Carleman's inequality

Carleman's inequality states that for nonnegative terms, the sum of successive geometric means is at most e times the sum of the terms. It is a sharp infinite-series inequality.

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Arithmetic–geometric mean inequality: The arithmetic mean of nonnegative numbers is at least their geometric mean. Applying this bound to each initial segment turns geometric means into weighted arithmetic sums.
Infinite series: A sum of infinitely many terms, defined through the limit of its finite partial sums. Carleman's inequality compares two infinite series through their partial-sum behavior.
Torsten Carleman: A Swedish mathematician known for work in analysis, including the inequality that bears his name. Carleman introduced the inequality in his work on inequalities for infinite series.
ℓ¹ space: The space of sequences whose absolute values have a finite sum. The right side is the ℓ¹ norm, so the inequality bounds geometric-mean sums by sequence mass.
Hölder's inequality: An inequality bounding sums or integrals of products using conjugate exponents. Hölder's inequality gives a direct finite-sum proof and exposes the role of the constant e.
Geometric mean: A mean of positive numbers defined as the root of their product. Each term on the left is the geometric mean of an initial segment.
Hardy–Littlewood–Pólya inequalities: A family of inequalities developed in classical analysis, including results on sequences, integrals, and majorization. Carleman's inequality belongs to the classical toolkit of inequalities associated with this tradition.
Sequence space: A vector space whose elements are sequences, often equipped with a norm or topology. Carleman's inequality is a tool for estimating operators and expressions on sequence spaces.
Jensen's inequality: A convex function of an average is no greater than the corresponding average of function values. The logarithm converts geometric means into averages, linking the inequality to convexity arguments.
Nonnegative real number: A real number greater than or equal to zero. The inequality's sequence terms are nonnegative, allowing products and sums without sign ambiguity.