Retkes identities

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

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

The notation is explained at Hermite–Hadamard inequality.

Particular cases

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

• ${\displaystyle 1<\alpha \quad \quad \quad \quad {\frac {1}{(\alpha +1)(\alpha +2)\cdots (\alpha +n-1)}}\sum _{i=1}^{n}{\frac {x_{i}^{\alpha +n-1}}{\Pi _{k}(x_{1},\ldots ,x_{n})}}<{\frac {1}{n!}}\sum _{i=1}^{n}x_{i}^{\alpha }}$
• ${\displaystyle \alpha =1\quad \quad \quad \quad \sum _{i=1}^{n}{\frac {x_{i}^{n}}{\Pi _{i}(x_{1},\ldots ,x_{n})}}=\sum _{i=1}^{n}x_{i}}$
• ${\displaystyle 0<\alpha <1\quad \quad {\frac {1}{(\alpha +1)(\alpha +2)\cdots (\alpha +n-1)}}\sum _{i=1}^{n}{\frac {x_{i}^{\alpha +n-1}}{\Pi _{k}(x_{1},\ldots ,x_{n})}}>{\frac {1}{n!}}\sum _{i=1}^{n}x_{i}^{\alpha }}$
• ${\displaystyle \alpha =0\quad \quad \quad \quad \sum _{i=1}^{n}{\frac {x_{i}^{n-1}}{\Pi _{i}(x_{1},\ldots ,x_{n})}}=1.}$

Consequences

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

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

• ${\displaystyle \quad \sum _{i=1}^{n}{\frac {x_{i}^{n}}{\Pi _{i}(x_{1},\ldots ,x_{n})}}=\sum _{i=1}^{n}x_{i}}$
• ${\displaystyle \quad \sum _{i=1}^{n}{\frac {x_{i}^{n-1}}{\Pi _{i}(x_{1},\ldots ,x_{n})}}=1}$
• ${\displaystyle \quad \sum _{i=1}^{n}{\frac {1}{x_{i}}}=(-1)^{n-1}\prod _{i=1}^{n}x_{i}\sum _{i=1}^{n}{\frac {1}{{x_{i}}^{2}\Pi _{i}(x_{1},\ldots ,x_{n})}}}$
• ${\displaystyle \quad \prod _{i=1}^{n}{\frac {1}{x_{i}}}=(-1)^{n-1}\sum _{i=1}^{n}{\frac {1}{x_{i}\Pi _{i}(x_{1},\ldots ,x_{n})}}}$

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.