An inequality for the function \(\pi(n)\) (Q2250857)

Let \(\pi(n)\) denote, as usual, the number of primes \(p\leq n\). The author proves that the inequality \(\pi^2(m) +\pi^2(n)\leq \frac{5}{4} \pi^2(m+n)\) holds for all integers \(m, n \geq 2\) and the constant \(5/4\) is sharp. The converse inequality \(\frac1{2} \pi^2(m+n)\leq \pi^2(m) + \pi^2(n)\) (\(m,n\geq 2\)) was obtained by \textit{L. Panaitopol} [Math. Inequal. Appl. 4, 327--330 (2001; Zbl 1013.11003)]. Note that the conjecture of Hardy and Littlewood (1923) stating that \(\pi(m + n) \leq \pi(m) + \pi(n)\) for every integers \(m,n \geq 2\) is still open.