A special Holder-Lipschitz function

Question

I am interested in extending classical inequalities in the domain of the real numbers to the appropriate setting within the complex numbers. Here is one example I couldn't solve yet. Let $\mathbf D$ be the closed unit disk, $0<\alpha<1$, and $L:= \displaystyle\sup_{z,w\in \mathbf D\atop z\not=w}\frac{|(1-z)^\alpha -(1-w)^\alpha|}{|z-w|^\alpha}$, where one takes the main branch of the $\alpha$-root on $\mathbb C\setminus\;]-\infty,0]$. Is $L<\infty$? Whereas this is easily seen to hold for real numbers $x,y\in [-1,1]$, that is one has $|(1-x)^\alpha-(1-y)^{\alpha}|\leq |x-y|^{\alpha}$, I see no elementary way to determine $L$. May be some one on this forum has an idea?

Answer 1

$L=\max\{1, 2^{1-\alpha}\sin(\alpha\pi/2)\}$. Proofs are in Comput. Methods Funct. Theory 20, 667–676 (2020) https://doi.org/10.1007/s40315-020-00344-7 and Elemente der Math. 75 (2020), 38--41.