FUNCTIONS WITH NEGATIVE COEFFICIENTS
|
|
- Annabella Reed
- 6 years ago
- Views:
Transcription
1 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 Department of Information Technology Salalah College of Technology PO Box 608, Salalah PC211, Sultanate of Oman C. RAMACHANDRAN S. SIVASUBRAMANIAN Department of Mathematics Department of Mathematics College of Engineering, Easwari Engineering College, Anna University Ramapuram, Chennai , Chennai , Tamilnadu, India Tamilnadu, India Received: 31 May, 2007 Accepted: 15 June, 2007 Communicated by: Th.M. Rassias 2000 AMS Sub. Class.: Primary 30C45. Key words: Analytic functions; Univalent functions; Coefficient inequalities and coefficient estimates; Starlike functions; Convex functions; -to-convex functions; k- Uniformly convex functions; k-uniformly starlike functions; Uniformly starlike functions; Hadamard product (or convolution); Extreme points; Radii of close-toconvexity, Starlikeness and convexity; Integral operators. Page 1 of 29
2 Abstract: Acknowledgements: The main object of this paper is to introduce and investigate a subclass U(λ, α, β, k) of normalized analytic functions in the open unit disk, which generalizes the familiar class of uniformly convex functions. The various properties and characteristics for functions belonging to the class U(λ, α, β, k) derived here include (for example) a characterization theorem, coefficient inequalities and coefficient estimates, a distortion theorem and a covering theorem, extreme points, and the radii of close-to-convexity, starlikeness and convexity. Relevant connections of the results, which are presented in this paper, with various known results are also considered. The present investigation was supported, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP Page 2 of 29
3 1 Introduction and Motivation 4 2 A Characterization Theorem and Resulting Coefficient Estimates 6 3 Distortion and Covering Theorems for the Function Class U(λ, α, β, k) 11 4 Extreme Points of the Function Class U(λ, α, β, k) 16 5 Modified Hadamard Products (or Convolution) 20 6 Radii of -to-convexity, Starlikeness and Convexity 23 7 Hadamard Products and Integral Operators 27 Page 3 of 29
4 1. Introduction and Motivation Let A denote the class of functions f normalized by (1.1) f(z) = z + a n z n, which are analytic in the open unit disk = {z : z C and z < 1}. As usual, we denote by S the subclass of A consisting of functions which are univalent in. Suppose also that, for 0 α < 1, S (α) and K(α) denote the classes of functions in A which are, respectively, starlike of order α in and convex of order α in (see, for example, [11]). Finally, let T denote the subclass of S consisting of functions f given by (1.2) f(z) = z a n z n (a n 0) with negative coefficients. Silverman [9] introduced and investigated the following subclasses of the function class T : (1.3) T (α) := S (α) T and C(α) := K(α) T (0 α < 1). Definition 1. A function f T is said to be in the class U(λ, α, β, k) if it satisfies the following inequality: ( ) zf (z) R > k zf (1.4) (z) F (z) F (z) 1 + β (0 α λ 1; 0 β < 1; k 0), Page 4 of 29
5 where (1.5) F (z) := λαz 2 f (z) + (λ α)zf (z) + (1 λ + α)f(z). The above-defined function class U(λ, α, β, k) is of special interest and it contains many well-known as well as new classes of analytic univalent functions. In particular, U(λ, α, β, 0) is the class of functions with negative coefficients, which was introduced and studied recently by Kamali and Kadıoğlu [3], and U(λ, 0, β, 0) is the function class which was introduced and studied by Srivastava et al. [12] (see also Aqlan et al. [1]). We note that the class of k-uniformly convex functions was introduced and studied recently by Kanas and Wiśniowska [4]. Subsequently, Kanas and Wiśniowska [5] introduced and studied the class of k-uniformly starlike functions. The various properties of the above two function classes were extensively investigated by Kanas and Srivastava [6]. Furthermore, we have [cf. Equation (1.3)] (1.6) U(0, 0, β, 0) T (α) and U(1, 0, β, 0) C(α). We remark here that the classes of k-uniformly starlike functions and k-uniformly convex functions are an extension of the relatively more familiar classes of uniformly starlike functions and uniformly convex functions investigated earlier by (for example) Goodman [2], Rønning [8], and Ma and Minda [7] (see also the more recent contributions on these function classes by Srivastava and Mishra [10]). In our present investigation of the function class U(λ, α, β, k), we obtain a characterization theorem, coefficient inequalities and coefficient estimates, a distortion theorem and a covering theorem, extreme points, and the radii of close-to-convexity, starlikeness and convexity for functions belonging to the class U(λ, α, β, k). Page 5 of 29
