## Abstract

The well known Bernstein inequality states that if D is a disk centered at the origin with radius R and if p(z) is a polynomial of degree n, then [formula omitted] with equality iff p(z) = AZ^{n}. However it is true that we have the following better inequality: [formula omitted] with equality iff p(z) = AZ^{n}.This is a consequence of a general equality that appears in Zygmund [7] (and which is due to Bernstein and Szegö): For any polynomial p(z) of degree n and for any 1 ≤ P < ∞ we have [formula omitted] where [formula omitted] with equality iff p(z) =AZ^{n} In this note we generalize the last result to domains different from Euclidean disks by showing the following: If geeix) is differentiable and if p(z) is a polynomial of degree n then for any 1 ≤ P < ∞ [formula omitted]we have with equality iff p(z) = Az^{n}. We then obtain some conclusions for Schlicht Functions.

Original language | English |
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Pages (from-to) | 289-296 |

Number of pages | 8 |

Journal | International Journal of Mathematics and Mathematical Sciences |

Volume | 16 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jan 1993 |

Externally published | Yes |

## Keywords

- Bernstein inequality
- Bernstein-Szegö inequality
- Dirichlet kernel
- Krzyz problem
- trigonometric interpolation

## ASJC Scopus subject areas

- Mathematics (miscellaneous)