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Comptes Rendus
Mathématique
Bujar Xh. Fejzullahu
On the maximum value of a confluent hypergeometric function
Volume 359, issue 10 (2021), p. 1217-1224
<https://doi.org/10.5802/crmath.256>
© Académie des sciences, Paris and the authors, 2021.
Some rights reserved.
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Comptes Rendus
Mathématique
2021, 359, n 10, p. 1217-1224
https://doi.org/10.5802/crmath.256
Harmonic analysis / Analyse harmonique
On the maximum value of a confluent
hypergeometric function
Bujar Xh. Fejzullahua
a Departament of Mathematics, University of Prishtina, Mother Teresa 5, 10000
Prishtinë, Kosovo
E-mail: bujar.fejzullahu@uni-pr.edu
Abstract. We study the maximum value of the confluent hypergeometric function with oscillatory conditions
of parameters. As a consequence, we obtain new inequalities for the Gauss hypergeometric function.
2000 Mathematics Subject Classification. 33C15, 33C20.
Manuscript received 2nd June 2021, accepted 8th August 2021.
1. Introduction and main results
The confluent hypergeometric function 1F1(a;b; x), which is defined as
1F1(a;b; x) =
∞∑
n=0
(a)n
(b)n
xn
n!
for b 6= 0,−1, . . . , is a particular solution of the linear differential equation
x y ′′+ (b −x)y ′−ay = 0. (1)
When a is a non-positive integer, the function 1F1(a;b; x) reduces to Laguerre polynomials, i.e.
L(b−1)n (x) =
(b)n
n!
1F1(−n;b; x), n = 0, 1, 2, . . . .
The function 1F1 is a special case of the generalized hypergeometric function
p Fq (a1, . . . , ap ;b1, . . . ,bq ; x) =
∞∑
n=0
∏p
i=1(ai )n∏q
j=1(b j )n
xn
n!
, (2)
p and q are non-negative integers, none of the numbers b j ( j = 1, . . . , q) is equal to zero or to a
negative integer. It is well known that the series p Fq (a1, . . . , ap ;b1, . . . ,bq ; x) converges absolutely
for all x if p ≤ q and for |x| < 1 if p = q +1, and it diverges for all x 6= 0 if p > q +1. If one of the
parameters ai equals zero or a negative integer, then the series (2) reduces to a polynomial.
The confluent hypergeometric function has been studied in great detail from its mathemat-
ical point of view (see, for instance, [12, 14, 18]). In particular, the estimate of the confluent hy-
pergeometric function 1F1(a;b; x) has been widely and deeply studied when x > 0 and b > a > 0
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1218 Bujar Xh. Fejzullahu
(see [3, 13], and references therein). For instance, Luke [13], among others, proved the following
inequalities
e
a
b x < 1F1(a;b; x) < 1− a
b
+ a
b
ex , x > 0,b > a > 0,
1F1(a;b; x) < (b −1)e
x
(b −a −1)(1+x) , x > 0,b −1 > a > 0.
We remark that when x > 0, b > a and b > 12 , Love [11, Corollary 2] showed that
|1F1(a;b; x)| ≤ Γ(b −a)e
x
|Γ(1−a)|Γ(b) . (3)
Unfortunately, the conditions given for (3) do not seem to be correct. For instance, for a ∈N and
b > a, (3) gives 1F1(a;b; x) = 0 for any x > 0.
When the parameters verify the so called oscillatory conditions a < 0 and b−a > 1, the estimate
of 1F1(a;b; x) is much more complicated and, to the author’s knowledge, has not been studied
in the literature, except for the case when a is a negative integer and b > 0 (see, for instance,
[9, 10, 17, 21, 23], and references therein).
It is well known that, for a < 0 and b > 0, 1F1(a;b; x) has a finite number of real zeros (see [18,
Section 13.9])
x1a,b < x2a,b < ·· · .
Based on the Kummer transformation
1F1(a;b; x) = ex 1F1(b −a;b;−x),
it follows that the real zeros of 1F1(a;b; x) are positive when a < 0 and b > 0.