6 2. A Characterization Theorem and Resulting Coefficient Estimates We employ the technique adopted by Aqlan et al. [1] to find the coefficient estimates for the function class U(λ, α, β, k). Our main characterization theorem for this function class is stated as Theorem 1 below. Theorem 1. A function f T given by (1.2) is in the class U(λ, α, β, k) if and only if (2.1) {n(k + 1) (k + β)} {(n 1)(nλα + λ α) + 1} a n (0 α λ 1; 0 β < 1; k 0). The result is sharp for the function f(z) given by (2.2) f(z) = z {n(k + 1) (k + β)}{(n 1)(nλα + λ α) + 1} zn (n 2). Proof. By Definition 1, f U(λ, α, β, k) if and only if the condition (1.4) is satisfied. Since it is easily verified that R(ω) > k ω 1 + β R ( ω(1 + ke iθ ) ke iθ) > β ( π θ < π; 0 β < 1; k 0), Page 6 of 29 the inequality (1.4) may be rewritten as follows: ( ) zf (z) (2.3) R F (z) (1 + keiθ ) ke iθ > β
7 or, equivalently, ( ) zf (z)(1 + ke iθ ) F (z)ke iθ (2.4) R > β. F (z) Now, by setting (2.5) G(z) = zf (z)(1 + ke iθ ) F (z)ke iθ, the inequality (2.4) becomes equivalent to G(z) + ()F (z) > G(z) (1 + β)f (z) (0 β < 1), where F (z) and G(z) are defined by (1.5) and (2.5), respectively. We thus observe that G(z) + ()F (z) (2 β)z (n β + 1){(n 1)(nλα + λ α) + 1}a n z n (n 1){(n 1)(nλα + λ α) + 1}a n z n keiθ (2 β) z k (n β + 1){(n 1)(nλα + λ α) + 1}a n z n (n 1){(n 1)(nλα + λ α) + 1}a n z n (2 β) z {(n(k + 1) (k + β) + 1}{(n 1)(nλα + λ α) + 1}a n z n. Page 7 of 29
8 Similarly, we obtain G(z) (1 + β)f (z) β z + {(n(k + 1) (k + β) 1}{(n 1)(nλα + λ α) + 1}a n z n. Therefore, we have G(z) + ()F (z) G(z) (1 + β)f (z) 2() z 2 {(n(k + 1) (k + β)}{(n 1)(nλα + λ α) + 1}a n z n 0, which implies the inequality (2.1) asserted by Theorem 1. Conversely, by setting 0 z = r < 1, and choosing the values of z on the positive real axis, the inequality (2.3) reduces to the following form: (1 β) {(n β) ke iθ (n 1)}{(n 1)(nλα+λ α)+1}a n r n 1 (2.6) R 1 0, (n 1){(n 1)(nλα+λ α)+1}a n r n 1 which, in view of the elementary identity: Page 8 of 29 R( e iθ ) e iθ = 1,
9 becomes (1 β) {(n β) k(n 1)}{(n 1)(nλα+λ α)+1}a n r n 1 (2.7) R 1 0. (n 1){(n 1)(nλα+λ α)+1}a n r n 1 Finally, upon letting r 1 in (2.7), we get the desired result. By taking α = 0 and k = 0 in Theorem 1, we can deduce the following corollary. Corollary 1. Let f T be given by (1.2). Then f U(λ, 0, β, 0) if and only if (n β){(n 1)λ + 1}a n. By setting α = 0, λ = 1 and k = 0 in Theorem 1, we get the following corollary. Corollary 2 (Silverman [9]). Let f T be given by (1.2). Then f C(β) if and only if n(n β)a n. The following coefficient estimates for f U(λ, α, β, k) is an immediate consequence of Theorem 1. Theorem 2. If f U(λ, α, β, k) is given by (1.2), then (2.8) a n {n(k + 1) (k + β)}{(n 1)(nλα + λ α) + 1} (0 α λ 1; 0 β < 1; k 0). Equality in (2.8) holds true for the function f(z) given by (2.2). (n 2) Page 9 of 29
10 By taking α = k = 0 in Theorem 2, we obtain the following corollary. Corollary 3. Let f T be given by (1.2). Then f U(λ, 0, β, 0) if and only if (2.9) a n (n β){(n 1)λ + 1} (n 2). Equality in (2.9) holds true for the function f(z) given by (2.10) f(z) = z (n β){(n 1)λ + 1} zn (n 2). Lastly, if we set α = 0, λ = 1 and k = 0 in Theorem 1, we get the following familiar result. Corollary 4 (Silverman [9]). Let f T be given by (1.2). Then f C(β) if and only if (2.11) a n n(n β) (n 2). Equality in (2.11) holds true for the function f(z) given by (2.12) f(z) = z n(n β) zn (n 2). Page 10 of 29
