Comptes Rendus Mathématique

Hüseyin Bor

Comptes Rendus
Mathématique
Hüseyin Bor
A new note on factored infinite series and trigonometric Fourier series
Volume 359, issue 3 (2021), p. 323-328.
<https://doi.org/10.5802/crmath.179>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2021, 359, n 3, p. 323-328
https://doi.org/10.5802/crmath.179
Harmonic analysis / Analyse harmonique
A new note on factored infinite series and
trigonometric Fourier series
Hüseyin Bora
a P. O. Box 121, TR-06502 Bahçelievler, Ankara, Turkey
E-mail: hbor33@gmail.com
Abstract. In this paper, we have proved two main theorems under more weaker conditions dealing with
absolute weighted arithmetic mean summability factors of infinite series and trigonometric Fourier series.
We have also obtained certain new results on the different absolute summability methods.
2020 Mathematics Subject Classification. 26D15, 40D15, 42A24, 46A45.
Manuscript received 14th November 2020, accepted 13th January 2021.
1. Introduction
Let
∑
an be a given infinite series with the partial sums (sn). By tαn we denote the nth Cesàro mean
of order α, with α>−1, of the sequence (nan), that is (see [21])
tαn =
1
Aαn
n∑
v=0
Aα−1n−v vav , (tn
1 = tn) (1)
where
Aαn =
(α+1)(α+2) . . . (α+n)
n!
=O(nα), Aα−n = 0 f or n > 0. (2)
The series
∑
an is said to be summable |C ,α|k ,k ≥ 1, if (see [23])
∞∑
n=1
1
n
∣∣tαn ∣∣k <∞. (3)
If we take α= 1, then |C ,α|k summability reduces to |C ,1|k summability. Let (pn) be a sequence
of positive real numbers such that
Pn =
n∑
v=0
pv →∞ as n →∞,
(
P−i = p−i = 0, i ≥ 1
)
. (4)
The sequence-to-sequence transformation (sn) → (vn) with
vn = 1
Pn
n∑
v=0
pv sv (5)
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
324 Hüseyin Bor
defines the sequence (vn) of weighted arithmetic mean or simply the (N , pn) mean of the
sequence (sn), generated by the sequence of coefficients (pn) (see [24]). If we write Xn =∑nv=0 pvPv ,
then (Xn) is a positive increasing sequence tending to infinity as n →∞. The series ∑an is said
to be summable |N , pn |k , k ≥ 1, if (see [3])
∞∑
n=1
(
Pn
pn
)k−1
|vn − vn−1|k <∞.
In the special case when pn = 1 for all n (resp. k = 1), |N , pn |k summability is the same as |C ,1|k
(resp. |N , pn | (see [35]) summability. Also if we take pn = 1n+1 and k = 1, then we obtain |R, logn,1|
summability (see [1]).
For any sequence (λn) we write that ∆2λn =∆λn −∆λn+1 and ∆λn =λn −λn+1. The sequence
(λn) is said to be of bounded variation, denoted by (λn) ∈BV , if ∑∞n=1 |∆λn | <∞.
2. The known result
Many works dealing with the absolute summability factors of infinite series and Fourier series
have been done (see [2,4–20,25–34,36–40]). Among them, in [12], the following theorem has been
proved.
Theorem 1. Let (Xn) be a positive increasing sequence and let (pn) be a sequence of positive
numbers such that
Pn =O(npn) as n →∞. (6)
If the conditions
λm = o(1) as m →∞, (7)
m∑
n=1
nXn |∆2λn | =O(1) as m →∞ (8)
m∑
n=1
pn
Pn
|tn |k
Xn k−1
=O(Xm) as m →∞, (9)
hold, then the series
∑
anλn is summable |N , pn |k , k ≥ 1.
Remark. It should be noted that, in Theorem 1, there is a restriction on the sequence (pn).
Therefore, due to restriction (6) on (pn) no result for pn = 1n+1 can be deduced from Theorem 1.
3. The main result
The aim of this paper is to obtain a further generalization of Theorem 1 under weaker conditions.
In this case, there is not any restriction on the sequence (pn). It is clear that (6) and (9) imply the
following formula
m∑
n=1
|tn |k
nXn k−1
=O(Xm) as m →∞. (10)
Also(6) implies that
m∑
n=1
Pn
n
=O(Pm) as m →∞. (11)
It should be remarked that (6) implies (11) but the converse needs not be true (see [27]).
Now we shall prove the following general theorem.
Theorem 2. If the sequences (Xn), (λn), and (pn) satisfy the conditions (7)–(11), then the series∑
anλn is summable |N , pn |k , k ≥ 1.
C. R. Mathématique — 2021, 359, n 3, 323-328
Hüseyin Bor 325
We need the following lemma for the proof of Theorem 2.
Lemma 3 (cf. [6]). Under the conditions of Theorem 2, we get
nXn |∆λn | =O(1), as n →∞ (12)
∞∑
n=1
Xn |∆λn | <∞ (13)
Xn |λn | =O(1) as n →∞. (14)
4. Proof of Theorem 2
Let (Tn) be the sequence of (N , pn) mean of the series
∑
anλn . Then, by definition, we have
Tn = 1
Pn
n∑
v=0
pv
v∑
r=0
arλr = 1
Pn
n∑
v=0
(Pn −Pv−1)avλv . (15)
Then, for n ≥ 1, we get
Tn −Tn−1 = pn
PnPn−1
n∑
v=1
Pv−1λv
v
vav . (16)
Applying Abel’s transformation to the right-hand side of (16), we have
Tn −Tn−1 = pn
PnPn−1
n−1∑
v=1
∆
(
Pv−1λv
v
) v∑
r=1
r ar + pnλn
nPn
n∑
v=1
vav
= (n +1)pn tnλn
nPn
− pn
PnPn−1
n−1∑
v=1
pv tvλv
v +1
v
+ pn
PnPn−1
n−1∑
v=1
Pv∆λv tv
v +1
v
+ pn
PnPn−1
n−1∑
v=1
Pvλv+1tv
1
v
= Tn,1 +Tn,2 +Tn,3 +Tn,4.
To complete the proof of Theorem 2, by Minkowski’s inequality, it is sufficient to show that
∞∑
n=1
(
Pn
pn
)k−1 ∣∣Tn,r ∣∣k <∞, for r = 1,2,3,4.
