Generalized Derivative Inequalities and Their Extensions in $L_p$ Normed Spaces

Abstract

In this work, we investigate Turán-type inequalities in the $L_q$-norm for generalized derivatives of polynomials. The study is further extended to encompass the generalized polar derivative. The findings not only broaden several established results but also yield additional inequalities as particular instances. The theorems presented herein provide a significant generalization of several established results in the theory of polynomial inequalities.