On a number theoretic inequality of Ramanujan (Q6615323)

In this short but interesting article, the authors prove that, for all integers \(n \geqslant 2\), we have \N\[\N\tau(n) \leqslant \left( 1+\frac{\log n}{\log \gamma(n)}\right)^{\omega(n)}\N\]\Nprovided that, if \(n = p_1^{\alpha_1} \dotsb p_r^{\alpha_r}\), then \(p_1 < \dotsb < p_r\) and \(\alpha_1 \leqslant \dotsb \leqslant\alpha_r\). Here, \(\tau(n)\), \(\gamma(n)\) and \(\omega(n)\) are respectively the number of divisors, the squarefree kernel and the number of distinct prime divisors of \(n\). This improves on an old result of Ramanujan [\textit{S. Ramanujan}, Proc. Lond. Math. Soc. (2) 14, 347--409 (1915; JFM 45.1248.01)]. The proof rests on Chebyshev's inequality dealing with the product of arithmetic means, which can be found for instance in [\textit{D. S. Mitrinović}, Analytic inequalities. In cooperation with P. M. Vasić. Berlin: Springer-Verlag (1970; Zbl 0199.38101)], Theorem 1 p. 36.