Special Issue in honor of Alan C. Lazer

Electronic Journal of Differential Equations, Special Issue 01 (2021), pp. 269–278.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu

THE MEAN VALUE PROPERTY AND ZEROS OF

HOLOMORPHIC FUNCTIONS (GAUSS, POISSON, BOLZANO,

AND CAUCHY MEET IN THE COMPLEX PLANE)

JEAN MAWHIN

Dedicated to the living memory of Alan C. Lazer

Abstract. An existence condition for a zero of holomorphic functions in a

disk is stated and proved in a very simple way using the mean value property.

It contains as special cases Bolzano’s theorem and Brouwer fixed point theorem
in a disk for holomorphic functions, the fundamental theorem of algebra and

an asymptotic condition for the existence of zeros of transcendental entire
functions. An elementary proof of the used mean value property is given.

1. Introduction

Alan Lazer is remembered for his numerous, deep and stimulating contributions
to ordinary differential equations, nonlinear analysis and critical point theory. It
may be less known that he contributed in two papers to the vast literature devoted
to proving the fundamental theorem of algebra, namely the existence of a complex
root to any complex algebraic equation. Not surprisingly for those who know Alan
and his work, his two approaches are both original and unusual.

Lazer [9] used a sufficient condition for a maximum of a function of two variables
together with an identity for the Laplacian of the square of the modulus of a
holomorphic function proved in an elementary way. In [10], the authors gave four
proofs of the fundamental theorem of algebra based upon a version of the Fourier
transform and inversion formula for continuous functions.

Good examples have to be followed and we propose in this paper to deduce
the fundamental theorem of algebra and other existence results for the zeros of
holomorphic functions from a new geometric condition for the existence of a zero
of a holomorphic function f in a closed disk DR of center 0 and radius R (Theorem
3.1 in Section 3). The proof of this result is based upon the mean value property for
a holomorphic function, telling that f(0) is equal to the average of f over any circle
of center 0 and radius r ≤ R, Proceeding by contradiction and assuming that f has
no zero in DR, the integral over [0, 2π] of some associated holomorphic function is
computed in two ways, one being the mean value property, and provides different

2010 Mathematics Subject Classification. 12D05, 30C15.
Key words and phrases. Mean value property; holomorphic functions; Bolzano’s theorem;

fundamental theorem of algebra.
©2021 This work is licensed under a CC BY 4.0 license.

Published December 27, 2021.

269



270 JEAN MAWHIN EJDE/SI/01

results. Section 2 proposes a proof of the used mean value property avoiding any
use of complex function techniques.

Theorem 3.1 contains as easy special cases or immediate consequences Shi Mau-
Hsiang’s extension to holomorphic functions of Bolzano’s intermediate value theo-
rem [17] (Corollary 4.1 in Section 4), Brouwer fixed point theorem for holomorphic
functions on a closed ball (Corollary 4.3 in Section 4), the fundamental theorem of
algebra (Corollary 5.1 in Section 5) and a recent necessary and sufficient asymp-
totic condition of Bao Qin Li [11] for the existence of a zero of an entire function
(Corollary 6.1 in Section 6). An Appendix gives an advanced calculus proof of
an extension property of holomorphic functions that is only used in proving the
necessary condition in Corollary 6.1. Examples are given in the various sections.

This is how Gauss, Poisson, Bolzano and Cauchy meet in the complex field.
A different and longer proof of the fundamental theorem of algebra (FTA) based

on the mean value property was given some years ago by Vyborny [18] and a much
shorter one recently by Schep [16]. The monographs [1, 4, 8] and the survey [15]
illustrate the richness and diversity of the proofs of the FTA, a good competitor,
with more than 250 papers, among the mathematical statements having received
the largest number N of (more or less) different proofs. Lazer’s correct estimate in
[10] is N > 80.

2. An elementary proof of the mean value property (or Gauss mean
value theorem)

Let DR ⊂ C denote the open disk of center 0 and radius R > 0, DR its closure,
∂DR its boundary.

