Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2007, Article ID 97135, 9 pages doi:10.1155/2007/97135 Research Article Integral Means Inequalities for Fractional Derivatives of a Unified Subclass of Prestarlike Functions with Negative Coefficients H. ¨ Ozlem G ¨ uney and Shigeyoshi Owa Received 24 May 2007; Revised 13 July 2007; Accepted 28 July 2007 Recommended by Narendra K. Govil Integral means inequalities are obtained for the fractional derivatives of order p + λ(0 ≤ p ≤ n,0≤ 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 © 2007 H. ¨ O. G ¨ uney and S. Owa. T his is an open access article distributed un- der the Creative Commons Attribution License, which permits unrestricted use, dist ribu- tion, 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 + ∞  n=2 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 − ∞  n=2 a n z n  a n ≥ 0  . (1.2) The Hadamard product (or convolution) of two functions f given by (1.1)andg given by g(z) = z + ∞  n=2 b n z n (1.3) 2 Journal of Inequalities and Applications is defined by ( f ∗g)(z) = z + ∞  n=2 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), 0 ≤ α<1, 0 ≤ β<1  , (1.5) where ᏿ ∗ (β) denotes the class of starlike functions of order β(0 ≤ β<1) and s α is the well-known extremal function for ᏿ ∗ (α)givenby s α (z) = z(1 − z) −2(1−α) (1.6) (cf. [1, 2]). Letting c n (α) =  n k =2 (k − 2α) (n − 1)! (n = 2,3, ), (1.7) s α can be written in the form s α (z) = z + ∞  n=2 c n (α)z n . (1.8) The class ᏾(α,β) was investigated by Sheil-Small et al. [3]. We also denote the subclass Ꮿ(α,β)of᏿, which was investigated by Owa and Uralegaddi [4], by Ꮿ(α,β) =  f ∈ ᏿ : zf  (z) ∈ ᏾(α,β)  . (1.9) In particular, the subclasses ᏾[α,β] = ᏾(α,β) ∩ ᐀, Ꮿ[α,β] = Ꮿ(α,β) ∩ ᐀ (1.10) were considered earlier by Srivastava and Aouf [5]. Let us define the unified class ᏼ(α, β,σ) of the classes ᏾[α,β]andᏯ[α,β]by ᏼ(α,β,σ) = (1 − σ)᏾[α,β]+σᏯ[α, β](0≤ σ ≤ 1), (1.11) so that ᏼ(α,β,0) = ᏾[α,β], ᏼ(α,β,1) = Ꮿ[α,β]. (1.12) The unified class ᏼ(α,β, σ) was studied by Raina and Srivastava [6]. H. ¨ O. G ¨ uney and S. Owa 3 We begin by recalling the following useful characterizations of the function class ᏼ(α,β,σ) due to Raina and Srivastava [6]. Lemma 1.1. Afunction f defined by (1.2)belongstotheclassᏼ(α,β,σ) if and only if ∞  n=2  (n − β)(1 − σ + σn) 1 − β  c n (α)a n ≤ 1, (1.13) for some α(0 ≤ α<1), β(0 ≤ β<1), σ(0 ≤ σ ≤ 1). We continue by proving the following lemma. Lemma 1.2. Let f 1 (z) = z, f k (z) = z − 1 − β (k − β)(1 − σ + σk)c k (α) z k (k = 2,3, ). (1.14) Then f ∈ ᏼ(α,β,σ) if and only if it can be expressed in the form f (z) = ∞  k=1 λ k f k (z), (1.15) where λ k ≥ 0 and  ∞ k=1 λ k = 1. Proof. Assume that f (z) = ∞  k=1 λ k f k (z). (1.16) Then f (z) = λ 1 f 1 (z)+ ∞  k=2 λ k f k (z) = λ 1 z + ∞  k=2 λ k  z − 1 − β (k − β)(1 − σ + σk)c k (α) z k  =  ∞  k=1 λ k  z − ∞  k=2 λ k 1 − β (k − β)(1 − σ + σk)c k (α) z k = z − ∞  k=2 λ k 1 − β (k − β)(1 − σ + σk)c k (α) z k . (1.17) Thus ∞  k=2 λ k  1 − β (k − β)(1 − σ + σk)c k (α)  (k − β)(1 − σ + σk)c k (α) 1 − β  = ∞  k=2 λ k = ∞  k=1 λ k − λ 1 = 1 − λ 1 ≤ 1. (1.18) Therefore, we have f ∈ ᏼ(α,β,σ).  