Firstly, we have that
m∑
n=1
(
Pn
pn
)k−1
|Tn,1|k =O(1)
m∑
n=1
|λn |k−1|λn |pn
Pn
|tn |k =O(1)
m∑
n=1
|λn |pn
Pn
|tn |k
Xn k−1
=O(1)
m−1∑
n=1
∆|λn |
n∑
v=1
pv
Pv
|tv |k
Xv k−1
+O(1)|λm |
m∑
n=1
pn
Pn
|tn |k
Xn k−1
=O(1)
m−1∑
n=1
|∆λn |Xn +O(1)|λm |Xm =O(1) as m →∞,
by the hypotheses of Theorem 2 and Lemma 3. Also, as in Tn,1, we have that
m+1∑
n=2
(
Pn
pn
)k−1
|Tn,2|k =O(1)
m+1∑
n=2
pn
PnPn−1
(
n−1∑
v=1
pv |tv |k |λv |k
)
×
(
1
Pn−1
n−1∑
v=1
pv
)k−1
=O(1)
m∑
v=1
|λv |k−1|λv |pv |tv |k
m+1∑
n=v+1
pn
PnPn−1
=O(1)
m∑
v=1
|λv |pv
Pv
|tv |k
Xv k−1
=O(1) as m →∞.
C. R. Mathématique — 2021, 359, n 3, 323-328
326 Hüseyin Bor
Again, by using (11), we obtain that
m+1∑
n=2
(
Pn
pn
)k−1
|Tn,3|k =O(1)
m+1∑
n=2
pn
PnP kn−1
{
n−1∑
v=1
Pv |∆λv ||tv |
}k
=O(1)
m+1∑
n=2
pn
PnP kn−1
(
n−1∑
v=1
Pv
v
v |∆λv ||tv |
)k
=O(1)
m+1∑
n=2
pn
PnPn−1
(
n−1∑
v=1
Pv
v
(v |∆λv |)k |tv |k
)
×
(
1
Pn−1
n−1∑
v=1
Pv
v
)k−1
=O(1)
m∑
v=1
Pv
v
(v |∆λv |)k−1v |∆λv ||tv |k
m+1∑
n=v+1
pn
PnPn−1
=O(1)
m∑
v=1
v |∆λv | |tv |
k
v Xv k−1
=O(1)
m−1∑
v=1
∆ (v |∆λv |)
v∑
r=1
|tr |k
r Xr k−1
+O(1)m|∆λm |
m∑
v=1
|tv |k
v Xv k−1
=O(1)
m−1∑
v=1
|∆ (v |∆λv |)|Xv +O(1)m|∆λm |Xm
=O(1)
m−1∑
v=1
v Xv |∆2λv |+O(1)
m−1∑
v=1
Xv |∆λv |+O(1)m|∆λm |Xm
=O(1) as m →∞,
by the hypotheses of Theorem 2 and Lemma 3. Finally by using (11), as in Tn,1, we have that
m+1∑
n=2
(
Pn
pn
)k−1 ∣∣Tn,4∣∣k ≤ m+1∑
n=2
pn
PnP kn−1
(
n−1∑
v=1
Pv
v
|λv+1||tv |
)k
=O(1)
m+1∑
n=2
pn
PnPn−1
(
n−1∑
v=1
Pv
v
|λv+1|k |tv |k
)
×
(
1
Pn−1
n−1∑
v=1
Pv
v
)k−1
=O(1)
m∑
v=1
Pv
v
|λv+1|k−1|λv+1||tv |k
m+1∑
n=v+1
pn
PnPn−1
=O(1)
m∑
v=1
|λv+1| |tv |
k
v Xv k−1
=O(1) as m →∞.
This completes the proof of Theorem 2.
If we take pn = 1 for all n, then we obtain a new result dealing with |C ,1|k summability
factors of infinite series. Also if we set k = 1, then we obtain a new result concerning the |N , pn |
summability factors of infinite series. Finally, if we take pn = 1n+1 and k = 1, then we obtain a new
result for |R, logn,1| summability of factored infinite series.
5. An application to trigonometric Fourier series
Let f be a periodic function with period 2π and Lebesgue integrable over (−π,π). The trigono-
metric Fourier series of f is defined as
f (x) ∼ 1
2
a0 +
∞∑
n=1
(an cosnx +bn sinnx) =
∞∑
n=0
An(x),
where
a0 = 1
π
∫ π
−π
f (x)dx, an = 1
π
∫ π
−π
f (x)cos(nx)dx, and bn = 1
π
∫ π
−π
f (x)sin(nx)dx.
C. R. Mathématique — 2021, 359, n 3, 323-328
Hüseyin Bor 327
Write φ(t ) = 12 { f (x + t )+ f (x − t )}, and φα(t ) = αtα
∫ t
0 (t −u)α−1φ(u)du, (α> 0).
It is known that if φ1(t ) ∈ BV (0,π), then tn(x) = O(1), where tn(x) is the (C ,1) mean of the
sequence (n An(x)) (see [22]). Using this fact, we have obtained the following theorem dealing
with trigonometric Fourier series.
Theorem 4 (cf. [12]). If φ1(t ) ∈ BV (0,π), and the sequences (pn), (λn) and (Xn) satisfy the
conditions of the Theorem 1, then the series
∑
An(x)λn is summable |N , pn |k , k ≥ 1.
Now, we can generalize Theorem 4 under weaker conditions in the following form.
Theorem 5. If φ1(t ) ∈ BV (0,π), and the sequences (pn), (λn), and (Xn) satisfy the conditions of
Theorem 2, then the series
∑
An(x)λn is summable |N , pn |k , k ≥ 1.
In the special cases of (pn) and k as in Theorem 2, we can obtain similar results from
Theorem 5 for the trigonometric Fourier series.
References
[1] S. N. Bhatt, “An aspect of local property of |R, logn,1| summability of Fourier series”, Tôhoku Math. J. 11 (1959),
p. 13-19.
[2] H. Bor, “On |N , pn |k summability factors”, Proc. Am. Math. Soc. 94 (1985), p. 419-422.
[3] ——— , “On two summability methods”, Math. Proc. Camb. Philos. Soc. 97 (1985), p. 147-149.
[4] ——— , “On quasi-monotone sequences and their applications”, Bull. Aust. Math. Soc. 43 (1991), no. 2, p. 187-192.
[5] ——— , “On absolute weighted mean summability methods”, Bull. Lond. Math. Soc. 25 (1993), no. 3, p. 265-268.
[6] ——— , “On the absolute Riesz summability factors”, Rocky Mt. J. Math. 24 (1994), no. 4, p. 1263-1271.
