ON A SUBCLASS OF UNIFORMLY CONVEX FUNCTIONS INVOLVING CHO-SRIVASTAVA OPERATOR

Novi Sad J. Math. Vol. 39, No. 1, 29, 47-56 ON A SUBCLASS OF UNIFORMLY CONVEX FUNCTIONS INVOLVING CHO-SRIVASTAVA OPERATOR G. Murugusundaramoorthy 1, S. Sivasubramanian 2, R. K. Raina 3 Abstract. The authors introduce a new subclass UH(α, β, γ, δ, λ, k) of functions which are analytic in the open disc = {z C : z < 1}. Various results studied include the coefficient estimates and distortion bounds, radii of close-to-convexity, starlikeness and convexity and integral means inequalities for functions belonging to the above class. Relevances of the main results are also briefly indicated. AMS Mathematics Subject Classification (2): 3C45 Key words and phrases: Analytic function, Starlike function, Convex function, Uniformly convex function, Convolution product, Cho-Srivastava operator 1. Introduction and Motivations Let A denote the class of functions of the form (1) f = z + a n z n, which are analytic in the open unit disc := {z C : z < 1}. Let S be a subclass of A consisting of univalent functions in. By S (β) and K(β), respectively, we mean the classes of analytic functions that satisfy the analytic conditions ( zf ) ( R > β, and R 1 + zf ) f f > β (z ) for β < 1. In particular, S = S () and K = K(), respectively, are the well-known standard classes of starlike and convex functions. Let T denote the subclass of S of functions of the form (2) f = z a n z n (a n ), 1 School of Science and Humanities, VIT University, Vellore-632 14, India, e-mail: gmsmoorthy@yahoo.com 2 Department of Mathematics, University College of Engineering, Tindivanam, Anna University-Chennai, Saram-64 37, India, e-mail: sivasaisastha@rediffmail.com 3 1/11, Ganpati Vihar, Opposite Sector 5, Udaipur 3132, Rajasthan, India, e-mail: rkraina 7@hotmail.com

48 G. Murugusundaramoorthy, S. Sivasubramanian, R. K. Raina which are analytic in the open unit disc, introduced and studied in [1]. Analogously to the subclasses S (β) and K(β) of S, respectively, the subclasses of T denoted by T (β) and C(β), β < 1, have also been investigated in [1]. For functions f A given by (1) and g A given by g = z + b nz n, we define the Hadamard product (or convolution) of f and g by (3) (f g) = z + a n b n z n (z ). Also, for functions f A, we recall the multiplier transformation I(λ, k) introduced by Cho and Srivastava [3] which is defined by (4) I(λ, k)f = z + Ψ n a n z n ( λ ; k Z), (5) Ψ n := and, obviously it follows that ( ) k n + λ 1 + λ (6) z (I(λ, k) f) = (1 + λ)i(λ, k + 1) f λi(λ, k) f. In the special case when λ = 1, the operators I(1, k) were studied earlier by Uralegaddi and Somanatha [15]. It may be observed that the operators I(λ, k) are closely related to the multiplier transformations studied by Flett [4] and also to the differential and integral operators investigated by Sălăgean [8]. For comprehensive details of various convolution operators which are related to the multiplier transformations of Flett [4], one may refer to the paper of Li and Srivastava [5] (as well as the references cited by therein). For the purpose of this paper, we now define a unified class of analytic functions which is based on the Cho-Srivastava operator (1.4). Definition 1. For δ γ 1, β < 1 and α, and for all z, we let the class UH(α, β, γ, δ, λ, k) consist of functions f T which satisfy the condition ( zf ) R F β > α zf (7) F 1, (8) and (9) (1) (11) F := F 1 + F 2 + F 3, F 1 := γδ (1 + λ) 2 I(λ, k + 2)f, F 2 := {γ δ γδ(1 + 2λ)} (1 + λ) I(λ, k + 1)f, F 3 := {1 (λ + 1)(γ δ γδλ)} I(λ, k)f, and I(λ, k)f is the Cho-Srivastava operator defined by (4).

