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14-18                               Advances in Systems Science and Applications (2010), Vol.10, No.1 

ISSN   1078-6236   International Institute for General Systems Studies, Inc. 

Extensions of Liouville’s Theorem 
 Jianfei Wang 

Applied Science College, Harbin University of Science and Technology, Harbin 150080, China 
Email: jianfei1965@sohu.com 

Abstract  In this paper, we first generalize Liouville’s theorem into the general forms based on power series 
representations for analytic functions. Second, in simply connected domains harmonic functions can be 
identified as real parts of analytic functions. Observing the relations between analytic functions and harmonic 
functions, we extend Liouville’s theorem to harmonic functions by the Harnack’s inequality. The generalized 
Liouville’s theorems obtained in this paper will help us to further study the properties of entire functions and 
harmonic functions. 

Keywords  Liouville’s theorem  Entire function  Extension  Harmonic function  Harnack’s inequality 
Analytic 

1. Introduction 

If f (z) is analytic on the whole complex plane, then it is said to be an entire function. For 
entire functions, there exists a beautiful theorem, known as Liouville’s theorem. It gives many 
important properties of entire functions, these properties have been widely applied to Complex 
Analysis. And it enables us to prove the Fundamental Theorem of Algebra. Therefore, it is 
necessary for us to further discuss Liouville’s theorem. In this paper, we first derive two 
extension forms of Liouville’s theorem by simplifying some conditions of classical Liouville’s 
theorem. Meanwhile we get two results for analytic functions. Second, in simply connected 
domains harmonic functions can be identified as real parts of analytic functions. And there are 
many consequences for analytic functions. Some of these are the infinite differentiability of 
analytic functions, Liouville’s theorem, and the maximum modulus theorem. Hence we think 
that these results have analogues for harmonic functions. In terms of the ideas, we extend 
Liouville’s theorem to harmonic functions. The generalized Liouville’s theorems provide 
theoretical basis for us to further study the properties of entire functions and harmonic functions. 

2. Inequalities and Lemmas 

We list some useful inequalities and lemmas before giving to the generalized Liouville’s 
theorems. 

Theorem 2.1 (Liouville’s Theorem) [1]  The only bounded entire functions are the constant 
functions. 

Lemma 2.1 [2]  Let ϕ  be harmonic on a simply connected domain D . Then there is an 
analytic function f such that Re fϕ = on D . 

Lemma 2.2 (mean-value theorem for harmonic function) [3]  Let ϕ  be harmonic in a 
domain containing the dish z R≤ . Then  

                      
2

0

1(0) ( )
2

i tRe dt
π

ϕ ϕ
π

= ∫                              (1) 

Lemma 2.3 (Poisson integral formula) [2]  Let ϕ  be harmonic in a domain containing 
the dish z R≤ . Then for 0 r R≤ < , we have  



Advances in Systems Science and Applications (2010), Vol.10, No.1                                 15

    ( ) ( )
( )

2 2 2

2 202 2 cos

i t
i

ReR rre dt
R r rR t

πθ
ϕ

ϕ
π θ
−

=
+ − −∫                  (2) 

Proof: Using Lemma 2.1, we have 
Re fϕ =  

Here f is analytic on a simply connected domain D . 
(Assuming the domain D includes the circle :RC z R=  as well as its interior) 
Applying Cauchy integral formula, we obtain 

        ( ) ( ) ( )1
2 RC

f
f z d z R

i z
ζ

ζ
π ζ

= <
−∫                           (3) 

For fixed z , with z R< , the function ( )
2

f z
R z

ζ
ζ−

 is an analytic function of ζ inside and 

on RC . Hence by Cauchy theorem  

( )
2

1 0
2 RC

f z
d

i R z
ζ

ζ
π ζ

=
−∫                              (4) 

We add it to Equation (Eq.) (3): 

( ) ( )2

1 1
2 RC

zf z f d
i z R z

ζ ζ
π ζ ζ

⎛ ⎞
= +⎜ ⎟⎜ ⎟− −⎝ ⎠

∫  

              
( )( ) ( )

22

2

1
2 RC

R z
f d

i z R z
ζ ζ

π ζ ζ

−
=

− −∫                     (5) 

If we parameterize RC by , 0 2 ,i tRe tζ π= ≤ ≤ Eq. (5) becomes  

( )
( )( ) ( )

22
2

20

1
2

i t i t
i t i t

R z
f z f Re R e dt

Re z R Re z

π

π
−

=
− −∫              (6) 

( )
( )( )

22
2

02

i t

i t i t

f ReR z
dt

Re z Re z

π

π −

−
=

− −∫                     (7) 

