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In this study, a broader class of rational functions r(u(z)) of degree mn, where u(z) is a polynomial of degree m is taken into consideration and obtain certain sharp compact generalization of well-known inequalities for rational functions.

By considering a fixed point in the punctured unit disk and using the $q$--derivative, a new subfamily of Meromorphic and univalent functions is defined. Also, the first and second order {$q$--derivative} of Meromorphic functions are introduced. Coefficient bounds, extreme points, radii of starlikeness and convexity are obtained. Furthermore, the convexity and preserving under convolution with some restrictions on parameters are investigated

Let 𝝨 be the class of Meromorphic bi-univalent functions f of the formf(z)=z+b_0+∑ _(n=1)^∞ ▒ b_n/z^n, which are univalent (analytic and one to one) on the domain Δ ={z∈ C∶ 1.