On a subclass of uniformly convex functions... 49 The function class UH(α, β, γ, δ, λ, k) unifies many well known classes of analytic univalent functions. To illustrate, we observe that the class UH(α, β, γ, δ,, ) was studied by Kamali and Kadioglu [7] and the class UH(, β,, γ,, ) was studied by Altintas in [1]. Also, many other classes including UH(, β,,,, ) and UH(, β, 1,,, ) were investigated by Srivastava et al. [14]. We further note that the class UH(α, β,, γ,, ) is the known class of α uniformly convex functions of order β studied by Aqlan et al. [2] (also see [13]). In the present paper we obtain a characterization property giving coefficients estimates, distortion theorem and covering theorem, extreme points and radii of close-to-convexity, starlikeness and convexity and integral means inequalities for functions belonging to the class UH(α, β, γ, δ, λ, k). 2. Coefficient estimates and Distortion bounds Theorem 1. Let f T be given by (2), then f UH(α, β, γ, δ, λ, k) if and only if (12) [{n(α + 1) (α + β)(n 1)(nγδ + γ δ) + 1 Ψ n a n, function (13) f = z δ γ 1, β < 1 and α. The result is sharp for the {n(α + 1) (α + β)(n 1)(nγδ + γ δ) + 1} zn ( n 2). Proof. Following [2], we assert that f UH(α, β, γ, δ, λ, k) if and only if the condition (7) is satisfied and this is equivalent to { zf (1 + αe iθ ) F αe iθ } (14) R > β ( π θ < π). F By putting G = zf (1 + αe iθ ) F αe iθ, (14) is equivalent to G + ()F > G (1 + β)f ( β < 1), F is as defined in (8). Simple computations readily give G + ()F (2 β) z [{ }{ n(α + 1) (α + β) + 1 (n 1)(nγδ + γ δ) + 1 Ψ n a n z n and G (1 + β)f β z + [{ + n(α + 1) (α + β) (n 1)(nγδ + γ δ) + 1 Ψ n a n z n.

5 G. Murugusundaramoorthy, S. Sivasubramanian, R. K. Raina It follows that G + ()F G (1 + β)f 2() z [{ 2 n(α + 1) (α + β) (n 1)(nγδ + γ δ) + 1 Ψ n a n z n, which implies that f UH(α, β, γ, δ, λ, k). On the other hand, for all π θ < π, we assume that { zf } R F (1 + αeiθ ) αe iθ > β. Choosing the values of z on the positive real axis such that z = r < 1, and using the fact that R { e iθ } e iθ = 1, the above inequality can be written as [{ }{ (1 β) n(α+1) (α+β) (n 1)(nγδ+γ δ)+1 Ψ n a n r n 1 R 1 [{ (n 1)(nγδ + γ δ) + 1 Ψ n a n r n 1 which on letting r 1 yields the desired inequality (2.1)., Theorem 2. If f UH(α, β, γ, δ, λ, k), then (15) a n [{ (n 2), n(α + 1) (α + β) (n 1)(nγδ + γ δ) + 1 Ψn δ γ 1, function β < 1 and α. Equality in (15) holds for the (16) f = z [{ z n. n(α + 1) (α + β) (n 1)(nγδ + γ δ) + 1 Ψn It may be observed that for λ = k = δ = γ 1 = α =, Theorem 1 corresponds to the following results due to Silverman [1]. Corollary 1. ([1]) If f T, then f K(β) if and only if n(n β)a n. ( ) 2 Corollary 2. ([1]) If f K(β), then f T. The result is sharp 3 β for the extremal function f = z 2(2 β) z2.

