EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 3, 2024, 1490-1496
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Riesz Inequality for Harmonic Quasiregular Mappings

Elver Bajrami

Department of Mathematics, University of Prishtina, Mother Teresa, No. 5, 10000,
Prishtina, Kosovo

Abstract. In this paper, we generalize the Riesz theorem for harmonic quasiregular mappings for
a special case (when p = 2) in the unit disc. Our results improve similar results in this field and
are proved with milder conditions. Moreover, we prove another variant forms of Riesz inequality
for harmonic quasiregular functions.

2020 Mathematics Subject Classifications: 30H10, 30H05

Key Words and Phrases: Harmonic mappings, Quasiregular mappings, Riesz theorem

1. Introduction

Let U = {z ∈ C : |z| < 1} be the unit disk and let T = {z ∈ C : |z| = 1} be the unit
circle in plane. For p > 1 we define the Hardy class hp as the class of harmonic mappings
f = g + h, where h and g are holomorphic mappings defined on unit disk U ⊂ C. Norm
in this space is defined

||f ||p = ||f ||hp = sup
0<r<1

Mp(f, r) < ∞,

where

Mp(f, r) =

(∫
T
|f(rζ)|pdσ(ζ)

)1/p

.

Here σ is probability measure on T. With Hp (Hp), we denote the subclass of holo-
morphic (quasiregular) mappings that belongs to the class hp. For the theory of Hardy
spaces in the unit disk we refer to [12], [4], [5] and [6].

For a given real-valued function u harmonic in U, let v be its harmonic conjugate,
normalized by v(0) = 0. Then f = u + iv is analytic in U. The following theorem was
proved by M. Riesz.

DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5281

Email address: elver.bajrami@uni-pr.edu (E. Bajrami)

https://www.ejpam.com 1490 © 2024 EJPAM All rights reserved.



E. Bajrami / Eur. J. Pure Appl. Math, 17 (3) (2024), 1490-1496 1491

Theorem A . ([4, Theorem 4.1] ) If u ∈ hp for some p, 1 < p < ∞, then its harmonic
conjugate v is also of class hp. Furthermore, there is a constant Ap, depending only on p,
such that

Mp(r, v) ≤ ApMp(r, u), 0 ≤ r < 1,

for all u ∈ hp.

Problem of finding the sharp constant Ap is very old and has been calculated for
several classes of functions. Pichorides in [10], prove that this constant is Ap = cot π

2p ,

where p = max
{
p, p

p−1

}
.

Later, Verbitsky improved the above inequality in [13] with the following sharp result:
if f = u+ iv ∈ Hp and v(0) = 0, then

1

cos π
2p

||v||p ≤ ||f ||p ≤
1

sin π
2p

||u||p.

Some other results for this constant are obtained by Kalaj in [9], for harmonic functions
with constant

Ap =

(
1− | cos π

2p
|
)− 1

2

.

Also, this inequality was generalized in several directions. Let us mention Beckenbach’s
results: the same inequality holds where in place of |f |p we have a positive logarithmically
subharmonic function. This kind of generalizations can be found in [1]. We refer interested
readers to [2], [3] and [9].

Our aim in this paper is to find similar sharp constant but for specific class of function
- quasiregular mapping. In case of planar harmonic K-quasiregular mappings it is defined
as below.

Given K ≥ 0, a sense-preserving harmonic function f = h + g in U is said to be
K-quasiregular if and only if

||µ||∞ = sup
|g′(z)|
|h′(z)|

≤ k < 1,

where k = K−1
K+1 .

Another definition of quasiregular mappings in domain of Rn is given in [14] as follows:
Let A ⊂ Rn be a domain, and let n ≥ 2. A mapping f : A → Rn is said to be

quasiregular, if satisfy next two conditions:

(a) f is an absolutely continuous functions in every line segment parallel to the coordi-
nate axis and there exists partial derivatives which are locally Ln integrable functions
on A.

(b) there exists a constant K ≥ 1 such that, a.e. in A,

Lf (x)
n ≤ KJf (x),



E. Bajrami / Eur. J. Pure Appl. Math, 17 (3) (2024), 1490-1496 1492

where Lf (x) is the maximum stretching for f at the point x, i.e.,

Lf (x) = lim
y→x

sup
|f(y)− f(x)|

|y − x|
,

and Jf denotes the Jacobian determinant.

The smallest constant K ≤ 1 in above definition is called the outer dilatation of f
and denote by KO(f). Also for quasiregular function f , the smallest constant K ≤ 1, for
which the inequality

Jf (x) ≤ Klf (x)
n,

where lf (x) = min{|f ′(x)h| : |h| = 1}, holds (in A), is called the inner dilation of f and
denoted by KI(f). The maximal dilation of f is the number K(f) = max{KI(f),KO(f)}.
If K(f) ≤ K, then f is said to be K-quasiregular. If f is not quasiregular, we set
K(f) = KI(f) = KO(f) = ∞.

