Geodesic
Inequalities for the arithmetical functions of Euler and Dedekind
Czechoslovak Mathematical Journal, Tome 70 (2020) no. 3, pp. 781-791 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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).
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).
DOI : 10.21136/CMJ.2020.0530-18
Classification : 11A25
Keywords: Euler's phi function; Dedekind's psi function; inequalities 
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     author = {Alzer, Horst and Kwong, Man Kam},
     title = {Inequalities for the arithmetical functions of {Euler} and {Dedekind}},
     journal = {Czechoslovak Mathematical Journal},
     pages = {781--791},
     year = {2020},
     volume = {70},
     number = {3},
     doi = {10.21136/CMJ.2020.0530-18},
     mrnumber = {4151705},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2020.0530-18/}
}
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Alzer, Horst; Kwong, Man Kam. Inequalities for the arithmetical functions of Euler and Dedekind. Czechoslovak Mathematical Journal, Tome 70 (2020) no. 3, pp. 781-791. doi: 10.21136/CMJ.2020.0530-18

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