Abstract
We characterize solutions of the mean value linear elliptic equation with constant coefficients in the complex plane in the case of regular polygon.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Kakutani and M. Nagumo, “On the functional equation \( {\sum}_{v=0}^{n-1}f\left(z+{e}^{2 v\pi i/n}\xi \right)= nf(z) \),” Zenkoku Sûgaku Danwakai, 66, 10–12 (1935).
J. L. Walsh, “A mean value theorem for polynomials and harmonic polynomials,” Bull. Amer. Math. Soc., 42, 923–930 (1936).
I. I. Privalov, Subharmonic Functions [in Russian], Moscow–Leningrad, ONTI, 1937.
T. Ramsey and Y. Weit, “Mean values and classes of harmonic functions,” Math. Proc. Camb. Phil. Soc., 96, 501–505 (1984).
V. V. Volchkov, Integral Geometry and Convolution Equations, Kluwer, Dordrecht, 2003.
O. Trofymenko, “Convolution equations and mean value theorems for solutions of linear elliptic equations with constant coefficients in the complex plane,” J. Math. Sci., 229(1), 96–107 (2018).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Ukrains’kiĭ Matematychnyĭ Visnyk, Vol. 17, No. 4, pp. 594–600, October–December, 2020.
The author was supported by the Fundamental Research Programme funded by the Ministry of Education and Science of Ukraine (project 0118U003138).
Rights and permissions
About this article
Cite this article
Trofymenko, O.D. Mean value theorems for polynomial solutions of linear elliptic equations with constant coefficients in the complex plane. J Math Sci 254, 439–443 (2021). https://doi.org/10.1007/s10958-021-05315-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-021-05315-4