Geometric features of general differential solutions

Abstract

This papers examines the general differential equation \[y''(z)+a(z) y'(z)+ b(z)y(z)=0\] in the unit disk of the complex plane, and finds conditions on the analytic functions $a$ and $b$ that ensures the solutions are Janowski starlike. Also studied is Janowski convexity of solutions to \[z (1-z)y''(z)+ a(z) y'(z)+ \alpha y(z) =0,\] where $\alpha$ is a given constant. Janowski starlikeness and Janowski convexity encompass various widely studied classes of classical starlikeness and convexity. As application, we give convexity and starlikeness geometric description of solutions to differential equations related to several important special functions.