4 Journal of Inequalities and Applications Conversely, suppose that f ∈ ᏼ(α,β,σ). Since |a k |≤ 1 − β (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) 1 − β (k = 2,3, ), λ 1 = 1 − ∞  k=1 λ k . (1.20) Then f (z) = z − ∞  k=2 a k z k = z − ∞  k=2 λ k 1 − β (k − β)(1 − σ + σk)c k (α) z k =  1 − ∞  k=2 λ k  z + ∞  k=2 λ k f k (z) = λ 1 f 1 (z)+ ∞  k=2 λ k f k (z) = ∞  k=1 λ k f k (z). (1.21) This completes the assertion of Lemma 1.2. Lemma 1.2 g ives us the following. Corollar y 1.3. The extreme points of ᏼ(α,β,σ) are given by f 1 (z) = z, f k (z) = z − 1 − β (k − β)(1 − σ + σk)c k (α) z k . (1.22) We will make use of the following definitions of fractional derivatives by Owa [7](also by Srivastava and Owa [8]). Definit ion 1.4. The fractional derivative of order λ is defined, for a function f ,by D λ z f (z) = 1 Γ(1 − λ) d dz  z 0 f (ξ) (z − ξ) λ dξ (0 ≤ λ<1), (1.23) 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) > 0. H. ¨ O. G ¨ uney and S. Owa 5 Definit ion 1.5. Under the hypothesis of Definition 1.4, the fractional derivative of order (n + λ) is defined, for a function f ,by D n+λ z f (z) = d n dz n D λ z f (z), (1.24) where 0 ≤ λ<1andn = 0,1,2, It readily follows from (1.23)inDefinition 1.4 that D λ z z k = Γ(k +1) Γ(k − λ +1) z k−λ (0 ≤ λ<1). (1.25) We will also need the concept of subordination between analytic functions and a subor- dination 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 w(0) = 0,   w(z)   < 1(z ∈ U), (1.26) such that f (z) = g  w(z)  (z ∈ U). (1.27) We denote this subordination by f (z) ≺ g(z). (1.28) Lemma 1.6. If the functions f and g are analytic in U with g(z) ≺ f (z), (1.29) then, for μ>0 and z = re iθ (0 <r<1),  2π 0   g  re iθ    μ dθ ≤  2π 0   f  re iθ    μ dθ. (1.30) 2. The main integral means inequalities We discuss the integral means inequalities for functions f in ᏼ(α,β,σ). Our main theo- rem is contained in the following. Theorem 2.1. Let f ∈ ᏼ(α,β,σ) and suppose that ∞  n=2 (n − p) p+1 a n ≤ (1 − β)Γ(k +1)Γ(3 − λ− p) (k − β)(1 − σ + σk)c k (α)Γ(k +1− λ − p)Γ(2 − p) (k ≥ 2) (2.1) for 0 ≤ λ<1,where(n − p) p+1 denotes the Pochhammer symbol defi ned by (n − p) p+1 = (n − p)(n − p +1)···n. (2.2) 6 Journal of Inequalities and Applications Also let the function f k be defined by f k (z) = z − 1 − β (k − β)(1 − σ + σk)c k (α) z k . (2.3) If there exists an analytic function w defined