Alternate definitions of $C^{1,\alpha}$ and $C^{1,\alpha}(\bar{D})$ maps To Pietro, thanks. If for the first question, I JUST need $C^{1,\alpha}-regularity$ near $1$, i.e. if I want to get $f'(a)-f'(1)=O(|a-1|)^{\alpha},$ can we get it from the alternate definition. I am unable to fill out the details.

Alternate definitions of $C^{1,\alpha}$ and $C^{1,\alpha}(\bar{D})$ maps (To Conor, continued): every point $\zeta \in S^1,$ I have $F(z)-F(\zeta)-l_{\zeta}=O(|z-\zeta|^{1+\alpha})$, where the constant of Holder continuity is locally uniform ? My second comment, passing from difference quotient to the derivative expresses my main concern. You can ignore the rest.