Subordinations and Norm Estimates for Functions Associated with Ma-Minda Subclasses

Abstract

For a function p analytic in the open unit disc and satisfying p(0)=1, we prove certain subordination implications of the first order differential subordination 1+zp′(z)≺1+Mz, which provides sufficient conditions for a function to belong to various subclasses of Ma-Minda starlike functions. The pre-Schwarzian norm estimate and inclusion criteria for certain subclasses of analytic function are also obtained. Additionally, using Gronwall’s inequality we give a sufficient condition for a normalized function to belong to a class of functions with bounded arguments that extends the class of strongly α-Bazilevič functions of order γ studied by Gao in 1996.

Keywords

• starlike
• convex and close-to-convex functions
• differential subordination
• norm estimate
• Schwarzian derivative
• pre-Schwarzian derivative