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Abstract and Applied Analysis
Volume 2011, Article ID 167160, 10 pages
http://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

Academic Editor: Marcia Federson

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: 𝑧=𝑥+𝑖𝑦, 𝑧=𝑥𝑖𝑦, 𝑝0=(𝑥0,𝑦0), 𝑝=(𝑥,𝑦), 𝐵𝑟(𝑝0)={𝑝dist(𝑝,𝑝0)𝑟},  𝜕𝐵𝑟(𝑝0)={𝑝dist(𝑝,𝑝0)=𝑟}. Using the chain rule of derivation, we have 𝜕=𝜕𝜕𝑧𝜕𝜕𝑥𝑖,𝜕𝜕𝑦𝜕𝑧=𝜕𝜕𝜕𝑥+𝑖𝜕𝜕𝑦,Δ=𝜕𝑧𝜕𝑧=𝜕𝜕=𝜕𝑧𝜕𝑧2𝜕𝑥2+𝜕2𝜕𝑦2.(1.1) The Cauchy-Riemann equation of analytic function 𝑓(𝑧)=𝑢(𝑥,𝑦)+𝑖𝑣(𝑥,𝑦) can be written as 𝜕𝑓(𝑧)𝜕𝑧=0.(1.2)

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

Definition 1.1 (see [10]). The functions 𝜑𝜀(𝑐𝑥)=𝜀𝑛𝜀exp2|𝑥|2𝜀2,|𝑥|<𝜀,0,|𝑥|𝜀,(1.3) with 𝑐𝑅 such that 𝑅𝑛𝜑𝜀(𝑥)𝑑𝑥=1, are called standard mollifiers.

From the definition, we see the functions 𝜑𝜀 are 𝐶.

Definition 1.2 (see [3]). Give a locally integrable function 𝑓𝑅𝑛𝑅𝑚 and a sequence of bounded mollifiers, and define the functions 𝑓𝜀 by the convolution 𝑓𝜀(𝑥)=𝑅𝑛𝑓(𝑥𝑦)𝜑𝜀(𝑦)𝑑𝑦=𝑅𝑛𝑓(𝑦)𝜑𝜀(𝑥𝑦)𝑑𝑦.(1.4) The sequence 𝑓𝜀(𝑥) is said to be a sequence of mollified functions.

Proposition 1.3 (Properties of mollifiers, see [10]). Suppose that ΩRn is open, 𝜀>0, write Ω𝜀={xΩdist(x,𝜕Ω)>𝜀}. Then,(i)𝑓𝜀𝐶(Ω𝜀), (ii)𝑓𝜀𝑓 a.e. as 𝜀0,(iii)if 𝑓𝐶(Ω), then 𝑓𝜀𝑓 uniformly on compact subsets of Ω,(iv)If 1𝑝< and 𝑓𝐿𝑝loc(Ω) then 𝑓𝜀𝑓 in 𝐿𝑝loc(Ω).

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 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 𝑧0Ω and {𝑧|𝑧𝑧0|𝑟}Ω, if 𝑓𝑧0=12𝜋𝑟||𝑧𝑧0||=𝑟𝑓(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 𝑤𝑝0=12𝜋𝑟𝜕𝐵𝑟(𝑝0)𝑤(𝑝)𝑑𝑠,(2.2) we say that 𝑤(𝑝) satisfies the first mean value condition.(ii)For any 𝐵𝑟(𝑝0)Ω, if 𝑤𝑝0=1𝜋𝑟2𝐵𝑟(𝑝0)𝑤(𝑝)𝑑𝑝,(2.3) we say that 𝑤(𝑝) satisfies the second mean value condition.

