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The Schwarzian norm estimates for Janowski convex functions
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 67 / Issue 2 / May 2024
- Published online by Cambridge University Press:
- 12 February 2024, pp. 299-315
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- Article
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For
$-1\leq B \lt A\leq 1$, let 
$\mathcal{C}(A,B)$ denote the class of normalized Janowski convex functions defined in the unit disk 
$\mathbb{D}:=\{z\in\mathbb{C}:|z| \lt 1\}$ 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 Dieudonné’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.
