The Schwarzian norm estimates for Janowski convex functions

For -1≤ B < A≤ 1 , let C(A,B) denote the class of normalized Janowski convex functions defined in the unit disk D:=:|z| < 1\ that satisfy the subordination relation 1+zf''(z)/f'(z) (1+Az)/(1+Bz) . In the present article, we determine the sharp estimate of the Schwarzian norm for functions in the class 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.