My question is about the precise definition regarding the following:

Let $f$ be an orientation-preserving $C^1$ diffeomorphism of the unit circle $S^1$. So $f'(b)$ exists and can be thought as a positive real number for every $b$ in $S^1$. Now I want to define when $f$ is $C^{1,\alpha}, 0<\alpha<1,$ near $1\in S^1$. I know one definition can be $f'(a)-f'(1)= O|a-1|^{\alpha}, i.e. |f'(a)-f'(1)|\le K.|a-1|^{\alpha}$. But I was wondering whether we can use the following alternate definition, using only information on $f$, but not on $f'$, motivated by $C^{1,\alpha}$-maps on $\mathbb{R}^1$:

1) Can we say $f$ is $C^{1,\alpha}$ near $1$ if $|f(a)-f(1)-f'(1)(a-1)|= O(|a-1|^{1+\alpha})$ ?

2) I have also another related question. let $F$ be a $C^1$ diffeomorphism on an open set containing the closed unit disk $\bar{D}$ and $F\in C^2({\mathbb{D}})$, and let $lim_{z\to1}F_z(z)=p, lim_{z\to1}F_{\bar{z}}(z)=q$. then can we say that $F$ is $C^{1,\alpha}$ near $1$(near in the sense of the topology of $\bar{D}$) if $|F(z)-F(1)-p(z-1)-q(\bar{z}-1)|=O|z-1|^{1+\alpha}$.

If the above are not correct, could you please give me or refer to me an alternate definition for each of the above ? Thank you.