Nesbitt's inequality

Nesbitt's inequality states that for positive real numbers x, y, and z, (x+y+z)(1/x+1/y+1/z) is at least 9, with equality when x=y=z.

Arithmetic mean–geometric mean inequality: An inequality stating that the arithmetic mean of nonnegative numbers is at least their geometric mean. It proves the equivalent bound x/y+y/z+z/x ≥ 3 after cyclic rearrangement.
Reciprocal: The reciprocal of a nonzero number x is the number 1/x, whose product with x is 1. The inequality pairs the sum of the variables with the sum of their reciprocals.
Olympiad inequality: An inequality used in mathematical olympiad problems to establish bounds among variables or expressions. Nesbitt's inequality is a standard entry point for olympiad proofs involving positive variables.
Mitrinović's inequality: A classical inequality comparing sums of powers of positive variables. It supplies a broader mean-comparison framework rather than Nesbitt's specific three-variable reciprocal bound.
Titu's lemma: A form of the Cauchy–Schwarz inequality stating that Σ(aᵢ²/bᵢ) ≥ (Σaᵢ)²/Σbᵢ for positive bᵢ. Setting each aᵢ to 1 and bᵢ to x, y, z bounds the reciprocal sum.
Positive real number: A real number greater than zero. Positivity ensures the reciprocals exist and allows the standard inequality arguments.
Shapiro cyclic inequality: A family of inequalities bounding cyclic sums of fractions in positive variables. Nesbitt's inequality is the three-variable base case of a well-known cyclic inequality pattern.
Hölder's inequality: An inequality bounding sums or integrals of products using conjugate exponents. Hölder generalizes the product-sum technique behind a Cauchy–Schwarz proof of Nesbitt.
Equality condition: The conditions under which the two sides of an inequality are equal. Equality in Nesbitt's inequality holds exactly when all three positive variables are equal.
Algebraic expansion: The process of rewriting a product of sums as a sum of individual products. Expanding the product reveals three terms x/y, y/z, z/x and their oppositely ordered counterparts.

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