Proposition 2.4. (1) The first and the second mean value conditions of 𝑤(𝑝) are equivalent.
(2) (i)The first mean value condition of 𝑤(𝑝) can be written as 𝑤𝑝0=12𝜋|𝜔|=1𝑤𝑝0+𝑟𝜔𝑑𝑠.(2.4)(ii)The second mean value condition of 𝑤(𝑝) can be written as 𝑤𝑝0=1𝜋|𝜔|1𝑤𝑝0+𝑟𝜔𝑑𝜔.(2.5)
(3) The mean value condition of complex function 𝑓(𝑧) can be written as 𝑓𝑧0=12𝜋02𝜋𝑓𝑧0+𝑟𝑒𝑖𝜃𝑑𝜃.(2.6)
(4) The complex function𝑓(𝑧)=𝑢(𝑥,𝑦)+𝑖𝑣(𝑥,𝑦) satisfies mean value condition if and only if real functions𝑢(𝑥,𝑦) and𝑣(𝑥,𝑦) satisfy mean value conditions.

Proof. (1) Differentiating both sides of 𝑤𝑝0=1𝜋𝑟2𝐵𝑟(𝑝0)1𝑤(𝑝)𝑑𝑝=𝜋𝑟2𝑟0𝑑𝜌𝜕𝐵𝜌(𝑝0)𝑤(𝑝)𝑑𝑠,(2.7) with respect to 𝑟, we have 20=𝜋𝑟3𝑟0𝑑𝜌𝜕𝐵𝜌(𝑝0)1𝑤(𝑝)𝑑𝑠+𝜋𝑟2𝜕𝐵𝑟(𝑝0)𝑤(𝑝)𝑑𝑠,(2.8) that is, 12𝜋𝜕𝐵𝑟(𝑝0)1𝑤(𝑝)𝑑𝑠=𝜋𝑟2𝐵𝑟(𝑝0)𝑝𝑤(𝑝)𝑑𝑝=𝑤0.(2.9) We write the first mean value condition as 𝑤𝑝01𝜌=2𝜋𝜕𝐵𝜌(𝑝0)𝑤(𝑝)𝑑𝑠(2.10) and get the second mean value condition by integrating the both sides of (2.10) with respect to 𝜌 on [0,𝑟].
(2) (i)Let 𝑝=𝑝0+𝑟𝜔. Then, by integral transform formula, we get 𝑤𝑝0=12𝜋𝑟𝜕𝐵𝑟(𝑝0)1𝑤(𝑝)𝑑𝑠=2𝜋|𝜔|=1𝑤𝑝0+𝑟𝜔𝑑𝑠.(2.11)(ii)In the same way, we get 𝑤𝑝0=1𝜋𝑟2𝐵𝑟(𝑝0)1𝑤(𝑝)𝑑𝑝=𝜋|𝜔|1𝑤𝑝0+𝑟𝜔𝑑𝜔.(2.12)
(3) Let 𝑧=𝑧0+𝑟𝑒𝑖𝜃. Then, by integral transform formula, we get 𝑓𝑧0=12𝜋𝑟||𝑧𝑧0||=𝑟1𝑓(𝑧)𝑑𝑠=2𝜋02𝜋𝑓𝑧0+𝑟𝑒𝑖𝜃𝑑𝜃.(2.13)
(4) Let 𝑧=𝑧0+𝑟(𝜔1+𝑖𝜔2) and 𝜔=(𝜔1,𝜔2). Then, by integral transform formula, we see 𝑢𝑥0,𝑦0𝑥+𝑖𝑣0,𝑦0𝑧=𝑓0=12𝜋𝑟||𝑧𝑧0||=𝑟[]=1𝑢(𝑥,𝑦)+𝑖𝑣(𝑥,𝑦)𝑑𝑠2𝜋|𝜔|=1𝑢𝑝0𝑝+𝑟𝜔+𝑖𝑣0+𝑟𝜔𝑑𝑠,(2.14) which implies 𝑢𝑥0,𝑦0=12𝜋|𝜔|=1𝑢𝑝0𝑥+𝑟𝜔𝑑𝑠,𝑣0,𝑦0=12𝜋|𝜔|=1𝑣𝑝0+𝑟𝜔𝑑𝑠.(2.15)

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 Δ𝑤(𝑝)=0,𝑝=(𝑥,𝑦)Ω, then, 𝑤(𝑥,𝑦) satisfies the mean value condition in domain Ω.