11 3. Distortion and Covering Theorems for the Function Class U(λ, α, β, k) Theorem 3. If f U(λ, α, β, k), then (3.1) r (2 + k β)(2λα + λ α) r2 f(z) r + (2 + k β)(2λα + λ α) r2 ( z = r < 1). Equality in (3) holds true for the function f(z) given by (3.2) f(z) = z (2 + k β)(2λα + λ α) z2. Proof. We only prove the second part of the inequality in (3), since the first part can be derived by using similar arguments. Since f U(λ, α, β, k), by using Theorem 1, we find that (2 + k β)(2λα + λ α + 1) = a n (2 + k β)(2λα + λ α + 1)a n {n(k + 1) (k + β)} {(n 1)(nλα + λ α) + 1} a n, Page 11 of 29
12 which readily yields the following inequality: (3.3) a n Moreover, it follows from (1.2) and (3.3) that f(z) = z a n z n (2 + k β)(2λα + λ α + 1). z + z 2 r + r 2 r + a n a n (2 + k β)(2λα + λ α + 1) r2, which proves the second part of the inequality in (3). Theorem 4. If f U(λ, α, β, k), then (3.4) 1 2() (2 + k β)(2λα + λ α) r f (z) 2() 1 + r ( z = r < 1). (2 + k β)(2λα + λ α) Equality in (3.4) holds true for the function f(z) given by (3.2). Page 12 of 29
13 Proof. Our proof of Theorem 4 is much akin to that of Theorem 3. Indeed, since f U(λ, α, β, k), it is easily verified from (1.2) that (3.5) f (z) 1 + and (3.6) f (z) 1 na n z n r na n z n r na n na n. The assertion (3.4) of Theorem 4 would now follow from (3.5) and (3.6) by means of a rather simple consequence of (3.3) given by (3.7) na n This completes the proof of Theorem 4. 2() (2 + k β)(2λα + λ α + 1). Theorem 5. If f U(λ, α, β, k), then f T (δ), where δ := 1 (2 + k β)(2λα + λ α) (). The result is sharp with the extremal function f(z) given by (3.2). Page 13 of 29 Proof. It is sufficient to show that (2.1) implies that (3.8) (n δ)a n 1 δ,
14 that is, that (3.9) n δ 1 δ {n(k + 1) (k + β)}{(n 1)(nλα + λ α) + 1} Since (3.9) is equivalent to the following inequality: (n 1)() δ 1 {n(k + 1) (k + β)}{(n 1)(nλα + λ α) + 1} () =: Ψ(n), and since (3.9) holds true for Ψ(n) Ψ(2) (n 2), n 2, 0 λ 1, 0 α 1, 0 β < 1 and k 0. This completes the proof of Theorem 5. By setting α = k = 0 in Theorem 5, we can deduce the following result. Corollary 5. If f U(λ, α, β, k), then ( ) λ(2 β) + β f T. λ(2 β) + 1 This result is sharp for the extremal function f(z) given by f(z) = z (λ + 1)(2 β) z2. (n 2), (n 2) Page 14 of 29
15 For the choices α = 0, λ = 1 and k = 0 in Theorem 5, we obtain the following result of Silverman [9]. Corollary 6. If f C(β), then ( ) 2 f T. 3 β This result is sharp for the extremal function f(z) given by f(z) = z 2(2 β) z2. Page 15 of 29
16 4. Extreme Points of the Function Class U(λ, α, β, k) Theorem 6. Let f 1 (z) = z and f n (z) = z {n(k + 1) (k + β)}{(n 1)(nλα + λ α) + 1} zn (n 2). Then f U(λ, α, β, k) if and only if it can be represented in the form: ( ) (4.1) f(z) = µ n f n (z) µ n 0; µ n = 1. n=1 Proof. Suppose that the function f(z) can be written as in (4.1). Then ) f(z) = µ n (z {n(k + 1) (k + β)}{(n 1)(nλα + λ α) + 1} zn n=1 ( ) = z µ n z n. {n(k + 1) (k + β)}{(n 1)(nλα + λ α) + 1} Now ( ) {n(k + 1) (k + β)}{(n 1)(nλα + λ α) + 1}() µ n (){n(k + 1) (k + β)}{(n 1)(nλα + λ α) + 1} = µ n n=1 Page 16 of 29 = 1 µ 1 1,
17 which implies that f U(λ, α, β, k). Conversely, we suppose that f U(λ, α, β, k). Then, by Theorem 2, we have a n {n(k + 1) (k + β)}{(n 1)(nλα + λ α) + 1} (n 2). Therefore, we may write and Then µ n = {n(k + 1) (k + β)}{(n 1)(nλα + λ α) + 1} µ 1 = 1 f(z) = µ n. µ n f n (z), with f n (z) given as in (4.1). This completes the proof of Theorem 6. n=1 a n (n 2) Corollary 7. The extreme points of the function class f U(λ, α, β, k) are the functions f 1 (z) = z and f n (z) = z {n(k + 1) (k + β)}{(n 1)(nλα + λ α) + 1} zn (n 2). Page 17 of 29 For α = k = 0 in Corollary 7, we have the following result.
18 Corollary 8. The extreme points of f U(λ, 0, β, 0) are the functions f 1 (z) = z and f n (z) = z {n β}{(n 1)λ + 1} zn (n 2). By setting α = 0, λ = 1 and k = 0 in Corollary 7, we obtain the following result obtained by Silverman [9]. Corollary 9. The extreme points of the class C(β) are the functions f 1 (z) = z and f n (z) = z n(n β) zn (n 2). Theorem 7. The class U(λ, α, β, k) is a convex set. Proof. Suppose that each of the functions f j (z) (j = 1, 2) given by (4.2) f j (z) = z a n,j z n (a n,j 0; j = 1, 2) is in the class U(λ, α, β, k). It is sufficient to show that the function g(z) defined by g(z) = µf 1 (z) + (1 µ)f 2 (z) (0 µ 1) is also in the class U(λ, α, β, k). Since g(z) = z [µa n,1 + (1 µ)a n,2 ]z n, Page 18 of 29
