Inequalities for the arithmetical functions of Euler and Dedekind

Alzer, Horst, and Kwong, Man Kam. "Inequalities for the arithmetical functions of Euler and Dedekind." Czechoslovak Mathematical Journal 70.3 (2020): 781-791. <http://eudml.org/doc/297346>.

@article{Alzer2020,
abstract = {For positive integers $n$, Euler’s phi function and Dedekind’s psi function are given by \[ \phi (n)= n \prod \_\{\begin\{array\}\{c\} p\mid n \\ p \ \{\rm prime\}\end\{array\}\} \Bigl (1-\frac\{1\}\{p\}\Bigr ) \quad \mbox\{and\} \quad \psi (n)=n\prod \_\{\begin\{array\}\{c\} p\mid n \\ p \ \{\rm prime\}\end\{array\}\} \Bigl (1+\frac\{1\}\{p\}\Bigr ), \] respectively. We prove that for all $n\ge 2$ we have \[ \Bigl (1-\frac\{1\}\{n\}\Bigr )^\{n-1\}\Bigl (1+\frac\{1\}\{n\}\Bigr )^\{n+1\} \le \Bigl (\frac\{\phi (n)\}\{n\} \Bigr )^\{\phi (n)\} \Bigl ( \frac\{\psi (n)\}\{n\}\Bigr )^\{\psi (n)\} \] and \[ \Bigl (\frac\{\phi (n)\}\{n\} \Bigr )^\{\psi (n)\} \Bigl ( \frac\{\psi (n)\}\{n\}\Bigr )^\{\phi (n)\} \le \Bigl (1-\frac\{1\}\{n\}\Bigr )^\{n+1\}\Bigl (1+\frac\{1\}\{n\}\Bigr )^\{n-1\}. \] 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).},
author = {Alzer, Horst, Kwong, Man Kam},
journal = {Czechoslovak Mathematical Journal},
keywords = {Euler's phi function; Dedekind's psi function; inequalities},
language = {eng},
number = {3},
pages = {781-791},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Inequalities for the arithmetical functions of Euler and Dedekind},
url = {http://eudml.org/doc/297346},
volume = {70},
year = {2020},
}

TY - JOUR
AU - Alzer, Horst
AU - Kwong, Man Kam
TI - Inequalities for the arithmetical functions of Euler and Dedekind
JO - Czechoslovak Mathematical Journal
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 70
IS - 3
SP - 781
EP - 791
AB - For positive integers $n$, Euler’s phi function and Dedekind’s psi function are given by \[ \phi (n)= n \prod _{\begin{array}{c} p\mid n \\ p \ {\rm prime}\end{array}} \Bigl (1-\frac{1}{p}\Bigr ) \quad \mbox{and} \quad \psi (n)=n\prod _{\begin{array}{c} p\mid n \\ p \ {\rm prime}\end{array}} \Bigl (1+\frac{1}{p}\Bigr ), \] respectively. We prove that for all $n\ge 2$ we have \[ \Bigl (1-\frac{1}{n}\Bigr )^{n-1}\Bigl (1+\frac{1}{n}\Bigr )^{n+1} \le \Bigl (\frac{\phi (n)}{n} \Bigr )^{\phi (n)} \Bigl ( \frac{\psi (n)}{n}\Bigr )^{\psi (n)} \] and \[ \Bigl (\frac{\phi (n)}{n} \Bigr )^{\psi (n)} \Bigl ( \frac{\psi (n)}{n}\Bigr )^{\phi (n)} \le \Bigl (1-\frac{1}{n}\Bigr )^{n+1}\Bigl (1+\frac{1}{n}\Bigr )^{n-1}. \] 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).
LA - eng
KW - Euler's phi function; Dedekind's psi function; inequalities
UR - http://eudml.org/doc/297346
ER -