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arXiv · 2212.06377

The Schwarzian norm estimates for Janowski convex functions

Md Firoz Ali·Sanjit Pal
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Abstract

For $-1\leq B<A\leq 1$, let $\mathcal{C}(A,B)$ denote the class of normalized Janowski convex functions defined in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$ that satisfy the subordination relation $1+zf''(z)/f'(z)\prec (1+Az)/(1+Bz)$. In the present article, we determine the sharp estimate of the Schwarzian norm for functions in the class $\mathcal{C}(A,B)$. The Dieudonn\'{e}'s lemma which gives the exact region of variability for derivatives at a point of bounded functions, plays the key role in this study, and we also use this lemma to construct the extremal functions for the sharpness by a new method.

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Md Firoz Ali, Sanjit Pal. 2022-12-13. The Schwarzian norm estimates for Janowski convex functions. https://doi.org/10.1017/s0013091524000014

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