Maximal area integral problem for certain class of univalent analytic functions

TL;DR

This paper determines the maximum Dirichlet integral for a class of univalent functions with specific subordination properties, solving an extremal problem and settling a conjecture related to the maximal area integral.

Contribution

It provides a solution to the extremal problem of the Dirichlet integral for functions in the class al S^*(A,B), including a resolution of Yamashita's conjecture.

Findings

Explicit formula for the maximum Dirichlet integral as a function of r.

Resolution of Yamashita's conjecture.

Extension of extremal problems in geometric function theory.

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 $\Delta(1,f)$, where $$\Delta(r,f)=\iint_{|z|<r}|f'(z)|^2 \, dxdy \quad (0<r\leq 1). $$ 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 in [11]. One of the particular cases includes solution to a conjecture of Yamashita which was settled recently by Obradovi\'{c} et. al [9].