Refinement of some inequalities concerning to $B_{n}$-operator of polynomials with restricted zeros

Abstract

If F(z) is a polynomial of degree n having all zeros in $|z|\le k,~k>0$ and f(z) is a polynomial of degree $m\le n$ such that $|f(z)|\le |F(z)|$ for $|z|=k$, then it was formulated by Rather and Gulzar (Adv Inequal Appl 2:16–30, 2013) that for every $|\delta |\le 1, |\beta |\le 1,~R>r\ge k$ and $|z|\ge 1,$

$$\begin{aligned} |B[fo\sigma ](z)+\psi B[fo\rho ](z)|\le |B[Fo\sigma ](z)+\psi B[Fo\rho ](z)|, \end{aligned}$$

where B is a $B_{n}$ operator, $\sigma (z){=}Rz, \rho (z){=}rz$ and $\psi {:=}\psi (R,r,\delta ,\beta ,k) {=}\beta \bigg \{\bigg (\frac{R+k}{r+k}\bigg )^{n}{-}|\delta |\bigg \}{-}\delta$. The authors have assumed that $B\in B_{n}$ is a linear operator which is not true in general. In this paper, besides discussing assumption of authors and their followers (see e.g, Rather et al. in Int J Math Arch 3(4):1533–1544, 2012), we present the correct proof of the above inequality. Moreover our result improves many prior results involving $B_{n}$ operators and a number of polynomial inequalities can also be deduced by a uniform procedure.