论文信息 - mean value property and zeros of holomorphic functions (Gauss, Poisson, Bolzano, and Cauchy meet in the complex plane)

mean value property and zeros of holomorphic functions (Gauss, Poisson, Bolzano, and Cauchy meet in the complex plane)

An existence condition for a zero of holomorphic functions in a disk is stated and proved in a very simple way using the mean value property. It contains as special cases Bolzano's theorem and Brouwer fixed point theorem in a disk for holomorphic functions, the fundamental theorem of algebra and an asymptotic condition for the existence of zeros of transcendental entire functions. An elementary proof of the used mean value property is given. For more information see https://ejde.math.txstate.edu/special/01/m2/abstr.html

J. Mawhin

[1] B. Chakraborty. A simple proof of the Fundamental Theorem of Algebra , 2022, Elemente der Mathematik.

[2] J. Mawhin. Bolzano’s theorems for holomorphic mappings , 2017 .

[3] A. Throop. The Fundamental Theorem of Algebra , 2015 .

[4] Bao Qin Li,et al. A Direct Proof and a Transcendental Version of the Fundamental Theorem of Algebra via Cauchy's Theorem , 2014, Am. Math. Mon..

[5] Alan C. Lazer,et al. The Fundamental Theorem of Algebra via the Fourier Inversion Formula , 2010, Am. Math. Mon..

[6] Anton R. Schep,et al. A Simple Complex Analysis and an Advanced Calculus Proof of the Fundamental Theorem of Algebra , 2009, Am. Math. Mon..

[7] A. Lazer. From the Cauchy-Riemann Equations to the Fundamental Theorem of Algebra , 2006 .

[8] R. Remmert,et al. Theory of Complex Functions , 1990 .

[9] Mau-Hsiang Shih. An Analog of Bolzano's Theorem for Functions of a Complex Variable , 1982 .

[10] J. Mawhin,et al. Brouwer Degree , 2021 .

[11] R. Výborný. A simple proof of the Fundamental Theorem of Algebra , 2010 .

[12] O. D. Kellogg,et al. Invariant points in function space , 1922 .

[13] G. B. M.,et al. Introduction à la théorie des fonctions d'une variable , 1911, Nature.