by  w(z)  k−1 = (k − β)(1 − σ + σk)c k (α) 1 − β Γ(k +1 − λ − p) Γ(k +1) ∞  n=2 (n − p) p+1 Φ(n)a n z n−1 (2.4) with Φ(n) = Γ(n − p) Γ(n +1− λ− p) (0 ≤ λ<1, n = 2,3, ), (2.5) then, for μ>0 and z = re iθ (0 <r<1),  2π 0   D p+λ z f (z)   μ dθ ≤  2π 0   D p+λ z f k (z)   μ dθ (0 ≤ λ<1, μ>0). (2.6) Proof. By virtue of the fractional derivative formula (1.25)andDefinition 1.5,wefind from (1.1)that D p+λ z f (z) = z 1−p−λ Γ(2 − λ− p)  1 − ∞  n=2 Γ(2 − λ− p)Γ(n +1) Γ(n +1− λ− p) a n z n−1  = z 1−p−λ Γ(2 − λ− p)  1 − ∞  n=2 Γ(2 − λ− p)(n − p) p+1 Φ(n)a n z n−1  , (2.7) where Φ(n) = Γ(n − p) Γ(n +1− λ− p) (0 ≤ λ<1, n = 2,3, ). (2.8) Since Φ is a decreasing function of n,wehave 0 < Φ(n) ≤ Φ(2) = Γ(2 − p) Γ(3 − λ− p) (0 ≤ λ<1, n = 2,3, ). (2.9) Similarly, from (2.3), (1.25), and Definition 1 .5,weobtain D p+λ z f k (z) = z 1−p−λ Γ(2 − λ− p)  1 − 1 − β (k − β)(1 − σ + σk)c k (α) Γ(2 − λ − p)Γ(k +1) Γ(k +1− λ − p) z k−1  . (2.10) H. ¨ O. G ¨ uney and S. Owa 7 For μ>0andz = re iθ (0 <r<1), we must show that  2π 0      1 − ∞  n=2 Γ(2 − λ− p)(n − p) p+1 Φ(n)a n z n−1      μ dθ ≤  2π 0      1 − 1 − β (k − β)(1 − σ + σk)c k (α) Γ(2 − λ − p)Γ(k +1) Γ(k +1− λ − p) z k−1      μ dθ. (2.11) Thus, by applying Lemma 1.6,itwouldsuffice to show that 1 − ∞  n=2 Γ(2 − λ− p)(n − p) p+1 Φ(n)a n z n−1 ≺ 1 − 1 − β (k − β)(1 − σ + σk)c k (α) Γ(2 − λ − p)Γ(k +1) Γ(k +1− λ − p) z k−1 . (2.12) If the subordination (2.12) holds true, then we have an analytic function w with w(0) = 0 and |w(z)| < 1suchthat 1 − ∞  n=2 Γ(2 − λ− p)(n − p) p+1 Φ(n)a n z n−1 = 1 − 1 − β (k − β)(1 − σ + σk)c k (α) Γ(2 − λ − p)Γ(k +1) Γ(k +1− λ − p)  w(z)  k−1 . (2.13) By the condition of the theorem, we define the function w by  w(z)  k−1 = (k − β)(1 − σ + σk)c k (α) 1 − β Γ(k +1 − λ − p) Γ(k +1) ∞  n=2 (n − p) p+1 Φ(n)a n z n−1 (2.14) which readily yields w(0) = 0. For such a function w,wehave   w(z)   k−1 ≤ (k − β)(1 − σ + σk)c k (α) 1 − β Γ(k +1 − λ − p) Γ(k +1) ∞  n=2 (n − p) p+1 Φ(n)a n |z| n−1 ≤|z| (k − β)(1 − σ + σk)c k (α) 1 − β Γ(k +1 − λ − p) Γ(k +1) Φ(2) ∞  n=2 (n − p) p+1 a n =|z| (k−β)(1−σ +σk)c k (α) 1−β Γ(k+1 −λ− p) Γ(k+1) Γ(2 − p) Γ(3−λ− p) ∞  n=2 (n− p) p+1 a n =|z| < 1, (2.15) by means of the hypothesis of the theorem.  This means that the subordination (2.12) holds true; therefore the theorem is proved. As special case p = 0, Theorem 2.1 readily yields. 