19 with the aid of Theorem 1, we have (4.3) {n(k + 1) (k + β)}{(n 1)(nλα + λ α) + 1} [µa n,1 + (1 µ)a n,2 ] µ {n(k + 1) (k + β)}{(n 1)(nλα + λ α) + 1}a n,1 + (1 µ) {n(k + 1) (k + β)}{(n 1)(nλα + λ α) + 1}a n,2 µ() + (1 µ)(), which implies that g U(λ, α, β, k). Hence U(λ, α, β, k) is indeed a convex set. Page 19 of 29
20 5. Modified Hadamard Products (or Convolution) For functions f(z) = a n z n and g(z) = n=0 b n z n, the Hadamard product (or convolution) (f g)(z) is defined, as usual, by (5.1) (f g)(z) := a n b n z n =: (g f)(z). On the other hand, for functions f j (z) = z n=0 n=0 a n,j z n (j = 1, 2) in the class T, we define the modified Hadamard product (or convolution) as follows: (5.2) (f 1 f 2 )(z) := z Then we have the following result. a n,1 a n,2 z n =: (f 2 f 1 )(z). Theorem 8. If f j (z) U(λ, α, β, k) (j = 1, 2), then (f 1 f 2 )(z) U(λ, α, β, k, ξ), where (2 β){2 + k β}{2λα + λ α + 1} 2()2 ξ := (2 β){2 + k β}{2λα + λ α + 1} (). 2 The result is sharp for the functions f j (z) (j = 1, 2) given as in (3.2). Page 20 of 29
21 Proof. Since f j (z) U(λ, α, β, k) (j = 1, 2), we have (5.3) {n(k+1) (k+β)}{(n 1)(nλα+λ α)+1}a n,j 1 β (j = 1, 2), which, in view of the Cauchy-Schwarz inequality, yields {n(k + 1) (k + β)}{(n 1)(nλα + λ α) + 1} (5.4) an,1 a n,2 1. We need to find the largest ξ such that {n(k + 1) (k + ξ)}{(n 1)(nλα + λ α) + 1} (5.5) a n,1 a n, ξ Thus, in light of (5.4) and (5.5), whenever the following inequality: n ξ an,1 a n,2 n β 1 ξ (n 2) holds true, the inequality (5.5) is satisfied. We find from (5.4) that (5.6) an,1 a n,2 {n(k + 1) (k + β)}{(n 1)(nλα + λ α) + 1} Thus, if ( ) ( ) n ξ 1 ξ {n(k + 1) (k + β)}{(n 1)(nλα + λ α) + 1} or, if n β (n 2). (n 2), Page 21 of 29 ξ (n β){n(k + 1) (k + β)}{(n 1)(nλα + λ α) + 1} n()2 (n β){n(k + 1) (k + β)}{(n 1)(nλα + λ α) + 1} () 2 (n 2),
22 then (5.4) is satisfied. Setting Φ(n) := (n β){n(k + 1) (k + β)}{(n 1)(nλα + λ α) + 1} n()2 (n β){n(k + 1) (k + β)}{(n 1)(nλα + λ α) + 1} () 2 (n 2), we see that Φ(n) is an increasing function for n 2. This implies that ξ Φ(2) = (2 β){2 + k β}{2λα + λ α + 1} 2()2 (2 β){2 + k β}{2λα + λ α + 1} () 2. Finally, by taking each of the functions f j (z) (j = 1, 2) given as in (3.2), we see that the assertion of Theorem 8 is sharp. Page 22 of 29
23 6. Radii of -to-convexity, Starlikeness and Convexity Theorem 9. Let the function f(z) defined by (1.2) be in the class U(λ, α, β, k). Then f(z) is close-to-convex of order ρ (0 ρ < 1) in z < r 1 (λ, α, β, ρ, k), where r 1 (λ, α, β, ρ, k) ( (1 ρ){n(k + 1) (k + β)}{(n 1)(nλα + λ α) + 1} := inf n n() The result is sharp for the function f(z) given by (2.2). Proof. It is sufficient to show that f (z) 1 1 ρ Since (6.1) f (z) 1 = na n z n 1 we have if (6.2) ) 1 n 1 ( 0 ρ < 1; z < r1 (λ, α, β, ρ, k) ). na n z n 1, f (z) 1 1 ρ (0 ρ < 1), ( ) n a n z n ρ Hence, by Theorem 1, (6.2) will hold true if ( ) n z n 1 {n(k + 1) (k + β)}{(n 1)(nλα + λ α) + 1}, 1 ρ (n 2). Page 23 of 29
24 that is, if ( (1 ρ){n(k+1) (k+β)}{(n 1)(nλα+λ α)+1} (6.3) z n(1 β) The assertion of Theorem 9 would now follow easily from (6.3). ) 1 n 1 (n 2). Theorem 10. Let the function f(z) defined by (1.2) be in the class U(λ, α, β, k). Then f(z) is starlike of order ρ (0 ρ < 1) in z < r 2 (λ, α, β, ρ, k), where r 2 (λ, α, β, ρ, k) ( (1 ρ){n(k + 1) (k + β)}{(n 1)(nλα + λ α) + 1} := inf n (n ρ)() The result is sharp for the function f(z) given by (2.2). Proof. It is sufficient to show that zf (z) f(z) 1 1 ρ ( 0 ρ < 1; z < r 2 (λ, α, β, ρ, k) ). Since (6.4) we have zf (z) f(z) 1 (n 1)a n z n 1 1, a n z n 1 zf (z) f(z) 1 1 ρ (0 ρ < 1), ) 1 n 1 (n 2). Page 24 of 29
25 if (6.5) ( ) n ρ a n z n ρ Hence, by Theorem 1, (6.5) will hold true if ( ) n ρ z n 1 {n(k + 1) (k + β)}{(n 1)(nλα + λ α) + 1}, 1 ρ that is, if ( (1 ρ){n(k+1) (k+β)}{(n 1)(nλα+λ α)+1} (6.6) z (n ρ)(1 β) The assertion of Theorem 10 would now follow easily from (6.6). ) 1 n 1 (n 2). Theorem 11. Let the function f(z) defined by (1.2) be in the class U(λ, α, β, k). Then f(z) is convex of order ρ (0 ρ < 1) in z < r 3 (λ, α, β, ρ, k), where r 3 (λ, α, β, ρ, k) ( (1 ρ){n(k + 1) (k + β)}{(n 1)(nλα + λ α) + 1} := inf n n(n ρ)() The result is sharp for the function f(z) given by (2.2). Proof. It is sufficient to show that zf (z) f (z) 1 ρ ( 0 ρ < 1; z < r3 (λ, α, β, ρ, k) ). ) 1 n 1 (n 2). Page 25 of 29
