Conversely, does this mean value property imply thatfis harmonic ?
First observe that (widetilde{mathscr{M}}) obviously admits the mean value property.
Another important observation is that, due to the mean value property, means are locally bounded functions.
The two remaining properties (mean value property, elimination principle) are obvious.
Mean value property: For all (n in mathbb {N}) and for all ((x,lambda ) in I^{n} times W_{n}(R)) min(x_{1},dots,x_{n}) le mathscr{M} x,lambda)le max(x_{1},dots,x_{n}).
(iii) Mean value property: For all (n in mathbb {N}) and for all ((x,lambda ) in I^{n} times W_{n}(R)) min(x_{1},dots,x_{n}) le mathscr{M} x,lambda)le max(x_{1},dots,x_{n}).
The main purpose of this paper is using the analytic method and the properties of the character sums to study the hybrid mean value properties of G ( χ, n ; p ) and S ( χ, p ), and to give an interesting mean value formula.
The main purpose of this paper is, using the properties of Gauss sums, the estimate for character sums and the analytic method, to study the mean value properties of the two-term Dedekind sums and give an interesting asymptotic formula for (1).
Let M ( a, p ) denote the number of all integers 1 ≤ b, c ≤ p − 1 such that b c ≡ a mod p and ( 2, b + c ) = 1, and E ( a, p ) = M ( a, p ) − p − 1 2. Then Zhang [4] also studied the mean value properties of E ( a, p ), and proved that ∑ a = 1 p − 1 E 2 ( a, p ) = 3 4 p 2 + O ( p ⋅ exp ( 3 ln p ln ln p ) ), where exp ( y ) = e y.
For example, Zhang [11] studied the hybrid mean value properties of Cochrane sums and Kloosterman sums and proved that for any prime p > 3, we have the asymptotic formula ∑ h = 1 p − 1 K ( h, 1 ; p ) C ( h, p ) = − 1 2 π 2 p 2 + O ( p ⋅ exp ( 3 ln p ln ln p ) ), where exp ( y ) = e y, K ( m, n ; q ) = ∑ ′ a = 1 q e ( m a + n a ¯ q ).
end{aligned}Then by the sub-mean value property for subharmonic functions, begin{aligned} log |T|^2 le lambda, quad text {i.e.}, quad |T|^2 e^{-lambda } le 1, end{aligned}and by the Poincaré Lelong Formula, begin{aligned} Delta lambda = frac{#(Gamma cap D_r z))}{pi r^2} end{aligned}Therefore by Theorem 5.14, we have the following result.