The function g : DR → C is said to be holomorphic in DR if, for each z ∈ DR,
the limit

g′(z) := lim
h→0

g(z + h)− g(z)

h
(2.1)

exists and the complex derivative g′ : DR → C of g is continuous. A holomorphic
function in DR is continuous in DR, and the usual rules of the calculus of functions
of one real variable immediately extend to the complex derivative. For example,
for any integer n ≥ 1, the function z → zn is holomorphic on C, and the same is
true for any polynomial.

If we consider g as a function of the two real variables (x, y) with x = <z, the
real part of z, and y = =z, the imaginary part of z, then, by taking respectively h
real and h purely imaginary in (2.1), we obtain the Cauchy-Riemann equations

g′(z) =
∂g

∂x
(z) =

1

i

∂g

∂y
(z). (2.2)

The last equality in (2.2) is necessary and sufficient for a complex function f of
class C1 on DR to be holomorphic in DR.

In particular, if eit = cos t + i sin t, the chain rule and the relations (2.2) imply
that, for all r ∈ [0, R) and t ∈ R,

∂

∂r
[g(a+ reit)] =

∂g

∂x
(a+ reit) cos t+

∂g

∂y
(a+ reit) sin t

= g′(reit)eit,



EJDE-2021/SI/01 MEAN VALUE PROPERTY 271

∂

∂t
[g(a+ reit)] =

∂g

∂x
(a+ reit)(−r sin t) +

∂g

∂y
(a+ reit)(r cos t)

= irg′(reit)eit

so that
∂

∂r
[g(a+ reit)] =

1

ir

∂

∂t
[g(a+ reit)]. (2.3)

For the reader not familiar with the theory of complex functions, we recall an
elementary proof of the mean value property for a holomorphic function in a disk,
also called Gauss mean value theorem because of Gauss’ similar result for harmonic
functions. It was first stated and proved in 1823 by Poisson [13] for the sum of
a power series, and is a special case of Cauchy’s integral formula (see e.g. [14, p.
203]).

Lemma 2.1. If the continuous function g : DR → C is holomorphic in DR \ {0, z}
for some z ∈ DR, then

g(0) =
1

2π

∫ 2π

0

g(reit) dt for all r ∈ [0, R].

Proof. We define I : [0, R]→ C by

I(r) =
1

2π

∫ 2π

0

g(reit) dt.

As the functions t 7→ g(reit) and t 7→ ∂
∂r [g(reit)] are continuous on [0, 2π] for

all r ∈ [0, R] and r ∈ (0, |z|) ∪ (|z|, R) respectively, I is continuous on [0, R] and
continuously differentiable on (0, |z|)∪ (|z|, R). We can apply Leibniz’ rule, formula
(2.3), the fundamental theorem of calculus and the 2π-periodicity of the function
g(reit) to obtain, for all r ∈ (0, |z|) ∪ (|z|, R),

I ′(r) =

∫ 2π

0

∂

∂r
[g(reit)] dt =

1

ir

∫ 2π

0

∂

∂t
[g(reit)] dt = 0.

Consequently, I(r) is constant on (0, |z|), and hence on [0, |z|] by continuity, and,
for r ∈ [0, |z|), I(r) = I(0) = g(0). Similarly, I(r) is constant on (|z|, R), and hence
on [|z|, R] by continuity, and, for r ∈ [|z|, R], I(r) = I(|z|) = g(0). �

3. A geometric condition for the existence of a zero of a
holomorphic function

Denoting by z the conjugate z = x− iy of z = x+ iy ∈ C, so that <z = <z and
=z = −=z, we state and prove the main theorem of this note.

Theorem 3.1. If the continuous function f : DR → C is holomorphic in DR \{0},
and if there exists a continuous function h : DR → C holomorphic in DR \ {0},
such that h(0) = 0 and such that <[hf ] or =[hf ] does not change sign on ∂DR,
then f has at least one zero in DR.

