In mathematics, the Retkes identities, named after Zoltán Retkes, are one of the most efficient applications of the Retkes inequality

Hermite–Hadamard inequality

In mathematics, the Hermite–Hadamard inequality, named after Charles Hermite and Jacques Hadamard and sometimes also called Hadamard's inequality, states that if a function ƒ : [a, b] → R is convex, then...

 when
 and . In this special settings one can have for the iterated integrals

Hermite–Hadamard inequality

In mathematics, the Hermite–Hadamard inequality, named after Charles Hermite and Jacques Hadamard and sometimes also called Hadamard's inequality, states that if a function ƒ : [a, b] → R is convex, then...

Since is strictly convex

Convex function

In mathematics, a real-valued function f defined on an interval is called convex if the graph of the function lies below the line segment joining any two points of the graph. Equivalently, a function is convex if its epigraph is a convex set...

 if , strictly concave

Concave function

In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap or upper convex.-Definition:...

 if , linear

Linear function

In mathematics, the term linear function can refer to either of two different but related concepts:* a first-degree polynomial function of one variable;* a map between two vector spaces that preserves vector addition and scalar multiplication....

 if hence the following inequalities and identities (mathematics) hold:

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One of the consequence of the case is the Retkes convergence criterion

Retkes convergence criterion

In mathematics, the Retkes convergence criterion, named after Zoltán Retkes, gives necessary and sufficient conditions for convergence of numerical series. Numerous criteria are known for testing convergence. The most famous of them is the so called Cauchy criterion, the only one that gives...

because of the right side of the equality is precisely the nth partial sum of

Assume henceforth that Under this condition substituting instead of in the second and fourth identities one can have two universal algebraic identities. These four identities are the so called Retkes identities:

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