Research Article Integral Means Inequalities for Fractional Derivatives of a Unified Subclass of Prestarlike Functions with Negative Coefficients

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1 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 of a Unified Subclass of Prestarlike Functions with Negative Coefficients H. Özlem Güney and Shigeyoshi Owa Received 24 May 27; Revised 13 July 27; Accepted 28 July 27 Recommended by Narendra K. Govil Integral means inequalities are obtained for the fractional derivatives of order p + λ( p n, λ<1) of functions belonging to a unified subclass of prestarlike functions. Relevant connections with various known integral means inequalities are also pointed out. Copyright 27 H. Ö. Güney and S. Owa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Let denote the class of (normalized) functions of the form f (z) = z + a n z n, (1.1) which are analytic and univalent in the open unit disk U ={z C : z < 1}. Also let denote the subclass of consisting of functions f of the form f (z) = z a n z n ( an ). (1.2) The Hadamard product (or convolution) of two functions f given by (1.1) andg given by g(z) = z + b n z n (1.3)

2 2 Journal of Inequalities and Applications is defined by ( f g)(z) = z + a n b n z n. (1.4) We denote the subclass (α,β)of consisting of α-prestarlike functions of order β by (α,β) = { f : ( f s α ) (z) (β), α<1, β<1 }, (1.5) where (β) denotes the class of starlike functions of order β( β<1) and s α is the well-known extremal function for (α)givenby (cf. [1, 2]). Letting s α (z) = z(1 z) 2(1 α) (1.6) c n (α) = n (k 2α) (n 1)! (n = 2,3,...), (1.7) s α can be written in the form s α (z) = z + c n (α)z n. (1.8) The class (α,β) was investigated by Sheil-Small et al. [3]. We also denote the subclass (α,β)of, which was investigated by Owa and Uralegaddi [4], by In particular, the subclasses (α,β) = { f : zf (z) (α,β) }. (1.9) [α,β] = (α,β), [α,β] = (α,β) (1.1) were considered earlier by Srivastava and Aouf [5]. Let us define the unified class (α, β,σ) of the classes [α,β]and [α,β]by so that (α,β,σ) = (1 σ) [α,β]+σ [α,β] ( σ 1), (1.11) (α,β,)= [α,β], (α,β,1)= [α,β]. (1.12) The unified class (α,β,σ) was studied by Raina and Srivastava [6].

3 H. Ö. Güney and S. Owa 3 We begin by recalling the following useful characterizations of the function class (α,β,σ) due to Raina and Srivastava [6]. Lemma 1.1. Afunction f defined by (1.2)belongstotheclass (α,β,σ) if and only if { } (n β)(1 σ + σn) c n (α)a n 1, (1.13) for some α( α<1), β( β<1), σ( σ 1). We continue by proving the following lemma. Lemma 1.2. Let f 1 (z) = z, f k (z) = z (k β)(1 σ + σk)c k (α) zk (k = 2,3,...). (1.14) Then f (α,β,σ) if and only if it can be expressed in the form where λ k and λ k = 1. Proof. Assume that Then Thus f (z) = λ 1 f 1 (z)+ f (z) = f (z) = λ k f k (z), (1.15) λ k f k (z). (1.16) λ k f k (z) ) = λ 1 z + λ k (z (k β)(1 σ + σk)c k (α) zk ( = λ k )z λ k (k β)(1 σ + σk)c k (α) zk = z λ k (k β)(1 σ + σk)c k (α) zk. ( )( ) (k β)(1 σ + σk)ck (α) λ k (k β)(1 σ + σk)c k (α) = λ k = λ k λ 1 = 1 λ 1 1. Therefore, we have f (α,β,σ). (1.17) (1.18)

4 4 Journal of Inequalities and Applications Conversely, suppose that f (α,β,σ). Since a k (k β)(1 σ + σk)c k (α) (k = 2,3,...), (1.19) we can set λ k = (k β)(1 σ + σk)c k(α) (k = 2,3,...), λ 1 = 1 λ k. (1.2) Then f (z) = z a k z k = z λ k (k β)(1 σ + σk)c k (α) zk ( = 1 λ k )z + λ k f k (z) = λ 1 f 1 (z)+ λ k f k (z) = λ k f k (z). (1.21) This completes the assertion of Lemma 1.2. Lemma 1.2 gives us the following. Corollary 1.3. The extreme points of (α,β,σ) are given by f 1 (z) = z, f k (z) = z (k β)(1 σ + σk)c k (α) zk. (1.22) We will make use of the following definitions of fractional derivatives by Owa [7](also by Srivastava and Owa [8]). Definition 1.4. The fractional derivative of order λ is defined, for a function f,by Dz λ 1 d f (z) = Γ(1 λ) dz z f (ξ) λ dξ ( λ<1), (1.23) (z ξ) where the function f is analytic in a simply connected region of the complex z-plane containing the origin, and the multiplicity of (z ξ) λ is removed by requiring log(z ξ) to be real when (z ξ) >.