On a subclass of uniformly convex functions... 51 Theorem 3. If f UH(α, β, γ, δ, λ, k), then f T (η), η = 1 [{ 2(α + 1) (α + β) 2γδ + γ δ + 1 Ψ2 (). This result is sharp with the extremal function given by Proof. f = z [{ 2(α + 1) (α + β) 2γδ + γ δ + 1 Ψ2 z 2. It is sufficient to show that (12) implies (n η)a n 1 η. In view of (2.4), we find that (17) [{ n η n(α + 1) (α + β) (n 1)(nγδ + γ δ) + 1 1 η Ψn For n 2, (17) is equivalent to (n 2). (n 1)() η 1 [{ }{ n(α + 1) (α + β) (n 1)(nγδ + γ δ) + 1 Ψn () = Φ(n), and since Φ(n) Φ(2)(n 2), therefore, (17) holds true for any δ γ 1, β < 1 and α. This completes the proof of Theorem 3. The following results give the growth and distortion bounds for the class of functions UH(α, β, γ, δ, λ, k) which can be established by adopting the well known methods of derivation and we omit the proof details. Theorem 4. If f UH(α, β, γ, δ, λ, k), then (z = re iθ ): (18) r B(α, β, γ, δ, λ)r 2 f r + B(α, β, γ, δ, λ)r 2, (19) B(α, β, γ, δ, λ) := [{ 2(α + 1) (α + β) 2γδ + γ δ + 1 and Ψ 2 is given by (5) Theorem 5. If f UH(α, β, γ, δ, λ, k), then ( z = r < 1): (2) 1 B(α, β, γ, δ, λ)r f 1 + B(α, β, γ, δ, λ)r, B(α, β, γ, δ, λ) is given by (2.8). The equality in Theorems 4 and 5 hold for the function given by f = z [{ z 2. 2(α + 1) (α + β) 2γδ + γ δ + 1 Ψ 2 Ψ 2

52 G. Murugusundaramoorthy, S. Sivasubramanian, R. K. Raina 3. Radii of close-to-convexity, starlikeness and convexity The following results giving the radii of convexity, starlikeness and convexity for a function f to belong to the class UH(α, β, γ, δ, λ, k) can be established by following similar lines of proof as given in [13] and [14]. We merely state here these results and omit their proof details. Theorem 6. Let the function f T be in the class UH(α, β, γ, δ, λ, k), then f is close-to-convex of order ρ ( ρ < 1) in z < r 1 (α, β, γ, δ, ρ), r 1 (α, β, γ, δ, ρ) = [ [ { (1 ρ) {n(α + 1) (α + β)} (n 1)(nγδ + γ δ) + 1 Ψn = inf n n() ] 1 n 1 for n 2 with Ψ n defined as in (5). The result is sharp for the function f given by (13). Theorem 7. Let the function f defined by (2) be in the class UH(α, β, γ, δ, λ, k), then f is starlike of order ρ ( ρ < 1) in z < r 2 (α, β, γ, δ, ρ), r 2 (α, β, γ, δ, ρ) = [ [ { (1 ρ) {n(α + 1) (α + β)} (n 1)(nγδ + γ δ) + 1 Ψn = inf n (n ρ)() ] 1 n 1 for n 2 with Ψ n defined as in (5). The result is sharp for the function f given by (13). Theorem 8. Let the function f defined by (2) be in the class UH(α, β, γ, δ, λ, k), then f is convex of order ρ ( ρ < 1) in z < r 3 (α, β, γ, δ, ρ), r 3 (α, β, γ, δ, ρ) = [ [{ (1 ρ) n(α + 1) (α + β) (n 1)(nγδ + γ δ) + 1 Ψn = inf n n(n ρ)() ] 1 n 1 for n 2 with Ψ n defined as in (5). The result is sharp for the function f given by (13). 4. Extreme points of the class UH(α, β, γ, δ, λ, k) Theorem 9. Let f 1 = z and (21) f n = z [ {n(α + 1) (α + β)} { (n 1)(nγδ + γ δ) + 1 Ψn z n,

