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 $\Delta (1,f)$, where $$\Delta (r,f)={\iint }_{|z|<r}|f^{\prime }(z){|}^{2}dxdy(0<r\le 1).$$ The class ${\mathcal{S}}^{\ast }(A,B)$ of normalized functions $f$ analytic in $|z|<1$ and satisfies the subordination condition $z{f}^{\prime }(z)/f(z)\prec (1+Az)/(1+Bz)$ in $|z|<1$ and for some $-1\le B\le 0$, $A\in \mathbb{C}$ with $A\ne B$, has been studied extensively. In this paper, we solve the extremal problem of determining the value of $$\underset{f\in {\mathcal{S}}^{\ast }(A,B)}{\max }\Delta (r,z/f)$$ 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 Obradović et. al [9].

16 pages, 8 figures, 3 tables