In this paper, we use the Sonin–Pólya theorem as well as the Watson–Glaeske product formula
for confluent hypergeometric functions to study the maximum value of 1F1(a;b; x) with oscilla-
tory conditions of parameters.
Here is our main results.
Theorem 1. For a < 0 and b > 1
max
x≥0 e
−x |1F1(a;b; x)| = 1. (4)
Moreover, this maximum value is attained only when x = 0.
Corollary 2. When a < 0 and b > 1, let ξk , k = 1, . . . , be the successive maxima of y(x) =
e−x 1F1(a;b; x) arranged in increasing order, and let jb,k be the k-th positive zero of the Bessel
function Jb(x). Then,
y2(ξi )− y2(ξ j ) < b −a
b2
(
2b −1
2
∆ξ2i j +
2
3
∆ξ3i j
)
, i < j ,
where ∆ξki j = ξkj −ξki and
ξk =
j 2b,k
2b −4a +2
(
1+
2(b2 −1)+ j 2b,k
3(2b −4a +2)2
)
+O
(
1
a5
)
, as a →−∞.
We remark that from the asymptotics (see [14, Section 6.8.2])
e−x 1F1(a;b; x) = Γ(b)
Γ(a)
xa−b[1+O(x−1)], x →+∞,
where a is not a negative integer or zero, it follows that e−x in Theorem 1 cannot be replaced by
any e−cx , 0 ≤ c < 1. In the case when a is a negative integer or zero, we have (see [23])
max
x≥0 e
− x2 |1F1(a;b; x)| = 1, (5)
where b > 1.
C. R. Mathématique — 2021, 359, n 10, 1217-1224
Bujar Xh. Fejzullahu 1219
On the other hand, in the oscillation region of 1F1(a;b; x), we have the following result
analogous to (5).
Theorem 3. Let a < 0 and b > 1. For 0 ≤ x < (2b−1)(b−2a)b
e−
x
2 |1F1(a;b; x)| ≤ 1. (6)
Furthermore, for 0 ≤ x < (2b−1)(b−2a)2b
e−
x
2 |1F1(a;b; x)| ≤
√
M (x,b), (7)
where
M (x,b) = b −1
π
∫ π
0
∫ π
0
|cos(x sinθcosψ)|(sinψ)2b−3(sinθ)2b−2 dψdθ
and it has the property
0 <M (x,b) <M (0, x) = 1, x > 0.
Consequently, by applying (4) and (5), we obtain the following inequalities for the Gauss
hypergeometric function 2F1.
Corollary 4. Let a < 0, b > 1 and ℜe(σ) > 0.
(i) If a is not a negative integer and 0 ≤ x <ℜe(z)∣∣∣2F1 (σ, a;b; x
z
)∣∣∣≤√cosh(πℑm(σ)) |zσ| [ℜe(z)−x]−ℜe(σ) . (8)
In particular, for σ> 0 and 0 ≤ x < 1
|2F1(σ, a;b; x)| ≤ (1−x)−σ.
(ii) If a is a negative integer and 0 ≤ x < 2ℜe(z)∣∣∣2F1 (σ, a;b; x
z
)∣∣∣≤√cosh(πℑm(σ)) |zσ|[ℜe(z)− x
2
]−ℜe(σ)
.
In particular, for σ> 0 and 0 ≤ x < 2
|2F1(−n,σ;b; x)| ≤
(
1− x
2
)−σ
, n ∈N∪ {0}.
Under condition a > 0, several lower and upper bound inequalities for 2F1(σ, a;b; x) have been
derived in the literature using different approaches (e.g. [2–4,6,13,22] and references therein). For
instance, in [13, Theorem 13], Luke gave the following two-sided bounds(
1− a
b
x
)−σ
< 2F1(σ, a;b; x) < 1− a
b
+ a
b
(1−x)−σ , 0 < x < 1,0 <σ,0 < a < b,
whereas Karp and Sitnik [6, Theorem 5] showed that
2F1(σ, a;b; x) <
(
1− a
b −1 x
)−σ
, 0 < x < 1,0 <σ≤ 1,1 < a +1 < b.
