Abstract Let P(z) be a polynomial of degree at most n. We consider an operator N, which carries a polynomial P(z) into N[P](z):=∑j=0mλj(nz2)jP(j)(z)j!,documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$begin{aligned} N[P](z):=sum limits _{j=0}^{m}lambda _jbigg (frac{nz}{2}bigg )^jfrac{P^{(j)}(z)}{j!}, end{aligned}$$end{document}where λ0,λ1,…,λmdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$lambda _0,lambda _1,ldots ,lambda _m$$end{document} are such that all the zeros of u(z)=∑j=0mnjλjzjdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$begin{aligned} u(z)=sum limits _{j=0}^{m}left( {begin{array}{c}n jend{array}}right) lambda _jz^j end{aligned}$$end{document}lie in the half plane |z|≤|z-n2|.documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$begin{aligned} |z|le bigg |z-frac{n}{2}bigg |. end{aligned}$$end{document}In this paper, we estimate the minimum and maximum modulii of N[P(z)] on |z|=1documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$|z|=1$$end{document} with restrictions on the zeros of P(z) and thereby obtain compact generalizations of some well known polynomial inequalities.
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