On a Generalization of an Operator Preserving Turán-Type Inequality for Complex Polynomials

Abstract

Abstract Let $$W(\zeta)=(a_{0}+a_{1}\zeta+...+a_{n}\zeta^{n})$$ be a polynomial of degree $$n$$ having all its zeros in $$\mathbb{T}_{k}\cup\mathbb{E}^{-}_{k}$$, $$k\geq 1$$, then for every real or complex number $$\alpha$$ with $$|\alpha|\geq 1+k+k^{n}$$, Govil and McTume [7] showed that the following inequality holds $$\max\limits_{\zeta\in\mathbb{T}_{1}}|D_{\alpha}W(\zeta)|\geq n\left(\frac{|\alpha|-k}{1+k^{n}}\right)||W||+n\left(\frac{|\alpha|-(1+k+k^{n})}{1+k^{n}}\right)\min\limits_{\zeta\in\mathbb{T}_{k}}|W(\zeta)|.$$ In this paper, we have obtained a generalization of this inequality involving sequence of operators known as polar derivatives. In addition, the problem for the limiting case is also considered.