Jensen's inequality

Jensen's inequality states that a convex function evaluated at an average is no greater than the average of its values; for a concave function, the inequality reverses.

Convex function: A function whose value along any line segment lies no higher than the segment joining its endpoint values. Its defining property supplies Jensen’s inequality for weighted averages.
Markov's inequality: A probability bound on the chance that a nonnegative random variable exceeds a positive threshold. Applying Jensen to an exponential moment helps derive exponential probability bounds.
Concave function: A function whose value along any line segment lies no lower than the segment joining its endpoint values. Jensen’s inequality reverses direction when the function is concave.
Hölder's inequality: An inequality bounding the integral or sum of a product using powers of the factors. It controls products through norms, rather than comparing a function of an average with an average of function values.
Epigraph: The set of points lying on or above the graph of a function. Convexity of this set gives a geometric route to the inequality.
Log-sum inequality: An inequality comparing sums of weighted logarithms with the logarithm of corresponding sums. It follows from convexity and underpins standard comparisons between probability distributions.
Weighted average: An average in which each value contributes according to a specified nonnegative weight. Jensen compares applying a function to this average with averaging the function values.
Cauchy–Schwarz inequality: An inequality bounding an inner product by the product of the lengths of its vectors. It gives a different route to moment bounds, with different assumptions and equality cases.
Finite convex combination: A weighted sum of points with nonnegative weights that sum to one. Jensen’s finite form compares the function of this weighted point with the weighted function values.
Kullback–Leibler divergence: A nonnegative measure of the information lost when one probability distribution approximates another. Jensen’s inequality proves its nonnegativity.

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