| Title:
|
Inequalities for the arithmetical functions of Euler and Dedekind (English) |
| Author:
|
Alzer, Horst |
| Author:
|
Kwong, Man Kam |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
70 |
| Issue:
|
3 |
| Year:
|
2020 |
| Pages:
|
781-791 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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). (English) |
| Keyword:
|
Euler's phi function |
| Keyword:
|
Dedekind's psi function |
| Keyword:
|
inequalities |
| MSC:
|
11A25 |
| idZBL:
|
07250689 |
| idMR:
|
MR4151705 |
| DOI:
|
10.21136/CMJ.2020.0530-18 |
| . |
| Date available:
|
2020-09-07T09:39:05Z |
| Last updated:
|
2022-10-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148328 |
| . |
| Reference:
|
[1] Apostol, T. M.: Introduction to Analytic Number Theory.Undergraduate Texts in Mathematics, Springer, New York (1976). Zbl 0335.10001, MR 0434929, 10.1007/978-1-4757-5579-4 |
| Reference:
|
[2] Atanassov, K. T.: Note on $\varphi$, $\psi$ and $\sigma$-functions III.Notes Number Theory Discrete Math. 17 (2011), 13-14. Zbl 1259.11009, MR 1418823 |
| Reference:
|
[3] Kannan, V., Srikanth, R.: Note on $\varphi$ and $\psi$ functions.Notes Number Theory Discrete Math. 19 (2013), 19-21. Zbl 1329.11006 |
| Reference:
|
[4] Mitrinović, D. S., Sándor, J., Crstici, B.: Handbook of Number Theory.Mathematics and Its Applications 351, Kluwer, Dordrecht (1996). Zbl 0862.11001, MR 1374329, 10.1007/1-4020-3658-2 |
| Reference:
|
[5] Sándor, J.: On certain inequalities for $\sigma$, $\varphi$, $\psi$ and related functions.Notes Number Theory Discrete Math. 20 (2014), 52-60. Zbl 1344.11008, MR 1417443 |
| Reference:
|
[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}}. |
| Reference:
|
[7] Sándor, J., Crstici, B.: Handbook of Number Theory II.Kluwer, Dordrecht (2004). Zbl 1079.11001, MR 2119686, 10.1007/1-4020-2547-5 |
| Reference:
|
[8] Solé, P., Planat, M.: Extreme values of Dedekind's $\psi$-function.J. Comb. Number Theory 3 (2011), 33-38. Zbl 1266.11107, MR 2908180 |
| . |