On the other hand, in [22] the authors derived some inequalities for the Gauss hypergeometric
function 2F1(σ, a;b, x) when −1 < a < 0, 1 < b < 2, 0 <σ< 1, and x ∈ (0,1). We remark that when a
is a negative integer or zero, the estimate of the polynomial 2F1(a,σ;b; x) has been considered in
several papers from different point of views (see for instance [7, 8] and references therein)
2. Proof of the main results
One of the main tool that we need for our purpose is the well-known Sonin–Pólya theorem
(see [21, footnote to Theorem 7.31.1]) in the following form given by Szegö. Notice that this
theorem was used by Szegö [21] in a similar context to study the successive relative maxima of
classical orthogonal polynomials.
C. R. Mathématique — 2021, 359, n 10, 1217-1224
1220 Bujar Xh. Fejzullahu
The Sonin–Pólya theorem
Suppose that a function y = y(x) satisfies on an interval I ⊂R the differential equation
(py ′)′+q y = 0, (9)
where p = p(x) > 0, q = q(x) > 0 and both p ′ and q ′ are continuous on that interval. Define Sonin’s
function by
S(x) := y2(x)+ p(x)
q(x)
y ′2(x), (10)
then we observe that
S′(x) =−[p (x)q(x)]′
[
y ′(x)
q(x)
]2
, (11)
by which successive relative maxima of y2 form an increasing or decreasing sequence according
as pq decreases or increases on the corresponding interval.
Now, we can prove our main results.
Proof of Theorem 1. From (1), the corresponding differential equation for y(x)=e−x 1F1(a;b;x) is
x y ′′+ (b +x)y ′+ (b −a)y = 0.
By writing it in the self-adjoint form
(xbex y ′)′+ (b −a)xb−1ex y = 0,
we see that
p(x) = xbex , q(x) = (b −a)xb−1ex
and
[p (x)q(x)]′ = (b −a)(2b −1+2x)x2b−1e2x .
Thus, if a < 0 and b > 1, the successive relative maxima of |e−x 1F1(a;b; x)| are decreasing on [0,∞)
and
[e−x 1F1(a;b; x)]2 ≤ S(x) ≤ S(0) = y2(0) = 1, x ≥ 0.
This proves (4). 
Proof of Corollary 2. We observe that, using the differential equation
d
dx
[e−x 1F1(a;b; x)] =−b −a
b
e−x 1F1(a;b +1; x), (12)
ξk = xka,b+1, for all k = 1, . . . .
Thus, from (10) and (11) one has
y2(ξ j )− y2(ξi ) =− 1
b −a
∫ ξ j
ξi
x(2b −1+2x)[y ′(x)]2 dx, i < j .
Now we can apply (4) and (12) to yield
y2(ξi )− y2(ξ j ) < b −a
b2
∫ ξ j
ξi
x(2b −1+2x)dx
= b −a
b2
(
2b −1
2
∆ξ2i j +
2
3
∆ξ3i j
)
,
where ∆ξki j = ξkj −ξki .
Finally, taking into account that the k-th positive zero xka,b can be approximated by (see [18,
Section 13.9])
j 2b−1,k
2b −4a
(
1+
2b(b −2)+ j 2b−1,k
3(2b −4a)2
)
+O
(
1
a5
)
, as a →−∞,
we can achieve the proof of the corollary. 
C. R. Mathématique — 2021, 359, n 10, 1217-1224
Bujar Xh. Fejzullahu 1221
Proof of Theorem 3. For the proof of (6), we proceed as in the proof of (4). According to (1), it is
straightforward to check that the function y(x) = e− x2 1F1(a;b; x) satisfies
x y ′′(x)+by ′(x)+ 2b −4a −x
4
y(x) = 0. (13)
In its self-adjoint form equation (13) becomes
(xb y ′(x))′+ 2b −4a −x
4
xb−1 y(x) = 0,
which corresponds to equation (9) with
p(x) = xb , q(x) = 2b −4a −x
4
xb−1,
and
[p (x)q(x)]′ = x
2b−2
2
[(2b −1)(b −2a)−bx] .
