Maximal Area Integral Problem for Certain Class of Univalent Analytic Functions

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

$$\Delta(r ,f) = \iint_{|z| < r} |f' (z)| ^ 2 \, {\rm d} x {\rm d}y \quad (0 < r \leq 1)$$

The class \({\mathcal{S} ^*(A,B)}\) of normalized functions f analytic in |z| < 1 and satisfies the subordination condition \({zf'(z)/f(z)\prec (1+Az)/(1+Bz)}\) in |z| < 1 and for some \({-1\leq B\leq 0}\) , \({A \in \mathbb{C}}\) with \({A\neq B}\) , has been studied extensively. In this paper, we solve the extremal problem of determining the value of

$$\max_{f\in \mathcal{S}^*(A,B)}\Delta(r,z/f)$$

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 Obradovi? et al. (Comput Methods Funct Theory 13:479–492, 2013).