Proof. For any 𝐵𝑟(𝑝0)Ω, using Green formula, we have 𝐵𝜌(𝑝0)Δ𝑤(𝑝)𝑑𝑝=𝜕𝐵𝜌(𝑝0)𝜕𝑤𝜕𝜈𝑑𝑠=𝜌|𝜔|=1𝜕𝑤𝑝𝜕𝜌0𝜕+𝜌𝜔𝑑𝑠=𝜌𝜕𝜌|𝜔|=1𝑤𝑝0+𝜌𝜔𝑑𝑠.(3.1) Since 𝑤 is harmonic in Ω, we obtain from (3.1) that 𝜕𝜕𝜌|𝜔|=1𝑤𝑝0+𝜌𝜔𝑑𝑠=0.(3.2) Integrating both sides of (3.2) with respect to 𝜌 on [0,𝑟], we get |𝜔|=1𝑤𝑝0+𝑟𝜔𝑑𝑠=|𝜔|=1𝑤𝑝0𝑝𝑑𝑠=2𝜋𝑤0,(3.3) that is, 𝑤𝑝0=12𝜋|𝜔|=1𝑤𝑝01+𝑟𝜔𝑑𝑠=2𝜋𝑟𝜕𝐵𝑟(𝑝0)𝑤(𝑝)𝑑𝑠.(3.4)

Lemma 3.2. Assume (i) 𝜓(𝜌)C[0,r] and A(r)=𝐵𝑟(𝑝0)𝜓(|𝑝𝑝0|)dp0; (ii) 𝑤(𝑝) satisfies the mean value condition in 𝐵𝑟(𝑝0) and 𝑤(𝑝)C(𝐵𝑟(𝑝0)). Then, 𝑤𝑝0=1𝐴(𝑟)𝐵𝑟(𝑝0)||𝑤(𝑝)𝜓𝑝𝑝0||𝑑𝑝.(3.5)

Proof. From (i) and (ii), we have 𝐴(𝑟)=𝐵𝑟(𝑝0)𝜓||𝑝𝑝0||𝑑𝑝=𝑟0𝑑𝜌𝜕𝐵𝜌(𝑝0)𝜓||𝑝𝑝0||𝑑𝑠=2𝜋𝑟0𝑤𝑝𝜓(𝜌)𝜌𝑑𝜌,(3.6)0=12𝜋𝜌𝜕𝐵𝜌(𝑝0)𝑤(𝑝)𝑑𝑠.(3.7) Multiplying the both sides of (3.7) by 2𝜋𝜌𝜓(𝜌) and integrating the result with respect to 𝜌 on [0,𝑟], we have 𝑝2𝜋𝑤0𝑟0𝜓(𝜌)𝜌𝑑𝜌=𝑟0𝜕𝐵𝜌(𝑝0)𝑤(𝑝)𝜓(𝜌)𝑑𝑠𝑑𝜌=𝐵𝑟(𝑝0)||𝑤(𝑝)𝜓𝑝𝑝0||𝑑𝑝.(3.8) Combining (3.6) and (3.8), we obtain the conclusion.

Lemma 3.3. If 𝑤(𝑝)C(Ω) satisfies the mean value condition, then, (i) 𝑤(𝑝)𝐶(Ω); (ii) Δ𝑤(𝑝)=0.