For a function f ∈ U, the norm in class Hp is provided by:

||f ||p = ||f ||Hp = sup
0<r<1

Mp(r, f).

For a more detailed observation of quasiregular mappings we refer to [8], [11] and [14].
Similar results about Riesz theorem in class of quasiregular functions, also are proved

by J. Li and J. Zhu, in [7], but there are used additional conditions about functions u and
v. They generalize this theorem for planar harmonic K-quasiregular mappings (1 < p ≤ 2)
provided that the real part does not vanish at the unit disk. Moreover, they extended this
results for invariant harmonic quasiconformal mappings in the unit ball assuming that the
first coordinate in non-vanishing.

In this paper, using different methods, we find similar constant as in [7], but without
restriction about conditions of functions u and v. These results are given with next
theorem.

Theorem 1. If f = u+ iv is harmonic K-quasiregular, with ℜf(0) = 0, then

∥u∥2 ≤ K∥v∥2,

and the constant K is sharp.

The following results easily follows from Theorem 1.

Corollary 1. Let f = h + g be K-quasiregular in U. If u ∈ H2, then its harmonic
conjugate v is also of class H2.

Based on Theorem 1 and Corollary 1, we can prove next theorem.

Theorem 2. Let f(z) = h(z)+g(z) =
∑∞

k=0 akz
k+

∑∞
k=1 bkz

k be harmonic K-quasiregular,
with g(0) = 0 . If f ∈ H2, then following sharp inequality holds

||f ||2 ≤ ck||h||2,

where ck = 1 + k2.



E. Bajrami / Eur. J. Pure Appl. Math, 17 (3) (2024), 1490-1496 1493

Let we say that this results coincide with results in Theorem 2 on [7], in case of n = 2.
Next theorem generalize previous theorem.

Theorem 3. Let f(z) = h(z)+g(z) =
∑∞

k=0 akz
k+

∑∞
k=1 bkz

k be harmonic K-quasiregular,
with g(0) = 0 . If f ∈ Hn and n > 2, then following inequality holds

||f ||n ≤ c(k, n)||h||n,

where c(k, n) = (2(1 + k2))n/2.

2. Proof of main results

Proof. [Proof of Theorem 1] Let

f = h+ ḡ =
∞∑
j=0

ajz
j +

∞∑
j=1

bj z̄
j .

We can assume that both of following integrals converge. If no, then we take the
dilatation f(rz) for r < 1.

If we integrate in the unit circle∫
T
|g′(z)|2|dz| ≤ k2

∫
U
|h′(z)|2|dz|

we get ∑
j=1

j2|bj |2 ≤ k2
∑
j=1

j2|aj |2.

Here k = K−1
K+1 .

Let u = ℜ(g + h) and v = ℜ(i(h− g)). Then

4
1

π

∫
U
|u|2dxdy = 4(ℜa0)2 +

∞∑
j=1

(|aj |2 + |bj |2 + 2ℜ(akbj))

and

4
1

π

∫
U
|v|2dxdy = 4(ℑa0)2 +

∞∑
j=1

(|aj |2 + |bj |2 − 2ℜ(akbj)).

Without loss of generality we assume that a0 = 0, because ℜ(a0) = 0 by assumption. We
will find the best constant ck in the inequality

1

π

∫
U
|u|2dxdy =

∞∑
j=0

(|aj |2 + |bj |2 + 2|aj ||bj |) ≤ ck
1

π

∫
U
|u|2dxdy

=

∞∑
j=0

(|aj |2 + |bj |2 − 2|aj ||bj |)



E. Bajrami / Eur. J. Pure Appl. Math, 17 (3) (2024), 1490-1496 1494

under the condition ∑
j=1

j2|bj |2 ≤ k2
∑
j=1

j2|aj |2.

Let

W (a, b) =

∑∞
j=0(|aj |2 + |bj |2 + 2|aj ||bj |)∑∞
j=0(|aj |2 + |bj |2 − 2|aj ||bj |)

.

We need to find the maximum of expression W under the condition

H(a, b) =
∑
j=1

j2|bj |2 − k2
∑
j=1

j2|aj |2 = 0.

It is equivalent of finding the maximum of expression M = U
V , where

U =

∞∑
j=0

|aj |2, V =

∞∑
j=0

|bj |2

under the condition H(a, b) = 0. The Lagrangian is L = M − λH. Assume without loss
of generality that aj ≥ 0 and bj ≥ 0. Also we have a0 = 0. Then Laj = 0 and Lbj = 0
imply that

bj
U

= λj2bj ,
ajV

U2
= k2λj2aj , j ≥ 1.

