Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID 683985, 9 pages doi:10.1155/2009/683985 Research Article Fractional Calculus and p-Valently Starlike Functions Osman Altıntas¸ 1 and ¨ Oznur ¨ Ozkan 2 1 Department of Matematics Education, Bas¸kent University, Ba ˘ glıca, TR-06530, Ankara, Turkey 2 Department of Statistics and Computer Sciences, Bas¸kent University, Ba ˘ glıca, TR 06530, Ankara, Turkey Correspondence should be addressed to ¨ Oznur ¨ Ozkan, oznur@baskent.edu.tr Received 18 November 2008; Accepted 28 February 2009 Recommended by Alberto Cabada In this investigation, the authors prove coefficient bounds, distortion inequalities for fractional calculus of a family of multivalent functions with negative coefficients, which is defined by means of a certain nonhomogenous Cauchy-Euler differential equation. Copyright q 2009 O. Altıntas¸and ¨ O. ¨ Ozkan. 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 and Definitions Let T n p denote the class of functions fz of the f orm fzz p − ∞  knp a k z k a k ≥ 0; n, p ∈ N  {1, 2, 3, }, 1.1 which are analytic and multivalent in the unit disk U  {z : z ∈ C and |z| < 1}. The fractional calculus are defined as follows e.g., 1, 2. Definition 1.1. The fractional integral of order δ is defined by D −δ z fz 1 Γδ  z 0 fξ z − ξ 1−δ dξ δ>0, 1.2 where fz is an analytic function in a simply-connected region of the z-plane containing the origin and the multiplicity of z − ξ δ−1 is removed by requiring log z − ξ to be real when z − ξ>0. 2 Journal of Inequalities and Applications Definition 1.2. The fractional derivative of order δ is defined by D δ z fz 1 Γ1 − δ d dz  z 0 fξ z − ξ δ dξ 0 ≤ δ<1, 1.3 where fz is constrained and multiplicity of z − ξ −δ is removed as in Definition 1.1. Definition 1.3. Under the hypotheses of Definition 1.1, the fractional derivative of order nδ is defined by D nδ z fz d n dz n D δ z fz  0 ≤ δ<1,n∈ N 0  N ∪{0}  . 1.4 a v denotes the Pochhammer symbol (or the shifted factorial),since 1 n  n!forn ∈ N 0 : N ∪{0}, 1.5 defined for a, v ∈ C and in terms of the Gamma function by a v : Γa  v Γa   1, v  0,a∈ C \{0}, aa  1 ···a  n − 1, v  n ∈ N; a ∈ C. 1.6 The earlier investigations by Goodman 3, 4 and Ruscheweyh 5, we define the n, p, ε- neighborhood of a function f ∈T n p by N ε n,p  D δ z f, D δ z g  :  g ∈T n p : gzz p − ∞  knp b k z k , ∞  knp k  1 − δ δ k   a k − b k   ≤ ε  , 1.7 so that, obviously, N ε n,p  D δ z h, D δ z g  :  g ∈T n p : gzz p − ∞  knp b k z k , ∞  knp k  1 − δ δ k   b k   ≤ ε  , 1.8 where hz : z p . 1.9 Journal of Inequalities and Applications 3 The class S δ n,p λ, α denote the subclass of T n p consisting of functions fz which satisfy Re zF  z Fz >α 0 ≤ α<p, p∈ N, 1.10 where FzλzD 1δ z fz1 − λD δ z fz0 ≤ λ ≤ 1, 0 ≤ δ<1. 