Proof. (i) Method 1: choose 𝜑(𝑝)𝐶0(𝐵1(0)) with𝐵1(0)||𝑝||.𝜑(𝑝)𝑑𝑝=1,𝜑(𝑝)=𝜓(3.9) Using integral transform formulas, we have 2𝜋10𝑟𝜓(𝑟)𝑑𝑟=1.(3.10)
Define 𝜑𝜀(𝑝)=(1/𝜀2)𝜑(𝑝/𝜀), with 𝜀<dist(𝑝,𝜕Ω),𝑝Ω. Using integral transform formulas and (3.10), we getΩ𝑤(𝑝)𝜑𝜀𝑝𝑝01𝑑𝑝=𝜀2𝑝<𝜀𝑤𝑝0𝜑𝑝+𝑝𝜀𝑑𝑝=𝑝<1𝑤𝑝0=+𝜀𝑝𝜑(𝑝)𝑑𝑝10𝑑𝑟𝜕𝐵𝑟(𝑝0)𝑤𝑝0=+𝜀𝑝𝜑(𝑝)𝑑𝑠10𝑟𝑑𝑟𝜕𝐵1(𝑝0)𝑤𝑝0𝜑=+𝜀𝑟𝜔(𝑟𝜔)𝑑𝑠10𝜓(𝑟)𝑟𝑑𝑟|𝜔|=1𝑤𝑝0𝑝+𝜀𝑟𝜔𝑑𝑠=2𝜋𝑤010𝑝𝜓(𝑟)𝑟𝑑𝑟=𝑤0,(3.11) that is, 𝑤𝑝0=𝜑𝜀𝑝𝑤0,𝑝0=𝑥0,𝑦0Ω𝜀=𝑝0𝑝0𝑝Ω,𝑑0,𝜕Ω>𝜀.(3.12) Applying (3.12) and Proposition 1.3, noticing the arbitrariness of 𝜀, we conclude that 𝑤(𝑝)𝐶(Ω).
Method 2: choose 𝜑(𝑝) as above. Define 𝜑𝜀(𝑝)=(1/𝜀2)𝜑(𝑝/𝜀), with 𝜀<dist(𝑝,𝜕Ω),𝑝Ω, then,𝐵𝑟(𝑝0)𝜑𝜀(𝑝)𝑑𝑝=1.(3.13) Using Lemma 3.2, we obtain 𝑤𝑝0=𝜑𝜀𝑝𝑤0,𝑝0=𝑥0,𝑦0Ω𝜀=𝑝0𝑝0𝑝Ω,𝑑0,𝜕Ω>𝜀.(3.14)
Applying Proposition 1.3, noticing the arbitrariness of 𝜀, we conclude that 𝑤(𝑝)𝐶(Ω).
(ii)Using (3.1) and Proposition 2.4, we get 𝐵𝑟(𝑝0)𝜕Δ𝑤(𝑝)𝑑𝑝=𝑟𝜕𝑟|𝜔|=1𝑤𝑝0𝜕+𝑟𝜔𝑑𝑠=𝑟𝑝𝜕𝑟2𝜋𝑤0=0,𝐵𝑟𝑝0Ω,(3.15) which implies Δ𝑤(𝑝)=0,𝑝=(𝑥,𝑦)Ω.

4. Main Results

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

Proposition 4.1. 𝑓(𝑧) satisfies the mean value condition in Ω if and only if Δ𝑓(𝑧)=0 in Ω.

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

Theorem 4.2. 𝑓(𝑧) satisfies the mean value condition in Ω and 𝜕𝑓(𝑧)/𝜕𝑧=0 if and only if 𝑓(𝑧) is analytic in Ω.

Proof. Denote 𝑓(𝑧)=𝑢(𝑥,𝑦)+𝑖𝑣(𝑥,𝑦).
Firstly, we prove the necessary condition. Employing the assumption and Lemma 3.3, we can assert 𝑢(𝑥,𝑦),𝑣(𝑥,𝑦)𝐶(Ω); hence, the partial derivatives of 𝑢(𝑥,𝑦) and 𝑣(𝑥,𝑦) are continuous in Ω. Combining with Cauchy-Riemann equation 𝜕𝑓(𝑧)/𝜕𝑧=0, we conclude that 𝑓(𝑧) is analytic in Ω.
Secondly, we prove the sufficient condition. The assumption that 𝑓(𝑧) is analytic in Ω implies(i)𝑢𝑥(𝑥,𝑦)=𝑣𝑦(𝑥,𝑦),𝑢𝑦(𝑥,𝑦)=𝑣𝑥(𝑥,𝑦), that is, 𝜕𝑓(𝑧)/𝜕𝑧=0,(ii)Δ𝑓(𝑧)=0. Combining with Proposition 4.1 implies 𝑓(𝑧) satisfies the mean value condition in Ω.

