Concave univalent functions and Dirichlet finite integral

Article Type

Research Article

Publication Title

Mathematische Nachrichten

Abstract

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.

First Page

649

Last Page

661

DOI

10.1002/mana.201500458

Publication Date

4-1-2017

Comments

Open Access, Green

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