Inequalities involving \(\pi (x)\) (Q2154576)

Summary: We present several inequalities involving the prime-counting function \(\pi (x)\). Here, we give two examples of our results. We show that \[ \frac{16}{9} \pi (x) \pi (y) \leq \pi^2 (x + y) \] is valid for all integers \(x, y \geq 2\). The constant factor \(16/9\) is the best possible. The special case \(x = y\) leads to \[ \frac{4}{3} \leq \frac{\pi (2x)}{ \pi (x)} \quad (x = 2, 3, \ldots), \] where the lower bound \(4/3\) is sharp. This complements Landau's well-known inequality \[ \frac{\pi (2 x)}{\pi (x)} \leq 2 \quad (x = 2, 3, \ldots). \] Moreover, we prove that the inequality \[ \left(2 \frac{ \pi (x + y)}{ x + y} \right)^s \leq \left( \frac{ \pi (x}{x} \right)^s + \left(\frac{ \pi (y)}{y} \right)^s \quad (0 < s \in \mathbb{R}) \] holds for all integers \(x, y \geq 2\) if and only if \(s \leq s_0 = 0.94745 \ldots\). Here, \( s_0\) is the only positive solution of \[ \left(\frac{16}{7} \right)^t - \left( \frac{6}{5} \right)^t = 1. \]