Titu's lemma

Titu's lemma states that for real numbers and positive denominators, the sum of squared numerators divided by denominators is at least the square of their sum divided by the sum of denominators. It is a direct form of the Cauchy–Schwarz inequality.

Cauchy–Schwarz inequality: An inequality bounding the squared inner product of two vectors by the product of their squared lengths. Applying it to numerator-over-root-denominator terms gives Titu's lemma.
Nesbitt's inequality: For positive numbers, the sum of pairwise fractions such as a/(b+c) is at least three halves. Titu's lemma proves this by bounding the sum below using the total numerator and denominator.
Inner product: A bilinear operation that assigns a scalar to two vectors and generalizes the dot product. Cauchy–Schwarz, and therefore Titu's lemma, bounds inner products.
AM–GM inequality: The arithmetic mean of nonnegative numbers is at least their geometric mean. AM–GM gives a different lower bound, while Titu's lemma compares a sum of fractions with a ratio of totals.
Equality condition in Cauchy–Schwarz: The condition that two vectors are linearly dependent, making Cauchy–Schwarz an equality. It yields equality when each numerator is proportional to its denominator.
Shapiro inequality: A family of inequalities bounding cyclic sums of fractions involving positive variables and their neighbors. Titu's lemma supplies a common lower-bound step in proofs of Shapiro-type inequalities.
Euclidean norm: The square root of the sum of squared coordinates of a vector. The Cauchy–Schwarz proof compares the lengths of vectors built from the terms.
Hölder's inequality: An inequality bounding sums of products using conjugate-power norms. Hölder generalizes the product estimate behind Cauchy–Schwarz to exponents beyond two.
Engel form of the Cauchy–Schwarz inequality: The inequality stating that a sum of squared terms over positive terms is at least the squared sum divided by the sum of the positive terms. This is the standard name for Titu's lemma in inequality texts.
Cauchy–Schwarz inequality in algebraic inequalities: The use of Cauchy–Schwarz and its variants to establish inequalities among algebraic expressions. Titu's lemma is one of the most reusable specialized forms in this proof toolkit.

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