26 Since (6.7) we have if (6.8) zf (z) f (z) zf (z) f (z) n(n 1)a n z n 1 1, na n z n 1 1 ρ (0 ρ < 1), ( ) n(n ρ) a n z n ρ Hence, by Theorem 1, (6.8) will hold true if ( ) n(n ρ) z n 1 {n(k + 1) (k + β)}{(n 1)(nλα + λ α) + 1}, 1 ρ that is, if ( (1 ρ){n(k+1) (k+β)}{(n 1)(nλα+λ α)+1} (6.9) z n(n ρ)(1 β) Theorem 11 now follows easily from (6.9). ) 1 n 1 (n 2). Page 26 of 29
27 7. Hadamard Products and Integral Operators Theorem 12. Let f U(λ, α, β, k). Suppose also that (7.1) g(z) = z + g n z n (0 g n 1). Then f g U(λ, α, β, k). Proof. Since 0 g n 1 (n 2), {n(k + 1) (k + β)}{(n 1)(nλα + λ α) + 1}a n g n (7.2) {n(k + 1) (k + β)}{(n 1)(nλα + λ α) + 1}a n, which completes the proof of Theorem 12. Corollary 10. If f U(λ, α, β, k), then the function F(z) defined by (7.3) F(z) := 1 + c z t c 1 f(t)dt (c > 1) z c is also in the class U(λ, α, β, k). Proof. Since F(z) = z + 0 ( ) ( c + 1 z n 0 < c + 1 ) c + n c + n < 1, the result asserted by Corollary 10 follows from Theorem 12. Page 27 of 29
28 References [1] E. AQLAN, J.M. JAHANGIRI AND S.R. KULKARNI, Classes of k-uniformly convex and starlike functions, Tamkang J. Math., 35 (2004), 1 7. [2] A.W. GOODMAN, On uniformly convex functions, Ann. Polon. Math., 56 (1991), [3] M. KAMALI AND E. KADIOĞLU, On a new subclass of certain starlike functions with negative coefficients, Atti Sem. Mat. Fis. Univ. Modena, 48 (2000), [4] S. KANAS AND A. WIŚNIOWSKA, Conic regions and k-uniform convexity, J. Comput. Appl. Math., 105 (1999), [5] S. KANAS AND A. WIŚNIOWSKA, Conic domains and starlike functions, Rev. Roumaine Math. Pures Appl., 45 (2000), [6] S. KANAS AND H.M. SRIVASTAVA, Linear operators associated with k- uniformly convex functions, Integral Transform. Spec. Funct., 9 (2000), [7] W. MA AND D. MINDA, Uniformly convex functions. II, Ann. Polon. Math., 58 (1993), [8] F. RØNNING, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc., 118 (1993), [9] H. SILVERMAN, Univalent functions with negative coefficients, Proc. Amer. Math. Soc., 51 (1975), Page 28 of 29
29 [10] H.M. SRIVASTAVA AND A.K. MISHRA, Applications of fractional calculus to parabolic starlike and uniformly convex functions, Comput. Math. Appl., 39 (3 4) (2000), [11] H.M. SRIVASTAVA AND S. OWA (Editors), Current Topics in Analytic Function Theory, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, [12] H.M. SRIVASTAVA, S. OWA AND S.K. CHATTERJEA, A note on certain classes of starlike functions, Rend. Sem. Mat. Univ Padova, 77 (1987), Page 29 of 29
SOME 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 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 informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied Mathematics INTEGRAL MEANS INEQUALITIES FOR FRACTIONAL DERIVATIVES OF SOME GENERAL SUBCLASSES OF ANALYTIC FUNCTIONS TADAYUKI SEKINE, KAZUYUKI TSURUMI, SHIGEYOSHI
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 informationA New Subclasses of Meromorphic p-valent Functions with Positive Coefficient Defined by Fractional Calculus Operators
Volume 119 No. 15 2018, 2447-2461 ISSN: 1314-3395 (on-line version) url: http://www.acadpubl.eu/hub/ http://www.acadpubl.eu/hub/ A New Subclasses of Meromorphic p-valent Functions with Positive Coefficient
More informationMeromorphic parabolic starlike functions with a fixed point involving Srivastava-Attiya operator
Malaya J. Mat. 2(2)(2014) 91 102 Meromorphic parabolic starlike functions with a fixed point involving Srivastava-Attiya operator G. Murugusundaramoorthy a, and T. Janani b a,b School of Advanced Sciences,
More informationOn neighborhoods of functions associated with conic domains
DOI: 0.55/auom-205-0020 An. Şt. Univ. Ovidius Constanţa Vol. 23(),205, 29 30 On neighborhoods of functions associated with conic domains Nihat Yağmur Abstract Let k ST [A, B], k 0, B < A be the class of
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 informationDIFFERENTIAL SANDWICH THEOREMS FOR SOME SUBCLASSES OF ANALYTIC FUNCTIONS INVOLVING A LINEAR OPERATOR. 1. Introduction
Acta Math. Univ. Comenianae Vol. LXXVI, 2(2007), pp. 287 294 287 DIFFERENTIAL SANDWICH THEOREMS FOR SOME SUBCLASSES OF ANALYTIC FUNCTIONS INVOLVING A LINEAR OPERATOR T. N. SHAMMUGAM, C. RAMACHANDRAN, M.
