New Properties of Complex Functions with Mean Value Conditions

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=𝑥−𝑖𝑦, 𝑝₀=(𝑥₀,𝑦₀), 𝑝=(𝑥,𝑦), 𝐵_𝑟(𝑝₀)={𝑝∣dist(𝑝,𝑝₀)≤𝑟},  𝜕𝐵_𝑟(𝑝₀)={𝑝∣dist(𝑝,𝑝₀)=𝑟}. Using the chain rule of derivation, we have 𝜕𝜕𝑧=𝜕𝜕𝑥−𝑖𝜕𝜕𝑦,𝜕𝜕𝑧=𝜕𝜕𝑥+𝑖𝜕𝜕𝑦,Δ=𝜕𝑧𝜕𝜕𝑧=𝜕𝑧𝜕𝜕𝑧=𝜕𝜕𝑥²²+𝜕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 𝜑_𝜀(𝑥)=⎧⎪⎨⎪⎩𝜀𝑐^𝑛exp𝜀²0,|𝑥|²−𝜀²,||𝑥𝑥||<≥𝜀𝜀,,(1.3) with 𝑐∈𝑅 such that ∫_𝑅𝑛𝜑_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 𝑓_𝜀(𝑥)∶=_𝑅𝑛𝑓(𝑥−𝑦)𝜑_𝜀(𝑦)𝑑𝑦=_𝑅𝑛𝑓(𝑦)𝜑_𝜀(𝑥−𝑦)𝑑𝑦.(1.4) The sequence 𝑓_𝜀(𝑥) is said to be a sequence of mollified functions.

