NEW INEQUALITIES FOR THE B-OPERATORS
Abstract
Let P_n be the class of polynomials P(z) of degree n and B_n a family of operators
that map P_n into itself. For B in B_n, we investigate the dependence of
| B[P(Rz)] - \alpha B[P(rz)] + \beta { ( \fraq{R + 1}{r + 1} )^n - | \alpha | } B[P(rz)] |
on the minimum and the maximum modulus of P(z) on |z| = 1 for arbitrary real or complex
numbers \alpha, \beta with |\alpha|<=1, |\beta|<=1 and R > r >= 1 with or without restriction on the zeros of
the polynomial P(z) and present some new inequalities for B-operators yielding certain sharp
compact generalizations of some well-known Bernstein-type inequalities for polynomials.
that map P_n into itself. For B in B_n, we investigate the dependence of
| B[P(Rz)] - \alpha B[P(rz)] + \beta { ( \fraq{R + 1}{r + 1} )^n - | \alpha | } B[P(rz)] |
on the minimum and the maximum modulus of P(z) on |z| = 1 for arbitrary real or complex
numbers \alpha, \beta with |\alpha|<=1, |\beta|<=1 and R > r >= 1 with or without restriction on the zeros of
the polynomial P(z) and present some new inequalities for B-operators yielding certain sharp
compact generalizations of some well-known Bernstein-type inequalities for polynomials.
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ISSN: 1229-1595 (Print), 2466-0973 (Online)
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