More informationA Certain Subclass of Multivalent Functions Involving Higher-Order Derivatives
Filomat 30:1 (2016), 113 124 DOI 10.2298/FIL1601113S Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat A Certain Subclass of Multivalent
More informationDIFFERENTIAL SANDWICH THEOREMS FOR SOME SUBCLASSES OF ANALYTIC FUNCTIONS INVOLVING A LINEAR OPERATOR. 1. Introduction
DIFFERENTIAL SANDWICH THEOREMS FOR SOME SUBCLASSES OF ANALYTIC FUNCTIONS INVOLVING A LINEAR OPERATOR T. N. SHAMMUGAM, C. RAMACHANDRAN, M. DARUS S. SIVASUBRAMANIAN Abstract. By making use of the familiar
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 informationJournal of Inequalities in Pure and Applied Mathematics
Journal of Inequalities in Pure and Applied Mathematics APPLICATIONS OF DIFFERENTIAL SUBORDINATION TO CERTAIN SUBCLASSES OF MEROMORPHICALLY MULTIVALENT FUNCTIONS H.M. SRIVASTAVA AND J. PATEL Department
More informationSome Further Properties for Analytic Functions. with Varying Argument Defined by. Hadamard Products
International Mathematical Forum, Vol. 10, 2015, no. 2, 75-93 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2015.412205 Some Further Properties for Analytic Functions with Varying Argument
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 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 informationSUBCLASSES OF P-VALENT STARLIKE FUNCTIONS DEFINED BY USING CERTAIN FRACTIONAL DERIVATIVE OPERATOR
Sutra: International Journal of Mathematical Science Education Technomathematics Research Foundation Vol. 4 No. 1, pp. 17-32, 2011 SUBCLASSES OF P-VALENT STARLIKE FUNCTIONS DEFINED BY USING CERTAIN FRACTIONAL
More informationACTA UNIVERSITATIS APULENSIS No 18/2009 SOME SUBCLASS OF ANALYTIC FUNCTIONS. Firas Ghanim and Maslina Darus
ACTA UNIVERSITATIS APULENSIS No 18/2009 SOME SUBCLASS OF ANALYTIC FUNCTIONS Firas Ghanim and Maslina Darus Abstract. In this paper we introduce a new class M (α, β, γ, A, λ) consisting analytic and univalent
More informationON CERTAIN SUBCLASSES OF UNIVALENT FUNCTIONS AND RADIUS PROPERTIES
ON CERTAIN SUBCLASSES OF UNIVALENT FUNCTIONS AND RADIUS PROPERTIES M. OBRADOVIĆ and S. PONNUSAMY Let S denote the class of normalied univalent functions f in the unit disk. One of the problems addressed
More informationSubclass of Meromorphic Functions with Positive Coefficients Defined by Frasin and Darus Operator
Int. J. Open Problems Complex Analysis, Vol. 8, No. 1, March 2016 ISSN 2074-2827; Copyright c ICSRS Publication, 2016 www.i-csrs.org Subclass of Meromorphic Functions with Positive Coefficients Defined
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 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 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 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 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 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 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 informationSOME CLASSES OF MEROMORPHIC MULTIVALENT FUNCTIONS WITH POSITIVE COEFFICIENTS INVOLVING CERTAIN LINEAR OPERATOR
International Journal of Basic & Alied Sciences IJBAS-IJENS Vol:13 No:03 56 SOME CLASSES OF MEROMORPHIC MULTIVALENT FUNCTIONS WITH POSITIVE COEFFICIENTS INVOLVING CERTAIN LINEAR OPERATOR Abdul Rahman S
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 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 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 informationOn a New Subclass of Meromorphically Multivalent Functions Defined by Linear Operator
International Mathematical Forum, Vol. 8, 213, no. 15, 713-726 HIKARI Ltd, www.m-hikari.com On a New Subclass of Meromorphically Multivalent Functions Defined by Linear Operator Ali Hussein Battor, Waggas
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 informationSome Geometric Properties of a Certain Subclass of Univalent Functions Defined by Differential Subordination Property
Gen. Math. Notes Vol. 20 No. 2 February 2014 pp. 79-94 SSN 2219-7184; Copyright CSRS Publication 2014 www.i-csrs.org Available free online at http://www.geman.in Some Geometric Properties of a Certain
More informationA Note on Coefficient Inequalities for (j, i)-symmetrical Functions with Conic Regions
BULLETIN OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) 2303-4874 ISSN (o) 2303-4955 wwwimviblorg /JOURNALS / BULLETIN Vol 6(2016) 77-87 Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA
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 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 informationNEW SUBCLASS OF MULTIVALENT HYPERGEOMETRIC MEROMORPHIC FUNCTIONS
Kragujevac Journal of Mathematics Volume 42(1) (2018), Pages 83 95. NEW SUBCLASS OF MULTIVALENT HYPERGEOMETRIC MEROMORPHIC FUNCTIONS M. ALBEHBAH 1 AND M. DARUS 2 Abstract. In this aer, we introduce a new
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 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 informationResearch Article Certain Subclasses of Multivalent Functions Defined by Higher-Order Derivative
Hindawi Function Spaces Volume 217 Article ID 5739196 6 pages https://doi.org/1.1155/217/5739196 Research Article Certain Subclasses of Multivalent Functions Defined by Higher-Order Derivative Xiaofei
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 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 informationarxiv: v1 [math.cv] 22 Apr 2009
arxiv:94.3457v [math.cv] 22 Apr 29 CERTAIN SUBCLASSES OF CONVEX FUNCTIONS WITH POSITIVE AND MISSING COEFFICIENTS BY USING A FIXED POINT SH. NAJAFZADEH, M. ESHAGHI GORDJI AND A. EBADIAN Abstract. By considering
