I need help to prove the inequalities in the following cases $ ||x|^p-|y|^p|\leq \begin{cases} |x-y|^p & \mathrm{if} \, 0<p<1\\ p|x-y|(x^{p-1}+y^{p-1}) & \mathrm{if} \, 1\leq p<\infty \end{cases}$
and for the case $ 1 \leq p <\infty $ and $\|f\|_p \leq R $ , $ \|g\|_p \leq R $ prove that $\int||f|^p-|g|^p| \, \mathrm{d}\mu \leq 2pR^{p-1}\|f-g\|_p $.
of course it's part of the problem 24 of Rudin's Real and Complex Analysis, Chapter 3.
To prove the first inequality it suffices to prove that $(x+y)^{p}\le x^{p}+y^{p}$ for $x,y\ge0$ and $p\in(0,1)$. Notice that we can use this as follows: $\vert x\rvert^{p}\le\lvert x-y\rvert^{p}+\lvert y\rvert^{p}$ then reversing the roles of $x$ and $y$. Notice that this Lemma can be proven by reducing to the case when $y=1$ then looking at $f(x)=(x+1)^{p}-x^{p}$.
For the second inequality notice that for $x>y$:
$\lvert\lvert x\rvert^{p}-\lvert y\rvert^{p}\rvert\le\lvert x^{p}-y^{p}\vert=\lvert\int_{y}^{x}pt^{p-1}dt\rvert\le\int_{y}^{x}pdt(\lvert x\rvert^{p-1}+\lvert y\vert^{p-1})=p\lvert x-y\rvert(\vert x\rvert^{p-1}+\lvert y\rvert^{p-1})$.