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

Abstract

Functionsf(z)=z+∑2∞‍anznthat are analytic in the unit disk and satisfy the differential equationf'(z)+αzf''(z)+γz2f'''(z)=g(z)are considered, wheregis subordinated to a normalized convex univalent functionh. These functionsfare given by a double integral operator of the formf(z)=∫01∫01‍G(ztμsν)t-μs-νds dtwithG'subordinated toh. 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 functionh.