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

Voir la notice de l'article

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 
@article{10_21136_CMJ_2020_0530_18,
     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/}
}
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
SP  - 781
EP  - 791
VL  - 70
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2020.0530-18/
DO  - 10.21136/CMJ.2020.0530-18
LA  - en
ID  - 10_21136_CMJ_2020_0530_18
ER  - 
%0 Journal Article
%A Alzer, Horst
%A Kwong, Man Kam
%T Inequalities for the arithmetical functions of Euler and Dedekind
%J Czechoslovak Mathematical Journal
%D 2020
%P 781-791
%V 70
%N 3
%U http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2020.0530-18/
%R 10.21136/CMJ.2020.0530-18
%G en
%F 10_21136_CMJ_2020_0530_18
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

[1] Apostol, T. M.: Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics, Springer, New York (1976). | DOI | MR | JFM

[2] Atanassov, K. T.: Note on $\varphi$, $\psi$ and $\sigma$-functions III. Notes Number Theory Discrete Math. 17 (2011), 13-14. | MR | JFM

[3] Kannan, V., Srikanth, R.: Note on $\varphi$ and $\psi$ functions. Notes Number Theory Discrete Math. 19 (2013), 19-21. | JFM

[4] Mitrinović, D. S., Sándor, J., Crstici, B.: Handbook of Number Theory. Mathematics and Its Applications 351, Kluwer, Dordrecht (1996). | DOI | MR | JFM

[5] Sándor, J.: On certain inequalities for $\sigma$, $\varphi$, $\psi$ and related functions. Notes Number Theory Discrete Math. 20 (2014), 52-60. | MR | JFM

[6] Sándor, J.: Theory of Means and Their Inequalities. (2018), Available at \let \relax\brokenlink { http://www.math.ubbcluj.ro/ jsandor/lapok/Sandor-Jozsef-Theory of Means {and Their Inequalities.pdf}}.

[7] Sándor, J., Crstici, B.: Handbook of Number Theory II. Kluwer, Dordrecht (2004). | DOI | MR | JFM

[8] Solé, P., Planat, M.: Extreme values of Dedekind's $\psi$-function. J. Comb. Number Theory 3 (2011), 33-38. | MR | JFM

Cité par Sources :