Proof. It suffices to prove the theorem under the stronger assumption that <[hf ]
or =[hf ] keeps a strict sign on ∂DR. Indeed, for k = 1, 2, . . ., if say the strict
sign is positive, the functions fk defined by fk(z) = k−1h(z) + f(z) verify the
stronger assumption when f satisfies the assumption of Theorem 3.1, and a sequence
(zk)k≥1 in DR verifying fk(zk) = 0 contains a subsequence converging to some



272 JEAN MAWHIN EJDE/SI/01

z∗ ∈ DR such that f(z∗) = 0. For a negative strict sign, it suffices to take fk(z) =
−k−1h(z) + f(z).

Proceeding by contradiction, if f(z) 6= 0 for all z ∈ DR and f satisfies the strong
assumption, the function h/f is continuous on DR, holomorphic in DR \ {0}, and
hence, by Lemma 2.1 applied to g = h/f , we obtain∫ 2π

0

h(Reit)

f(Reit)
dt = 2π

h(0)

f(0)
= 0. (3.1)

On the other hand,∫ 2π

0

h(Reit)

f(Reit)
dt =

∫ 2π

0

h(Reit)f(Reit)

|f(Reit)|2
dt

=

∫ 2π

0

{<[h(Reit)f(Reit)]

|f(Reit)|2
− i=[h(Reit)f(Reit)]

|f(Reit)|2
}
dt 6= 0,

because the real part or the imaginary part of the integrated function keeps a strict
sign for all t ∈ [0, 2π], a contradiction with (3.1). �

Remark 3.2. As

<[h(z)f(z)] = <h(x, y)<f(x, y) + =h(x, y)=h(x, y),

=[h(z)f(z)] = −=h(x, y)<f(x, y) + <h(x, y)=f(x, y)

are respectively the inner product in R2 of the mappings v(x, y) = (<h(x, y),
=h(x, y)) and w(x, y) = (<f(x, y),=f(x, y)), and of the mappings Jv(x, y) and
w(x, y), with

J =

(
0 −1
1 0

)
the symplectic matrix, the assumptions of Theorem 3.1 are geometric conditions
on those mappings on ∂DR and are local (not asymptotic). Furthermore, Theorem
3.1 provides a localization z ∈ DR for the obtained zeros.

Remark 3.3. In the frame of Brouwer’s degree theory [5], Theorem 3.1 is the
consequence of the fact that the (strict) assumption implies that v and w are non
zero and homotopic on ∂DR and that the mapping v associated to the holomorphic
mapping h vanishes at 0 and hence has a positive Brouwer degree (see e.g. [5]). The
simplicity and shortness of the proof given in this paper with respect to a proof
based upon Brouwer degree is a consequence of using the supplementary product
structure in C and replacing the estimation of the Brouwer degrees d[w,DR, 0] and

d[v,DR, 0] by the much more easy evaluation of 1
2π

∫ 2π

0
h(Reit)
f(Reit) dt.

Remark 3.4. There is no uniqueness conclusion in Theorem 3.1. Indeed, the
function f(z) = zn − 1, with n ≥ 2 an integer, is such that, for |z| = R ≥ 1,
<[zn(zn − 1)] = R2n − <zn ≥ Rn(Rn − 1) ≥ 0 and we know that f(z) has n zeros
in DR.

Example 3.5. Let us define the holomorphic function f : C→ C by

f(z) = znez + a cosh z,

where n ≥ 1 is an integer and a ∈ C. When z ∈ ∂DR, we have, if we take
h(z) = znez, so that h(z) = znez,

<[znez(znez + a cosh z)] = |z|2ne2<z + <[aznez cosh z]



EJDE-2021/SI/01 MEAN VALUE PROPERTY 273

≥ R2ne−2R − |a|RneR cosh z|
≥ Rn(Rne−2R − |a|eR coshR) ≥ 0

if

|a| ≤ Rne−3R

coshR
=

2Rne−4R

1 + e−2R
.

As e−2R ≤ 1 if R > 0, this is surely satisfied if |a| ≤ Rne−4R, and the expression in
the right-hand member has its maximum value for R = n

4 . Therefore, f has a zero

in Bn/4 when

|a| ≤ en(logn−2 log 2−1).

This implies in particular that, given any a ∈ C, the function f(z) = znez+a cosh z
has a zero in Bn/4 for all sufficiently large n.

