On Chebyshev inequality for arithmetic functions (Q1825221)

Using a multidimensional discrete version of the Chebyshev inequality the author obtains some inequalities concerning arithmetic functions illustrated with examples. One of his results is the following: An arithmetic function f is called d-non-decreasing (d-non-increasing) if for all positive integers m, n the inequality f(m)\(\leq f(mn)\) (f(m)\(\geq f(mn))\) holds. Now, let f and g be arithmetic functions, both d-non- decreasing or both d-non-increasing and let h be a multiplicative and non-negative arithmetic function. Then \[ \sum_{d | n}h(d)\sum_{d | n}h(d)f(d)g(d)\quad \geq \quad \sum_{d | n}h(d)f(d)\sum_{d | n}h(d)g(d) \] for every positive integer n. If one of the functions f and g is d-non-decreasing and the other d-non-increasing, then the sign \(\geq\) should be reversed.