5 H. Ö. Güney and S. Owa 5 Definition 1.5. Under the hypothesis of Definition 1.4, the fractional derivative of order (n + λ) is defined, for a function f,by Dz n+λ f (z) = dn dz n Dλ z f (z), (1.24) where λ<1andn =,1,2,... It readily follows from (1.23)in Definition1.4 that Dzz λ k Γ(k +1) = Γ(k λ +1) zk λ ( λ<1). (1.25) We will also need the concept of subordination between analytic functions and a subordination theorem of Littlewood [9] in our investigation. Given two functions f and g, which are analytic in U, the function f is said to be subordinate to g in U if there exists a function w analytic in U with such that w() =, We denote this subordination by Lemma 1.6. If the functions f and g are analytic in U with then, for μ> and z = re iθ ( <r<1), w(z) < 1 (z U), (1.26) f (z) = g ( w(z) ) (z U). (1.27) g ( re iθ) μ dθ 2. The main integral means inequalities f (z) g(z). (1.28) g(z) f (z), (1.29) f ( re iθ) μ dθ. (1.3) We discuss the integral means inequalities for functions f in (α,β,σ). Our main theorem is contained in the following. Theorem 2.1. Let f (α,β,σ) and suppose that ()Γ(k +1)Γ(3 λ p) a n (k β)(1 σ + σk)c k (α)γ(k +1 λ p)γ(2 p) (k 2) (2.1) for λ<1,where denotes the Pochhammer symbol defined by = (n p)(n p +1) n. (2.2)

6 6 Journal of Inequalities and Applications Also let the function f k be defined by f k (z) = z If there exists an analytic function w defined by { } k 1 (k β)(1 σ + σk)c k (α) Γ(k +1 λ p) w(z) = Γ(k +1) with (k β)(1 σ + σk)c k (α) zk. (2.3) Φ(n)a n z n 1 (2.4) Φ(n) = Γ(n p) Γ(n +1 λ p) ( λ<1, n = 2,3,...), (2.5) then, for μ> and z = re iθ ( <r<1), D p+λ f (z) μ dθ z Dz p+λ f k (z) μ dθ ( λ<1, μ>). (2.6) Proof. By virtue of the fractional derivative formula (1.25) and Definition 1.5, we find from (1.1)that ( ) Dz p+λ z f (z) 1 p λ Γ(2 λ p)γ(n +1) = 1 a n z n 1 Γ(2 λ p) Γ(n +1 λ p) ( z 1 p λ = 1 Γ(2 λ p)(n p) Γ(2 λ p) p+1 Φ(n)a n z ), n 1 (2.7) where Φ(n) = Γ(n p) Γ(n +1 λ p) ( λ<1, n = 2,3,...). (2.8) Since Φ is a decreasing function of n,wehave < Φ(n) Φ(2) = Γ(2 p) Γ(3 λ p) ( λ<1, n = 2,3,...). (2.9) Similarly, from (2.3), (1.25), anddefinition1.5, we obtain D p+λ z f k (z) = z 1 p λ ( 1 Γ(2 λ p) (k β)(1 σ + σk)c k (α) Γ(2 λ p)γ(k +1) Γ(k +1 λ p) z k 1 ). (2.1)