On a subclass of uniformly convex functions... 53 for n 2 and Ψ n be as defined in (5). Then f UH(α, β, γ, δ, λ, k) if and only if it can be represented in the form (22) f = µ n f n (µ n ), n=1 µ n = 1. n=1 Proof. Suppose f is expressible in the form (22). Then { } f = z µ n [ { z n. {n(α + 1) (α + β)} (n 1)(nγδ + γ δ) + 1 Ψn Since µ n [ {n(α + 1) (α + β)} { (n 1)(nγδ + γ δ) + 1 Ψn () () [ {n(α + 1) (α + β) (n 1)(nγδ + γ δ) + 1 Ψ n = µ n = 1 µ 1 1, which implies that f UH(α, β, γ, δ, λ, k). Conversely, suppose f UH(α, β, γ, δ, λ, k). Using (15), we may write [ { {n(α + 1) (α + β)} (n 1)(nγδ + γ δ) + 1 Ψn µ n = a n (n 2) and µ 1 = 1 µ n. This gives f = µ n f n, f n is given by (21). n=1 Corollary 3. The extreme points of f UH(α, β, γ, δ, λ, k) are the functions f 1 = z and f n = z [ { z n (n 2). {n(α + 1) (α + β)} (n 1)(nγδ + γ δ) + 1 Ψn Theorem 1. The class UH(α, β, γ, δ, λ, k) is a convex set. Proof. (23) Suppose the functions f j = z a n, j z n (a n, j ; j = 1, 2) be in the class UH(α, β, γ, δ, λ, k). It sufficient to show that the function g defined by g = µf 1 + (1 µ)f 2 ( µ 1)

54 G. Murugusundaramoorthy, S. Sivasubramanian, R. K. Raina is in the class UH(α, β, γ, δ, λ, k). Since g = z and applying Theorem 1, we get [µa n,1 + (1 µ)a n,2 ]z n, [ { {n(α + 1) (α + β)} ( (n 1)(nγδ + γ δ) + 1 Ψ n [µa n,1 +(1 µ)a n,2 ] µ() + (1 µ)(), which asserts that g UH(α, β, γ, δ, λ, k). Hence UH(α, β, γ, δ, λ, k) is convex. 5. Integral Means Inequalities Lemma 1. ([6]) If the functions f and g are analytic in with g f, then for η >, and < r < 1: (24) 2π g(re iθ ) 2π η dθ f(re iθ ) η dθ. In [1], Silverman found that the function f 2 = z z2 2 is often extremal over the family T. He applied this function to obtain the following integral means inequality (which was conjectured in [11] and settled in [12]): 2π f(re iθ ) 2π η dθ f 2 (re iθ ) η dθ for all f T, η > and < r < 1. In [12], he also proved his conjecture for the subclasses T (β) and C(β) of T. In this section, we obtain integral means inequalities for the functions in the family UH(α, β, γ, δ, λ, k). By assigning appropriate values to the parameters α, β, γ, δ, λ, k, we can deduce various integral means inequalities for various known as well as new subclasses. We prove the following result. Theorem 11. Suppose f UH(α, β, γ, δ, λ, k) and η >. If f 2 is defined by f 2 = z Φ(α, β, δ, λ, γ, k, 2) z2, (25) Φ(α, β, δ, λ, γ, k, 2) = (2 β) [{ 2(α + 1) (α + β) (2γδ + γ δ) + 1 Ψ 2

On a subclass of uniformly convex functions... 55 (26) Proof. then for z = re iθ ( < r < 1): For (26) is equivalent to proving that Ψ 2 := ( ) k 2 + λ, 1 + λ 2π 2π f η dθ f 2 η dθ. f = z a n z n, 2π 1 η 2π a n z n 1 dθ 1 (1 γ) Φ(α, β, δ, λ, γ, k, 2) z By Lemma 1, it suffices to show that Setting (27) 1 1 a n z n 1 1 γ 1 Φ(α, β, δ, λ, γ, k, 2) z. and using (12), we obtain w = z a n z n 1 1 γ = 1 Φ(α, β, δ, λ, γ, k, 2) w, z, Φ(α, β, δ, λ, γ, k, n) a n z n 1 1 γ Φ(α, β, δ, λ, γ, k, n) a n 1 γ Φ(α, β, δ, λ, γ, k, n) = [{n(α + 1) (α + β)(n 1)(nγδ + γ δ) + 1 Ψ n η dθ. and Ψ n is given by (5). This completes the proof of Theorem 11. Finally, we conclude this paper by remarking that by suitably specializing the values of the parameters λ, k, δ, γ, and α in the various results mentioned in this paper, we would be led to some interesting results including those which were obtained in [7], [9], [1] and [12].

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