On amalgamated Banach algebras

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 \(|z|\ge 1,\)

$$\begin{aligned} |Bfo\sigma ](z)+\psi Bfo\rho ](z)|\le |BFo\sigma ](z)+\psi BFo\rho ](z)|, \end{aligned}$$

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.