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 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 informationCoefficient Inequalities for Classes of Uniformly. Starlike and Convex Functions Defined by Generalized. Ruscheweyh Operator
International Mathematical Forum, Vol. 6, 2011, no. 65, 3227-3234 Coefficient Inequalities for Classes of Uniformly Starlike Convex Functions Defined by Generalized Ruscheweyh Operator Hesam Mahzoon Department
More informationConvolution Properties of Convex Harmonic Functions
Int. J. Open Problems Complex Analysis, Vol. 4, No. 3, November 01 ISSN 074-87; Copyright c ICSRS Publication, 01 www.i-csrs.org Convolution Properties of Convex Harmonic Functions Raj Kumar, Sushma Gupta
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 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 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 informationCoefficients estimates of some subclasses of analytic functions related with conic domain
DOI: 10.2478/auom-2013-0031 An. Şt. Univ. Ovidius Constanţa Vol. 212),2013, 181 188 Coefficients estimates of some subclasses of analytic functions related with conic domain Abstract In this paper, the
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 informationOn a new class of (j, i)-symmetric function on conic regions
Available online at wwwisr-publicationscom/jnsa J Nonlinear Sci Appl 10 (2017) 4628 4637 Research Article Journal Homepage: wwwtjnsacom - wwwisr-publicationscom/jnsa On a new class of (j i)-symmetric function
More informationResearch Article Integral Means Inequalities for Fractional Derivatives of a Unified Subclass of Prestarlike Functions with Negative Coefficients
Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 27, Article ID 97135, 9 pages doi:1.1155/27/97135 Research Article Integral Means Inequalities for Fractional Derivatives
More informationResearch Article A New Class of Meromorphically Analytic Functions with Applications to the Generalized Hypergeometric Functions
Abstract and Applied Analysis Volume 20, Article ID 59405, 0 pages doi:0.55/20/59405 Research Article A New Class of Meromorphically Analytic Functions with Applications to the Generalized Hypergeometric
More informationConvolution and Subordination Properties of Analytic Functions with Bounded Radius Rotations
International Mathematical Forum, Vol., 7, no. 4, 59-7 HIKARI Ltd, www.m-hikari.com https://doi.org/.988/imf.7.665 Convolution and Subordination Properties of Analytic Functions with Bounded Radius Rotations
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 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 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 informationStarlike Functions of Complex Order
Applied Mathematical Sciences, Vol. 3, 2009, no. 12, 557-564 Starlike Functions of Complex Order Aini Janteng School of Science and Technology Universiti Malaysia Sabah, Locked Bag No. 2073 88999 Kota
More informationResearch Article A Subclass of Analytic Functions Related to k-uniformly Convex and Starlike Functions
Hindawi Function Spaces Volume 2017, Article ID 9010964, 7 pages https://doi.org/10.1155/2017/9010964 Research Article A Subclass of Analytic Functions Related to k-uniformly Convex and Starlike Functions
More informationarxiv: v2 [math.cv] 24 Nov 2017
adii of the β uniformly convex of order α of Lommel and Struve functions Sercan Topkaya, Erhan Deniz and Murat Çağlar arxiv:70.0975v [math.cv] 4 Nov 07 Department of Mathematics, Faculty of Science and
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 informationa n z n, (1.1) As usual, we denote by S the subclass of A consisting of functions which are also univalent in U.
MATEMATIQKI VESNIK 65, 3 (2013), 373 382 September 2013 originalni nauqni rad researh paper CERTAIN SUFFICIENT CONDITIONS FOR A SUBCLASS OF ANALYTIC FUNCTIONS ASSOCIATED WITH HOHLOV OPERATOR S. Sivasubramanian,
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 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 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 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 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 informationOn Certain Class of Meromorphically Multivalent Reciprocal Starlike Functions Associated with the Liu-Srivastava Operator Defined by Subordination
Filomat 8:9 014, 1965 198 DOI 10.98/FIL1409965M Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat On Certain Class of Meromorphically
More informationSECOND HANKEL DETERMINANT FOR A CERTAIN SUBCLASS OF ANALYTIC AND BI-UNIVALENT FUNCTIONS
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
More informationDIFFERENTIAL SUBORDINATION FOR MEROMORPHIC MULTIVALENT QUASI-CONVEX FUNCTIONS. Maslina Darus and Imran Faisal. 1. Introduction and preliminaries
FACTA UNIVERSITATIS (NIŠ) SER. MATH. INFORM. 26 (2011), 1 7 DIFFERENTIAL SUBORDINATION FOR MEROMORPHIC MULTIVALENT QUASI-CONVEX FUNCTIONS Maslina Darus and Imran Faisal Abstract. An attempt has been made
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 informationResearch Article A Study of Cho-Kwon-Srivastava Operator with Applications to Generalized Hypergeometric Functions
International Mathematics and Mathematical Sciences, Article ID 374821, 6 pages http://dx.doi.org/10.1155/2014/374821 Research Article A Study of Cho-Kwon-Srivastava Operator with Applications to Generalized
More informationAn Application of Wilf's Subordinating Factor Sequence on Certain Subclasses of Analytic Functions
International Journal of Applied Engineering Research ISS 0973-4562 Volume 3 umber 6 (208 pp. 2494-2500 An Application of Wilf's Subordinating Factor Sequence on Certain Subclasses of Analytic Functions