[7] ——— , “On the local property of factored Fourier series”, Z. Anal. Anwend. 16 (1997), no. 3, p. 769-773.
[8] ——— , “A study on local properties of Fourier series”, Nonlinear Anal., Theory Methods Appl. 57 (2004), no. 2, p. 191-
197.
[9] ——— , “A note on local property of factored Fourier series”, Nonlinear Anal., Theory Methods Appl. 64 (2006), no. 3,
p. 513-517.
[10] ——— , “Some new results on infinite series and Fourier series”, Positivity 19 (2015), no. 3, p. 467-473.
[11] ——— , “On absolute weighted mean summability of infinite series and Fourier series”, Filomat 30 (2016), no. 10,
p. 2803-2807.
[12] ——— , “Some new results on absolute Riesz summability of infinite series and Fourier series”, Positivity 20 (2016),
no. 3, p. 599-605.
[13] ——— , “Absolute weighted arithmetic mean summability factors of infinite series and trigonometric Fourier series”,
Filomat 31 (2017), no. 15, p. 4963-4968.
[14] ——— , “An application of power increasing sequences to infinite series and Fourier series”, Filomat 31 (2017), no. 6,
p. 1543-1547.
[15] ——— , “An application of quasi-monotone sequences to infinite series and Fourier series”, Anal. Math. Phys. 8
(2018), no. 1, p. 77-83.
[16] ——— , “On absolute summability of factored infinite series and trigonometric Fourier series”, Results Math. 73
(2018), no. 3, article no. 116 (9 pages).
[17] ——— , “On absolute Riesz summability factors of infinite series and their application to Fourier series”, Georgian
Math. J. 26 (2019), no. 3, p. 361-366.
[18] ——— , “Certain new factor theorems for infinite series and trigonometric Fourier series”, Quaest. Math. 43 (2020),
no. 4, p. 441-448.
[19] H. Bor, B. Kuttner, “On the necessary conditions for absolute weighted arithmetic mean summability factors”, Acta
Math. Hung. 54 (1989), no. 1-5, p. 57-61.
[20] H. Bor, D. Yu, P. Zhou, “On local property of absolute summability of factored Fourier series”, Filomat 28 (2014), no. 8,
p. 1675-1686.
[21] E. Cesàro, “Sur la multiplication des séries”, Bull. Sci. Math. 14 (1890), p. 114-120.
[22] K.-K. Chen, “Functions of bounded variation and the Cesàro means of Fourier series”, Acad. Sinica Sci. Record 1
(1945), p. 283-289.
[23] T. M. Flett, “On an extension of absolute summability and some theorems of Littlewood and Paley”, Proc. Lond. Math.
Soc. 7 (1957), p. 113-141.
[24] G. H. Hardy, Divergent Series, Clarendon Press, 1949.
C. R. Mathématique — 2021, 359, n 3, 323-328
328 Hüseyin Bor
[25] J.-O. Lee, “On the summability of infinite series and Hüseyin Bor”, J. Hist. Math. 30 (2017), p. 353-365, in Korean,
English translation available at https://sites.google.com/site/hbor33/paper-of-prof-lee-english.
[26] L. Leindler, “A new application of quasi power increasing sequences”, Publ. Math. 58 (2001), no. 4, p. 791-796.
[27] S. M. Mazhar, “Absolute summability factors of infinite series”, Kyungpook Math. J. 39 (1999), no. 1, p. 67-73.
[28] H. S. Özarslan, “Local properties of generalized absolute matrix summability of factored Fourier series”, Southeast
Asian Bull. Math. 43 (2019), no. 2, p. 263-272.
[29] ——— , “A new factor theorem for absolute matrix summability”, Quaest. Math. 42 (2019), no. 6, p. 803-809.
[30] ——— , “On the localization of factored Fourier series”, J. Comput. Anal. Appl. 29 (2021), p. 344-354.
[31] H. S. Özarslan, Ş. Yıldız, “Local properties of absolute matrix summability of factored Fourier series”, Filomat 31
(2017), no. 15, p. 4897-4903.
[32] M. A. Sarıgöl, “On local properties of summability of Fourier series”, J. Comput. Anal. Appl. 12 (2010), no. 4, p. 817-
820.
[33] ——— , “On the local properties of factored Fourier series”, Appl. Math. Comput. 216 (2010), no. 11, p. 3386-3390.
[34] M. A. Sarıgöl, H. Bor, “Characterization of absolute summability factors”, J. Math. Anal. Appl. 195 (1995), no. 2, p. 537-
545.
[35] G. Sunouchi, “Notes on Fourier analysis. XVIII. Absolute summability of series with constant terms”, Tôhoku Math.
J. 1 (1949), p. 57-65.
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670.
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no. 14, p. 4343-4351.
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C. R. Mathématique — 2021, 359, n 3, 323-328


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Arnaud Beauville

Comptes Rendus
Mathématique
Arnaud Beauville
A non-hyperelliptic curve with torsion Ceresa class
Volume 359, issue 7 (2021), p. 871-872
<https://doi.org/10.5802/crmath.226>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2021, 359, n 7, p. 871-872
https://doi.org/10.5802/crmath.226
Algebraic geometry / Geometrie algébrique
A non-hyperelliptic curve with torsion Ceresa
class
Arnaud Beauville∗, a
a Université Côte d’Azur, CNRS – Laboratoire J.-A. Dieudonné, Parc Valrose, F-06108
Nice cedex 2, France.
E-mail: arnaud.beauville@unice.fr
Abstract. We exhibit a non-hyperelliptic curve C of genus 3 such that the class of the Ceresa cycle [C ]− [−C ]
in the intermediate Jacobian of JC is torsion.
Manuscript received 21st May 2021, accepted 24th May 2021.
1. Introduction
Let C be a complex curve of genus g ≥ 3, and p a point of C . We embed C into its Jacobian J
by the Abel–Jacobi map x 7→ [x]− [p]. The Ceresa cycle zp (C ) is the cycle [C ]− [(−1J )∗C ] in the
Chow group C H1(J )hom of homologically trivial 1-cycles. The Ceresa class cp (C ) is the image of
zp (C ) in the intermediate Jacobian J1(J ) parameterizing 1-cycles under the Abel–Jacobi map
C H1(J )hom → J1(J ).
