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