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WITHDRAWN: A proof of the Krzyz conjecture

Denis Stupin

Abstract

WITHDRAWN: The proof contains an uncorrectable gap in the proof of theorem 7 on page 11. A proof of the Krzyz conjecture is presented, based on the application of the variational method, as well as on the use of two classical results and some of their consequences. The mentioned results are the Caratheodory-Toeplitz criterion of continuing a polynomial to a Caratheodory class function, and the Riesz-Fejer theorem about trigonometric polynomials. This is an English translation of a preprint originally published in Russian: https://preprints.ru/article/1799

WITHDRAWN: A proof of the Krzyz conjecture

Abstract

WITHDRAWN: The proof contains an uncorrectable gap in the proof of theorem 7 on page 11. A proof of the Krzyz conjecture is presented, based on the application of the variational method, as well as on the use of two classical results and some of their consequences. The mentioned results are the Caratheodory-Toeplitz criterion of continuing a polynomial to a Caratheodory class function, and the Riesz-Fejer theorem about trigonometric polynomials. This is an English translation of a preprint originally published in Russian: https://preprints.ru/article/1799

Paper Structure

This paper contains 16 theorems, 50 equations.

Key Result

Theorem 1

Let $n\in\mathbb{N}$, $\{h\}_0>0$, $\{h\}_1,\ldots,\{h\}_{n}\in\mathbb{C}$. The polynomial can be extended to a function if and only if the determinants are either all positive, or positive up to some index $m\leqslant n$ after which they all equal zero. In the latter case, the extension is unique and there exist numbers $\alpha_k>0$, $k=1,\ldots,m$, and $0\leqslant\varphi_1<\ldots<\varphi_m<2\

Theorems & Definitions (16)

  • Theorem 1: Caratheodory, Toeplitz
  • Corollary 1
  • Lemma 1
  • Theorem 2
  • Theorem 3
  • Corollary 2
  • Lemma 2
  • Theorem 4: Riesz, Fejer
  • Corollary 3
  • Theorem 5
  • ...and 6 more