4. Hadamard-Shi’s existence theorem and Brouwer’s fixed point
theorem for a holomorphic function

The special case h(z) = z of Theorem 3.1, proved in [12] by a similar argument,
using Cauchy’s integral theorem instead of the mean value property, generalizes
a result that Mau-Hsiang Shi [17] had obtained under the stronger assumption
<[zf(z)] > 0 on ∂DR, using Rouché’s theorem in complex analysis (see e.g. [14, p.
390]).

Corollary 4.1. If the continuous function f : DR → C is holomorphic in DR, and
if <[zf(z)] or =[zf(z)] does not change sign on ∂DR, then f has at least one zero
in DR.

Corollary 4.1 extends to holomorphic functions Bolzano’s condition “f(−R) and
f(R) have opposite signs” or, equivalently, “−Rf(−R) and Rf(R) have the same
sign” for the existence of a zero in [−R,R] of the continuous function f : [−R,R]→
R. As

<[zf(z)] = 〈(x, y), (<f(x, y),=f(x, y))〉,
we have named it Hadamard-Shi’s theorem in [12], by reference to the early use of
this condition by Hadamard of [7] in his proof of the n-dimensional Brouwer fixed
point theorem for a continuous self mapping on a closed n-ball (see e.g. [5, Chapter
8]).

Remark 4.2. According to Remark 3.2, the assumption on <[zf(z)] means that
the vector field corresponding to f qualitatively behaves like the outer or the inner
normal vector field on ∂BR, and the assumption on =[zf(z)] means that the vector
field corresponding to f qualitatively behaves like the tangent vector field on ∂DR.

Hadamard’s argument and Corollary 4.1 provide a version for holomorphic func-
tions of a slightly generalized Brouwer’s fixed point theorem first obtained by Birkhoff
and Kellogg in 1922 [2].

Corollary 4.3. If the continuous function c : DR → C is holomorphic in DR and
such that c(∂DR) ⊆ DR, then c has at least one fixed point in DR.

Proof. Defining f : DR → C by f(z) = z − c(z), we have, for all z ∈ ∂DR,

<[zf(z)] = |z|2 −<[zc(z)] ≥ R2 − |zc(z)| = R2 −R|c(z)| ≥ 0,

and the result follows from Corollary 4.1. �



274 JEAN MAWHIN EJDE/SI/01

Example 4.4. Given a ∈ C and c(z) = a cos z, we have, if |z| = R,

|c(z)| = |a cos z| = |a|
∣∣eiz + e−iz

2

∣∣
≤ |a|e

−=z + e=z

2
= |a| cosh=z

≤ |a| coshR ≤ R

if |a| ≤ R/ coshR. Now, the function R 7→ R/ coshR reaches its maximum on
(0,+∞) whenR = 1/ tanhR, namelyR∗ ≈ 1.199 . . .. Hence, for |a| ≤ R∗/ coshR∗ ≈
0.662 . . ., c(∂BR∗) ⊆ BR∗ and c has a fixed point in BR∗ by Corollary 4.3.

5. The fundamental theorem of algebra

The fundamental theorem of algebra is an easy consequence of Theorem 3.1.

Corollary 5.1. Any polynomial

p(z) =

n∑
k=0

akz
k (an 6= 0)

of effective degree n ≥ 1 with coefficients ak ∈ C (k = 0, 1, . . . , n) has at least one
zero in C.

Proof. Without loss of generality, we can assume that an = 1. We take f(z) = p(z)
and h(z) = zn in Theorem 3.1. If z ∈ ∂Dr,

<[znp(z)] = |z|2n +

n−1∑
k=1

<[akz
kzn] ≥ r2n −

n−1∑
k=0

|<[akz
kzn]|

≥ r2n −
n−1∑
k=0

|ak|rk+n = r2n
(

1−
n−1∑
k=0

|ak|rk−n
)
.

Now, there exists R > 0 such that
∑n−1
k=0 |ak|Rk−n ≤ 1, and p satisfies the assump-

tions of Theorem 3.1 on DR. �

Remark 5.2. The fundamental theorem of algebra can also be directly deduced
from Lemma 2.1, as shown by Schep [16]. If p has no zero, 1/p is holomorphic on
C and, by Lemma 2.1 with g = 1/p, we have

0 6= 1

p(0)
=

1

2π

∫ 2π

0

dt

p(reit)
for all r > 0.

