Starlike functions associated with the generalized Koebe function

Abstract

In this paper we study some properties of functions f which are analytic and normalized (i.e. f(0)=0=f'(0)-1) such that satisfy the following subordination relation zf′(z)f(z)-1≺z(1-pz)(1-qz),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\left( \\frac{zf'(z)}{f(z)}-1\\right) \\prec \\frac{z}{(1-pz)(1-qz)}, \\end{aligned}$$\\end{document}where (p,q) in [-1,1] times [-1,1]. These types of functions are starlike related to the generalized Koebe function. Some of the features are: radius of starlikeness of order gamma in [0,1), image of fleft( {z:|z|<r}right) where rin (0,1), radius of convexity, estimation of initial and logarithmic coefficients, and Fekete–Szegö problem.