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 informationOn a class of analytic functions related to Hadamard products
General Mathematics Vol. 15, Nr. 2 3(2007), 118-131 On a class of analytic functions related to Hadamard products Maslina Darus Abstract In this paper, we introduce a new class of analytic functions which
More informationON POLYHARMONIC UNIVALENT MAPPINGS
ON POLYHARMONIC UNIVALENT MAPPINGS J. CHEN, A. RASILA and X. WANG In this paper, we introduce a class of complex-valued polyharmonic mappings, denoted by HS pλ, and its subclass HS 0 pλ, where λ [0, ]
More informationHarmonic multivalent meromorphic functions. defined by an integral operator
Journal of Applied Mathematics & Bioinformatics, vol.2, no.3, 2012, 99-114 ISSN: 1792-6602 (print), 1792-6939 (online) Scienpress Ltd, 2012 Harmonic multivalent meromorphic functions defined by an integral
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 informationDifferential subordination theorems for new classes of meromorphic multivalent Quasi-Convex functions and some applications
Int. J. Adv. Appl. Math. and Mech. 2(3) (2015) 126-133 (ISSN: 2347-2529) Journal homepage: www.ijaamm.com International Journal of Advances in Applied Mathematics and Mechanics Differential subordination
More informationSANDWICH-TYPE THEOREMS FOR A CLASS OF INTEGRAL OPERATORS ASSOCIATED WITH MEROMORPHIC FUNCTIONS
East Asian Mathematical Journal Vol. 28 (2012), No. 3, pp. 321 332 SANDWICH-TYPE THEOREMS FOR A CLASS OF INTEGRAL OPERATORS ASSOCIATED WITH MEROMORPHIC FUNCTIONS Nak Eun Cho Abstract. The purpose of the
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 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 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 informationDifferential subordination related to conic sections
J. Math. Anal. Appl. 317 006 650 658 www.elsevier.com/locate/jmaa Differential subordination related to conic sections Stanisława Kanas Department of Mathematics, Rzeszów University of Technology, W. Pola,
More informationNotes on Starlike log-harmonic Functions. of Order α
Int. Journal of Math. Analysis, Vol. 7, 203, no., 9-29 Notes on Starlike log-harmonic Functions of Order α Melike Aydoğan Department of Mathematics Işık University, Meşrutiyet Koyu Şile İstanbul, Turkey
More informationResearch Article Arc Length Inequality for a Certain Class of Analytic Functions Related to Conic Regions
Complex Analysis Volume 13, Article ID 47596, 4 pages http://dx.doi.org/1.1155/13/47596 Research Article Arc Length Inequality for a Certain Class of Analytic Functions Related to Conic Regions Wasim Ul-Haq,
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 informationA NEW CLASS OF MEROMORPHIC FUNCTIONS RELATED TO CHO-KWON-SRIVASTAVA OPERATOR. F. Ghanim and M. Darus. 1. Introduction
MATEMATIQKI VESNIK 66, 1 14, 9 18 March 14 originalni nauqni rad research paper A NEW CLASS OF MEROMORPHIC FUNCTIONS RELATED TO CHO-KWON-SRIVASTAVA OPERATOR F. Ghanim and M. Darus Abstract. In the present
More informationON A SUBCLASS OF HARMONIC UNIVALENT FUNCTIONS DEFINED BY CONVOLUTION AND INTEGRAL CONVOLUTION
International Journal of Pure and pplied Mathematics Volume 69 No. 3 2011, 255-264 ON SUBCLSS OF HRMONIC UNIVLENT FUNCTIONS DEFINED BY CONVOLUTION ND INTEGRL CONVOLUTION K.K. Dixit 1,.L. Patha 2, S. Porwal
More informationSome properties for α-starlike functions with respect to k-symmetric points of complex order
doi: 1.17951/a.217.71.1.1 A N N A L E S U N I V E R S I T A T I S M A R I A E C U R I E - S K Ł O D O W S K A L U B L I N P O L O N I A VOL. LXXI, NO. 1, 217 SECTIO A 1 9 H. E. DARWISH, A. Y. LASHIN and
More informationRadius of close-to-convexity and fully starlikeness of harmonic mappings
Complex Variables and Elliptic Equations, 014 Vol. 59, No. 4, 539 55, http://dx.doi.org/10.1080/17476933.01.759565 Radius of close-to-convexity and fully starlikeness of harmonic mappings David Kalaj a,
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 RESULTS ASSOCIATED WITH FRACTIONAL CALCULUS OPERATORS INVOLVING APPELL HYPERGEOMETRIC FUNCTION
Volume 29), Issue, Article 4, 7 pp. SOME RESULTS ASSOCIATED WITH FRACTIONAL CALCULUS OPERATORS INVOLVING APPELL HYPERGEOMETRIC FUNCTION R. K. RAINA / GANPATI VIHAR, OPPOSITE SECTOR 5 UDAIPUR 332, RAJASTHAN,
More informationResearch Article Properties of Certain Subclass of Multivalent Functions with Negative Coefficients
International Journal of Mathematics and Mathematical Sciences Volume 2012, Article ID 362312, 21 pages doi:10.1155/2012/362312 Research Article Properties of Certain Subclass of Multivalent Functions
More informationResearch Article A New Class of Meromorphic Functions Associated with Spirallike Functions
Applied Mathematics Volume 202, Article ID 49497, 2 pages doi:0.55/202/49497 Research Article A New Class of Meromorphic Functions Associated with Spirallike Functions Lei Shi, Zhi-Gang Wang, and Jing-Ping
More informationPROPERTIES AND CHARACTERISTICS OF A FAMILY CONSISTING OF BAZILEVIĆ (TYPE) FUNCTIONS SPECIFIED BY CERTAIN LINEAR OPERATORS
Electronic Journal of Mathematical Analysis and Applications Vol. 7) July 019, pp. 39-47. ISSN: 090-79X online) http://math-frac.org/journals/ejmaa/ PROPERTIES AND CHARACTERISTICS OF A FAMILY CONSISTING
More informationOn Multivalent Functions Associated with Fixed Second Coefficient and the Principle of Subordination
International Journal of Mathematical Analysis Vol. 9, 015, no. 18, 883-895 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ijma.015.538 On Multivalent Functions Associated with Fixed Second Coefficient
More information