Hermite–Hadamard inequality

For a continuous convex function on $[a,b]$, where $a<b$, its integral average lies between its midpoint value and the average of its endpoint values. For a concave function both inequalities reverse. The bound involving endpoints follows by comparison with the affine function joining them and monotonicity of the Riemann integral. The midpoint bound follows from the Jensen inequality on equal-weight midpoint tags and the Riemann sum convergence theorem.