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

In (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 (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 (Convex and starlike univalent functions, Trans. Amer. Math. Soc. 135, 429–446, 1969) and given by (1) Lγ(f)(z) = F(z) = γ + 1z γ Z z 0 f(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 (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,

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Bernardi, S. D. Convex and starlike univalent functions, Trans. Amer. Math. Soc. 135, 429–446, 1969.
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Pascu, N. N. Alpha-close-to-convex functions, Romanian-Finish Seminar on Complex Anal- ysis, Springer Berlin, 331–335, 1979.