New Properties of Complex Functions with Mean Value Conditions

and Applied Analysis 3 This paper is organized as follows. In Section 2, we give the definitions of mean value conditions and their equivalent forms. Applying mollifiers, we show some properties of real functions with mean value conditions in Section 3. Section 4 contains our main results for complex functions satisfying mean value condition, that is, the new equivalent condition of complex analytic function and the new properties of complex functions. At last, we present two problems with their answers. 2. Mean Value Conditions Definition 2.1 Mean value condition . LetΩ be a domain in complex number field bounded or unbounded and f z u x, y iv x, y a continuous complex function defined inΩ. For any z0 ∈ Ω and {z | |z − z0| ≤ r} ⊂ Ω, if f z0 1 2πr ∫ |z−z0| r f z ds, 2.1 we say that f z satisfies the mean value condition in domain Ω. Remark 2.2. If f z is an analytic function in domain Ω, then f z satisfies the mean value condition in domain Ω see 1 , the converse is wrong. For example, f z 1 iy satisfies the mean value condition in the complex number field, but it is not analytic. Hence, mean value condition is weaker than analytic condition. Definition 2.3 Mean value condition . Set w p ∈ C Ω . i For any Br p0 ⊂ Ω, if


Introduction
There are many good properties of complex analytic function.In the references on complex function theory see 1 and the references therein , we see that analytic function satisfies mean value theorem but the converse is wrong.Hence, mean value condition is weaker than analytic condition.
The mean value problem has been a very active area in recent years.The mean value theorem for real-valued differentiable functions defined on an interval is one of the most fundamental results in analysis.However, the theorem is incorrect for complex-valued functions even if the function is differentiable throughout the complex plane.Qazi 2 illustrated that by examples and presented three results of a positive nature.A mean value theorem for continuous vector functions was introduced by mollified derivatives and smooth approximations in 3 .Crespi et al. 4 and La Torre 5 gave some characterizations of convex functions by means of second-order mollified derivatives.Second-order necessary optimality conditions for nonsmooth vector optimization problems were given by smooth approximations in 6 .Eberhard and Mordukhovich 7 mainly concerned deriving firstorder and second-order necessary and partly sufficient optimality conditions for a general class of constrained optimization problems via convolution smoothing.Eberhard et al. 8 demonstrated that second-order subdifferentials were constructed via the accumulation of local Hessian information provided by an integral convolution approximation of the function.In 9 , Aimar et al. showed the parabolic mean value formula.
In this paper, we will apply mollifiers to study the properties of real functions which satisfy mean value conditions and present new equivalent conditions for complex analytic functions.New properties of complex functions with mean value conditions will be given.
We introduce the notations: z x iy, z x − iy, p 0 x 0 , y 0 , p x, y , B r p 0 {p | dist p, p 0 ≤ r}, ∂B r p 0 {p | dist p, p 0 r}.Using the chain rule of derivation, we have 1.1 The Cauchy-Riemann equation of analytic function f z u x, y iv x, y can be written as We will use the following classical definitions and results of functional analysis.
Definition 1.1 see 10 .The functions with c ∈ R such that R n ϕ ε x dx 1, are called standard mollifiers.
From the definition, we see the functions ϕ ε are C ∞ .
Definition 1.2 see 3 .Give a locally integrable function f : R n → R m and a sequence of bounded mollifiers, and define the functions f ε by the convolution The sequence f ε x is said to be a sequence of mollified functions.
This paper is organized as follows.In Section 2, we give the definitions of mean value conditions and their equivalent forms.Applying mollifiers, we show some properties of real functions with mean value conditions in Section 3. Section 4 contains our main results for complex functions satisfying mean value condition, that is, the new equivalent condition of complex analytic function and the new properties of complex functions.At last, we present two problems with their answers.

Mean Value Conditions
Definition 2.1 Mean value condition .Let Ω be a domain in complex number field bounded or unbounded and f z u x, y iv x, y a continuous complex function defined in Ω.For any we say that f z satisfies the mean value condition in domain Ω.
Remark 2.2.If f z is an analytic function in domain Ω, then f z satisfies the mean value condition in domain Ω see 1 , the converse is wrong.For example, f z 1 iy satisfies the mean value condition in the complex number field, but it is not analytic.Hence, mean value condition is weaker than analytic condition.i The first mean value condition of w p can be written as ii The second mean value condition of w p can be written as w p 0 1 π |ω|≤1 w p 0 rω dω.

2.5
(3) The mean value condition of complex function f z can be written as

Preliminaries
In this section, we give the properties of real functions satisfying the mean value conditions.These properties will be used to prove our main results in Section 4.
Proof.For any B r p 0 ⊂ Ω, using Green formula, we have Using integral transform formulas, we have 3.12 Applying 3.12 and Proposition 1.

Main Results
In this section, we give the main results for the complex functions which satisfy the mean value conditions.which implies u and v are constants, that is, f z is a constant in Ω.
Theorem 4.6.Suppose 1 f z u x, y iv x, y satisfies the mean value condition in Ω; 2 f z is continuous on Ω; 3 f z is not a constant.Then, max Ω |f z | can be obtained only on the boundary of Ω.
For any B ρ z 0 ⊂ Ω, the mean value condition implies that, for all z ∈ ∂B ρ , |f z | M. Hence |f z | is a constant in the neighborhood of M 0 .Theorem 4.3 implies f z is a constant in this neighborhood of M 0 .Applying the circular chain method, we have f z is a constant in Ω, which is a contradiction.
In the following, we present two problems.Problem 1. Suppose 1 f z u x, y iv x, y satisfies the mean value condition in Ω; 2 f z is continuous on Ω; 3 f z is not a constant; 4 for all z ∈ Ω, f z / 0. Can one confirm that min Ω |f z | is obtained only on the boundary of Ω?

Answer
One can't confirm.For example, f z 1 iy in Ω {z | |z| < 1}.This example shows that the minimal module principle doesn't hold for complex function satisfying mean value condition.But analytic complex function has minimal module principle.Problem 2. If f z satisfies the mean value condition in Ω, can one confirm that f z is infinitely differentiable in Ω?

Answer
One can not confirm.For example, f z 1 iy in Ω {z | |z| < 1} does not satisfy the Cauchy-Riemann equation.This example shows that mean value condition can not imply the differential property of complex function.But analytic complex function is infinitely differentiable.

From 4 . 3 , we get 2uu x 2vv x 0 and u 2
Definition 2.3 Mean value condition .Set w p ∈ C Ω .
4)The complex function f z u x, y iv x, y satisfies mean value condition if and only if real functions u x, y and v x, y satisfy mean value conditions.