Trouble in a proof , functional analysis So, here is the point in the proof that I don't understand, he uses that it holds for f and g which are boundered(limited, e.g. there is some M such that $|f|<M$ and $|g|<M$) then for all p, $1\leq p < +\infty$ and $\alpha \in (0,1)$ it holds that:
$|f+g|^p \leq (1-\alpha)^{1-p} |f|^p + \alpha^{1-p} |g|^p$. Please help me why is this true?? This has to be true, it is rewritten form the book, and the rest of the proof is completely clear to me, I have to know why this??

This is due to convexity: Because of $1 \leq p <\infty$, the map $[0,\infty) \to [0,\infty), x \mapsto x^p$ is convex (check that the first derivative is increasing).
Hence,
\begin{eqnarray*}
|f+g|^p &\leq& \bigg( (1-\alpha) \frac{|f|}{1-\alpha} + \alpha \frac{|g|}{\alpha}\bigg)^p
\\ & \leq & (1 - \alpha) \cdot (|f|/(1-\alpha))^p + \alpha \cdot (|g|/\alpha)^p
\\ & = &    (1-\alpha)^{1-p} |f|^p + \alpha^{1-p} |g|^p.
\end{eqnarray*}
