Holder inequality Proof Explanation

Question

I don't understand why

$$ptf(x)h(x)^{p-1}\leq (h(x)+tf(x))^p-h(x)^p$$.

Answer 1

As Thomas Shelby's question comment states, it follows from the lemma of

$$(a + tb)^p \ge a^p + ptba^{p-1} \implies ptba^{p-1} \le (a + tb)^p - a^p \tag{1}\label{eq1A}$$

Using $a = h(x)$ and $b = f(x)$ in \eqref{eq1A} gives what you're asking about, i.e.,

$$ptf(x)h(x)^{p-1}\leq (h(x)+tf(x))^p-h(x)^p \tag{2}\label{eq2A}$$