On Lemniscate Starlikeness of the Solution of General Differential Equations

Abstract

In this article, we derived conditions on the coefficient functions a(z) and b(z) of the differential equations y″(z)+a(z)y′(z)+b(z)y(z)=0 and z2y″(z)+a(z)zy′(z)+b(z)y(z)=0, such their solution f(z) with normalization f(0)=0=f′(0)−1 is starlike in the lemniscate domain, equivalently zf′(z)/f(z)≺1+z. We provide several examples with graphical presentations for a clear view of the obtained results.