( )22
2

202

i t

i t

f ReR z
dt

Re z

π

π
−

=
−

∫                             (8) 

Writing z in the polar form iz re θ= , we have  

( ) ( )2 2 2

202

i t
i

i t i

f ReR rf re dt
Re re

πθ

θπ
−

=
−

∫                        (9) 

( )
( )

2 2 2

2 202 2 cos

i tf ReR r dt
R r rR t

π

π θ
−

=
+ − −∫              (10) 

Finally, by taking the real part of this equation, we arrive at Poisson integral formula 

( ) ( )
( )

2 2 2

2 20

,
,

2 2 cos
R tR rr dt

R r rR t
π ϕ

ϕ θ
π θ
−

=
+ − −∫                (11) 

or, equivalently,   

                   ( ) ( )
( )

2 2 2

2 202 2 cos

i t
i

ReR rre dt
R r rR t

πθ
ϕ

ϕ
π θ
−

=
+ − −∫               (12) 

Poisson integral formula expresses the values of a harmonic function in a region is 



Wang: Extensions of Liouville’s Theorem 16 

completely determined by its values on the boundary. Using Poisson integral formula and mean - 
value theorem for harmonic function, we may derive the following important inequality. 

Theorem 2.2 (Harnack’s inequality) [2]  Let ϕ  be harmonic and nonnegative in a 
domain containing the dish z R≤ . Then for 0 r R≤ < , we have 

( ) ( ) ( )0 0iR r R rre
R r R r

θϕ ϕ ϕ− +
≤ ≤

+ −
                     (13) 

Proof: Applying Lemma 2.3 to harmonic functionϕ . Then for 0 r R≤ < , 

                  ( ) ( )
( )

2 2 2

2 202 2 cos

i t
i

ReR rre dt
R r rR t

πθ
ϕ

ϕ
π θ
−

=
+ − −∫                (14) 

Observing that  
( ) ( ) ( )2 22 2 2 cosR r R r rR t R rθ− ≤ + − − ≤ +                (15) 

Since ϕ  is nonnegative, we have  

( )
( )

( )
( )

( )
( )

2 2 2

2 2 2 20 0 02 cos

i t i t i tRe Re Re
dt dt dt

R r rR tR r R r

π π πϕ ϕ ϕ

θ
≤ ≤

+ − −+ −∫ ∫ ∫           (16) 

Hence 

( ) ( ) ( )2 2

0 0

1 1
2 2

i t i i tR r R rRe dt re Re dt
R r R r

π πθϕ ϕ ϕ
π π

− +
⋅ ≤ ≤ ⋅

+ −∫ ∫          (17) 

Finally, by Lemma 2.2, the proof is completed.     

3. The Generalized Liouville’s Theorems   

In this section, we consider the extension problem of classical Liouville’s theorem. This is 
the main work of this paper.  

Firstly, we obtain two extension forms of Liouville’s theorem by reducing the condition of 
classical Liouville’s theorem. These theorems are stated below. 

Theorem 3.1 I f f is an entire function, and suppose there are a nonnegative integer n , and 
two positive constants ,R M such that ( ) nf z M z≤  when z R≥ , then f is a polynomial 
with ( )deg f n≤  or constant. 

Proof:  The case 0n = will be treated first. 
In this case, ( )f z M≤  when z R≥ . Since f is an entire function, it must be continuous 

on the closed domain z R≤ .Under such circumstances it is known from calculus that the 
function must be bounded there. In other words, there exists a positive constant G such that 

( )f z G≤  when z R≤ . By taking { }max , 0N M G= > , we have  
( )f z N≤                               (18) 

when z < +∞ . Thus from Theorem 2.1 there follows f is constant. This completes the 
proof. 

Then the general case 1n ≥  will be dealt with. 
Since f  is an entire function, it must have a Maclaurin series representation, namely  

( )2
0 1 2( ) n

nf z c c z c z c z z= + + + + + < +∞                (19) 
where 

( ) ( ) ( )
( )

1

0 1 0; 0,1, 2,
! 2

n

n nz r

f f
c d r n

n i
ζ

ζ
π ζ +=

= = > =∫              (20) 

For an arbitrary integer 1p ≥ and a large enough 0R R>  , we consider that 



Advances in Systems Science and Applications (2010), Vol.10, No.1                                 17

( )
0

1

1
2n p n Pz R

f
c d

i
ζ

ζ
π ζ+ + +=

= ∫                     (21) 

Notice that  
( )