Proposition 1.3 (Properties of mollifiers, see [10]). Suppose that Ω⊂Rⁿ is open, 𝜀>0, write Ω_𝜀={x∈Ω∣dist(x,𝜕Ω)>𝜀}. Then,(i)𝑓_𝜀∈𝐶^∞(Ω_𝜀), (ii)𝑓_𝜀→𝑓 a.e. as 𝜀→0,(iii)if 𝑓∈𝐶(Ω), then 𝑓_𝜀→𝑓 uniformly on compact subsets of Ω,(iv)If 1≤𝑝<∞ and 𝑓∈𝐿^𝑝loc(Ω) then 𝑓_𝜀→𝑓 in 𝐿^𝑝loc(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 𝑧₀∈Ω and {𝑧∣|𝑧−𝑧₀|≤𝑟}⊂Ω, if 𝑓𝑧₀=12𝜋𝑟_{||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 𝐵_𝑟(𝑝₀)⊂Ω, if 𝑤𝑝₀=12𝜋𝑟_{𝜕𝐵𝑟(𝑝0)}𝑤(𝑝)𝑑𝑠,(2.2) we say that 𝑤(𝑝) satisfies the first mean value condition.(ii)For any 𝐵_𝑟(𝑝₀)⊂Ω, if 𝑤𝑝₀=1𝜋𝑟²_{𝐵𝑟(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 𝑤𝑝₀=12𝜋_|𝜔|=1𝑤𝑝₀+𝑟𝜔𝑑𝑠.(2.4)(ii)The second mean value condition of 𝑤(𝑝) can be written as 𝑤𝑝₀=1𝜋_|𝜔|≤1𝑤𝑝₀+𝑟𝜔𝑑𝜔.(2.5)
(3) The mean value condition of complex function 𝑓(𝑧) can be written as 𝑓𝑧₀=12𝜋^02𝜋𝑓𝑧₀+𝑟𝑒^𝑖𝜃𝑑𝜃.(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 𝑤𝑝₀=1𝜋𝑟²_{𝐵𝑟(𝑝0)}𝑤(𝑝)𝑑𝑝=1𝜋𝑟²^𝑟0𝑑𝜌_{𝜕𝐵𝜌(𝑝0)}𝑤(𝑝)𝑑𝑠,(2.7) with respect to 𝑟, we have 0=−2𝜋𝑟³^𝑟0𝑑𝜌_{𝜕𝐵𝜌(𝑝0)}𝑤(𝑝)𝑑𝑠+1𝜋𝑟²_{𝜕𝐵𝑟(𝑝0)}𝑤(𝑝)𝑑𝑠,(2.8) that is, 12𝜋_{𝜕𝐵𝑟(𝑝0)}𝑤(𝑝)𝑑𝑠=1𝜋𝑟²_{𝐵𝑟(𝑝0)}𝑤(𝑝)𝑑𝑝=𝑤𝑝₀.(2.9) We write the first mean value condition as 𝑤𝑝₀𝜌=12𝜋_{𝜕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 𝑤𝑝₀=12𝜋𝑟_{𝜕𝐵𝑟(𝑝0)}𝑤(𝑝)𝑑𝑠=12𝜋_|𝜔|=1𝑤𝑝₀+𝑟𝜔𝑑𝑠.(2.11)(ii)In the same way, we get 𝑤𝑝₀=1𝜋𝑟²_{𝐵𝑟(𝑝0)}𝑤(𝑝)𝑑𝑝=1𝜋_|𝜔|≤1𝑤𝑝₀+𝑟𝜔𝑑𝜔.(2.12)
(3) Let 𝑧=𝑧₀+𝑟𝑒^𝑖𝜃. Then, by integral transform formula, we get 𝑓𝑧₀=12𝜋𝑟_{||𝑧−𝑧0||=𝑟}𝑓(𝑧)𝑑𝑠=12𝜋^02𝜋𝑓𝑧₀+𝑟𝑒^𝑖𝜃𝑑𝜃.(2.13)
(4) Let 𝑧=𝑧₀+𝑟(𝜔₁+𝑖𝜔₂) and 𝜔=(𝜔₁,𝜔₂). Then, by integral transform formula, we see 𝑢𝑥₀,𝑦₀+𝑖𝑣𝑥₀,𝑦₀=𝑓𝑧₀=12𝜋𝑟_{||𝑧−𝑧0||=𝑟}=1[𝑢(𝑥,𝑦)+𝑖𝑣(𝑥,𝑦)]𝑑𝑠2𝜋_|𝜔|=1𝑢𝑝₀+𝑟𝜔+𝑖𝑣𝑝₀+𝑟𝜔𝑑𝑠,(2.14) which implies 𝑢𝑥₀,𝑦₀=12𝜋_|𝜔|=1𝑢𝑝₀+𝑟𝜔𝑑𝑠,𝑣𝑥₀,𝑦₀=12𝜋_|𝜔|=1𝑣𝑝₀+𝑟𝜔𝑑𝑠.(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 𝐵_𝑟(𝑝₀)⊂Ω, using Green formula, we have _{𝐵𝜌(𝑝0)}Δ𝑤(𝑝)𝑑𝑝=_{𝜕𝐵𝜌(𝑝0)}𝜕𝑤𝜕𝜈𝑑𝑠=𝜌_|𝜔|=1𝜕𝑤𝜕𝜌𝑝⁰+𝜌𝜔𝑑𝑠=𝜌𝜕𝜕𝜌_|𝜔|=1𝑤𝑝₀+𝜌𝜔𝑑𝑠.(3.1) Since 𝑤 is harmonic in Ω, we obtain from (3.1) that 𝜕𝜕𝜌_|𝜔|=1𝑤𝑝₀+𝜌𝜔𝑑𝑠=0.(3.2) Integrating both sides of (3.2) with respect to 𝜌 on [0,𝑟], we get _|𝜔|=1𝑤𝑝₀+𝑟𝜔𝑑𝑠=_|𝜔|=1𝑤𝑝₀𝑑𝑠=2𝜋𝑤𝑝₀,(3.3) that is, 𝑤𝑝₀=12𝜋_|𝜔|=1𝑤𝑝₀+𝑟𝜔𝑑𝑠=12𝜋𝑟_{𝜕𝐵𝑟(𝑝0)}𝑤(𝑝)𝑑𝑠.(3.4)

Lemma 3.2. Assume (i) 𝜓(𝜌)∈C[0,r] and A(r)∶=∫_{𝐵𝑟(𝑝0)}𝜓(|𝑝−𝑝₀|)dp≠0; (ii) 𝑤(𝑝) satisfies the mean value condition in 𝐵_𝑟(𝑝₀) and 𝑤(𝑝)∈C(𝐵_𝑟(𝑝₀)). Then, 𝑤𝑝₀=1𝐴(𝑟)_{𝐵𝑟(𝑝0)}𝑤(𝑝)𝜓||𝑝−𝑝₀||𝑑𝑝.(3.5)

