I'm reading Kai Diethelm's "The analysis of fractional differential equations" and there's a part I haven't been able to figure out. It's from page 93.
Let $p >0.$ Why does the following inequality hold?
$\frac{L}{\Gamma(p)}\sup_{0\leq w \leq x}\int_{0}^{w}(w-t)^{p-1}|A^{j-1}y_{1}(t)-A^{j-1}y_{2}(t)|dt\\ \leq \frac{L}{\Gamma(p)}\int_{0}^{x}(x-t)^{p-1}\sup_{0\leq w \leq t}|A^{j-1}y_{1}(w)-A^{j-1}y_{2}(w)|dt$