Hermite–Hadamard inequality

For a convex function on an interval, its value at the midpoint is no greater than its integral average, which is no greater than the mean of its endpoint values.

Trapezoidal rule: A numerical integration method that approximates an integral by the area under a piecewise linear interpolant. The endpoint upper bound is the trapezoidal approximation, exact when the function is affine.
Definite integral: A limit of sums that measures signed accumulation of a function over an interval. The central quantity is the integral divided by the interval's length.
Charles Hermite: A French mathematician whose work ranged across number theory, algebra, and analysis. His name is attached to the inequality alongside Hadamard's.
Numerical integration: The approximation of definite integrals using finite computations or sampled function values. The endpoint average offers a simple upper estimate for integrals of convex functions.
Convex function: A function whose value along each line segment is no greater than the corresponding linear interpolation of endpoint values. The inequality follows from this defining chord property.
Average value of a function: The integral of a function over an interval divided by the interval's length. Hermite–Hadamard bounds this quantity using only the midpoint and endpoints.
Jacques Hadamard: A French mathematician known for contributions to analysis, geometry, and the theory of differential equations. His name identifies the classical integral bounds for convex functions.
Probability measure: A measure assigning total mass one to outcomes of a probabilistic model. Uniform probability on an interval turns the integral average into an expectation.
Integral mean value theorem: A theorem guaranteeing an intermediate function value equals the average value over an interval under continuity conditions. It translates the integral average into a function value that convexity can bound.
Strict convexity: A stronger form of convexity in which the chord inequality is strict between distinct points. Strict convexity generally makes the bounds strict on nondegenerate intervals.

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