I'm having trouble with the following problem:
Consider $\beta\in(\frac{1}{2},1]$, $\xi\in\mathbb{R}$, $f:\mathbb{R}\longrightarrow \mathbb{R}$ bounded with first and second derivatives bounded, $X\in \mathcal{C}^{\beta}[0,T]$ (Space of Hölder continuous functions of order $\beta$). Let $\alpha\in (\frac{1}{2},\beta)$ and define $$\mathcal{M}_T:\mathcal{C}^{\alpha}[0,T]\longrightarrow \mathcal{C}^{\alpha}[0,T]$$ $$Y\mapsto \xi + \int_{0}^{t}f(Y(r))dX(r), \quad t\in[0,T]$$ (Usual Riemann-Stieltjes integral). Show that there exist $T_0\in[0,T]$ such that for every $Y,\tilde{Y}\in\mathcal{C}^{\alpha}[0,T_0]$ $$\|\mathcal{M}_{T_0}(Y)-\mathcal{M}_{T_0}(\tilde{Y})\|_{\alpha, [0,T_0]}\leq \frac{1}{2}\|Y-\tilde{Y}\|_{\alpha,[0,T_0]},$$ where $$\|X\|_{\alpha,[0,T]}:=\sup_{s\neq t \in [0,T]}\dfrac{|X(t)-X(s)|}{|t-s|^{\alpha}}.$$
I don't see how to use the boundedness of the second derivative and to obtain the inequality. What is more, I don't even know if the function is well-defined. (I just know that the integral exists, and here I use the boundedness of the first derivative).