Suppose you have a Hoelder-continuous function $f \colon \mathbb{R} \to \mathbb{R}_{+}$ of any order $\gamma \in (0,1/2)$. I would like to replace the "complicated" difference $\frac{1}{h} \int_x^{x+h} (f(u)-f(x)) \, \mathrm{d}u $, which by the Mean value theorem is equal to $f(x+\xi_{h})-f(x)$ for some $\xi_h \in (x,x+h)$, by the simpler one $f(x+h)-f(x)$. To be more specific, I'm looking for conditions (not involving differentiability which would make things a lot easier but is unfortunately not given in my case) for the following two statements to be true:
Statement 1: $$ \operatorname{limsup}_{h \to 0} \left|\frac{\frac{1}{h} \int_x^{x+h} (f(u)-f(x)) \, \mathrm{d}u }{f(x+h)-f(x)}\right|\leq 1$$.
Statement 2: $$ \operatorname{lim}_{h \to 0} \frac{\frac{1}{h} \int_x^{x+h} (f(u)-f(x)) \, \mathrm{d}u }{f(x+h)-f(x)} = 1$$.
Thanks for your help!