1.11 We note that the class S 0 1,1 λ, α was investigated by Altıntas¸ 6 and the class S 0 n,p λ, α was studied by Alt ıntas¸etal.7, 8 . We donote by S 0 n,p 0,αS ∗ n p, α, S 0 n,p 1,αC n p, α1.12 the classes of p-valently starlike functions of order α in U 0 ≤ α<p and p-valently convex functions of order α in U 0 ≤ α<p, respectively see, 2, 9. Finally K δ n,p λ, α, μ denote the subclass of the general class T n p consisting of functions fz ∈T n p satisfying the following nonhomogeneous Cauchy-Euler differential equation: z 2 D 2δ z ω  21  μzD 1δ z ω  μ1  μD δ z ω p − δ  μp − δ  μ  1D δ z g, 1.13 where ω  fz,fz ∈T n p,g  gz ∈S δ n,p λ, α and μ>δ− p. The main object of the present paper is to give coefficients bounds and distortion inequalities for functions in the classes S δ n,p λ, α and K δ n,p λ, α, μ. 2. Coefficient Bounds and Distortion Inequalities We begin by proving the following result. Lemma 2.1. Let the function fz ∈T n p be defined by 1.1.Thenfz is in the class S δ n,p λ, α if and only i f ∞  knp k  1 − δ δ k − α − δ1  λk − 1 − δa k ≤ p  1 − δ δ p − α − δ1  λp − 1 − δ 0 ≤ λ ≤ 1; 0 ≤ α<p− δ;0≤ δ<1; p ∈ N. 2.1 The result is sharp for the function fz given by fzz p − p  1 − δ δ p − α − δ1  λp − 1 − δ n  p  1 − δ δ n  p − α − δ1  λn  p − 1 − δ z np . 2.2 4 Journal of Inequalities and Applications Proof. Let fz ∈T n p and Fz be defined by 1.11. Suppose that fz ∈S δ n,p λ, α. Then, in conjunction with 1.10 and 1.11 yields Re p1−δ δ p−δ1λp−1−δz p−δ −  ∞ knp k1−δ δ k − δ1λk−1−δa k z k−δ p  1 − δ δ 1  λp − 1 − δz p−δ −  ∞ knp k  1 − δ δ 1  λk − 1 − δa k z k−δ >α. 2.3 By letting z → 1 − along the real axis, we arrive easily at the inequality in 2.1. Lemma 2.2. Let the function fz given by 1.1 be in the class S δ n,p λ, α. Then ∞  knp k  1 − δ δ a k ≤ p  1 − δ δ p − α − δ1  λp − 1 − δ n  p − α − δ1  λn  p − 1 − δ , 2.4 ∞  knp k  1 − δ δ ka k ≤ p  1 − δ δ p − α − δ1  λp − 1 − δn  p n  p − α − δ1  λn  p − 1 − δ . 2.5 Proof. By using Lemma 2.1,wefindfrom2.1 that n  p − α − δ1  λn  p − 1 − δ ∞  knp k  1 − δ δ a k ≤ ∞  knp k  1 − δ δ k − α − δ1  λk − 1 − δa k ≤ p  1 − δ δ p − α − δ1  λp − 1 − δ, 2.6 which immediately yields the first assertion 2.4 of Lemma 2.2. For the proof of second assertion, by appealing to 2.1, we also have 1  λn  p − δ − 1  ∞  knp k  1 − δ δ ka k − α  δ ∞  knp k  1 − δ δ a k  ≤ p  1 − δ δ p − α − δ1  λp − 1 − δ, 2.7 by using 2.4 in 2.7, we can easily get the assertion 2.5 of Lemma 2.2. The distortion inequalities for functions in the class K δ n,p λ, α, μ are given by Theorem 2.3 below. Journal of Inequalities and Applications 5 Theorem 2.3. Let a function fz ∈T n p be in the class K δ n,p λ, α, μ.Then |fz|≤|z| p  p  1 − δ δ p − α − δp − δ  μp − δ  μ  1 n  p  1 − δ δ n  p − α − δn  p − δ  μ 1  λp − δ − 1 1  λn  p − δ − 1 |z| np , 2.8 |fz|≥|z| p − p  1 − δ δ p − α − δp − δ  μp − δ  μ  1 n  p  1 − δ δ n  p − α − δn  p − δ  μ 1  λp − δ − 1 1  λn  p − δ − 1 |z| np . 2.9 Proof. Suppose that a function fz ∈T n p is given by 1.1 and also let the function gz ∈ S δ n,p λ, α occurring in the nonhomogenous differential equation 1.13 be given as in the Definitions 1.2 or 1.3 with of course b k ≥ 0 k  n  p, n  p  1, . 2.10 Then we easily see from 1.13 that a k  p − δ  μp − δ  μ  1 k − δ  μk − δ  μ  1 b k k  n  p, n  p  1, . 2.11 So that fzz p − ∞  knp a k z k  z p − ∞  knp p − δ  μp − δ  μ  1 k − δ  μk − δ  μ  1 b k z k , 2.12 |fz|≤|z| p  |z| np ∞  knp p − δ  μp − δ  μ  1 k − δ  μk − δ  μ  1 b k . 