Alternate definitions of $C^{1,\alpha}$ and $C^{1,\alpha}(\bar{D})$ maps 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.
 A: It is a good exercise to show that for a function $u: \mathbb{R}^n \rightarrow \mathbb{R}$,  $C^{1,\alpha}$ regularity is equivalent to the following: there exists a linear function $l_x$ such that 
$$\|u-l_x\|_{L^{\infty}(B_r(x))} \leq Cr^{1+\alpha}$$
where $C$ is uniform. The forward direction is clear; for the other direction, observe that for points distance $r$ apart, the linear approximations differ by $Cr^{1+\alpha}$ nearby these points, so (up to changing $C$ by constants depending only on $n$ and $\alpha$) their slopes can differ by at most $Cr^{\alpha}$.
Actually, in the proof above I only needed that I can choose $C$ uniform on neighborhoods. This is crucial; as Pietro notes, we can get this property at a point (say 0) by taking an arbitrary bounded function and multiplying by $|x|^{1+\alpha}$, which clearly doesn't give $C^{1,\alpha}$ regularity away from $0$.
The equivalence of "pointwise $C^{1,\alpha}$ regularity" with uniform constant and $C^{1,\alpha}$ is the basis of most approaches I know to proving $C^{1,\alpha}$ estimates in PDE. This is usually done by estimating a solution with a linear function, rescaling, and improving the estimate geometrically, i.e. finding $r_k \rightarrow 0$ and $l_k$ so that 
$$\|u-l_k\|_{L^{\infty}(B_r(x))} \leq r_k^{1+\alpha}.$$
One can analogously define "pointwise $C^{k,\alpha}$" by approximation with $k^{th}$ order polynomials to get Holder estimates on higher derivatives.
A: The latter property does not implies the former: any $C^1$ diffeo of the form $f(a)=a+O(|a-1|^{\alpha+1})$, satisfies the latter property, but can have a derivative oscillating so strongly, that the former property does not hold.
However, the former property implies the latter, since
$|f(a)-f(1)-f'(1)(a-1)|\le \int_1^a |f'(t)-f'(1)| dt\le K \int_1^a |t-1|^\alpha dt \le K_1 |a-1|^{\alpha+1}\, .$