When C is general, zp (C ) is not algebraically trivial [2]. On the other hand, if C is hyperelliptic
zp (C ) is algebraically trivial – in fact it is zero if one chooses for p a Weierstrass point. Not much
is known besides these two extreme cases. There are few curves for which zp (C ) is known to be
not algebraically trivial: Fermat curves of degree ≤ 1000 [4], and the Klein quartic [5]. An essential
ingredient of these results is the fact that cp (C ) is not a torsion class.
It is an open question whether there are non-hyperelliptic curves with zp (C ) algebraically
trivial. As observed in [3, Remark 2.4], this condition is equivalent to a number of interesting
properties: in particular the existence of a multiplicative Chow–Künneth decomposition modulo
algebraic equivalence, or the fact that the class [C ] ∈C H1(J )⊗Q is algebraically equivalent to the
minimal class θ
g−1
(g−1)! , where θ ∈C H 1(J ) is the class of the principal polarization.
In this note we exhibit a curve C of genus 3 with the weaker property that the Ceresa class cp (C )
is torsion (under the Bloch–Beilinson conjectures, this actually implies the algebraic triviality of
zp (C ) up to torsion). The construction is very simple: the curve C has an automorphism σ which
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
872 Arnaud Beauville
fixes a point p, and therefore preserves cp (C ); we just have to check that the fixed point set of σ
acting on J1(J ) is finite.
A similar example, based on a much more sophisticated approach, appears in [1, Remark 3.6].
2. The result
Proposition 1. Let C ⊂P2 be the genus 3 curve defined by X 4+X Z 3+Y 3Z = 0, and let p = (0,0,1).
The Ceresa class cp (C ) is torsion.
Proof. Let ω be a primitive 9th root of unity. We consider the automorphism σ of C defined by
σ(X ,Y , Z ) = (X ,ω2Y ,ω3Z ). We have σ(p) = p; therefore σ preserves the Ceresa cycle zp (C ), and
also its class cp (C ) in J := J1(J ).
Thus it suffices to prove that σ has finitely many fixed points on J; equivalently, that the
eigenvalues of σ acting on the tangent space T0(J) are 6= 1.
Now T0(J) is identified with H 0,3(J ) ⊕ H 1,2(J ) = ∧3 V ∗ ⊕ (∧2 V ∗ ⊗ V ), where V = H 1,0(J )
= H 0(C ,KC ). We first compute the eigenvalues of σ on V . The elements of V are of the form
L · X d Z−Z d X
Y 2 Z
, with L ∈ H 0(P2,OP(1)); it follows that the eigenvalues of σ on V are ω5,ω7,ω8.
Therefore the eigenvalue on
∧3 V ∗ is ω7, and the eigenvalues on ∧2 V ∗ are ω3,ω5,ω6. Thus each
product of an eigenvalue on
∧2 V ∗ and one on V is 6= 1, hence the Proposition. 
References
[1] D. Bisogno, W. Li, D. Litt, P. Srinivasan, “Group-theoretic Johnson classes and non-hyperelliptic curves with torsion
Ceresa class”, https://arxiv.org/abs/2004.06146, 2020.
[2] G. Ceresa, “C is not algebraically equivalent to C− in its Jacobian”, Ann. Math. 117 (1983), no. 2, p. 285-291.
[3] L. Fu, R. Laterveer, C. Vial, “Multiplicative Chow–Künneth decompositions and varieties of cohomological K3 type”,
Ann. Mat. Pura Appl. (4) 200 (2021), no. 5, p. 2085-2126.
[4] N. Otsubo, “On the Abel–Jacobi maps of Fermat Jacobians”, Math. Z. 270 (2012), no. 1-2, p. 423-444.
[5] Y. Tadokoro, “A nontrivial algebraic cycle in the Jacobian variety of the Klein quartic”, Math. Z. 260 (2008), no. 2, p. 265-
275.
C. R. Mathématique — 2021, 359, n 7, 871-872


A non-hyperelliptic curve with torsion Ceresa class

Nikolay Kuznetsov

Comptes Rendus
Mathématique
Nikolay Kuznetsov
Characterization of balls via solutions of the modified Helmholtz
equation
Volume 359, issue 8 (2021), p. 945-948
<https://doi.org/10.5802/crmath.250>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2021, 359, n 8, p. 945-948
https://doi.org/10.5802/crmath.250
Partial differential equations / Équations aux dérivées partielles
Characterization of balls via solutions of the
modified Helmholtz equation
Nikolay Kuznetsova
a Laboratory for Mathematical Modelling of Wave Phenomena, Institute for Problems
in Mechanical Engineering, Russian Academy of Sciences, V.O., Bol’shoy pr. 61,
St Petersburg 199178, Russian Federation
E-mail: nikolay.g.kuznetsov@gmail.com
Abstract. A theorem characterizing analytically balls in the Euclidean space Rm is proved. For this purpose
positive solutions of the modified Helmholtz equation are used instead of harmonic functions applied in
previous results. The obtained Kuran type theorem is based on the volume mean value property of solutions
to this equation.
Manuscript received 15th June 2021, revised 19th July 2021, accepted 26th July 2021.
1. Introduction and main result
In 1972 Kuran [5] proved the following inverse of the volume mean value theorem for harmonic
functions:
Let D be a domain (= connected open set) of finite (Lebesgue) measure in the
Euclidean space Rm where m ≥ 2. Suppose that there exists a point P0 in D such
that, for every function h harmonic in D and integrable over D, the volume mean
of h over D equals h(P0). Then D is an open ball (disk when m = 2) centred at P0.
The result was originally obtained by Epstein [2] for a simply connected two-dimensional D .
Armitage and Goldstein [1] proved this result assuming that the mean value equality holds only
for positive harmonic functions which are Lp -integrable, p ∈ (0,n/(n−2)). Hansen and Netuka [3]
considered some particular class of potentials as the set of test harmonic functions in Kuran’s
theorem. A slight modification of his considerations shows that Kuran’s theorem is valid even if D
is disconnected; see [8, p. 377].
In the survey article [8], one finds also a discussion of applications of Kuran’s theorem and
a possibility of similar results involving some kinds of average over ∂D , where D is a bounded
domain. One of them (due to Kosmodem’yanskii [4]) is based on the relation similar to that
between the mean values over balls and spheres and reads as follows:
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
946 Nikolay Kuznetsov
Let D ⊂R2 be a bounded, convex C 2-domain. If the equality
1
|D|
∫
D
u(x)dx = 1|∂D|
∫
∂D
u(x)dSx
holds for every function u ∈ C 2(D)∩C 1(D) which is harmonic in D, then D is an
open disc.