2.13 Since gz ∈S δ n,p λ, α, the first assertion 2.4 of Lemma 2.2 yields the following inequality:   b k   ≤ p  1 − δ δ p − α − δ1  λp − δ − 1 n  p  1 − δ δ n  p − α − δ1  λn  p − δ − 1 . 2.14 From 2.13 and 2.14 we have |fz|≤|z| p  |z| np p  1 − δ δ p − α − δ1  λp − δ − 1 n  p  1 − δ δ n  p − α − δ1  λn  p − δ − 1 · ∞  knp 1 k − δ  μk − δ  μ  1 , 2.15 6 Journal of Inequalities and Applications andalsonotethat ∞  knp 1 k − δ  μk − δ  μ  1  ∞  knp  1 k − δ  μ − 1 k − δ  μ  1   1 n  p − δ  μ , 2.16 where μ ∈ R \{−n− p, − n− p − 1, }. The assertion 2.8 of Theorem 2.3 follows at once from 2.15. The assertion 2.9  of Theorem 2.3 can be proven by similarly applying 2.12, 2.14, and 2.15,andalso2.16. By setting δ : 0inTheorem 2.3, we obtain the following Corollary 2.4. Corollary 2.4 See Altıntas¸etal.8, Theorem 1. If the functions f and g satisfy the nonhomoge- neous Cauchy-Euler differential equation 1.13,then |fz|≤|z| p  p − αp  μp  μ  11  λp − 1 n  p − αn  p  μ1  λn  p − 1 |z| np , |fz|≥|z| p − p − αp  μp  μ  11  λp − 1 n  p − αn  p  μ1  λn  p − 1 |z| np . 2.17 By letting δ : 0, λ : 0andδ : 0, λ : 1inTheorem 2.3. We arrive at Corollaries 2.5 and 2.6 see, 8. Corollary 2.5. If the functions f and g satisfy the nonhomogeneous Cauchy-Euler differential equa- tion 1.13 with g ∈S ∗ n p, α, then |fz|≤|z| p  p − αp  μp  μ  1 n  p − αn  p  μ |z| np , |fz|≥|z| p − p − αp  μp  μ  1 n  p − αn  p  μ |z| np . 2.18 Corollary 2.6. If the functions f and g satisfy the nonhomogeneous Cauchy-Euler differential equa- tion 1.13 with g ∈C n p, α, then |fz|≤|z| p  pp − αp  μp  μ  1 n  p − αn  p  μn  p |z| np , |fz|≥|z| p − pp − αp  μp  μ  1 n  p − αn  p  μn  p |z| np . 2.19 3. Neighborhoods for the Classes S δ n,p λ, α and K δ n,p λ, α, μ In this section, we determine inclusion relations for the classes S δ n,p λ, α and K δ n,p λ, α, μ concerning the n, p, ε-neighborhoods is defined by 1.7 and 1.8. Journal of Inequalities and Applications 7 Theorem 3.1. Let a function fz ∈T n p be in the class S δ n,p λ, α.Then S δ n,p λ, α ⊂N ε n,p  D δ z h, D δ z f  , 3.1 where hz is given by 1.9 and the parameter ε is the given by ε : n  pp − δ δ p − α − δ1  λp − 1 − δ n  p − α − δ1  λn  p − 1 − δ . 3.2 Proof. Assertion 3.1 would follow easily from the definition of N ε n,p D δ z h, D δ z f, which is given by 1.8 with gz replaced by fz and the second assertion 2.5 of Lemma 2.2. Theorem 3.2. Let a function fz ∈T n p be in the class K δ n,p λ, α, μ.Then K δ n,p λ, α, μ ⊂N ε n,p  D δ z g,D δ z f  , 3.3 where gz is given by 1.13 and the parameter ε is the given by ε : n  pp − δ δ p − α − δ1  λp − 1 − δn p − δ  μp − δ  μ  2 n  p − α − δn  p − δ  μ1  λn  p − 1 − δ . 3.4 Proof. Suppose that fz ∈K δ n,p λ, α, μ. Then, upon substituting from 2.11 into the follo- wing coefficient inequality: ∞  knp k − δ δ k   b k − a k   ≤ ∞  knp k − δ δ kb k  ∞  knp k − δ δ ka k , 3.5 where a k ≥ 0andb k ≥ 0, we obtain that ∞  knp k − δ δ k   b k − a k   ≤ ∞  knp k − δ δ kb k  ∞  knp p − δ  μp − δ  μ  1 k − δ  μk − δ  μ  1 k − δ δ kb k . 