Theorem 4.3. Suppose that 𝑓(𝑧) satisfies the mean value condition in Ω, and |𝑓(𝑧)| is bounded. Then 𝑓(𝑧) is a constant in Ω.

Proof. Since 𝑓(𝑧)=𝑢(𝑥,𝑦)+𝑖𝑣(𝑥,𝑦) satisfies the mean value condition in Ω, using Proposition 2.4 and Lemma 3.3, we get Δ𝑢=0,Δ𝑣=0,(𝑥,𝑦)Ω.
Since |𝑓(𝑧)| is bounded, we obtain 𝑢 and 𝑣 are bounded, respectively, Without loss of generality, we assume that 𝑢0. For all 𝑀02, one can choose 𝐵𝑅(𝑂) with 𝑀0𝐵𝑅(𝑂). Denote 𝑅0=𝑑(𝑀0,𝑂). The Harnack inequality (see [11]) implies 𝑅𝑅0𝑅+𝑅0𝑀𝑢(𝑂)𝑢0𝑅+𝑅0𝑅𝑅0𝑢(𝑂).(4.1) Letting 𝑅+, we conclude 𝑢(𝑀0)=𝑢(𝑂). In the similar way, we conclude 𝑣(𝑀0)=𝑣(𝑂). Since 𝑀0 is arbitrary, we conclude 𝑓(𝑧)=𝑢(𝑥,𝑦)+𝑖𝑣(𝑥,𝑦) is a constant in Ω.

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 𝑓(𝑧) satisfies the mean value condition in Ω, and |𝑓(𝑧)| is a constant. Then, 𝑓(𝑧) is a constant in Ω.

Proof. Since 𝑓(𝑧)=𝑢+𝑖𝑣 satisfies the mean value condition in Ω, using Proposition 2.4 and Lemma 3.3, we obtain Δ𝑢(𝑥,𝑦)=Δ𝑣(𝑥,𝑦)=0.(4.2)
Since |𝑓(𝑧)| is a constant, we get the following in Ω𝑢2(𝑥,𝑦)+𝑣2(𝑥,𝑦)constant.(4.3) From (4.3), we get 2𝑢𝑢𝑥+2𝑣𝑣𝑥=0 and 𝑢2𝑥+𝑣2𝑥+𝑢𝑢𝑥𝑥+𝑣𝑣𝑥𝑥=0.(4.4) In the similar way, we have 𝑢2𝑦+𝑣2𝑦+𝑢𝑢𝑦𝑦+𝑣𝑣𝑦𝑦=0.(4.5) Adding (4.4) to (4.5) and noting (4.2), we obtain 𝑢2𝑥+𝑢2𝑦+𝑣2𝑥+𝑣2𝑦=0,(4.6) 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, maxΩ|𝑓(𝑧)| can be obtained only on the boundary of Ω.

Proof. Denoting 𝑀=maxΩ|𝑓(𝑧)|, then, we have 0<𝑀<+. Suppose there is 𝑧0Ω such that |𝑓(𝑧0)|=𝑀. For any 𝐵𝜌(𝑧0)Ω, the mean value condition implies that, for all 𝑧𝜕𝐵𝜌,|𝑓(𝑧)|=𝑀. Hence |𝑓(𝑧)| is a constant in the neighborhood of 𝑀0. Theorem 4.3 implies 𝑓(𝑧) is a constant in this neighborhood of 𝑀0. 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 𝑧Ω,𝑓(𝑧)0. Can one confirm that minΩ|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.

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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