Inequalities for the arithmetical functions of Euler and Dedekind

Alzer, Horst ; Kwong, Man Kam

Czechoslovak Mathematical Journal, Tome 70 (2020) no. 3, pp. 781-791.

Résumé

For positive integers $n$, Euler's phi function and Dedekind's psi function are given by $$ \phi (n)= n \prod _{\substack { p\mid n \\ p \ {\rm prime}}} \Bigl (1-\frac {1}{p}\Bigr ) \quad \mbox {and} \quad \psi (n)=n\prod _{\substack { p\mid n \\ p \ {\rm prime}}} \Bigl (1+\frac {1}{p}\Bigr ), $$ respectively. We prove that for all $n\geq 2$ we have $$ \Bigl (1-\frac {1}{n}\Bigr )^{n-1}\Bigl (1+\frac {1}{n}\Bigr )^{n+1} \leq \Bigl (\frac {\phi (n)}{n} \Bigr )^{\phi (n)} \Bigl ( \frac {\psi (n)}{n}\Bigr )^{\psi (n)} $$ and $$ \Bigl (\frac {\phi (n)}{n} \Bigr )^{\psi (n)} \Bigl ( \frac {\psi (n)}{n}\Bigr )^{\phi (n)} \leq \Bigl (1-\frac {1}{n}\Bigr )^{n+1}\Bigl (1+\frac {1}{n}\Bigr )^{n-1}. $$ \endgraf The sign of equality holds if and only if $n$ is a prime. The first inequality refines results due to Atanassov (2011) and Kannan \ Srikanth (2013).

Détail

DOI : 10.21136/CMJ.2020.0530-18

Classification : 11A25
Keywords: Euler's phi function; Dedekind's psi function; inequalities