Thus, for a < 0 and b > 1, the successive relative maxima of |e− x2 1F1(a;b; x)| are decreasing if
0 < x < (2b−1)(b−2a)b and increasing if (2b−1)(b−2a)b < x < 2b −4a. This completes the proof of (6).
We now continue with the proof of (7). Our starting point in the proof is the Glaeske [5] product
formula for Laguerre functions, which in terms of the confluent hypergeometric functions can be
written as
1F1(a;b; x) · 1F1(a;b; y) = Γ(b)p
π
∫ π
0
e−
p
x y cosθ(sinθ)2b−2Jb− 32 (
p
x y sinθ)
× 1F1(a;b; x + y +2px y cosθ)dθ, (14)
where x, y ≥ 0, ℜe(b) > 12 , and Jν(z) :=
( z
2
)−ν Jν(z). For a = −n, n ∈ N, equation (14) was
first obtained by Watson [23] and later on by several authors using quite different methods
(see [5, 7, 16, 19, 20]), whereas in [15] Markett gave another, analytic proof of Glaeske’s result. In
Appendix A, we give another simple proof of (14).
Using Poisson’s integral (see [23, 3.3])
Jν(z) = 1p
πΓ
(
ν+ 12
) ∫ π
0
e i z cosψ(sinψ)2νdψ, ℜe(ν) >−1
2
,
the product formula (14) becomes
1F1(a;b; x) · 1F1(a;b; y) = b −1
π
∫ π
0
∫ π
0
e−
p
x y cosθ+ipx y sinθcosψ(sinψ)2b−3(sinθ)2b−2
× 1F1(a,b, x + y +2px y cosθ)dψdθ
= b −1
π
∫ π
0
∫ π
0
e−
p
x y cosθ cos(
p
x y sinθcosψ)(sinψ)2b−3
× (sinθ)2b−21F1(a,b, x + y +2px y cosθ)dψdθ.
(15)
We put x = y in (15) and multiply the obtained relation by e−x . As a result, we obtain[
e−
x
2 1F1(a;b; x)
]2 = b −1
π
∫ π
0
∫ π
0
cos(x sinθcosψ)(sinψ)2b−3(sinθ)2b−2
×e−x(1+cosθ)1F1(a,b,2x(1+cosθ))dψdθ.
Then, taking into account (6), for 0 ≤ x < (2b−1)(b−2a)2b[
e−
x
2 1F1(a;b; x)
]2 ≤ b −1
π
∫ π
0
∫ π
0
|cos(x sinθcosψ)|(sinψ)2b−3(sinθ)2b−2 dψdθ =M (x,b).
Finally, based on equation (20), we have
0 <M (x,b) <M (0,b), x > 0.
The proof of Theorem 3 is completed. 
C. R. Mathématique — 2021, 359, n 10, 1217-1224
1222 Bujar Xh. Fejzullahu
Proof of Corollary 4. (i). Applying inequality (4) to the confluent hypergeometric function ap-
pearing in the Laplace transform of the Gauss hypergeometric function (see [12, p. 59])
2F1
(
σ, a;b;
x
z
)
= z
σ
Γ(σ)
∫ ∞
0
e−zt tσ−11F1(a;b; xt )dt ,
where 0 ≤ x <ℜe(z) and ℜe(σ) > 0, we have∣∣∣2F1(σ, a;b; x
z
)∣∣∣≤ |zσ||Γ(σ)|
∫ ∞
0
e−t [ℜe(z)−x]tℜe(σ)−1 dt
= Γ(ℜe(σ))|Γ(σ)| |z
σ| [ℜe(z)−x]−ℜe(σ) .
Finally, using inequality (see [18, Section 5.6])
|Γ(p + i q)| ≥ |Γ(p)|√
cosh(qπ)
we get (8).
(ii). By making use of (5), the proof of case (ii) can be completed by following the proof of
case (i). 
