Inequalities involving Higher Degree Polynomial Functions in $\pi (x)$

The primary purpose of this article is to study the asymptotic and numerical estimates in detail for higher degree polynomials in $\pi (x)$ having a general expression of the form, \begin{align*} P(π(x)) - \frac{e x}{\log x} Q(π(x/e)) + R(x) \end{align*} $P$, $Q$ and $R$ are arbitrarily chosen polynomials and $\pi (x)$ denotes the $\mathit{\text{Prime}}\mathit{\text{Counting}}\mathit{\text{Function}}$. The proofs require specific order estimates involving $\pi (x)$ and the $\mathit{\text{Second}}\mathit{\text{Chebyshev}}\mathit{\text{Function}}$ $\psi (x)$, as well as the famous $\mathit{\text{Prime}}\mathit{\text{Number}}\mathit{\text{Theorem}}$ in addition to certain meromorphic properties of the $\mathit{\text{Riemann}}\mathit{\text{Zeta}}\mathit{\text{Function}}$ $\zeta (s)$ and results regarding its non-trivial zeros. A few generalizations of these concepts have also been discussed in detail towards the later stages of the paper, along with citing some important applications.