Pre-Schwarzian and Schwarzian norm estimates, growth, and distortion theorems for the generalized Robertson class

Abstract

Abstract In this paper, we investigate the geometric and analytic properties of the generalized Robertson class S P α ( β ) ${\mathcal{S}\mathcal{P}}_{\alpha }\left(\beta \right)$ , consisting of normalized analytic functions f on the unit disk D $\mathbb{D}$ that satisfy the subordination condition 1 + z f ″ ( z ) f ′ ( z ) ≺ 1 + e − i α ( e − i α − 2 β ⁡ cos ⁡ α ) z 1 − z . $$1+\frac{z{f}^{{\prime\prime}}\left(z\right)}{{f}^{\prime }\left(z\right)}\prec \frac{1+{\text{e}}^{-\text{i}\alpha }\left({\text{e}}^{-\text{i}\alpha }-2\beta \mathrm{cos}\alpha \right)z}{1-z}.$$ For the subclass S P α 0 ( β ) ${\mathcal{S}\mathcal{P}}_{\alpha }^{0}\left(\beta \right)$ defined by the additional normalization f ″(0) = 0, we establish a sharp characterization in terms of both the pre-Schwarzian derivative and a differential inequality. Using differential subordination and the Schwarz–Pick lemma, we derive sharp growth and distortion theorems for this class. Furthermore, we obtain explicit extremal functions and sharp norm estimates for the pre-Schwarzian and Schwarzian derivatives; specifically, we prove that if f ∈ S P α 0 ( β