Appendix A. Proof of the Watson–Glaeske formula
Substituting the integral representation for 1F1 (see [14, Section 6.5])
1F1(a;b; z) = 2
1−bΓ(b)e
z
2
Γ(a)Γ(b −a)
∫ 1
−1
e
zt
2 (1− t )b−2
(
1+ t
1− t
)a−1
dt , (16)
where ℜe(b) > ℜe(a) > 0, into Bailey’s product formula for confluent hypergeometric functions
(see [1])
1F1(a;b; x) · 1F1(a;b; y) =
∞∑
k=0
(−1)k (a)k (b −a)k
k !(b)k (b)2k
(x y)k 1F1(a +k;b +2k; x + y)
we get, after interchanging the order of summation and integration
1F1(a;b; x) · 1F1(a;b; y) = 2
1−b[Γ(b)]2e
x+y
2
Γ(a)Γ(b −a)
∫ 1
−1
e
x+y
2 t (1− t )b−2
(
1+ t
1− t
)a−1
×
 ∞∑k=0
(−1)k
(p
x y(1−t 2)
2
)2k
k !Γ(b +k)
 dt .
Using the series expansion
Jν(z) =
∞∑
k=0
(−1)k ( z2 )2k
k !Γ(ν+k +1) ,
we obtain
1F1(a;b; x) · 1F1(a;b; y) = 2
1−b[Γ(b)]2e
x+y
2
Γ(a)Γ(b −a)
∫ 1
−1
e
x+y
2 t (1− t )b−2
(
1+ t
1− t
)a−1
×Jb−1(
√
x y(1− t 2))dt . (17)
On the other hand, from Gegenbauer’s double integral representation for Jν (see [23, Sec-
tion 3.33])
Jν(ω)) = 1
πΓ(ν)
∫ π
0
∫ π
0
e i Z cosθ−i z(cosΦcosθ+sinΦsinθcosψ)(sinψ)2ν−1(sinθ)2νdψdθ,
C. R. Mathématique — 2021, 359, n 10, 1217-1224
Bujar Xh. Fejzullahu 1223
where ω2 = Z 2 + z2 − z Z cosΦ and ν> 0, for Z =−i tpx y , z =−px y ,Φ= π2 and ν= b −1 we have
Jb−1
(√
x y(1− t 2)
)
= 1
πΓ(b −1)
∫ π
0
∫ π
0
e t
p
x y cosθ+ipx y sinθcosψ
× (sinψ)2b−3(sinθ)2b−2 dψdθ. (18)
Now substituting (18) into (17) and taking into account (16) yields
1F1(a;b; x) · 1F1(a;b; y) = 2
1−b(b −1)Γ(b)e x+y2
πΓ(a)Γ(b −a)
∫ π
0
∫ π
0
e i
p
x y sinθcosψ(sinψ)2b−3(sinθ)2b−2
×
[∫ 1
−1
e
(
x+y
2 +
p
x y cosθ
)
t
(1− t )b−2
(
1+ t
1− t
)a−1
dt
]
dψdθ
= b −1
π
∫ π
0
∫ π
0
e−
p
x y cosθ+ipx y sinθcosψ(sinψ)2b−3(sinθ)2b−2
× 1F1(a,b, x + y +2px y cosθ)dψdθ.
(19)
By using analytic continuation, equation (19) can be extended to a ∈C and ℜe(b) > 1. This proves
equation (15) and completes the proof of Glaeske’s result.
In particular, putting x = y = 0 in (19) yields
1 = b −1
π
∫ π
0
∫ π
0
(sinψ)2b−3(sinθ)2b−2 dψdθ. (20)
Acknowledgements
The author thanks the referee for a careful reading of the manuscript and for his suggestions.
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C. R. Mathématique — 2021, 359, n 10, 1217-1224


On the maximum value of a confluent hypergeometric function

Maria Malin

Cristinel Mardare

Comptes Rendus
Mathématique
Maria Malin and Cristinel Mardare
A nonlinear Korn inequality on a surface with an explicit estimate of the
constant
Volume 359, issue 2 (2021), p. 105-111.
