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 Δ(r, f) denote the area of the image of the subdisk |z| < r under f ,w here 0< r 1. In 1990, Yamashita conjectured that Δ(r,z/f) πr 2 for convex functions f and it was finally settled in 2013 by Obradovica nd et. al.. In this paper, we consider a class of analytic functions in the unit disk satisfying the subordination relation zf � (z)/f(z) ≺ (1+(1 −2β)αz)/(1 − αz) for 0 β < 1a nd 0< α 1. We prove Yamashita's conjecture problem for functions in this class, which provides a partial solution to an open problem posed by Ponnusamy and Wirths. 1. Introduction, preliminaries, and main result