`Abstract and Applied AnalysisVolume 2011, Article ID 167160, 10 pageshttp://dx.doi.org/10.1155/2011/167160`
Research Article

New Properties of Complex Functions with Mean Value Conditions

1School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
2Xingtan College, Qufu Normal University, Qufu 273100, China

Received 6 August 2011; Accepted 24 September 2011

Copyright © 2011 Yuzhen Bai and Lei Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We 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 are given.

1. 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 first-order 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: , , , , ,  . Using the chain rule of derivation, we have The Cauchy-Riemann equation of analytic function can be written as

We will use the following classical definitions and results of functional analysis.

Definition 1.1 (see [10]). The functions with such that , are called standard mollifiers.

From the definition, we see the functions are .

Definition 1.2 (see [3]). Give a locally integrable function and a sequence of bounded mollifiers, and define the functions by the convolution The sequence is said to be a sequence of mollified functions.

Proposition 1.3 (Properties of mollifiers, see [10]). Suppose that is open, , write . Then,(i), (ii) a.e. as ,(iii)if , then uniformly on compact subsets of ,(iv)If and then in .

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). Let be a domain in complex number field (bounded or unbounded) and a continuous complex function defined in . For any and , if we say that satisfies the mean value condition in domain .

Remark 2.2. If is an analytic function in domain , then satisfies the mean value condition in domain (see [1]), the converse is wrong. For example, 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 .(i)For any , if we say that satisfies the first mean value condition.(ii)For any , if we say that satisfies the second mean value condition.

Proposition 2.4. (1) The first and the second mean value conditions of are equivalent.
(2) (i)The first mean value condition of can be written as (ii)The second mean value condition of can be written as
(3) The mean value condition of complex function can be written as
(4) The complex function satisfies mean value condition if and only if real functions and satisfy mean value conditions.

Proof. (1) Differentiating both sides of with respect to , we have that is, We write the first mean value condition as and get the second mean value condition by integrating the both sides of (2.10) with respect to on .
(2) (i)Let . Then, by integral transform formula, we get (ii)In the same way, we get
(3) Let . Then, by integral transform formula, we get
(4) Let and . Then, by integral transform formula, we see which implies

3. 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.

Lemma 3.1. If , then, satisfies the mean value condition in domain .

Proof. For any , using Green formula, we have Since is harmonic in , we obtain from (3.1) that Integrating both sides of (3.2) with respect to on , we get that is,

Lemma 3.2. Assume (i) and ; (ii) satisfies the mean value condition in and . Then,

Proof. From (i) and (ii), we have Multiplying the both sides of (3.7) by and integrating the result with respect to on , we have Combining (3.6) and (3.8), we obtain the conclusion.

Lemma 3.3. If satisfies the mean value condition, then, (i) ; (ii) .

Proof. (i) Method 1: choose with Using integral transform formulas, we have
Define , with . Using integral transform formulas and (3.10), we get that is, Applying (3.12) and Proposition 1.3, noticing the arbitrariness of , we conclude that .
Method 2: choose as above. Define , with , then, Using Lemma 3.2, we obtain
Applying Proposition 1.3, noticing the arbitrariness of , we conclude that .
(ii)Using (3.1) and Proposition 2.4, we get which implies .

4. Main Results

In this section, we give the main results for the complex functions which satisfy the mean value conditions.

Proposition 4.1. satisfies the mean value condition in if and only if in .

Proof. Proposition 2.4, Lemmas 3.1, and 3.3 yield the assertion.

Theorem 4.2. satisfies the mean value condition in and if and only if is analytic in .

Proof. Denote .
Firstly, we prove the necessary condition. Employing the assumption and Lemma 3.3, we can assert ; hence, the partial derivatives of and are continuous in . Combining with Cauchy-Riemann equation , we conclude that is analytic in .
Secondly, we prove the sufficient condition. The assumption that is analytic in implies(i), that is, ,(ii). Combining with Proposition 4.1 implies satisfies the mean value condition in .

Theorem 4.3. Suppose that satisfies the mean value condition in , and is bounded. Then is a constant in .

Proof. Since satisfies the mean value condition in , using Proposition 2.4 and Lemma 3.3, we get .
Since is bounded, we obtain and are bounded, respectively, Without loss of generality, we assume that . For all , one can choose with . Denote . The Harnack inequality (see [11]) implies Letting , we conclude . In the similar way, we conclude . Since is arbitrary, we conclude is a constant in .

Remark 4.4. This theorem may be proved by the local estimates for harmonic functions too. On the local estimates for harmonic functions, one can see [10].

Theorem 4.5. Suppose that satisfies the mean value condition in , and is a constant. Then, is a constant in .

Proof. Since satisfies the mean value condition in , using Proposition 2.4 and Lemma 3.3, we obtain
Since is a constant, we get the following in From (4.3), we get and In the similar way, we have Adding (4.4) to (4.5) and noting (4.2), we obtain which implies and are constants, that is, is a constant in .

Theorem 4.6. Suppose (1)   satisfies the mean value condition in ; (2)   is continuous on ; (3)   is not a constant. Then, can be obtained only on the boundary of .

Proof. Denoting , then, we have . Suppose there is such that . For any , the mean value condition implies that, for all . Hence is a constant in the neighborhood of . Theorem 4.3 implies is a constant in this neighborhood of . Applying the circular chain method, we have is a constant in , which is a contradiction.

In the following, we present two problems.

Problem 1. Suppose (1) satisfies the mean value condition in ; (2)   is continuous on ; (3)   is not a constant; (4) for all . Can one confirm that is obtained only on the boundary of ?

One can't confirm. For example, in . 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 satisfies the mean value condition in , can one confirm that is infinitely differentiable in ?

One can not confirm. For example, in 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.

Acknowledgments

The authors gratefully acknowledge the referees who pointed out the important references and gave some valuable suggestions which improved the results in this paper. Project is supported by the Natural Science Foundation of Shandong Province of China (ZR2011AQ006, ZR2011AM008) and STPF of University in Shandong Province of China (J10LA13, J09LA04).

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