On the theorem of S. Kakutani-M. Nagumo and J. L. Walsh for the mean value property of harmonic and complex polynomials

Shigeru Haruki

Vol. 94 (1981), No. 1, 113–123

DOI: 10.2140/pjm.1981.94.113

Abstract

Let K be either the field of complex numbers C or the field of real numbers R. Let n be a fixed integer > 2, and 𝜃 denote the number exp(2πi∕n). Let f,f_j : C → K for j = 0,⋯,n. Define Λ_n and Ω_n by

Λn(x,y)	= n−1∑ j=0n−1f(x + 𝜃jy)− f(x),
Ωn(x,y)	= n−1∑ j=0n−1f j(x + 𝜃jy)− fn(x),

for all x,y ∈ C. Our main result is the following. If (n + 1) unknown functions f_j : C → K for j = 0,1,⋯,n satisfy the quasi mean value property Ω_n(x,y) = 0 for all x,y ∈ C, then (n + 1) unknown functions f_j satisfy the difference functional equation Δ_uⁿf_j(x) = 0 for all u,x ∈ C and for each j = 0,1,⋯,n, where the usual difference operator Δ_u is defined by Δ_uf(x) = f(x + u) −f(x). By using this result we prove somewhat stronger results than the theorem of S. Kakutani-M. Nagumo (Zenkoku, Sūgaku Danwakai, 66 (1935), 10–12) and J. L. Walsh (Bull. Amer. Math. Soc., 42 (1936), 923–930) for the mean value property Λ_n(x,y) = 0 of harmonic and complex polynomials.

Mathematical Subject Classification 2000

Primary: 30D05

Secondary: 30C10, 39A10

Milestones

Received: 10 November 1978

Published: 1 May 1981