform convexity, J. Comput. Appl. Math., [8 S. Kanas, Subordinations for domains bounded by conic sections, Bull. Belg. Math. Soc. Simon Stevin., [9 R. J. Libera and E. J. Z lotkiewic, Early coefficients of the inverse of a regular convex functions, Proc. Amer. Math. Soc , no., [10 W. Ma and D. Minda, A unified treatment of some special classes of univalent functions, In: Proceedings of the Conference on Complex Analysis Tianjin, 199 Z. Li, F. Y. Ren, L. Yang, S. Y. Zhang, eds., Conf. Proc. Lecture Notes Anal., Vol. 1, Int. Press, Massachusetts,

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