A Third‐Order Differential Equation and Starlikeness of a Double Integral Operator

Abstract

Functions that are analytic in the unit disk and satisfy the differential equation f ′( z ) + α zf ”( z ) + γ z 2 f ′”( z ) = g ( z ) are considered, where g is subordinated to a normalized convex univalent function h . These functions f are given by a double integral operator of the form with G ′ subordinated to h . The best dominant to all solutions of the differential equation is obtained. Starlikeness properties and various sharp estimates of these solutions are investigated for particular cases of the convex function h .