The Schwarzian norm estimates for Janowski convex functions

For $-1\le B<A\le 1$, let $\mathcal{C}(A,B)$ denote the class of normalized Janowski convex functions defined in the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }<1\}$ that satisfy the subordination relation $1+z{f}^{\mathrm{\prime }\mathrm{\prime }}(z)\mathrm{/}{f}^{\mathrm{\prime }}(z)\prec (1+Az)\mathrm{/}(1+Bz)$. 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.