Convexity and starlikeness of functions defined by a class of integral operators

Abstract

For A : [0,1] → R real-valued monotone decreasing function on [0,1] satisfying A(l)=0tA(t)→ 0 as t→0+ and tA′(t)/(l−t2 ) increasing on (0,1), we show that MA(f) ≥ 0 for f close-to-convex where This is analogous to a recent result of Fournier and Ruscheweyh [2]. Analogously we obtain least value of β so that for g analytic in , the functions and are convex. Here 2F1 is the Gaussian hypergeometric function. These results are extended to convexity and order of convexity of convex combinations of the form ρz +(1−ρ)F(z)ρ< 1. Corresponding starlikeness results in [2] are also extended to such convex combinations.