The Research Of Some Properties For Certain Classes Of Univalent Functions Defined By Differential Inequalities

The univalent function theory is an important part of the theory of complex anal-ysis.So far,it has been over one hundred years old.In 1916,Bieberbach presented the famous Bieberbach conjecture.For the next seventy years,Bieberbach conjecture had always been one of the most important problems of the field of univalent function,the conjecture was finally settled until 1984.In the process of proving the conjecture,many new methods and theories were produced,all these methods and theories became the frame of the univalent function theory.Because Bieberbach conjecture was very challenging before being finally settled,many scholars proved the conjecture on var-ious subfamilies of univalent function first.After that,the research of properties for various subfamilies of univalent function has become an important subject of univalent function theory.In recent years,some subfamilies of univalent function have attracted the attention of many scholars at home and abroad,one of them is the function family u,f∈u if and only if f is analytic in the unit disk |z|<1,f(0)= f’(0)-1 = 0,and|f’(z)(2/f(z))2-1|<1.Inspired by the function family u,this article defines the function family Ω,f∈Ω if and only if f is analytic in the unit disk |z|<1,f(0)= f’(0)-1 = 0,and|zf’(z)-f(z)|<1/2(|z|<1).Firstly,this paper further studies the properties of the function family u,then,it main-ly focuses on various properties of the function family Ω,including growth theorem,distortion theorem,radius of convexity,convolution,extreme point and support point.