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 $0<r\le 1$. 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 $zf'(z)/f(z)\prec (1+(1-2\beta)\alpha z)/(1-\alpha z)$ for $0\le \beta<1$ and $0<\alpha\le 1$. We prove Yamashita's conjecture problem for functions in this class, which solves a partial solution to an open problem posed by Ponnusamy and Wirths.