Let P(z) be a polynomial of degree at most n, and \(\tilde{\mathbb {S}}_a[P(z)]= (1+az)P'(z)-naP(z)\) , and denote the modified Smirnov operator, where a is a real or complex number in \(|z| \le 1\) . In this paper, we consider the operators B and \(\tilde{\mathbb {S}}_a\) such that operator B carries \(\tilde{\mathbb {S}}_a[P(z)]\) into \(\begin{aligned} B[\tilde{\mathbb {S}}_a[P(z)]] = \lambda _0\tilde{\mathbb {S}}_a[P(z)] + \lambda _1 \frac{mz}{2}\tilde{\mathbb {S}}_a[P'(z)] + \lambda _2\left( \frac{mz}{2}\right) ^2\frac{\tilde{\mathbb {S}}_a[P''(z)]}{2!}, \end{aligned}\) where \(0\le m \le n-1, \lambda _0, \lambda _1\) , and \( \lambda _2\) are such that all the zeros of \(\begin{aligned} u(z) = \lambda _0 + \left( {\begin{array}{c}m\\ 1\end{array}}\right) \lambda _1 z + \left( {\begin{array}{c}m\\ 2\end{array}}\right) \lambda _2 z^2, \end{aligned}\) lie in the half-plane \(Re \ z \le \frac{m}{4}\) and obtain some compact generalizations of some well-known polynomial inequalities.
