Retkes identities

In mathematics, the Retkes Identities, named after Zoltán Retkes, are one of the most efficient applications of the Retkes inequality, when $f(u)=u^{\alpha}$, $0\leq u <\infty$, and $0\leq\alpha$. In this special setting, one can have for the iterated integrals

$F^{(n-1)}(s)=\frac{s^{\alpha+n-1}}{(\alpha+1)(\alpha+2) \cdots (\alpha+n-1)}.$

The notation is explained at Hermite–Hadamard inequality.

Particular cases

Since $f$ is strictly convex if $\alpha >1$, strictly concave if $0<\alpha<1$, linear if $\alpha=0,1$, the following inequalities and identities hold:

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

• $\alpha=1\quad\quad\quad\quad\sum_{i=1}^n\frac{x_i^n}{\Pi_i(x_1,\ldots,x_n)}=\sum_{i=1}^n x_i$

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

• $\alpha=0\quad\quad\quad\quad\sum_{i=1}^n\frac{x_i^{n-1}}{\Pi_i(x_1,\ldots,x_n)}=1.$

Consequences

One of the consequences of the case $\quad\alpha=1$ is the Retkes convergence criterion because of the right side of the equality is precisely the nth partial sum of $\quad\sum_{i=1}^{\infty}x_i.$

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

• $\quad\sum_{i=1}^n\frac{x_i^n}{\Pi_i(x_1,\ldots,x_n)}=\sum_{i=1}^n x_i$

• $\quad\sum_{i=1}^n\frac{x_i^{n-1}}{\Pi_i(x_1,\ldots,x_n)}=1$

• $\quad \sum_{i=1}^n\frac{1}{x_i} = (-1)^{n-1} \prod_{i=1}^n x_i \sum_{i=1}^n \frac{1}{{x_i}^2 \Pi_i(x_1,\ldots,x_n)}$

• $\quad\prod_{i=1}^n\frac{1}{x_i}=(-1)^{n-1}\sum_{i=1}^n\frac{1}{x_i\Pi_i(x_1,\ldots,x_n)}$

References

• Zoltán Retkes (2008). "An extension of the Hermite—Hadamard inequality". Acta Sci. Math. (Szeged) 74: 95–106.
• Zoltán Retkes (2006). "Applications of the extended Hermite—Hadamard inequality". Journal of Inequalities in Pure and Applied Mathematics (JIPAM) 7 (1): article 24.