Here and below |D| is the domain’s area (volume if D ⊂Rm , m ≥ 3), whereas |∂D| is the boundary’s
length (area if D ⊂ Rm , m ≥ 3), and |Br | = ωmr m is the volume of a ball Br of radius r ; here
ωm = 2πm/2/[mΓ(m/2)] is the volume of unit ball, whereas Γ denotes the Gamma function.
In this note, we prove new analytic characterization of balls. Like Kuran’s theorem, it is based,
on the m-dimensional volume mean value equality, but instead of harmonic functions uses
solutions of the modified Helmholtz equation:
∇2u −λ2u = 0, λ ∈R\ {0}; (1)
∇= (∂1, . . . ,∂m) is the gradient operator and ∂i = ∂/∂xi . Solutions are assumed to be real; indeed,
the obtained results can be extended to complex-valued functions by considering the real and
imaginary part separately.
Before giving the precise formulation of the main result, let us introduce some notation. By
Br (x) = {y : |y − x| < r } we denote the open ball of radius r centred at x ∈ Rm ; if D ⊂ Rm is a
domain of finite Lebesgue measure and a function f is integrable over D , then
M( f ,D) = 1|D|
∫
D
f (x)dx
is its volume mean value over D . Also, we need the following function
a(t ) = Γ
(m
2
+1
) Im/2(t )
(t/2)m/2
, (2)
where Iν stands for the modified Bessel function of order ν. The relation
[z−νIν(z)]′ = z−νIν+1(z) (see [12, p. 79]), (3)
where the right-hand side is positive for z > 0 and vanishes at z = 0, implies that the function
a increases monotonically on [0,∞) from a(0) = 1 to infinity; the latter is a consequence of the
asymptotic formula valid as |z|→∞:
Iν(z) = e
z
p
2πz
[
1+O(|z|−1)] , |arg z| <π/2 (see [12, p. 80]) .
The function a arises in the m-dimensional mean value formula for balls
a(λr )u(x) = M(u,Br (x)) , x ∈ D, (4)
which holds, for example, if u ∈ C 0(D) is a solution of (1) in D and Br (x) ⊂ D . This equality was
obtained by the author recently; see [7, p. 95]. Before that only the three-dimensional mean value
formula for spheres had been derived by C. Neumann (see his book [9, Chapter 9, Section 3],
published in 1896), whereas the m-dimensional formula for spheres was given without proof
in [11]; its derivation see in the author’s note [6].
Now, we are in a position to formulate the main result.
Theorem 1. Let for a bounded domain D ⊂ Rm , m ≥ 2, a positive r be such that |Br | = |D|, and
let x0 ∈ D. If the equality u(x0) a(λr ) = M(u,D) holds for all solutions of equation (1) in an open
domain containing D, then D = Br (x0).
C. R. Mathématique — 2021, 359, n 8, 945-948
Nikolay Kuznetsov 947
2. Proof of Theorem 1 and discussion
Prior to proving Theorem 1, we introduce the following function
U (x) = Γ
(m
2
) I(m−2)/2(λ|x|)
(λ|x|/2)(m−2)/2 , x ∈R
m , (5)
where the coefficient is chosen so that U (0) = 1. Let us consider some of its properties. According
to (3), this spherically symmetric function monotonically increases as |x| goes from zero to
infinity. Also, it solves equation (1) in Rm ; indeed, the representation
U (x) = 2Γ(m/2)p
πΓ((m −1)/2)
∫ 1
0
(1− s2)(m−3)/2 cosh(λ|x|s)ds , (6)
is easy to differentiate, thus verifying (1). This formula for U is a consequence of Poisson’s integral
(see [10], p. 223):
Iν(z) = (z/2)
ν
p
πΓ(ν+1/2)
∫ 1
−1
(1− s2)ν−1/2 cosh zs ds .
Moreover, (6) takes particularly simple form for m = 3, namely, U (x) = (λ|x|)−1 sinhλ|x|.
Proof of Theorem 1. Without loss of generality, we suppose that the domain D is located so that
x0 coincides with the origin. If we assume that D 6= Br (0), then Gi = D \ Br (0) and Ge = Br (0) \ D
are bounded open sets such that |Ge | = |Gi | 6= 0 by the definition of r .
To obtain a contradiction from this assumption we write the volume mean equality for U over
D as follows:
|D|a(λr ) =
∫
D
U (y)dy ; (7)
here the condition U (0) = 1 is taken into account. Since property (4) holds for U over Br (0), we
write it in the same way:
|Br (0)|a(λr ) =
∫
Br (0)
U (y)dy . (8)
Subtracting (8) from (7) and using the definition of r , we obtain
0 =
∫
Gi
U (y)dy −
∫
Ge
U (y)dy > 0.
Here the difference is positive because U (y) (positive and monotonically increasing with |y |)
is greater (less) than [U (y)]|y |=r in Gi (Ge , respectively), whereas |Gi | = |Ge |. This contradiction
proves the theorem. 
In the limit λ→ 0, equation (1) turns into Laplace’s, whose solutions are harmonic functions;
moreover, the assumption about r becomes superfluous in this case. Thus, letting λ → 0 in
Theorem 1 leads to an improved formulation of Kuran’s theorem because only positive harmonic
functions are involved; see also [1].
In the case of sufficiently smooth ∂D , the integral
∫
D u(y)dy can be replaced by the flux
integral
∫
∂D ∂u/∂ny dSy in the formulation of Theorem 1; here n is the exterior unit normal.
Indeed, we have ∫
D
u(y)dy =λ−2
∫
D
∇2u (y)dy =λ−2
∫
∂D
∂u/∂ny dSy .
This suggests that the following mean flux equality
λ2r Γ(m/2)
2Γ
( m
2 +1
) a(λr )u(x0) = 1|∂D|
∫
∂D
∂v
∂ny
dSy
(cf. [7, formula (31)]) may also characterize balls provided there exists x0 ∈ D such that this
equality is valid for all sufficiently smooth solutions of equation (1) in D with a smooth boundary,
whereas λ,r > 0 satisfy assumptions similar to those in Theorem 1.
C. R. Mathématique — 2021, 359, n 8, 945-948
948 Nikolay Kuznetsov
In conclusion we notice that the equality (see [7, Theorem 8])
mIm/2(λr )
∫
∂Br (x)
u(y)dSy =λr I(m−2)/2(λr )
∫
Br (x)
u(y)dy
holds for every point x belonging to a domain D ⊂Rm and all r such that Br (x) ⊂ D if and only if u
is a solution of equation (1) in D . This is analogous to the equality of the mean values over spheres
and balls for harmonic functions. In view of Kosmodem’yanskii’s theorem, one might expect that
this equality with Br (x) changed to D characterizes balls inRm provided it is valid for all solutions
of equation (1) in D .
