On an inequality of Ramanujan concerning the prime counting function

We discuss on the sign of $\mathcal{R}_{\alpha }(x):=\pi(x)^{2}-\frac{\alpha x}{\log x}\pi(\frac{x}{\alpha })$ for x sufficiently large, and for various values of ??>0. The case ??=e refers to a result due to Ramanujan asserting that $\mathcal{R}_{e}(x)<0$ . Related by this inequality, we obtain a conditional result that gives the number N>530.2 such that $\mathcal{R}_{e}(x)<0$ is valid for x>e ^N . Moreover, we show that under assumption of validity of the Riemann hypothesis, the inequality $\mathcal{R}_{e}(x)<0$ holds for x>138,766,146,692,471,228. Then, in various cases for ??, we find numerical values of x _?? in which $\mathcal{R}_{\alpha }(x)$ is strictly positive or negative for x??x _?? .