Levinson's inequality

Levinson's inequality compares sums of a function at paired points on opposite sides of a midpoint. It bounds the difference between the two sums when the function has nonnegative third derivative.

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Convex function: A function whose value along each line segment does not exceed the corresponding linear interpolation of endpoint values. Nonnegative third derivative makes the first derivative convex, enabling the paired-sum comparison.
Third derivative: The derivative obtained by differentiating a function three times, when those derivatives exist. Its nonnegativity is the central shape condition in the stated form of Levinson's inequality.
Popoviciu's inequality: An inequality for convex functions relating their values at interval endpoints, the midpoint, and the average of sampled points. It belongs to the family of refinements and companions to classical convexity inequalities.
Norman Levinson: An American mathematician known for work in analysis, differential equations, and inequalities. Levinson introduced the inequality that bears his name.
Majorization: A preorder on vectors comparing how concentrated their components are through partial sums of sorted entries. Majorization provides a general framework for inequalities comparing sums over differently dispersed points.
Midpoint: The point exactly halfway between two points on a line or interval. Each comparison pairs one argument below this point with another the same distance above it.
Mercer’s inequality: An inequality for convex functions that bounds function values at reflected points using endpoint values. Its reflected-point structure offers a related way to exploit convexity in sum estimates.
Jensen's inequality: An inequality placing a convex function of an average below or equal to the average of its function values. Levinson's result extends the tradition of deriving sum bounds from convexity.
Hermite-Hadamard inequality: For a convex function on an interval, its integral average lies between its value at the midpoint and the endpoint average. It illustrates how convexity controls averages, while Levinson compares discrete paired sums.
Convexity: A property of a function whose graph lies below every chord joining two points on the graph. The third-derivative condition implies convexity of the first derivative.