ON SOME PROPERTIES OF A GENERALIZED CLASS OF CLOSE-TO-STARLIKE FUNCTIONS

Abstract

In this paper, we consider a new class of close-to-starlike functions denoted by CSα,β∗CS^{*}_{\alpha,\beta}CSα,β∗​, defined by the Carlson-Shaffer operator L(α,β)\mathcal{L}(\alpha,\beta)L(α,β). Let SSS denote the class of analytic univalent functions fff defined by

f(z)=z+∑n=2∞anzn,f(z)=z+\sum_{n=2}^{\infty}a_nz^n,f(z)=z+n=2∑∞​an​zn,

then f∈CSα,β∗f\in CS^{*}_{\alpha,\beta}f∈CSα,β∗​ if fff satisfies the condition

Re⁡{L(α,β)(f(z))g(z)}>0,z∈E,\operatorname{Re}\left\{\frac{\mathcal{L}(\alpha,\beta)\big(f(z)\big)}{g(z)}\right\}>0,\qquad z\in E,Re{g(z)L(α,β)(f(z))​}>0,z∈E,

where

L(α,β)f(z)=z+∑n=2∞(α)n−1(β)n−1anzn,\mathcal{L}(\alpha,\beta)f(z) = z+\sum_{n=2}^{\infty} \frac{(\alpha)_{n-1}}{(\beta)_{n-1}} a_nz^n,L(α,β)f(z)=z+n=2∑∞​(β)n−1​(α)n−1​​an​zn,

and g(z)g(z)g(z) is a starlike function. Properties of the class CSα,β∗CS^{*}_{\alpha,\beta}CSα,β∗​ such as the coefficient bounds, growth and distortion theorems, and radius results are investigated.