Proof. From (i) and (ii), we have 𝐴(𝑟)∶=_{𝐵𝑟(𝑝0)}𝜓||𝑝−𝑝₀𝑤||𝑑𝑝𝑝=^𝑟0𝑑𝜌_{𝜕𝐵𝜌(𝑝0)}𝜓||𝑝−𝑝₀||𝑑𝑠=2𝜋^𝑟0𝜓(𝜌)𝜌𝑑𝜌,(3.6)₀=2𝜋1𝜌_{𝜕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 2𝜋𝑤𝑝₀^𝑟0𝜓(𝜌)𝜌𝑑𝜌=^𝑟0_{𝜕𝐵𝜌(𝑝0)}𝑤(𝑝)𝜓(𝜌)𝑑𝑠𝑑𝜌=_{𝐵𝑟(𝑝0)}𝑤(𝑝)𝜓||𝑝−𝑝₀||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)∈𝐶^∞0(𝐵₁(0)) with_𝐵1(0)𝜑(𝑝)𝑑𝑝=1,𝜑(𝑝)=𝜓||𝑝||.(3.9) Using integral transform formulas, we have 2𝜋¹⁰𝑟𝜓(𝑟)𝑑𝑟=1.(3.10)
Define 𝜑_𝜀(𝑝)=(1/𝜀²)𝜑(𝑝/𝜀), with 𝜀<dist(𝑝,𝜕Ω),𝑝∈Ω. Using integral transform formulas and (3.10), we get_Ω𝑤(𝑝)𝜑_𝜀𝑝−𝑝₀𝑑𝑝=1𝜀²_{∣𝑝∣<𝜀}𝑤𝑝₀+𝑝𝜑𝑝𝜀𝑑𝑝=_∣𝑝∣<1=𝑤𝑝₀+𝜀𝑝𝜑(𝑝)𝑑𝑝₁₀=𝑑𝑟_{𝜕𝐵𝑟(𝑝0)}𝑤𝑝₀+𝜀𝑝𝜑(𝑝)𝑑𝑠₁₀=𝑟𝑑𝑟_{𝜕𝐵1(𝑝0)}𝑤𝑝₀+𝜀𝑟𝜔𝜑(𝑟𝜔)𝑑𝑠₁₀𝜓(𝑟)𝑟𝑑𝑟_|𝜔|=1𝑤𝑝₀+𝜀𝑟𝜔𝑑𝑠=2𝜋𝑤𝑝₀¹⁰𝜓(𝑟)𝑟𝑑𝑟=𝑤𝑝₀,(3.11) that is, 𝑤𝑝₀=𝜑_𝜀∗𝑤𝑝₀,∀𝑝₀=𝑥₀,𝑦₀∈Ω_𝜀=𝑝₀∣𝑝₀∈Ω,𝑑𝑝₀,𝜕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 𝜑_𝜀(𝑝)=(1/𝜀²)𝜑(𝑝/𝜀), with 𝜀<dist(𝑝,𝜕Ω),𝑝∈Ω, then,_{𝐵𝑟(𝑝0)}𝜑_𝜀(𝑝)𝑑𝑝=1.(3.13) Using Lemma 3.2, we obtain 𝑤𝑝₀=𝜑_𝜀∗𝑤𝑝₀,∀𝑝₀=𝑥₀,𝑦₀∈Ω_𝜀=𝑝₀∣𝑝₀∈Ω,𝑑𝑝₀,𝜕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|=1𝑤𝑝₀+𝑟𝜔𝑑𝑠=𝑟𝜕𝜕𝑟2𝜋𝑤𝑝⁰=0,∀𝐵_𝑟𝑝₀⊂Ω,(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)𝑢_𝑥(𝑥,𝑦)=𝑣_𝑦(𝑥,𝑦),𝑢_𝑦(𝑥,𝑦)=−𝑣_𝑥(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 𝑢≥0. For all 𝑀₀∈ℝ², one can choose 𝐵_𝑅(𝑂) with 𝑀₀∈𝐵_𝑅(𝑂). Denote 𝑅₀=𝑑(𝑀₀,𝑂). The Harnack inequality (see [11]) implies 𝑅−𝑅₀𝑅+𝑅₀𝑢(𝑂)≤𝑢𝑀₀≤𝑅+𝑅⁰𝑅−𝑅₀𝑢(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 2𝑢𝑢_𝑥+2𝑣𝑣_𝑥=0 and 𝑢^2𝑥+𝑣^2𝑥+𝑢𝑢_𝑥𝑥+𝑣𝑣_𝑥𝑥=0.(4.4) In the similar way, we have 𝑢^2𝑦+𝑣^2𝑦+𝑢𝑢_𝑦𝑦+𝑣𝑣_𝑦𝑦=0.(4.5) Adding (4.4) to (4.5) and noting (4.2), we obtain 𝑢^2𝑥+𝑢^2𝑦+𝑣^2𝑥+𝑣^2𝑦=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, max_Ω|𝑓(𝑧)| can be obtained only on the boundary of Ω.

Proof. Denoting 𝑀=max_Ω|𝑓(𝑧)|, then, we have 0<𝑀<+∞. Suppose there is 𝑧₀∈Ω such that |𝑓(𝑧₀)|=𝑀. For any 𝐵_𝜌(𝑧₀)⊂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 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 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.

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).