Acknowledgements
The author thanks an anonymous referee for a valuable comment.
References
[1] D. H. Armitage, M. Goldstein, “The volume mean-value property of harmonic functions”, Complex Variables, Theory
Appl. 13 (1990), no. 3-4, p. 185-193.
[2] B. Epstein, “On the mean-value property of harmonic functions”, Proc. Am. Math. Soc. 13 (1962), p. 830.
[3] W. Hansen, I. Netuka, “Inverse mean value property of harmonic functions”, Math. Ann. 297 (1993), no. 1, p. 147-156,
corrigendum in ibid. 303 (1995), no. 2, p.373-375.
[4] A. A. Kosmodem′yanskij, Jr, “A converse of the mean value theorem for harmonic functions”, Russ. Math. Surv. 36
(1981), no. 5, p. 159-160.
[5] Ü. Kuran, “On the mean value property of harmonic functions”, Bull. Lond. Math. Soc. 4 (1972), p. 311-312.
[6] N. Kuznetsov, “Mean value properties of solutions to the Helmholtz and modified Helmholtz equations”, to appear
in Russian, 2021, https://arxiv.org/abs/2105.09916v1.
[7] ——— , “Metaharmonic functions: mean flux theorem, its converse and related properties”, Algebra Anal. 33 (2021),
p. 82-97.
[8] I. Netuka, J. Veselý, “Mean value property and harmonic functions”, in Classical and modern potential theory and
applications, NATO ASI Series. Series C. Mathematical and Physical Sciences, vol. 430, Kluwer Academic Publishers,
1994, p. 359-398.
[9] C. Neumann, Allgemeine Untersuchungen über das Newton’sche Princip der Fernwirkungen, Teubner, 1896.
[10] A. F. Nikiforov, V. B. Uvarov, Special Functions of Mathematical Physics: A Unified Introduction with Applications,
Birkhäuser, 1988.
[11] H. Poritsky, “Generalizations of the Gauss law of the spherical mean”, Trans. Am. Math. Soc. 43 (1938), p. 199-225.
[12] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1944.
C. R. Mathématique — 2021, 359, n 8, 945-948


Characterization of balls via solutions of the modified Helmholtz equation

Francesca Bucci

Matthias Eller

The Cauchy–Dirichlet problem for the Moore–Gibson–Thompson equation

Iason Efraimidis

Comptes Rendus
Mathématique
Iason Efraimidis
Quasiconformal extension for harmonic mappings on finitely connected
domains
Volume 359, issue 7 (2021), p. 905-909
<https://doi.org/10.5802/crmath.233>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2021, 359, n 7, p. 905-909
https://doi.org/10.5802/crmath.233
Complex analysis and geometry / Analyse et géométrie complexes
Quasiconformal extension for harmonic
mappings on finitely connected domains
Iason Efraimidis ∗, a
a Department of Mathematics and Statistics, Texas Tech University, Box 41042,
Lubbock, TX 79409, United States.
E-mail: iason.efraimidis@ttu.edu
Abstract. We prove that a harmonic quasiconformal mapping defined on a finitely connected domain in
the plane, all of whose boundary components are either points or quasicircles, admits a quasiconformal
extension to the whole plane if its Schwarzian derivative is small. We also make the observation that a
univalence criterion for harmonic mappings holds on uniform domains.
Mathematical subject classification (2010). 30C55, 30C62, 31A05.
Manuscript received 4th February 2021, accepted 6th June 2021.
1. Introduction
Let f be a harmonic mapping in a planar domain D and let ω= fz / fz be its dilatation. According
to Lewy’s theorem the mapping f is locally univalent if and only if its Jacobian J f = | fz |2 − | fz |2
does not vanish. Duren’s book [4] contains valuable information about the theory of planar
harmonic mappings.
The Schwarzian derivative of f was defined by Hernández and Martín [9] as
S f = ρzz −
1
2
(ρz )
2, where ρ = log J f . (1)
When f is holomorphic this reduces to the classical Schwarzian derivative. Another definition,
introduced by Chuaqui, Duren and Osgood [3], applies to harmonic mappings which admit a
lift to a minimal surface via the Weierstrass–Enneper formulas. However, focusing on the planar
theory in this note we adopt the definition (1).
We assume that C\D contains at least three points, so that D is equipped with the hyperbolic
metric, defined by
λD (π(z))
∣∣π′(z)∣∣ |d z| = λD(z)|d z| = |d z|
1−|z|2 , z ∈D,
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
906 I. Efraimidis
where D is the unit disk and π : D→ D is a universal covering map. The size of the Schwarzian
derivative of a mapping f in D is measured by the norm∥∥S f ∥∥D = sup
z∈D
λD (z)
−2 ∣∣S f (z)∣∣ .
A domain D in C is a K -quasidisk if it is the image of the unit disk under a K -quasiconformal
self-map of C, for some K ≥ 1. The boundary of a quasidisk is called a quasicircle.
According to a theorem of Ahlfors [1], if D is a K -quasidisk then there exists a constant c > 0,
depending only on K , such that if f is analytic in D with ‖S f ‖D ≤ c then f is univalent in
D and has a quasiconformal extension to C. This has been generalized by Osgood [12] to the
case when D is a finitely connected domain whose boundary components are either points or
quasicircles. Further, the univalence criterion was generalized to uniform domains (see Section 4
for a definition) by Gehring and Osgood [7] and, subsequently, the quasiconformal extension
criterion was generalized to uniform domains by Astala and Heinonen [2].
For harmonic mappings and the definition (1) of the Schwarzian derivative, a univalence and
quasiconformal extension criterion in the unit disk D was proved by Hernández and Martín [8].
This was recently generalized to quasidisks by the present author in [5]. Moreover, in [5] it was
shown that if all boundary components of a finitely connected domain D are either points or
quasicircles then any harmonic mapping in D with sufficiently small Schwarzian derivative is
injective. The main purpose of this note is to prove the following theorem.
Theorem 1. Let D be a finitely connected domain whose boundary components are either points
or quasicircles and let also d ∈ [0,1). Then there exists a constant c > 0, depending only on the
domain D and the constant d, such that if f is harmonic in D with ‖S f ‖D ≤ c and with dilatation
ω satisfying |ω(z)| ≤ d for all z ∈ D then f admits a quasiconformal extension to C.