8 Journal of Inequalities and Applications Corollar y 2.2. Let f ∈ ᏼ(α,β,σ) and suppose that ∞  n=2 n   a n   ≤ (1 − β)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  w(z)  k−1 = (k − β)(1 − σ + σk)c k (α) 1 − β Γ(k +1 − λ) Γ(k +1) ∞  n=2 nΦ(n)a n z n−1 (2.17) with Φ(n) = Γ(n) Γ(n +1− λ) (0 ≤ λ<1, n = 2,3, ), (2.18) then, for μ>0 and z = re iθ (0 <r<1),  2π 0   D λ z f (z)   μ dθ ≤  2π 0   D λ z f k (z)   μ dθ (0 ≤ λ<1, μ>0). (2.19) Letting p = 1inTheorem 2.1, we have the following. Corollar y 2.3. Let f ∈ ᏼ(α,β,σ) and suppose that ∞  n=2 n(n − 1)   a n   ≤ (1 − β)Γ(k +1)Γ(2 − λ) (k − β)(1 − σ + σk)c k (α)Γ(k − λ) (k ≥ 2). (2.20) If there exists an analytic function w given by  w(z)  k−1 = (k − β)(1 − σ + σk)c k (α) 1 − β Γ(k − λ) Γ(k +1) ∞  n=2 (n − 1) 2 Φ(n)a n z n−1 (2.21) with Φ(n) = Γ(n − 1) Γ(n − λ) (0 ≤ λ<1, n = 2, 3, ), (2.22) then, for μ>0 and z = re iθ (0 <r<1),  2π 0   D 1+λ z f (z)   μ dθ ≤  2π 0   D 1+λ z f k (z)   μ dθ (0 ≤ λ<1, μ>0). (2.23) References [1] P. L. Duren, Univalent Functions, vol. 259 of Grundlehren der Mathematischen Wissenschaften, Springer, New York, NY, USA, 1983. [2] H.M.SrivastavaandS.Owa,Eds.,Current Topics in Analytic Function Theory, World Scientific, River Edge, NJ, USA, 1992. [3] T. Sheil-Small, H. Silverman, and E. Silvia, “Convolution multipliers and starlike functions,” Journal d’Analyse Math ´ ematique, vol. 41, pp. 181–192, 1982. H. ¨ O. G ¨ uney 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. 77–85, 1984. [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. 30, no. 1, pp. 53–61, 1995. [6] R. K. Raina and H. M. Srivastava, “A unified presentation of certain subclasses of prestar- like functions with negative coefficients,” Computers & Mathematics with Applications, vol. 38, no. 11-12, pp. 71–78, 1999. [7] S. Owa, “On the distortion theorems. I,” Kyungpook Mathematical Journal,vol.18,no.1,pp. 53–59, 1978. [8] H.M.SrivastavaandS.Owa,Eds.,Univalent Functions, Fractional Calculus, and Their Applica- tions, Ellis Horwood Series: Mathematics and Its Applications, Ellis Horwood, Chichester, UK; John Wiley & Sons, New York, NY, USA, 1989. [9] J. E. Littlewood, “On inequalities in the theory of functions,” Proceedings of the London Mathe- matical Society, vol. 23, no. 1, pp. 481–519, 1925. H. ¨ Ozlem G ¨ uney: Department of Mathematics, Faculty of Science and Letters, University of Dicle, 21280 Diyarbakır, Turkey Email address: ozlemg@dicle.edu.tr Shigeyoshi Owa: Department of Mathematics, Kinki University, Osaka 577-8502, Higashi-Osaka, Japan Email address: owa@math.kindai.ac.jp . Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2007, Article ID 97135, 9 pages doi:10.1155/2007/97135 Research Article Integral Means Inequalities for Fractional Derivatives. by Narendra K. Govil Integral means inequalities are obtained for the fractional derivatives of order p + λ(0 ≤ p ≤ n,0≤ λ<1) of functions belonging to a unified subclass of prestarlike functions. Relevant. and M. K. Aouf, “Some applications of fractional calculus operators to certain subclasses of prestarlike functions with negative coefficients,” Computers & Mathematics with Applications, vol.