Vol. 94, No. 1, 1981

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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
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,fj : C K for j = 0,,n. Define Λn and Ωn by

Λn(x,y) = n1[ j=0n1f(x + 𝜃jy)]f(x),
Ωn(x,y) = n1[ j=0n1f j(x + 𝜃jy)]fn(x),
for all x,y C. Our main result is the following. If (n + 1) unknown functions fj : C K for j = 0,1,,n satisfy the quasi mean value property Ωn(x,y) = 0 for all x,y C, then (n + 1) unknown functions fj satisfy the difference functional equation Δunfj(x) = 0 for all u,x C and for each j = 0,1,,n, where the usual difference operator Δu is defined by Δuf(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 Λn(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
Authors
Shigeru Haruki