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In mathematics, the '''Retkes Identities''', named after Zoltán Retkes, are one of the most efficient applications of the [[Hermite–Hadamard inequality#Theorem (Retkes inequality)|Retkes inequality]], when <math>f(u)=u^{\alpha}</math>, <math>0\leq u <\infty </math>, and <math>0\leq\alpha</math>. In this special setting, one can have for the [[Hermite–Hadamard inequality|iterated integrals]]

: <math>F^{(n-1)}(s)=\frac{s^{\alpha+n-1}}{(\alpha+1)(\alpha+2) \cdots (\alpha+n-1)}.</math>

The notation is explained at [[Hermite–Hadamard inequality#Theorem (Retkes inequality)|Hermite–Hadamard inequality]].

== Particular cases ==
Since <math>f</math> is strictly [[convex function|convex]] if <math>\alpha >1</math>, strictly [[concave function|concave]] if <math>0<\alpha<1</math>, [[linear function|linear]] if <math>\alpha=0,1</math>, the following [[inequalities]] and [[identities (mathematics)|identities]] hold:

* <math>1<\alpha\quad\quad\quad\quad\frac{1}{(\alpha+1)(\alpha+2)\cdots(\alpha+n-1)}\sum_{i=1}^n\frac{x_i^{\alpha+n-1}}{\Pi_k(x_1,\ldots,x_n)}<\frac{1}{n!}\sum_{i=1}^n x_i^{\alpha} </math>

* <math>\alpha=1\quad\quad\quad\quad\sum_{i=1}^n\frac{x_i^n}{\Pi_i(x_1,\ldots,x_n)}=\sum_{i=1}^n x_i</math>

* <math>0<\alpha<1\quad\quad\frac{1}{(\alpha+1)(\alpha+2) \cdots (\alpha+n-1)} \sum_{i=1}^n\frac{x_i^{\alpha+n-1}}{\Pi_k(x_1,\ldots,x_n)}>\frac{1}{n!}\sum_{i=1}^n x_i^{\alpha} </math>

* <math>\alpha=0\quad\quad\quad\quad\sum_{i=1}^n\frac{x_i^{n-1}}{\Pi_i(x_1,\ldots,x_n)}=1.</math>

== Consequences ==
One of the consequences of the case <math>\quad\alpha=1</math> is the [[Retkes convergence criterion]] because of the right side of the equality is precisely the ''n''th partial sum of <math>\quad\sum_{i=1}^{\infty}x_i.</math>

Assume henceforth that <math>x_k\neq 0\quad k=1,\ldots,n.</math> Under this condition substituting <math>\quad\frac{1}{x_k}</math> instead of <math>\quad x_k</math> in the second and fourth identities one can have two universal algebraic identities. These four identities are the so-called '''Retkes identities''':

* <math>\quad\sum_{i=1}^n\frac{x_i^n}{\Pi_i(x_1,\ldots,x_n)}=\sum_{i=1}^n x_i</math>

* <math>\quad\sum_{i=1}^n\frac{x_i^{n-1}}{\Pi_i(x_1,\ldots,x_n)}=1</math>

* <math>\quad \sum_{i=1}^n\frac{1}{x_i} = (-1)^{n-1} \prod_{i=1}^n x_i \sum_{i=1}^n \frac{1}{{x_i}^2 \Pi_i(x_1,\ldots,x_n)}</math>

* <math>\quad\prod_{i=1}^n\frac{1}{x_i}=(-1)^{n-1}\sum_{i=1}^n\frac{1}{x_i\Pi_i(x_1,\ldots,x_n)} </math>

==References ==

* {{cite journal |author=Zoltán Retkes |title=An extension of the Hermite—Hadamard inequality |journal=Acta Sci. Math. (Szeged) |volume=74 |year=2008 |pages=95—106}}
* {{cite journal |author=Zoltán Retkes |title=Applications of the extended Hermite—Hadamard inequality |journal=Journal of Inequalities in Pure and Applied Mathematics (JIPAM) |volume=7 |issue=1 |pages=article 24 |year=2006 |url=http://www.emis.de/journals/JIPAM/article631.html?sid=631}}

[[Category:Mathematical identities]]
[[Category:Mathematical series]]
[[Category:Inequalities]]

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183.173.54.211: /* Definition */

en>David Eppstein: unreferenced, stub sort