Maximal Area Integral Problem for Certain Class of Univalent Analytic Functions

Abstract

One of the classical problems concerns the class of analytic functions f on the open unit disk |z| < 1 which have finite Dirichlet integral Δ(1, f), where (Formula presented.). The class (Formula presented.) of normalized functions f analytic in |z| < 1 and satisfies the subordination condition (Formula presented.) in |z| < 1 and for some (Formula presented.), (Formula presented.) with (Formula presented.), has been studied extensively. In this paper, we solve the extremal problem of determining the value of (Formula presented.) as a function of r. This settles the question raised by Ponnusamy and Wirths (Ann Acad Sci Fenn Ser AI Math 39:721–731, 2014). One of the particular cases includes solution to a conjecture of Yamashita which was settled recently by Obradović et al. (Comput Methods Funct Theory 13:479–492, 2013). © 2015, Springer Basel.