Uniformly Convex and Bazilevic Functions

Abstract

Uniformly Convex and Bazilevic Functions This work is in the field of Geometric Function Theory in which we study geometric properties of analytic functions. It was originated around the turn of 20th century and has many applications in the field of applied sciences such as engineering, physics, electronics, medicines. In this dissertation, we define and discuss some new subclasses of normalized analytic functions by using some integral operators such as the operator Q a given by Q a f(z) z a ( )za 1 ta 0 2 log z t 1 f (t)dt, where  denotes gamma function, f(z) is analytic in open unit disc, a>0 and 0. We also study generalized Bazilevič functions. The functions in these classes generalize the idea of Bazilevič functions, k-uniformly convexity and bounded boundary and bounded radius rotations. The subordination and convolution tools are used to investigate the geometric properties of the functions in these classes and several inclusion results with some interesting consequences have been proved. We have investigated the univalency condition and coefficient bounds, arc length problem, integral preserving properties and rate of growth of Hankel determinant for these functions. The most of our results are sharp and they have been connected with previously known results.