As mentioned above, for the case when D is a (simply connected) quasidisk this was shown
in [5] while, on the other hand, for the case d = 0 (when f is analytic) this was proved by Osgood
in [12]. Osgood’s proof amounts to proving a univalence criterion in f (D). Such an approach does
not seem to work here since for a holomorphicφ on f (D) the compositionφ◦ f is not, in general,
harmonic.
Since isolated boundary points are removable for quasiconformal mappings (see [10, Ch. I,
§ 8.1]), we may assume for the proof of Theorem 1 that ∂D consists of n non-degenerate
quasicircles. Our proof will be based on the following theorem of Springer [13] (see also [10, Ch.II,
§ 8.3]).
Theorem 2 ([13]). Let D and D ′ be two n-tuply connected domains whose boundary curves are
quasicircles. Then every quasiconformal mapping of D onto D ′ can be extended to a quasiconfor-
mal mapping of the whole plane.
Hence, to prove Theorem 1 it suffices that we show that the boundary components of f (D) are
quasicircles. We prove this in Section 3. It relies on Osgood’s [12] quasiconformal decomposition,
which we briefly present in Section 2. In Section 4 we give a univalence criterion on uniform
domains.
2. Quasiconformal Decomposition
Let D be a domain in C. A collection D of domains ∆ ⊂ D is called a K -quasiconformal decom-
position of D if each ∆ is a K -quasidisk and any two points z1, z2 ∈ D lie in the closure of some
∆ ∈D. This definition was introduced by Osgood in [12], along with the following lemma.
Lemma 3 ([12]). If D is a finitely connected domain and each component of ∂D is either a point
or a quasicircle then D is quasiconformally decomposable.
C. R. Mathématique — 2021, 359, n 7, 905-909
I. Efraimidis 907
We now present, almost verbatim, the construction proving Lemma 3. We focus on the parts of
the construction we will be needing, maintaining the notation of [12] and skipping all the relevant
proofs. The interested reader should consult [12] for further details.
As we mentioned earlier, we may assume that ∂D consists of non-degenerate quasicircles
C0,C1, . . . , Cn−1, for n ≥ 2. Let F be a conformal mapping of D onto a circle domain D ′. Then, with
an application of Theorem 2 to F−1, it will be sufficient to find a quasiconformal decomposition of
D ′. Hence we may assume that D itself is a circle domain with boundary circles C j , j = 0, . . . , n−1.
If n = 2 then we may assume that D is the annulus 1 < |z| < R. Then the domains
∆1 =
{
z ∈ D : 0 < arg(z) < 4π
3
}
, ∆2 = e2πi /3∆1, ∆3 = e4πi /3∆1
make a quasiconformal decomposition of D .
Let n ≥ 3. Then there exists a conformal mapping Ψ of the circle domain D onto a domain
D ′ consisting of the entire plane minus n finite rectilinear slits lying on rays emanating from the
origin. The mapping can be chosen so that no two distinct slits lie on the same ray. The boundary
behavior ofΨ is the following: it can be analytically extended to D , and the two endpoints of the
slit C ′j =Ψ(C j ) correspond to two points on the circle C j which partition C j into two arcs, each of
which is mapped onto C ′j in a one-to-one fashion.
Let ξ j be the endpoint of C ′j furthest from the origin and let Q
′
j be the part of the ray that joins
ξ j to infinity. Let also S′j be the sector between C
′
j and C
′
j+1. Let ω
′
j be the midpoint of C
′
j and let
P ′j be a polygonal arc joiningω
′
j toω
′
j+1 that, except for its endpoints, lies completely in S
′
j . Then
P ′ =
n−1⋃
j=0
P ′j
is a closed polygon separating 0 from ∞ that does not intersect any of the Q ′j . Let G ′0 and G ′1 be
the components of D ′\P ′ that contain 0 and ∞, respectively. Now define
∆′0 j =G ′0 ∪S′j , ∆′j = D ′\∆′0 j
and
D′ =
{
∆′0 j : j = 0,1, . . . , n −1
}
∪
{
∆′j : j = 0,1, . . . , n −1
}
.
This collection has the covering property for D ′.
We denote the various parts of D corresponding underΨ−1 to those of D ′ by the same symbol
without the prime. Then
D= {∆0 j : j = 0,1, . . . , n −1}∪{∆ j : j = 0,1, . . . , n −1}
is a quasiconformal decomposition of D .
3. Proof of Theorem 1
Let f be a mapping in D as in Theorem 1. By [5, Theorem 2], f is injective if c is sufficiently small.
Also, f extents continuously to ∂D since every boundary point of D belongs to ∂∆ for some ∆
in the collection D and, by [5, Theorem 1], the restriction of f on ∆ admits a homeomorphic
extension to C.
LetΨ be a conformal mapping of D onto the slit domain D ′ of the previous section. Let C j be
a boundary quasicircle of D . We first prove that f (C j ) is a Jordan curve. The slit C ′j is divided by
its midpoint ω′j into two line segments, which we denote by Σ
′
j (m),m = 1,2, so that
Σ′j (1) =
{
z ∈C ′j : |z| ≤
∣∣∣ω′j ∣∣∣} and Σ′j (2) = {z ∈C ′j : |z| ≥ ∣∣∣ω′j ∣∣∣} .
C. R. Mathématique — 2021, 359, n 7, 905-909
908 I. Efraimidis
Figure 1. The slit C ′j and the three distinguished domains.
Let Σ′j (m)
± denote the two sides of Σ′j (m), so that a point z0 on Σ
′
j (m)
− is reached only by points
z ∈ S′j−1, meaning that arg z → (arg z0)− when z → z0. Similarly, a point z0 on Σ′j (m)+ is reached
only by points z ∈ S′j , so that arg z → (arg z0)+ when z → z0. Corresponding under Ψ−1 are four
disjoint -except for their endpoints- arcs on the quasicircle C j , denoted without the prime by
Σ j (m)±,m = 1,2. Now consider the domains ∆0, j−1,∆0 j and ∆k in the collection D, for some
k 6= j − 1, j ; see Figure 1 for their images under Ψ. By [5, Theorem 1] f is injective up to the
boundary of each ∆ ∈D. Note that the arcs Σ j (1)−,Σ j (1)+ and Σ j (2)− are subsets of ∂∆0, j−1, so
that their images under f , except for their endpoints, are disjoint. It remains to show that the
images of these three arcs under f are not intersected by the remaining image f (Σ j (2)+). Note
that the arcs Σ j (1)−,Σ j (1)+ and Σ j (2)+ are subsets of ∂∆0 j , so that f (Σ j (2)+) does not intersect
f (Σ j (1)−) nor f (Σ j (1)+). What remains to be seen is that f (Σ j (2)−) and f (Σ j (2)+) are disjoint and
this follows from the fact that the arcs Σ j (2)− and Σ j (2)+ are subsets of ∂∆k .
