The following inequality appeared in the analysis of a random approximation algorithm: $$ \int_u^{u+1} x^p\ \mathrm{dx} \leq \sqrt{u^p(u+1)^p}\text{, for } -1\leq p\leq 0, u\geq 1. $$

This resembles the well-known Hermite-Hadamard inequality for convex functions $$ \int_a^b f(x)\ \mathrm{dx} \leq (b-a)\frac{f(a)+f(b)}{2} $$ but with the right-hand side being the geometric mean, rather than the arithmetic mean $(u^p+(u+1)^p)/2$.

Computationally, this inequality appears to hold, but I have been unable to find a proof. Is this a known result? If so, is there a broader class of convex functions for which it holds?