Studying some Inequalities involving Polynomial Functions in π(x)

This article serves as a extensive research into proposing a few significant inequalities involving polynomial functions in π(x), the prime counting function, with an intention of exploring the behaviour of π(x) for increasing x. The general case for such polynomials has also been discussed in detail towards the later section of the article, where the primary focus was to study the order of polynomials of the form, {align*} P(π(x)) - e x/log x Q(π(x/e)) + R(x) {align*} P, Q and R being arbitrary polynomials, and establish for a particular case that, the polynomial yields negative values for sufficiently large values of x. Furthermore, the error term in such estimations is of order O(xᵈ(log x)ᵈ⁺¹), d depends heavily upon $deg(P)$ and $deg(Q)$.