Let $\varphi_{f,\delta}$ be a function defined by $\varphi_{f,\delta}(u)=\sup\left\{\left|f(y)-f(x)\right|:x,y\in\left[u-\delta,u+\delta\right]\right\}$ for a bounded and symmetric function $f:\left[-1,1\right]\rightarrow\mathbb{R}$ with compact support. Furthermore, the following holds. $\int_{\mathbb{R}}^{}\varphi_{f,\delta}(u)du=\mathcal{O}(\delta)$ as $\delta\rightarrow 0$. Question: How can I calculate upper bounds for following quantities using the functions $\varphi_{f,\delta}$.
(a) $\int_{-c}^{c}f(x+p)-f(y+p)dp$ and
(b) $\int_{-c}^{c}(f(x+p)-f(y+p))^{k}dp$ with $k\in\mathbb{N}$.
My own effort:
(a) It obviously holds that $f(x+p)-f(y+p)\leq\left|f(x+p)-f(y+p)\right|\leq \sup\left\{\left|f(z)-f(z')\right|:z,z'\in\left[x+p,y+p\right]\right\}$. In order to apply the functions $\varphi_{f,\delta}$ for a suitable $\delta$, we have to symmetrize the interval $\left[x+p,y+p\right]$. Therefore, we could replace the interval by $\left[\xi_{1},\xi_{2}\right]$ with $\xi_{1}=\frac{x+y+2p}{2}-(\frac{x+y+2p}{2}-(x+p))$ and $\xi_{2}=\frac{x+y+2p}{2}+(\frac{x+y+2p}{2}-(x+p))$. Where is my mistake?
(b) This should follow from (a) via Cauchy Schwarz or Jensen's inequality.