On maximal area integral problem for analytic functions in the starlike family

For an analytic function $f$ defined on the unit disk $|z|<1$, let $Delta(r,f)$ denote the area of the image of the subdisk $|z|<r$ under $f$, where $0<rle 1$. In 1990, Yamashita conjectured that $Delta(r,z/f)le pi r^2$ for convex functions $f$ and it was finally settled in 2013 by Obradovi{c} and et. al.. In this paper, we consider a class of analytic functions in the unit disk satisfying the subordination relation $zf(z)/f(z)prec (1+(1-2beta)alpha z)/(1-alpha z)$ for $0le beta<1$ and $0<alphale 1$. We prove Yamashitas conjecture problem for functions in this class, which solves a partial solution to an open problem posed by Ponnusamy and Wirths.