# Institute of Mathematics

## Inequalities for the arithmetical functions of Euler and Dedekind

#### Horst Alzer, Man Kam Kwong

###### Received December 2, 2018.   Published online January 27, 2020.

Abstract:  For positive integers $n$, Euler's phi function and Dedekind's psi function are given by $\varphi(n)= n \prod_{p\mid n, p\ {\rm prime}} (1-\frac{1}{p})$ and $\psi(n)=n\prod_{p\mid n, p \ {\rm prime}} (1+\frac{1}{p})$, respectively. We prove that for all $n\geq2$ we have $(1-\frac{1}{n})^{n-1} (1+\frac{1}{n})^{n+1} \leq (\frac{\varphi(n)}{n})^{\varphi(n)} (\frac{\psi(n)}{n})^{\psi(n)}$ and (\frac{\varphi(n)}{n} )^{\psi(n)} ( \frac{\psi(n)}{n})^{\varphi(n)} \leq (1-\frac{1}{n})^{n+1}(1+\frac{1}{n})^{n-1}$. The sign of equality holds if and only if$n$is a prime. The first inequality refines results due to Atanassov (2011) and Kannan \& Srikanth (2013). Keywords: Euler's phi function; Dedekind's psi function; inequalities Classification MSC: 11A25 DOI: 10.21136/CMJ.2020.0530-18 PDF available at: Springer Institute of Mathematics CAS References: [1] T. M. Apostol: Introduction to Analytic Number Theory. Undergraduate Texts in Mathematics, Springer, New York (1976). DOI 10.1007/978-1-4757-5579-4 | MR 0434929 | Zbl 0335.10001 [2] K. T. Atanassov: Note on$\varphi$,$\psi$and$\sigma$-functions III. Notes Number Theory Discrete Math. 17 (2011), 13-14. Zbl 1259.11009 [3] V. Kannan, R. Srikanth: Note on$\varphi$and$\psi$functions. Notes Number Theory Discrete Math. 19 (2013), 19-21. Zbl 1329.11006 [4] D. S. Mitrinović, J. Sándor, B. Crstici: Handbook of Number Theory. Mathematics and Its Applications 351, Kluwer, Dordrecht (1996). DOI 10.1007/1-4020-3658-2 | MR 1374329 | Zbl 0862.11001 [5] J. Sándor: On certain inequalities for$\sigma$,$\varphi$,$\psi$and related functions. Notes Number Theory Discrete Math. 20 (2014), 52-60. Zbl 1344.11008 [6] J. Sándor: Theory of Means and Their Inequalities. (2018), Available at http://www.math.ubbcluj.ro/~jsandor/lapok/Sandor-Jozsef-Theory of Means and Their Inequalities.pdf. [7] J. Sándor, B. Crstici: Handbook of Number Theory II. Kluwer, Dordrecht (2004). DOI 10.1007/1-4020-2547-5 | MR 2119686 | Zbl 1079.11001 [8] P. Solé, M. Planat: Extreme values of Dedekind's$\psi\$-function. J. Comb. Number Theory 3 (2011), 33-38. MR 2908180 | Zbl 1266.11107

Affiliations:   Horst Alzer (corresponding author), Morsbacher Strasse 10, 51545 Waldbröl, Germany, e-mail: h.alzer@gmx.de; Man Kam Kwong, Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, P. R. China, e-mail: mankwong@connect.polyu.hk

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