Concave univalent functions and Dirichlet finite integral
The article deals with the class Fα consisting of non-vanishing functions f that are analytic and univalent in D such that the complement C\f(D) is a convex set, f(1) = ∞, f(0) = 1 and the angle at ∞ is less than or equal to απ for some α∈ (1,2]. Related to this class is the class CO(α) of concave univalent mappings in D, but this differs from Fα with the standard normalization f(0) = 0 = f′(0) = 1. A number of properties of these classes are discussed which includes an easy proof of the coefficient conjecture for CO(2) settled by Avkhadiev et al. Moreover, another interesting result connected with the Yamashita conjecture on Dirichlet finite integral for CO(α) is also presented.
Abu Muhanna, Yusuf and Ponnusamy, Saminathan, "Concave univalent functions and Dirichlet finite integral" (2017). Journal Articles. 2632.