Shapiro inequality

For an odd number of positive real numbers arranged cyclically, the sum of each number divided by the sum of its two neighbors is at least half the number of terms.

Nesbitt's inequality: For positive numbers, the sum of pairwise ratios such as a/(b+c) is at least 3/2 for three variables. Its three-variable case is the first instance of Shapiro’s cyclic bound.
Cyclic permutation: A permutation that moves each position to the next, with the last position returning to the first. The denominator at each position uses precisely the next and previous entries in this cycle.
Olympiad inequality: An inequality problem designed for mathematical competitions, often requiring an elementary proof. Shapiro’s inequality is a standard model for competition problems involving cyclic ratios.
Shapiro's cyclic inequality for six variables: The six-variable case of Shapiro’s cyclic ratio sum, whose sharp lower bound is 2. It is a concrete even case showing that the odd-count bound does not persist.
Cauchy–Schwarz inequality: An inequality relating inner products to the product of vector lengths. Cauchy–Schwarz gives a direct lower bound for the cyclic sum, though not always the sharp one.
Positive real number: A real number greater than zero. Positivity keeps every denominator nonzero and makes the ratio expression meaningful.
AM–GM inequality: The arithmetic mean of positive numbers is at least their geometric mean. AM–GM estimates on neighboring sums can help derive lower bounds for Shapiro-type expressions.
Shapiro's cyclic inequality for three variables: The inequality a/(b+c)+b/(c+a)+c/(a+b) ≥ 3/2 for positive a, b, and c. This smallest odd case is sharp at equality and also known as Nesbitt’s inequality.
Rearrangement inequality: A theorem bounding sums of products according to how two ordered sequences are paired. Rearrangement arguments can exploit how cyclic denominators align with the numerators.
Harmonic mean: The reciprocal of the arithmetic mean of the reciprocals of a collection of nonzero numbers. Reciprocal-sum estimates related to harmonic means appear in proofs of ratio inequalities.

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