OICC Press Communications in Nonlinear Analysis 2371-7920 13 1 2025 06 30 Integral Bounds for Generalized Polar Derivatives via Bernstein and Tur´an Inequalities 10.57647/cna.2025.fyph-17u2.8 EN Shahadat Ali Department of Mathematics, University of Kashmir, South Campus, Anantnag 192101, Jammu and Kashmir, India Arshad Usmani Department of Mathematics, Monad University, N.H. 9, Delhi Hapur Road, Village & Post Kastla, Kasmabad, P.O Pilkhuwa - 245304, Dist. Hapur (U.P), India Rajbahadur Singh Department of Mathematics, Monad University, N.H. 9, Delhi Hapur Road, Village & Post Kastla, Kasmabad, P.O Pilkhuwa - 245304, Dist. Hapur (U.P), India Journal Article 2025 06 30 Let $P(z) = \sum_{\mu=0}^{n}a_v z^{v}$ be a polynomial degree $n$, and $\beta$ be any complex number. Then $nP(z) + ({\beta} - z)P^{\prime}(z)$ is the polar derivative of $P(z)$, usually denoted by $D_{\beta} P(z)$. The derivative $D_{\beta} P(z)$ is the polynomial of degree at most $n-1$. The ordinary derivative follows from polar derivative. In this work, we establish integral mean inequalities of Tur\'an and Erd\H os--Lax type for the Generalized polar derivative of a polynomial, incorporating specific coefficients of the polynomial. Our findings refine several earlier results in the literature, and notably, one of our theorems extends and sharpens the well-known inequality by Khangembam Babina Devi and Barchand Chanam as a particular case. These improvements take into account both the location of the zeros and the structure of the coefficients of the polynomial. To illustrate the effectiveness and enhanced sharpness of our results, we also include numerical simulations and graphical comparisons with existing inequalities.