As p(z)→∞ as z →∞, the right-hand member tends to 0 when r →∞ uniformly
in t ∈ [0, 2π], leading to a contradiction by going to the limit under the integral
sign in the formula above.

If n ≥ 1 is an integer, and c : DR → C, we say that z ∈ DR is a n-branch point
of c if zn = c(z). A 1-branch point of c is a fixed point of c. We have the following
n-branch point theorem.

Corollary 5.3. If the continuous function c : DR → C is holomorphic in DR and
if there exists an integer n ≥ 1 such that c(∂DR) ⊆ DRn , then c has an n-branch
point in DR.



EJDE-2021/SI/01 MEAN VALUE PROPERTY 275

Proof. We apply Theorem 3.1 to f(z) = zn − c(z) and h(z) = zn and have, for all
z ∈ ∂DR,

<[zn(zn − c(z))] = R2n −<[znc(z)] ≥ R2n −Rn|c(z)| ≥ 0.

�

Remark 5.4. The fundamental theorem of algebra is also a consequence of the
n-branch point theorem applied to c(z) = −

∑n−1
k=0 akz

k. Indeed,

|c(z)| ≤
n−1∑
k=0

|ak|Rk ≤ Rn for all z ∈ ∂DR,

if R > 0 is taken so large that
∑n−1
k=0 |ak|Rk−n ≤ 1.

6. Li’s asymptotic condition for the existence of a zero for an
entire function

Recall that an entire function f : C→ C is a function which is holomorphic on
C. It is either a polynomial or a transcendental entire function, like exp z, sin z or
cosh z for example.

Theorem 3.1 provides a very simple proof of the sufficiency part of a simpler
equivalent statement of a recent result of Bao Qin Li [11].

Corollary 6.1. The entire function f : C→ C has a zero if and only if there exists
an entire function h : C→ C such that h(0) = 0 and limz→∞ h(z)/f(z) exists and
is not zero.

Proof. Sufficiency. Let h : C → C be an entire function such that h(0) = 0 and

limz→∞
h(z)
f(z) = b+ ic 6= 0. As

lim
r→+∞

<
[h(reit)

f(reit)

]
= lim
r→∞

<[h(reit)f(reit)]

|f(reit)|2
= b,

lim
r→+∞

=
[h(reit)

f(reit)

]
= − lim

r→∞

=[h(r(eit)f(reit)]

|f(reit)|2
= −c,

uniformly in t ∈ [0, 2π], and b2+c2 6= 0, there existsR > 0 such that <[h(Reit)f(Reit)]

or =[h(Reit)f(Reit)] keeps a constant sign for all t ∈ [0, 2π]. Using Theorem 3.1, f
has a zero in DR.

Necessity. Let a ∈ C such that f(a) = 0. The function h̃ : C→ C defined by

h̃(z) =
z + a

z
f(z + a) if z 6= 0, h(0) = af ′(a)

is entire (see e.g.’ [14, p. 212] or see Corollary 7.4 in the Appendix). Hence the
function h : C→ C defined by

h(z) = h̃(z − a) =
z

z − a
f(z) if z 6= a, h(a) = af ′(a),

is entire and such that

lim
z→∞

h(z)

f(z)
= lim
z→∞

z

z − a
= 1.

�



276 JEAN MAWHIN EJDE/SI/01

Remark 6.2. As shown in [11], the fundamental theorem of algebra is also a
consequence of Corollary 6.1 with the choice of h(z) = zn.

7. Appendix: an elementary proof for the removability of an
apparent singularity

In this appendix, for the reader not familiar with the techniques of the theory
of complex functions, we give an elementary proof of the holomorphic continuation
property used only in the necessity part of Corollary 6.1. We start with an easy
consequence of Lemma 2.1.

