Some refinements on lower bound estimates for polynomial with restricted zeros

Abstract

By using the principle of mathematical induction, it was shown by Singh and Chanam (J. Math. Inequal 15:1663–1675, 2021) that if p(z) is a polynomial of degree n having all its zeros in $|z|\le 1$, then for all z on $|z|=1$ for which $p(z)\ne 0$,

$$\begin{aligned} \Re {\left( z\frac{p'(z)}{p(z)}\right) }\ge \frac{n+1}{2}-\frac{1}{2}\frac{\sqrt{|a_0|}}{\sqrt{|a_n|}}. \end{aligned}$$

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 obtain improvements of the well-known result due to Malik (J. Lond. Math. Soc 1(2):57–60, 1969). Further, we obtain some sharp refinements of a result due to Aziz and Rather (J. Math. Inequal. Appl 1:231–238, 1998). These results take into account the placement of the coefficients of the underlying polynomial. Moreover, a concrete numerical example is presented in order to graphically illustrate and compare the obtained inequalities with a classical result, showing that in some situations, the bounds obtained by our results can be considerably sharper than the ones previously known.