<https://doi.org/10.5802/crmath.122>
© 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 2, p. 105-111
https://doi.org/10.5802/crmath.122
Mathematical Problems in Mechanics, Differential Geometry / Problèmes
mathématiques en mécanique, Géométrie différentielle
A nonlinear Korn inequality on a surface with
an explicit estimate of the constant
Une inégalité de Korn non linéaire sur une surface avec
une majoration explicite de la constante
Maria Malina and Cristinel Mardare∗, b
a Department of Mathematics, University of Craiova, Craiova, Romania
b Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong
E-mails: malinmaria@yahoo.com (M. Malin), cmardare@cityu.edu.hk (C. Mardare)
Abstract. A nonlinear Korn inequality on a surface estimates a distance between a surface θ(ω) and another
surface φ(ω) in terms of distances between their fundamental forms in the space Lp (ω), 1 < p <∞.
We establish a new inequality of this type. The novelty is that the immersion θ belongs to a specific set of
mappings of class C 1 from ω into R3 with a unit vector field also of class C 1 over ω.
Résumé. Une inégalité de Korn non linéaire sur une surface estime une distance entre une surfaceθ(ω) et une
autre surfaceφ(ω) en fonction des distances entre leur formes fondamentales dans l’espace Lp (ω), 1 < p <∞.
Nous établissons une nouvelle inégalité de ce type. La nouveauté réside dans l’appartenance de l’immer-
sion θ à un ensemble particulier d’applications de classe C 1 de ω dans R3 avec un champ de vecteurs nor-
maux unitaires aussi de classe C 1 dans ω.
Funding. The work of the second author was substantially supported by a grant from City University of Hong
Kong (Project No. 7005495).
Manuscript received 18th September 2020, accepted 23rd September 2020.
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
106 Maria Malin and Cristinel Mardare
1. Notation and definitions
Vector and matrix fields are denoted by boldface letters.
Given any open set Ω ⊂ Rn , n > 1, any subset V ⊂ Y of a finite-dimensional vector space Y ,
and any integer `> 0, the notation C `(Ω;V ) designates the set of all fields v = (vi ) :Ω→ Y such
that v (x) ∈ V for all x ∈ Ω and vi ∈ C `(Ω). Likewise, given any real number p > 1, the notation
Lp (Ω;V ), resp. W `, p (Ω;V ), designates the set of all fields v = (vi ) :Ω→ Y such that v (x) ∈ V for
almost all x ∈Ω and vi ∈ Lp (Ω), resp. vi ∈W `, p (Ω).
The space of all real matrices with k rows and ` columns is denotedMk×`. We also let
Mk :=Mk×k ,Sk :=
{
A ∈Mk ; A = AT
}
,
Sk> :=
{
A ∈Sk ; A is positive-definite
}
, and Ok+ :=
{
A ∈Mk ; A AT = I and det A = 1
}
.
A k × ` matrix whose column vectors are the vectors v 1, . . . , v` ∈ Rk is denoted (v 1| . . . |v`). If
A ∈Sk>, there exists a unique matrix U ∈Sk> such that U 2 = A; this being the case, we let A1/2 :=U .
The Euclidean norm in R3 is denoted | · |. Spaces of matrices are equipped with the Frobenius
norm, also denoted | · |. The spaces Lp (Ω), Lp (Ω;Rk ), and Lp (Ω;Mk×`), are respectively equipped
with the norms denoted and defined by
‖u‖Lp (Ω) :=
(∫
Ω
|u(x)|p d x
)1/p
, ‖v‖Lp (Ω) :=
(∫
Ω
|v (x)|p d x
)1/p
,
and ‖A‖Lp (Ω) :=
(∫
Ω
|A(x)|p d x
)1/p
.
A domain Ω in Rn , n > 2, is a connected and open subset of Rn that is bounded and has
a Lipschitz-continuous boundary, the set Ω being locally on the same side of its boundary
(cf. Adams [1], Maz’ya [10], or Nečas [11]).
