New Results Related to the Convexity and Starlikeness of the Bernardi Integral Operator

Authors : G İ OROS, Georgia İrina OROS

Pages : 137-143

View : 21 | Download : 15

Publication Date : 2009-02-01

Article Type : Research Paper

Abstract :In insert ignore into journalissuearticles values(Lewandowski, Z., Miller, S. S. and Zlotkiewicz, E. Generating functions for some classes of univalent functions, Proc. Amer. Math. Soc. 56, 111–117, 1976); and insert ignore into journalissuearticles values(Pascu, N. N. Alpha-close-to-convex functions, Romanian–Finish Seminar on Complex Analysis, Springer Berlin, 331– 335, 1979); it has been proved that the integral operator defined by S. D. Bernardi insert ignore into journalissuearticles values(Convex and starlike univalent functions, Trans. Amer. Math. Soc. 135, 429–446, 1969); and given by insert ignore into journalissuearticles values(1); Lγinsert ignore into journalissuearticles values(f);insert ignore into journalissuearticles values(z); = Finsert ignore into journalissuearticles values(z); = γ + 1 z γ Z z 0 finsert ignore into journalissuearticles values(t);t γ−1 dt, z ∈ U preserves certain classes of univalent functions, such as the class of starlike functions, the class of convex functions and the class of closeto-convex functions. In this paper we determine conditions that a function f ∈ A needs to satisfy in order that the function F given by insert ignore into journalissuearticles values(1); be convex. We alsoprove two duality theorems between the classes K−12γand S∗, andbetween K−12γand S∗−12γ, respectively.
Keywords : Analytic function, Univalent function, Integral operator, Convex function, Starlike function, Close to convex function