AI Summary & Key Takeaways
AI InsightWe alsoprove two duality theorems between the classes K−12γand S∗, andbetween K−12γand S∗−12γ, respectively.
📌 3 Key Takeaways
Abstract
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.