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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 143869, 4 pages doi:10.1155/2008/143869 Research Article A Convexity Property for an Integral Operator on the Class S P β Daniel Breaz Department of Mathematics, “1 Decembrie 1918” University, Alba Iulia 510009, Romania Correspondence should be addressed to Daniel Breaz, dbreaz@uab.ro Received 30 October 2007; Accepted 30 December 2007 Recommended by Narendra Kumar K. Govil We consider an integral operator, F n z, for analytic functions, f i z, in the open unit disk, U.The object of this paper is to prove the convexity properties for the integral operator F n z,ontheclass S p β. Copyright q 2008 Daniel Breaz. 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 U  {z ∈ C, |z| < 1} be the unit disc of the complex plane and denote by HU the class of the holomorphic functions in U.LetA  {f ∈ HU,fzz a 2 z 2  a 3 z 3  ···,z∈ U} be the class of analytic functions in U and S  {f ∈ A : f is univalent in U}. Denote with K the class of convex functions in U, defined by K   f ∈ A : Re  zf  z f  z  1  > 0,z∈ U  . 1.1 A function f ∈ S is the convex function of order α, 0 ≤ α<1, and denote this class by Kα if f verifies the inequality Re  zf  z f  z  1 >α, z∈ U  . 1.2 Consider the class S p β, which was introduced by Ronning 1 and which is defined by f ∈ S p β ⇐⇒     zf  z fz − 1     ≤ Re  zf  z fz − β  , 1.3 where β is a real number with the property −1 ≤ β<1. 2 Journal of Inequalities and Applications For f i z ∈ A and α i > 0,i∈{1, ,n}, we define the integral operator F n z given by F n z  z 0  f 1 t t  α 1 ·····  f n t t  α n dt. 1.4 This integral operator was first defined by B. Breaz and N. Breaz 2. It is easy to see that F n z ∈ A. 2. Main results Theorem 2.1. Let α i > 0,fori ∈{1, ,n},letβ i be real numbers with the property −1 ≤ β i < 1,and let f i ∈ S p β i  for i ∈{1, ,n}. If 0 < n  i1 α i  1 − β i  ≤ 1, 2.1 then the function F n given by 1.4 is convex of order 1   n i1 α i β i − 1. Proof. We calculate for F n the derivatives of first and second orders. From 1.4 we obtain F  n z  f 1 z z  α 1 ·····  f n z z  α n , F  n z n  i1 α i  f i z z  α i  zf  i z − f i z zf i z  n  j1 j / i  f j z z  α j . 2.2 After some calculus, we obtain that F  n z F  n z  α 1  zf  1 z − f 1 z zf 1 z   ···  α n  zf  n z − f n z zf n z  . 2.3 This relation is equivalent to F  n z F  n z  α 1  f  1 z f 1 z − 1 z   ···  α n  f  n z f n z − 1 z  . 2.4 If we multiply the relation 2.4 with z,thenweobtain zF  n z F  n z  n  i1 α i  zf  i z f i z − 1   n  i1 α i zf  i z f i z − n  i1 α i . 2.5 The relation 2.5 is equivalent to zF  n z F  n z  1  n  i1 α i zf  i z f i z − n  i1 α i  1. 2.6 Daniel Breaz 3 This relation is equivalent to zF  n z F  n z  1  n  i1 α i  zf  i z f i z − β i   n  i1 α i β i − n  i1 α i  1. 2.7 We calculate the real part from both terms of the above equality and obtain Re  zF  n z F  n z  1   n  i1 α i Re  zf  i z f i z − β i   n  i1 α i β i − n  i1 α i  1. 2.8 Because f i ∈ S p β i  for i  {1, ,n}, we apply in the above relation inequality 1.3 and obtain Re  zF  n z F  n z  1  > n  i1 α i     zf  i z f i z − 1      n  i1 α i  β i − 1   1. 2.9 Since α i |zf  i z/f i z − 1| > 0 for all i ∈{1, ,n},weobtainthat Re  zF  n z F  n z  1  > n  i1 α i  β i − 1   1. 2.10 So, F n is convex of order  n i1 α i β i − 11. Corollary 2.2. Let α i ,i∈{1, ,n} be real positive numbers and f i ∈ S p β for i ∈{1, ,n}. If 0 < n  i1 α i ≤ 1 1 − β , 2.11 then the function F n is convex of order β − 1  n i1 α i  1. Proof. In Theorem 2.1, we consider β 1  β 2  ···  β n  β. Remark 2.3. If β  0and  n i1 α i  1, then Re  zF  n z F  n z  1  > 0, 2.12 so F n is a convex function. Corollary 2.4. Let γ bearealnumber,γ>0. Suppose that the functions f ∈ S p β and 0 <γ≤ 1/1 − β. In these conditions, the function F 1 z  z 0 ft/t γ dt is convex of order β − 1γ  1. Proof. In Corollary 2.2, w e consider n  1. Corollary 2.5. Let f ∈ S p β and consider the integral operator of Alexander, Fz  z 0 ft/tdt. In this condition, F is convex by the order β. Proof. We have zF  z F  z  zf  z fz − 1. 2.13 4 Journal of Inequalities and Applications From 2.13,wehave Re  zF  z F  z  1   Re  zf  z fz − β   β>     zf  z fz − 1      β>β. 2.14 So, the relation 2.14 implies that the Alexander operator is convex. References 1 F. Ronning, “Uniformly convex functions and a corresponding class of starlike functions,” Proceedings of the American Mathematical Society, vol. 118, no. 1, pp. 189–196, 1993. 2 D. Breaz and N. Breaz, “Two integral operators,” Studia Universitatis Babes¸-Bolyai, Mathematica, vol. 47, no. 3, pp. 13–19, 2002. . implies that the Alexander operator is convex. References 1 F. Ronning, “Uniformly convex functions and a corresponding class of starlike functions,” Proceedings of the American Mathematical Society,. Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 143869, 4 pages doi:10.1155/2008/143869 Research Article A Convexity Property for an Integral Operator on. prove the convexity properties for the integral operator F n z,ontheclass S p β. Copyright q 2008 Daniel Breaz. This is an open access article distributed under the Creative Commons Attribution