λ cannot be zero, because in that case a = 0 and b = 0. If λ ̸= 0, then there exists
j0 so that aj0 ̸= 0 and bj0 ̸= 0, aj = bj = 0 for j ̸= j0 and 1

U = k2V
U2 . In this case we get

M = k2. This implies that W ≤ 1+k2+2k
1+k2−2k

= K2.

In order to prove Theorem 2, we will need next result.

Lemma 1. Let f be analytic function with condition f(0) = 0. Then∫
U
ℜ(f(z))|dz| = 0.

Proof. Since f(0) = 0, this function can be expressed as f(z) =
∑∞

k=1 akz
k. Integrating

last expression in unit circle, we get∫
U
f(z)|dz| =

∞∑
k=1

ak

∫ 1

0
rkdr

∫ 2π

0
eiktdt.

Since the integral
∫ 2π
0 eiktdt = 0, for each k ∈ N, we get∫

U
f(z)|dz| = 0.

Similarly
∫
U f(z)|dz| = 0. Now, identity ℜ(f(z)) = f(z)+f(z)

2 , follows that
∫
Uℜ(f(z))|dz| =

0.



REFERENCES 1495

Proof. [Proof of Theorem 2] Let f(z) = h(z) + g(z) =
∑∞

k=0 akz
k +

∑∞
k=1 bkz

k, then∫
U
|f(z)|2|dz| =

∫
U
|h(z)|2|dz|+

∫
U
|g(z)|2|dz|+ 2

∫
U
ℜ(h(z)g(z))|dz|.

As in the proof of Theorem 1, using Cauchy-Schwartz inequality, Lemma 1, we get∫
U
|f(z)|2|dz| ≤ (1 + k2)

∫
T
|h(z)|2|dz|+ 2

∫
T
ℜ(h(z)g(z))|dz|

= ck

∫
T
|h(z)|2|dz|.

Which give required results.

Proof. [Proof of Theorem 3] Let f(z) = h(z) + g(z) =
∑∞

k=0 akz
k +

∑∞
k=1 bkz

k be
harmonic K-quasiregular, with g(0) = 0. We have∫

U
|f(z)|n =

∫
U
(|g + h|2)

n
2 =

∫
U
(|g|2 + |h|2 + 2ℜ(gh))

n
2 .

Using Theorem 2 and inequality ℜ(gh) ≤ |hg| ≤ 1
2(|h|

2 + |g|2), we get∫
U
|f(z)|n ≤

∫
U
(2(|h|2 + |g|2))

n
2 ≤ (2(1 + k2))

n
2

∫
U
|h|n.

3. Conclusion

In this paper we generalize the Riesz theorem for harmonic quasiregular mappings for
a special case in the unit disc. This result is given thought Theorem 1. In order to proving
this result we use Lagrange multipliers. Probably this method can be used to prove also
for other cases, to make a generalization of Riesz theorem. Moreover, thought Theorem
2 and 3, we prove another variant forms of Riesz inequality for harmonic quasiregular
functions.

References

[1] E. F. Beckenbach. On a theorem of Fejér and Riesz. J. London Math. Soc., 13:82–86,
1938.

[2] M. Stein C. Efferman. Hp spaces of several variables. Acta Math., 129:137–193, 1972.

[3] T. Tao S. Wainger C. Efferman, A. Lonescu. Analysis and its applications: the
mathematical work of elias stein. Bull. Amer. Math. Soc. New Ser., 57:523–594,
2020.

[4] P. L. Duren. Theory of Hp spaces. Pure and Applied Mathematics, 38:xii+25, 1970.



REFERENCES 1496

[5] P. L. Duren. Harmonic mappings in the plane. Cambridge Univ. Press, New York,
2004.

[6] J. B. Garnett. Bounded Analytic Functions. Springer, New York, 2007.

[7] J. Zhu J. Liu. Riesz conjugate functions theorem for harmonic quasiconformal map-
pings. Advances in Mathematics, 434:109321, 2023.

[8] P. Koskela K. Astala. Hp theory for quasiconformal mappings. Pure Appl. Math.,
1:19–50, 2011.

[9] D. Kalaj. On Riesz type inequalities for harmonic mappings in the unit disk. Trans.
Amer. Math. Soc., 372(6):4031–4051, 2019.

[10] S. K. Pichorides. On the best value of the constants in the theorem of M. Riesz,
Zygmund and Kolmogorov. Stud. Math., 44:165–179, 1972.

[11] S. Rickman. Quasiregular Mappings. Springer, Berlin, 1993.

[12] W. Ramey S. Axler, P. Bourdon. Harmonic function theory. Springer Verlag, New
York, 1992.

[13] I. E. Verbitsky. Esimate of the norm of a function in hardy space in terms of a norms
of its real and imaginary parts. Linear operators. Mat. Issled., 54:16–20, 164–165,
1980.

[14] M. Vuorinen. Conformal geometry and quasiregular mappings. Springer, Berlin,
Heidelberg,, 1988.