On Proving Ramanujan’s Inequality Using a Sharper Bound for the Prime Counting Function π(x)

This article provides a proof that the Ramanujan's Inequality given by, \begin{align*} \pi(x)^2 < \frac{e x}{\log x} \pi\left(\frac{x}{e}\right) \end{align*} holds unconditionally for every $x\geq \exp(43.5102146)$. In case for an alternate proof of the result stated above, we shall exploit certain estimates involving the Chebyshev Theta Function, $\vartheta(x)$ in order to derive appropriate bounds for $\pi(x)$, which'll lead us to a much improved condition for the inequality proposed by Ramanujan to satisfy unconditionally.