Czechoslovak Mathematical Journal, Vol. 70, No. 3, pp. 781-791, 2020
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).
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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
