Sharpening of Turán-Type Inequality for Polynomials

Abstract

For the polynomial \(P(z) = \sum\nolimits_{j = 0}^n {{c}_{j}}{{z}^{j}}\) of degree n having all its zeros in \({\text{|}}z{\text{|}} \leqslant k\), \(k \geqslant 1\), Jain in “On the derivative of a polynomial,” Bull. Math. Soc. Sci. Math. Roumanie Tome 59, 339–347 (2016) proved that \(\mathop {\max }\limits_{|z| = 1} {\text{|}}P'(z){\text{|}} \geqslant n\left( {\frac{{{\text{|}}{{c}_{0}}{\text{|}} + \,{\text{|}}{{c}_{n}}{\text{|}}{{k}^{{n + 1}}}}}{{{\text{|}}{{c}_{0}}{\text{|}}(1 + {{k}^{{n + 1}}}) + \,{\text{|}}{{c}_{n}}{\text{|}}({{k}^{{n + 1}}} + {{k}^{{2n}}})}}} \right)\mathop {\max }\limits_{|z| = 1} {\text{|}}P(z){\text{|}}.\) In this paper we strengthen the above inequality and other related results for the polynomials of degree \(n \geqslant 2\).