Extremal problems of Turán-type involving the location of all zeros of a polynomial

Abstract

If P ( z ) = a n ∏ v = 1 n ( z − z v ) is a polynomial of degree n having all its zeros in | z | ≤ k , k ≥ 1 then Aziz [Inequalities for the derivative of a polynomial. Proc Am Math Soc. 1983;89(2):259–266] proved that max | z | = 1 | P ′ ( z ) | ≥ 2 1 + k n ∑ v = 1 n k k + | z v | max | z | = 1 | P ( z ) | . Recently, Kumar [On the inequalities concerning polynomials. Complex Anal Oper Theory. 2020;14(6):1–11 (Article ID 65)] established a generalization of this inequality and proved under the same hypothesis for a polynomial P ( z ) = a 0 + a 1 z + a 2 z 2 + ⋯ + a n z n = a n ∏ v = 1 n ( z − z v ) , that max | z | = 1 | P ′ ( z ) | ≥ ( 2 1 + k n + ( | a n | k n − | a 0 | ) ( k − 1 ) ( 1 + k n ) ( | a n | k n + k | a 0 | ) ) ∑ v = 1 n k k + | z v | max | z | = 1 | P ( z ) | . In this paper, we sharpen the above inequalities and further extend the obtained results to the polar derivative of a polynomial. As a consequence, our results also sharpens considerably some results of Dewan and Upadhye [Inequalities for the polar derivative of a polynomial. J Ineq Pure Appl Math. 2008;9:1–9 (Article ID 119)].