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^α, 0 ≤ u < ∞ and 0 ≤ α. In this special settings one can have for the iterated integrals

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

Since   f is strictly convex if   α > 1, strictly concave if   0 < α < 1, linear if   α = 0, 1 hence the following inequalities and identities (mathematics) hold:

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

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

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

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

One of the consequence of the case   α = 1 is the Retkes convergence criterion because of the right side of the equality is precisely the nth partial sum of $\quad\sum_{k=1}^{\infty}x_k.$

Assume henceforth that x_k ≠ 0  k = 1, …, n. Under this condition substituting $\quad\frac{1}{x_k}$ instead of   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, "An extension of the Hermite–Hadamard inequality", Acta Sci. Math. (Szeged), 74 (2008), pages 95–106.
• Zoltán Retkes, "Applications of the extended Hermite–Hadamard inequality", Journal of Inequalities in Pure and Applied Mathematics (JIPAM), Vol 7, issue 1, article 24, (2006)