a convexity property for an integral operator on the class sp

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 143869, pages doi:10.1155/2008/143869 Research Article A Convexity Property for an Integral Operator on the Class SP 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, Fn z , for analytic functions, fi z , in the open unit disk, U The object of this paper is to prove the convexity properties for the integral operator Fn z , on the class Sp 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 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 Let A {f ∈ H U , f z z a2 z2 a3 z3 · · · , 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 > 0, z ∈ U 1.1 A function f ∈ S is the convex function of order α, ≤ α < 1, and denote this class by K α if f verifies the inequality Re zf z f z > α, z ∈ U 1.2 Consider the class Sp β , which was introduced by Ronning and which is defined by f ∈ Sp β ⇐⇒ zf z zf z − ≤ Re −β , f z f z where β is a real number with the property −1 ≤ β < 1.3 Journal of Inequalities and Applications For fi z ∈ A and αi > 0, i ∈ {1, , n}, we define the integral operator Fn z given by z α1 f1 t t Fn z αn fn t t ····· dt 1.4 This integral operator was first defined by B Breaz and N Breaz It is easy to see that Fn z ∈ A Main results Theorem 2.1 Let αi > 0, for i ∈ {1, , n}, let βi be real numbers with the property −1 ≤ βi < 1, and let fi ∈ Sp βi for i ∈ {1, , n} If n 0< i αi − βi ≤ 1, 2.1 n i αi then the function Fn given by 1.4 is convex of order βi − Proof We calculate for Fn the derivatives of first and second orders From 1.4 we obtain α1 f1 z z Fn z n Fn z i αn , zfi z − fi z zfi z αi fi z z αi fn z z ····· n j 2.2 αj fj z z j /i After some calculus, we obtain that Fn z Fn z α1 zf1 z − f1 z zf1 z ··· αn zfn z − fn z zfn z ··· αn fn z − fn z z 2.3 This relation is equivalent to Fn z Fn z f1 z − f1 z z α1 2.4 If we multiply the relation 2.4 with z, then we obtain zFn z Fn z n αi i zfi z −1 fi z n αi i zfi z − fi z n αi 2.5 i The relation 2.5 is equivalent to zFn z Fn z n αi i zfi z − fi z n αi i 1 2.6 Daniel Breaz This relation is equivalent to n zFn z Fn z n zfi z − βi fi z αi i i αi β i − n αi 2.7 i We calculate the real part from both terms of the above equality and obtain Re n zFn z Fn z i Because fi ∈ Sp βi for i obtain Re zFn z Fn z n zfi z − βi fi z αi Re i αi βi − n αi 2.8 i {1, , n}, we apply in the above relation inequality 1.3 and n > n zfi z −1 fi z αi i i αi βi − 1 2.9 Since αi |zfi z /fi z − 1| > for all i ∈ {1, , n}, we obtain that Re So, Fn is convex of order n zFn z Fn z n i αi αi βi − > i βi − 1 2.10 Corollary 2.2 Let αi , i ∈ {1, , n} be real positive numbers and fi ∈ Sp β for i ∈ {1, , n} If n αi ≤ 0< i then the function Fn is convex of order β − n i αi Proof In Theorem 2.1, we consider β1 ··· Remark 2.3 If β and n i αi β2 , 1−β 2.11 βn β > 0, 1, then Re zFn z Fn z 2.12 so Fn is a convex function Corollary 2.4 Let γ be a real number, γ > Suppose that the functions f ∈ Sp β and < γ ≤ z 1/ − β In these conditions, the function F1 z f t /t γ dt is convex of order β − γ Proof In Corollary 2.2, we consider n Corollary 2.5 Let f ∈ Sp β and consider the integral operator of Alexander, F z In this condition, F is convex by the order β z f t /t dt Proof We have zF z F z zf z − f z 2.13 Journal of Inequalities and Applications From 2.13 , we have Re zF z F z Re zf z −β f z β> zf z −1 f z β > β 2.14 So, the relation 2.14 implies that the Alexander operator is convex References 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 D Breaz and N Breaz, “Two integral operators,” Studia Universitatis Babes¸-Bolyai, Mathematica, vol 47, no 3, pp 13–19, 2002 ... 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 D Breaz and N Breaz,... Journal of Inequalities and Applications For fi z ∈ A and αi > 0, i ∈ {1, , n}, we define the integral operator Fn z given by z α1 f1 t t Fn z αn fn t t ····· dt 1.4 This integral operator was... Proof In Corollary 2.2, we consider n Corollary 2.5 Let f ∈ Sp β and consider the integral operator of Alexander, F z In this condition, F is convex by the order β z f t /t dt Proof We have zF z F