3.6 Since gz ∈S δ n,p λ, α, the second assertion 2.5 of Lemma 2.2 yields that k − δ δ kb k ≤ n  pp − δ δ p − α − δ1  λp − 1 − δ n  p − α − δ1  λn  p − 1 − δ k  n  p, n  p  1, . 3.7 8 Journal of Inequalities and Applications Finally, by making use of 2.5 as well as 3.7 on the right-hand side of 3.6,wefindthat ∞  knp k − δ δ k   b k − a k   ≤ n  pp − δ δ p − α − δ1  λp − 1 − δ n  p − α − δ1  λn  p − 1 − δ  1  p − δ  μp − δ  μ  1 k − δ  μk − δ  μ  1  , 3.8 which, by virtue of the identity 2.16, immediately yields that ∞  knp k − δ δ k   b k − a k   ≤ n  pp − δ δ p − α − δ1  λp − 1 − δ n  p − α − δ1  λn  p − 1 − δ · n p − δ  μp − δ  μ  2 n  p − δ  μ : ε. 3.9 Thus, by definition 1.7 with gz interchanged by fz, fz ∈N ε n,p D δ z g,D δ z f.This evidently completes the proof of Theorem 3.2. By setting δ  0inTheorem 3.2, we receive the following result. Corollary 3.3. If the function fz ∈T n p is in the class K 0 n,p λ, α, μ.Then K 0 n,p λ, α, μ ⊂N ε n,p g,f, 3.10 where gz is given by 1.13 and the parameter ε is the given by ε : n  pp − α1  λp − 1n p  μp  μ  2 n  p − αn  p  μ1  λn  p − 1 . 3.11 Acknowledgment This present investigation was supported by Bas¸kent University Ankara, TURKEY. References 1 S. Owa, “On the distortion theorems—I,” Kyungpook Mathematical Journal, vol. 18, no. 1, pp. 53–59, 1978. 2 H. M. Srivastava and S. Owa, Eds., Univalent Functions, Fractional Calculus, and Their Applications, Ellis Horwood Series: Mathematics and Its Applications, Ellis Horwood, Chichester, UK, 1989. 3 A. W. Goodman, Univalent Functions. Vol. I, Mariner, Tampa, Fla, USA, 1983. 4 A. W. Goodman, UnivalentFunctions. Vol. II, Mariner, Tampa, Fla, USA, 1983. 5 S. Ruscheweyh, “Neighborhoods of univalent functions,” Proceedings of the American Mathematical Society, vol. 81, no. 4, pp. 521–527, 1981. 6 O. Altıntas¸, “On a subclass of certain starlike functions with negative coefficients,” Mathematica Japonica, vol. 36, no. 3, pp. 489–495, 1991. 7 O. Altıntas¸, H. Irmak, and H. M. Srivastava, “Fractional calculus and certain starlike functions with negative coefficients,” Computers & Mathematics with Applications, vol. 30, no. 2, pp. 9–15, 1995. Journal of Inequalities and Applications 9 8 O. Altıntas¸, ¨ O. ¨ Ozkan, and H. M. Srivastava, “Neighborhoods of a certain family of multivalent functions with negative coefficients,” Computers & Mathematics with Applications, vol. 47, no. 10-11, pp. 1667–1672, 2004. 9 P. L. Duren, Univalent Functions, vol. 259 of Grundlehren der Mathematischen Wissenschaften,Springer, New York, NY, USA, 1983. . Inequalities and Applications Volume 2009, Article ID 683985, 9 pages doi:10.1155/2009/683985 Research Article Fractional Calculus and p-Valently Starlike Functions Osman Altıntas¸ 1 and ¨ Oznur ¨ Ozkan 2 1 Department. certain starlike functions with negative coefficients,” Mathematica Japonica, vol. 36, no. 3, pp. 489–495, 1991. 7 O. Altıntas¸, H. Irmak, and H. M. Srivastava, Fractional calculus and certain starlike. pp. 53–59, 1978. 2 H. M. Srivastava and S. Owa, Eds., Univalent Functions, Fractional Calculus, and Their Applications, Ellis Horwood Series: Mathematics and Its Applications, Ellis Horwood,