Knowra Karamata's inequality Karamata's inequality Karamata's inequality states that if one sequence majorizes another, then the sum of a convex function over the first is at least its sum over the second. It converts a comparison of spread into a comparison of function values.
Majorization : A partial-sum ordering of two vectors with equal total sum, comparing how unevenly their entries are distributed. Karamata's inequality applies exactly when one sequence majorizes the other.
AM-GM inequality : The arithmetic mean of nonnegative numbers is at least their geometric mean. Karamata proves this by applying the logarithm to positive entries and comparing equal-sum sequences.
Jovan Karamata : A Serbian mathematician known for work in analysis, including inequalities and summability methods. The inequality bears his name and arose from his work on convexity and sequences.
Rearrangement inequality : An inequality bounding sums of products according to how two sequences are paired. It compares pairings of entries, rather than convex sums under majorization.
Power sum : A sum of powers of sequence entries, commonly written as the sum of each entry raised to a fixed exponent. For convex powers, majorization directly orders these sums.
Convex function : A function whose value at a weighted average is no greater than the corresponding average of its values. Convexity makes greater spread increase the sum of function values.
Power mean inequality : For positive inputs, power means increase as their exponent increases. Majorization gives a common route to comparing sums of powers and related means.
Hardy–Littlewood–Pólya inequality : A classical inequality relating majorization, convex functions, and sums of function values. This closely related formulation helped establish the central role of majorization in Karamata-type results.
Chebyshev's sum inequality : An inequality relating the average of paired products to the product of averages for similarly ordered sequences. Its ordering condition concerns paired sequences, not majorization and convexity.
Variance : The average squared deviation of values from their mean. Squared deviation is convex, so equal-mean vectors ordered by majorization have ordered sums of squares and variances.
Show all 27