Some Compact Generalization of Bernstein-Type Inequalities Preserved by Modified Smirnov Operator

Let $P(z)$ be a polynomial of degree $n$. In $2004$, Aziz and Rather \cite{aziz2004some} investigated the dependence of $$\mid P(Rz)-\alpha P(z)+\beta \left\{\right(\frac{R+1}{2}{)}^n-\mathrm{\mid }\alpha \mathrm{\mid }\}P(z)\mid ,\text{ for }z\in B(\mathbb{D}),$$ on ${\max }_{z\in B(\mathbb{D})}\mathrm{\mid }P(z)\mathrm{\mid }$, for every real and complex number $\alpha ,\beta$ satisfying $\mathrm{\mid }\alpha \mathrm{\mid }\le 1$, $\mathrm{\mid }\beta \mathrm{\mid }\le 1$, and $R\ge 1$. This paper presents a compact generalization of several well-known polynomial inequalities using modified Smirnov operator, demonstrating that the operator preserves inequalities between polynomials.