What inequality was applied?

Question

I'm reading some probabilistic paper and have a trouble with understanding some part.

Here is this part:

mu is a Lipschitz function and M is the Lipschitz constant, and:

What inequality was applied in the first line?

Thanks, Marcin

Answer 1

The context is not probabilistic since one asks how to pass from the identity $$G(t)=G(1)+\int_1^tH(s)\,\mathrm ds,$$ to $$|G(t)|^p\leqslant|G(1)|^p+p\int_1^t|G(s)|^{p-1}\cdot|H(s)|\,\mathrm ds.$$ This follows from the fact that both sides coincide when $t=1$ and that the derivative of the LHS is at most $$p\,|G'(t)|\,|G(t)|^{p-1}=p\,|H(t)|\,|G(t)|^{p-1}.$$