0
1

1
2n p n pz R

f
c ds

ζ
π ζ

+ + +=
≤ ∫  

     
0

1

1
2

n

n pz R

M
d s

ζ
π ζ + +=

≤ ∫  

       01
0 0

1 2
2 p p

M MR
R R

π
π += ⋅ ⋅ =                     (22) 

we can get ( )0 1,2,n pc p+ = = . Thus  

( )2
0 1 2( ) n

nf z c c z c z c z z= + + + + < +∞                   (23) 

 It says that f is a polynomial with ( )deg f n≤ . 
By modifying the condition of Theorem 3.1, we easily obtain the following results. 
Corollary 3.1 I f f is an entire function, and suppose there are a nonnegative integer n , 

and three positive constants ,R M and N such that ( ) nf z N M z≤ +  when z R≥ , then f is 
a polynomial with ( )deg f n≤  or constant. 

Corollary 3.2 I f f is an entire function, and there is a positive integer n  such 

that ( )lim 0nz

f z k
z→∞

= > , then f is a polynomial with ( )deg f n≤ . 

Theorem 3.2 Let f  be analytic in the extended complex plane. Then f is constant. 
Proof: Since f  is analytic in the extended complex plane, it must have a Maclaurin series 

representation, namely 

            ( )
0

( ) n
n

n

f z c z z
∞

=

= < +∞∑                        (24) 

Meanwhile, z = ∞ is a removable singularity of f . Hence ( )f z has a finite limit 
as z approaches 0z . Recalling (24), the conclusion is obtained. 

Secondly, in simply connected domains harmonic functions can be identified as real parts of 
analytic functions. Based on the relations between analytic functions and harmonic functions, 
harmonic function in 2R  is considered [4]  in generalizing Liouville’s theorem. We state 
Liouville’s theorem for harmonic functions as follows. 

Theorem 3.3 (Liouville’s theorem for harmonic functions) [5]  Let ϕ  be harmonic in 
the whole real plane 2R  and bounded from above or below there. Thenϕ is constant. 

Proof: Assume that ϕ is bounded from above there, namely, there exists a constant 
M such that Mϕ ≤  for any 2z R∈ . 

Clearly, Mψ ϕ= −  is harmonic and nonnegative in the whole real plane 2R . Using 
Theorem 2.2, for 0 r R≤ < < +∞ , we have 

                       ( ) ( ) ( )0 0iR r R rre
R r R r

θψ ψ ψ− +
≤ ≤

+ −
                   (25) 

Letting R →+∞ , deduce that 
( ) ( )0ire θψ ψ=                             (26) 

where [ )0,r∈ +∞ .  



Wang: Extensions of Liouville’s Theorem 18 

Notice that r is an arbitrary nonnegative real number hence Mψ ϕ= −  is constant. It 
follows immediately that ϕ is constant. 

4. Conclusions 

Summing up, we first transform some useful inequalities and lemmas to generalize 
Liouville’s theorem. Then using power series representations for analytic functions [6] , we derive 
two extension forms of Liouville’s theorem by simplifying some conditions of classical 
Liouville’s theorem. Finally, observing the relations between analytic functions and harmonic 
functions, we extend Liouville’s theorem to harmonic functions by Harnack’s inequality. In this 
course, we know that the generalized Liouville’s theorems will help us to further study the 
properties of entire functions. Meanwhile, we can get some important conclusions for harmonic 
functions [7]  based on the consequences for analytic functions in the future. 

References 

[1] Yuquan Zhong. Functions of Complex Variable. Higher Education Press, Beijing, 2004: 
127-130. 

[2] Saff, E. B. Fundamental of Complex Analysis with Applications to Engineering and 
Science. China Machine Press, Beijing, 2004: 221-226. 

[3] Jiarong Yu. Functions of Complex Variable. Higher Education Press, Beijing, 2000: 
156-157. 

[4] Weixing Dai, Shaobo Zhou. Attraction and Stability for Neutral Stochastic Differential 
Delay Equations. Advances in Systems Science and Applications, 2008, 8(2): 212-219. 

[5] Zhenhua Jiao. A Note on Liouville’s Theorem. Journal of Hangzhou Dianzi University, 
2006, 26(2): 96-98. 

[6] Jiancong Chen, Zhihong Guan and Hu Chen. Fuzzy Association Analysis of Complex 
System in Correlation. Advances in Systems Science and Applications, 2008, 8(2): 251-257. 

[7] Elias Zafiris. Categorical Modeling of Natural Complex Systems. Part I: Functorial Process 
of Representation. Advances in Systems Science and Applications, 2008, 8(2): 187-200.