To see that the Jordan curve f (C j ) is actually a quasicircle note that each point of f (C j )
belongs to some open subarc of f (C j ) which is entirely included in the boundary of either
f (∆0, j−1), f (∆0 j ) or f (∆k ). These three domains are quasidisks by [5, Theorem 3]. Now the
assertion that f (C j ) is a quasicircle follows by an application of Theorem 8.7 in [10, Ch. II, § 8.9].
4. Remarks on uniform domains
A domain D in C is called uniform if there exist positive constants a and b such that each pair of
points z1, z2 ∈ D can be joined by an arc γ⊂ D so that for each z ∈ γ it holds
`(γ) ≤ a |z1 − z2|
and
min
j=1,2
`(γ j ) ≤ b dist(z,∂D) ,
where γ1,γ2 are the components of γ\{z}, dist(z,∂D) denotes the euclidean distance from z to
the boundary of D and `(·) denotes euclidean length. Uniform domains were introduced by
Martio and Sarvas [11]; see also, e.g., [7] for this equivalent definition. In [11] it was shown that
all boundary components of a uniform domain are either points or quasicircles. The converse
of this is also true for finitely connected domains, but not, in general, for domains of infinite
connectivity; see [6, § 3.5]. The following univalence criterion was proved in [11].
Theorem 4 ([7, 11]). If D is a uniform domain then there exists a constant c > 0 such that every
analytic function f in D with ‖S f ‖D ≤ c is injective.
Gehring and Osgood [7] gave a different proof of Theorem 4 by providing a characterization of
uniform domains. They showed that a domain D is uniform if and only if it is quasiconformally
decomposable in the following weaker (than the one we saw in Section 2) sense: there exists a
constant K with the property that for each z1, z2 ∈ D there exists a K -quasidisk ∆ ⊂ D for which
C. R. Mathématique — 2021, 359, n 7, 905-909
I. Efraimidis 909
z1, z2 ∈ ∆. Note that, in contrast to Osgood’s [12] decomposition, here ∆ depends on the points
z1, z2. However, this can readily be used to generalize the implication (i) ⇒ (iii) of [5, Theorem 2],
according to which a univalence criterion for harmonic mappings holds on finitely connected
uniform domains. The following theorem extends it to all uniform domains.
Theorem 5. Let D be a uniform domain in C. Then there exists a constant c > 0 such that if f is
harmonic in D with ‖S f ‖D ≤ c then f is injective.
Proof. Assume that there exist distinct points z1, z2 ∈ D for which f (z1) = f (z2). By [7], there
exists a K -quasidisk ∆ ⊂ D for which z1, z2 ∈ ∆. The domain monotonicity for the hyperbolic
metric shows that ∥∥S f ∥∥∆ ≤ ∥∥S f ∥∥D ≤ c.
But the homeomorphic extension of [5, Theorem 1] shows that if c is sufficiently small then f is
injective up to the boundary of ∆, a contradiction. 
Regarding quasiconformal extension, Astala and Heinonen [2] proved the following theorem.
Theorem 6 ([2]). If D is a uniform domain then there exists a constant c > 0 such that every
analytic function f in D with ‖S f ‖D ≤ c admits a quasiconformal extension to C.
This evidently implies Theorem 4 and was also proved in substantially greater generality, but
we omit it here. It is not clear how to generalize Theorem 6 to the setting of harmonic mappings.
Therefore, we propose the following problem.
Problem. Let D be a uniform domain. Does there exist a constant c > 0 such that if f is
harmonic in D with ‖S f ‖D ≤ c and with dilatation ω satisfying supz∈D |ω(z)| < 1 then f admits a
quasiconformal extension to C?
References
[1] L. V. Ahlfors, “Quasiconformal reflections”, Acta Math. 109 (1963), p. 291-301.
[2] K. Astala, J. Heinonen, “On quasiconformal rigidity in space and plane”, Ann. Acad. Sci. Fenn., Math. 13 (1988), no. 1,
p. 81-92.
[3] M. Chuaqui, P. L. Duren, B. G. Osgood, “The Schwarzian derivative for harmonic mappings”, J. Anal. Math. 91 (2003),
p. 329-351.
[4] P. L. Duren, Harmonic Mappings in the Plane, Cambridge Tracts in Mathematics, vol. 156, Cambridge University
Press, 2004.
[5] I. Efraimidis, “Criteria for univalence and quasiconformal extension for harmonic mappings on planar domains”,
https://arxiv.org/abs/2009.14766, to appear in the Annales Fennici Mathematici.
[6] F. W. Gehring, K. Hag, The ubiquitous quasidisk, Mathematical Surveys and Monographs, vol. 184, American Mathe-
matical Society, 2012.
[7] F. W. Gehring, B. G. Osgood, “Uniform domains and the quasihyperbolic metric”, J. Anal. Math. 36 (1979), p. 50-74.
[8] R. Hernández, M. J. Martín, “Criteria for univalence and quasiconformal extension of harmonic mappings in terms
of the Schwarzian derivative”, Arch. Math. 104 (2015), no. 1, p. 53-59.
[9] ——— , “Pre-Schwarzian and Schwarzian derivatives of harmonic mappings”, J. Geom. Anal. 25 (2015), no. 1, p. 64-
91.
[10] O. E. Lehto, K. I. Virtanen, Quasiconformal mappings in the plane, second ed., Grundlehren der Mathematischen
Wissenschaften, vol. 126, Springer, 1973.
[11] O. Martio, J. Sarvas, “Injectivity theorems in plane and space”, Ann. Acad. Sci. Fenn., Math. 4 (1979), no. 2, p. 383-401.
[12] B. G. Osgood, “Univalence criteria in multiply-connected domains”, Trans. Am. Math. Soc. 260 (1980), no. 2, p. 459-
473.
[13] G. Springer, “Fredholm eigenvalues and quasiconformal mapping”, Acta Math. 111 (1964), p. 121-142.
C. R. Mathématique — 2021, 359, n 7, 905-909


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