In Royden's real analysis, the proof for the Hölder inequality (on pg. 121) is stated as follows:
If $p$ and $q$ are nonnegative extended real numbers such that
$\frac{1}{p} + \frac{1}{q} = 1,$
and if $f \in L^p$ and $g \in L^q$, then $f \cdot g \in L^1$ and
$\int |fg| \leq ||f||_p \cdot ||g_q||.$
The proof is trivial for $p=\infty$ or $q = \infty$ so assume $1 < p < \infty$ and $1 < q < \infty$.
In the proof of this, the function $h(x) = g(x)^{q-1} = g(x)^{\frac{q}{p}}$ is defined. This yields $g(x) = h(x)^{\frac{p}{q}}$.
After defining $h$, the book says, without explanation,
$ptf(x)g(x) = ptf(x)h(x)^{p-1} \leq (h(x)+tf(x))^p - h(x)^p.$
Where does this inequality come from? I want to say that somehow it involves convexity, but I am not sure.