ROTARU ALPHA - CONVEX FUNCTIONS

Abstract

Let $S^*(a,b)$ denote the class of analytic functions $f$ in the unit disc $E$, with $f(0) =f'(0) - 1 =0$, satisfying the condition $|(zf'(z)/f(z))- a|<b$, $a\in C$, $|a- 1|<b\le Re(a)$, $z\in E$. In this paper the class $S^*(\alpha, a, b)$ of functions $f$ analytic in $E$, with $f(0) = f'(0)- 1 =0$, $f(z)f'(z)/z\neq 0$ for $z$ in $E$ and satisfying in $E$ the condition $|J(\alpha,f)- a|<b$, $a \in C$, $|a-1|<b\le Re(a)$, where $J(\alpha, f) =(1- \alpha)(zf'(z)/f(z)) +\alpha((zf'(z))'/f'(z))$, $\alpha$ a non-negative real number is introduced. It is proved that $S^*(\alpha, a,b)\subset S^*(a,b)$, if $a> (4b/c)|Im(a)|$, $c=(b^2- |a- 1|^2)/b$. Further a representation formula for $f \in S^*(\alpha, a, b)$ and an inequality relating the coefficients of functions in $S^*(\alpha, a, b)$ are obtained.