Functions with real part greater than $\alpha$

Abstract:

Let ${\mathcal {P}_b}(\alpha )$ denote the class of functions $P(z) = 1 + b(1 - \alpha )z + \cdots$ which are analytic and satisfy $\operatorname {Re} \{ P(z)\} > \alpha$ for $|z| < 1$ where $\alpha \in [0,1)$ and $b \in [0,2]$. We demonstrate some inequalities involving $|P’(z)|$ and $|P’(z)/P(z)|$ dependent on b and $\alpha$ which are subsequently applied to the class of functions whose derivative lies in ${\mathcal {P}_b}(\alpha )$ to obtain distortion, covering, and radius of convexity properties.