Some inequalities for polynomials with restricted zeros

By using the boundary Schwarz lemma, it was shown by Dubinin (J Math Sci 143:3069–3076, 2007) that if P(z) is a polynomial of degree n having all its zeros in | z| ≤ 1 , then for all z on | z| = 1 for which P(z) ≠ 0 , Re(zP′(z)P(z))≥n2+12(|an|-|a0||an|+|a0|).In this paper, by using simple techniques we generalize the above inequality, thereby give a simple proof of the above inequality. As an application of our result, we also obtain sharp refinements of some known results due to Malik (J Lond Math Soc 1:57–60, 1969), Aziz and Rather (Math Ineq Appl 1:231–238, 1998). These results take into account the size of the constant term and the leading coefficient of the polynomial P(z). © 2020, Università degli Studi di Ferrara.