7 H. Ö. Güney and S. Owa 7 For μ>andz = re iθ ( <r<1), we must show that 1 Γ(2 λ p) Φ(n)a n z μdθ n 1 2π 1 Γ(2 λ p)γ(k +1) z μdθ. (k β)(1 σ + σk)c k (α) Γ(k +1 λ p) k 1 (2.11) Thus, by applying Lemma 1.6, it would suffice to show that 1 Γ(2 λ p) Φ(n)a n z n 1 1 (k β)(1 σ + σk)c k (α) Γ(2 λ p)γ(k +1) z Γ(k +1 λ p) k 1. (2.12) If the subordination (2.12) holds true, then we have an analytic function w with w() = and w(z) < 1suchthat 1 Γ(2 λ p) Φ(n)a n z n 1 = 1 (k β)(1 σ + σk)c k (α) By the condition of the theorem, we define the function w by { } k 1 (k β)(1 σ + σk)c k (α) Γ(k +1 λ p) w(z) = Γ(k +1) which readily yields w() =. Forsuch a functionw,we have w(z) k 1 (k β)(1 σ + σk)c k(α) z (k β)(1 σ + σk)c k(α) = z (k β)(1 σ +σk)c k(α) 1 β = z < 1, by means of the hypothesis of the theorem. (2.13) Γ(2 λ p)γ(k +1) { } k 1. w(z) Γ(k +1 λ p) Φ(n)a n z n 1 (2.14) Γ(k +1 λ p) Γ(k +1) Φ(n)a n z n 1 Γ(k +1 λ p) Γ(k +1) Γ(k+1 λ p) Γ(k+1) Φ(2) a n Γ(2 p) (n p) Γ(3 λ p) p+1 a n (2.15) This means that the subordination (2.12) holds true; therefore the theorem is proved. As special case p =, Theorem 2.1 readily yields.

8 8 Journal of Inequalities and Applications Corollary 2.2. Let f (α,β,σ) and suppose that n a n ()Γ(k +1)Γ(3 λ) (k β)(1 σ + σk)c k (α)γ(k +1 λ) (k 2). (2.16) If there exists an analytic function w given by { } k 1 (k β)(1 σ + σk)c k (α) Γ(k +1 λ) w(z) = Γ(k +1) nφ(n)a n z n 1 (2.17) with Φ(n) = Γ(n) Γ(n +1 λ) ( λ<1, n = 2,3,...), (2.18) then, for μ> and z = re iθ ( <r<1), Dz λ f (z) μ dθ Letting p = 1inTheorem 2.1, we have the following. Corollary 2.3. Let f (α,β,σ) and suppose that Dz λ f k (z) μ dθ ( λ<1, μ>). (2.19) n(n 1) a n ()Γ(k +1)Γ(2 λ) (k β)(1 σ + σk)c k (α)γ(k λ) (k 2). (2.2) If there exists an analytic function w given by { } k 1 (k β)(1 σ + σk)c k (α) Γ(k λ) w(z) = Γ(k +1) (n 1) 2 Φ(n)a n z n 1 (2.21) with Φ(n) = Γ(n 1) Γ(n λ) ( λ<1, n = 2,3,...), (2.22) then, for μ> and z = re iθ ( <r<1), References D 1+λ f (z) μ dθ z Dz 1+λ f k (z) μ dθ ( λ<1, μ>). (2.23) [1] P. L. Duren, Univalent Functions, vol. 259 of Grundlehren der Mathematischen Wissenschaften, Springer, New York, NY, USA, [2] H. M. Srivastavaand S. Owa, Eds.,Current Topics in Analytic Function Theory, World Scientific, River Edge, NJ, USA, [3] T. Sheil-Small, H. Silverman, and E. Silvia, Convolution multipliers and starlike functions, Journal d Analyse Mathématique, vol. 41, pp , 1982.

9 H. Ö. Güney and S. Owa 9 [4] S.Owa and B.A.Uralegaddi, A class of functions α-prestarlike of order β, Bulletin of the Korean Mathematical Society, vol. 21, no. 2, pp , [5] H. M. Srivastava and M. K. Aouf, Some applications of fractional calculus operators to certain subclasses of prestarlike functions with negative coefficients, Computers & Mathematics with Applications, vol. 3, no. 1, pp , [6] R. K. Raina and H. M. Srivastava, A unified presentation of certain subclasses of prestarlike functions with negative coefficients, Computers & Mathematics with Applications, vol. 38, no , pp , [7] S. Owa, On the distortion theorems. I, Kyungpook Mathematical Journal, vol. 18, no. 1, pp , [8] H.M.SrivastavaandS.Owa,Eds.,Univalent Functions, Fractional Calculus, and Their Applications, Ellis Horwood Series: Mathematics and Its Applications, Ellis Horwood, Chichester, UK; John Wiley & Sons, New York, NY, USA, [9] J. E. Littlewood, On inequalities in the theory of functions, Proceedings of the London Mathematical Society, vol. 23, no. 1, pp , H. Özlem Güney: Department of Mathematics, Faculty of Science and Letters, University of Dicle, 2128 Diyarbakır, Turkey address: ozlemg@dicle.edu.tr Shigeyoshi Owa: Department of Mathematics, Kinki University, Osaka , Higashi-Osaka, Japan address: owa@math.kindai.ac.jp