Given an open subset Ω of Rn and any integer ` > 0, the notation C `(Ω) designates the
space of all functions u ∈ C `(Ω) such that u and all their partial derivatives up to order `
possess continuous extensions to the closure Ω of Ω. If Ω ⊂ Rn is a domain, then C `(Ω) = { f |Ω;
f ∈C `(Rn)}, where f |Ω denotes the restriction of the function f to the setΩ, thanks to Whitney’s
extension theorem: cf. Whitney [12]; see also Ciarlet & Mardare [5].
Given a connected open subset Ω of Rn , the geodesic distance between two points x, y ∈Ω is
defined by
distΩ(x, y) := inf
{
` ∈R; there exists a path c ∈C 1 ([0,`];Rn)
such that c(0) = x, c(`) = y, c(s) ∈Ω and ∣∣c ′(s)∣∣= 1 for all s ∈ [0,`]}. (1)
IfΩ⊂Rn is a domain, then there exists a constant CΩ> 1 such that
distΩ(x, y)6CΩ
∣∣x − y∣∣ for all x, y ∈Ω; (2)
see e.g. Anicic, Le Dret & Raoult [2, Proposition 5.1] .
A generic point in an open subset ω of R2 is denoted y = (y1, y2) and partial derivative
operators with respect to y1 and y2 are denoted ∂1 and ∂2.
2. Main result
An immersion of class C 1 from a two-dimensional domain ω ⊂ R2 into the three-dimensional
Euclidean space R3 is a mapping φ : ω→ R3 of class C 1 such that the two vector fields ∂1φ : ω
→ R3 and ∂2φ : ω→ R3 are linearly independent at each point of ω. This means that the image
C. R. Mathématique, 2021, 359, n 2, 105-111
Maria Malin and Cristinel Mardare 107
of ω by φ is a surface in R3 whose tangent plane at its point φ(y), y ∈ ω, is spanned by the two
vectors ∂1φ(y) and ∂2φ(y). Consequently,
ν(φ) := ∂1φ∧∂2φ∣∣∂1φ∧∂2φ∣∣ (3)
is a continuous unit vector field from ω into R3 that is normal to the surface φ(ω).
Given an immersion φ :ω→R3 of class C 1, we let
∇φ := (∂1φ ∣∣ ∂2φ) and A(φ) :=∇φT ∇φ. (4)
Note that ∇φ is field of 3 × 2 matrices whose column vectors are the partial derivatives of φ
and that A(φ) is a field of 2×2 positive-definite symmetric matrices whose components are the
covariant components of the first fundamental form associated with the immersion φ.
Given an immersion φ :ω→R3 of class C 1 such that the unit vector field ν(φ) :ω→R3 is also
of class C 1, we let
∇ν(φ) = (∂1ν(φ) ∣∣ ∂2ν(φ)) and B (φ) :=∇φT ∇ν(φ). (5)
Note that ∇ν(φ) is field of 3×2 matrices whose column vectors are the partial derivatives of the
vector field ν(φ) and that B (φ) is a field of 2×2 symmetric matrices whose components are the
covariant components of the second fundamental form associated with the immersion φ.
The above definitions and notations apply as well to immersions φ : ω → R3 and their
associated unit vector fields ν(φ) : ω → R3 that are both of class C 1 up to the boundary of ω,
or of class W 1, p in ω, 1 < p <∞. This being the case, we let
C 1+
(
ω;R3
)
:= {φ ∈C 1 (ω;R3) ; ∣∣∂1φ∧∂2φ∣∣> 0 in ω, ν(φ) ∈C 1 (ω;R3)} (6)
and
W 1, p+
(
ω;R3
)
:= {φ ∈W 1, p (ω;R3) ; ∣∣∂1φ∧∂2φ∣∣> 0 a.e. in ω, ν(φ) ∈W 1, p (ω;R3)} . (7)
The objective of this Note is to indicate how to establish a nonlinear Korn inequality with an
explicit estimate of the constant that appears in it for mappingsφ ∈W 1, p+ (ω;R3) andθ ∈C 1+(ω;R3);
see Theorem 1 below. We will show in particular that the estimate for the constant depends
on θ only via two scalar parameters, denoted ρ and δ in what follows, which are related to the
assumption that θ is an immersion such that θ and ν(θ) are continuously differentiable vector
fields over ω.