Corollary 7.1. If the continuous function g : DR → C is holomorphic in DR \{0}
then, for each z ∈ DR,∫ 2π

0

[g(reit)− g(z)]
reit

reit − z
dt = 0 for all r ∈ [0, R].

Proof. We define the function h : DR → C by

h(u) = [g(u)− g(z)]
u

u− z
if u 6= z, h(z) = g′(z)z.

Clearly h is continuous on DR, holomorphic in DR \ {0, z}, and h(0) = 0. Lemma
2.1 applied to h gives the result. �

Lemma 7.2. Given z ∈ C, one has, for each r > |z|,∫ 2π

0

reit

reit − z
dt = 2π.

Proof. The statement of this lemma is equivalent to∫ 2π

0

z

reit − z
dt = 0.

If we define J : (|z|,+∞)→ C by

J(r) =

∫ 2π

0

z

reit − z
dt

then we have
∂

∂r

( z

reit − z

)
= − z

(reit − z)2
=

1

ri

∂

∂t

( z

reit − z

)
,

and, using the easily justified Leibniz rule, the fundamental theorem of calculus
and the 2π-periodicity of the integrated function,

J ′(r) =

∫ 2π

0

∂

∂r

( z

reit − z

)
dt =

1

ri

∫ 2π

0

∂

∂t

( z

reit − z

)
dt = 0.

Hence, J(r) is constant on (|z|,+∞) and

J(r) = lim
r→+∞

I(r) =

∫ 2π

0

lim
r→+∞

( z

reit − z

)
dt = 0.

�

We now prove a generalized version of the mean value property due to Cauchy
(Cauchy’s integral formula on a disc) [3], expressing, for each z ∈ DR, g(z) as the
average of its values on ∂DR with respect to the complex measure [Reit/(Reit −
z)] dt. It reduces to (2.1) for z = 0.



EJDE-2021/SI/01 MEAN VALUE PROPERTY 277

Lemma 7.3. If g ∈ C(DR) ∩H(DR \ {0}), then, for each z ∈ DR one has

g(z) =
1

2π

∫ 2π

0

g(Reit)
Reit

Reit − z
dt.

Proof. By Corollary 7.1, we have

g(z)

∫ 2π

0

Reit

Reit − z
dt =

∫ 2π

0

g(Reit)
Reit

Reit − z
dt,

and the result follows from Lemma 7.2. �

Finally, we prove that a continuous function in DR which is holomorphic in
DR \ {0} is holomorphic in DR.

Corollary 7.4. If the continuous function g : DR → C is holomorphic in DR \{0},
then g is holomorphic in DR.

Proof. From Lemma 7.3 and the Leibniz rule, we have, for all z ∈ DR,

g′(z) =
1

2π

∫ 2π

0

g(Reit)
Reit

(Reit − z)2
dt

and, for z ∈ DR \ {0},

g(z)− g(0) =
1

2π

∫ 2π

0

g(Reit)
( Reit

Reit − z
− 1
)
dt

=
1

2π

∫ 2π

0

g(Reit)
( z

Reit − z

)
dt,

so that

g′(0) = lim
z→0

g(z)− g(0)

z

=
1

2π

∫ 2π

0

g(Reit) lim
z→0

( 1

Reit − z

)
dt

=
1

2πR

∫ 2π

0

g(Reit)e−it dt

=
1

2π

∫ 2π

0

g(Reit) lim
z→0

Reit

(Reit − z)2
dt

= lim
z→0

g′(z).

Hence, g′ exists and is continuous in DR. �

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Jean Mawhin
Institut de Recherche en Mathématique et Physique, Université Catholique de Louvain,

Chemin du Cyclotron, 2, 1348, Louvain-la-Neuve, Belgium

Email address: jean.mawhin@uclouvain.be


	1. Introduction
	2. An elementary proof of the mean value property (or Gauss mean value theorem)
	3. A geometric condition for the existence of a zero of a holomorphic function
	4. Hadamard-Shi's existence theorem and Brouwer's fixed point theorem for a holomorphic function
	5. The fundamental theorem of algebra
	6. Li's asymptotic condition for the existence of a zero for an entire function
	7. Appendix: an elementary proof for the removability of an apparent singularity
	References