Note that the nonlinear Korn inequality of Theorem 1 constitutes an improvement, when
n = 3, over two previous results by the authors about hypersurfaces in Rn , n > 3: see [7,
Theorem 3.1 and Lemma 3.2], or [6, Lemma 2].
The definition of the constant Cω in the next statement is justified by relations (1)-(2) in
Section 1.
Theorem 1. Given any domain ω ⊂ R2 and any real numbers p > 1, 1 > ρ > 0 and δ > 0, there
exists a constant C =C (ω, p,ρ,δ) such that
inf
R ∈O3+
(∥∥ν(φ)−Rν(θ)∥∥Lp (ω) +∥∥∇φ−R∇θ∥∥Lp (ω) +∥∥∇ν(φ)−R∇ν(θ)∥∥Lp (ω))
6C
∥∥∥∥∥ infR ∈O3+
(∣∣ν(φ)−Rν(θ)∣∣+ ∣∣∇φ−R∇θ∣∣+ ∣∣∇ν(φ)−R∇ν(θ)∣∣)∥∥∥∥∥
Lp (ω)
6C
p
3
(∥∥A(φ)1/2 − A(θ)1/2∥∥
Lp (ω) +
∥∥A(φ)−1/2B (φ)− A(θ)−1/2B (θ)∥∥
Lp (ω)
)
C. R. Mathématique, 2021, 359, n 2, 105-111
108 Maria Malin and Cristinel Mardare
for all mappings φ ∈W 1,p+ (ω;R3) and θ ∈C 1ρ,δ,µ(ω;R3), where
µ> 0 is any real number such that µ6 ρ
11
468(1+Cω)
,
Cω> 1 is any constant such that distω
(
y, ỹ
)
6Cω
∣∣y − ỹ∣∣ for all y, ỹ ∈ω,
and
C 1ρ,δ,µ
(
ω;R3
)
:={
θ ∈C 1+
(
ω;R3
)
; inf
y ∈ω
∣∣∂1θ(y)∧∂2θ(y)∣∣> ρ, sup
y ∈ω
|∇θ(y)|6 1
ρ
, sup
y ∈ω
|∇ν(θ)(y)|6 1
ρ
,
sup{
y, ỹ ∈ω,
|y − ỹ |6 δ
∣∣∇θ(y)−∇θ(ỹ)∣∣6µ, sup{
y, ỹ ∈ω,
|y − ỹ |6 δ
∣∣∇ν(θ)(y)−∇ν(θ)(ỹ)∣∣6µ}.
The proof of the Theorem 1 is sketched in Section 3 below; the details are given in [9].
The restriction in Theorem 1 that θ belongs to the subset C 1
ρ,δ,µ(ω;R
3) of the set C 1+(ω;R3),
rather than to the set C 1+(ω;R3) itself, is essential (i.e., not merely an artefact of the proof).
However, this inconvenient is alleviated by the fact that, as ρ → 0+ and δ → 0+, the subset
C 1
ρ,δ,µ(ω;R
3) becomes as large in C 1+(ω;R3) as one wants. More specifically, for each µ> 0,
C 1+
(
ω;R3
)= lim
ρ→0+
(
lim
δ→0+
C 1ρ,δ,µ
(
ω;R3
))
,
where the limits above are defined as the union of an increasing sequence of sets.
3. Sketch of the proof of Theorem 1
The proof is broken for clarity into six steps, numbered (i) to (vi).
Proof. As in the statement of the Theorem 1, let there be given a domain ω ⊂ R2, a constant Cω
such that
distω(y, ỹ)6Cω
∣∣y − ỹ∣∣ for all y, ỹ ∈ω,
four real numbers p > 1, 1> ρ > 0, δ> 0, µ> 0, and two mappings
θ ∈C 1ρ,δ,µ
(
ω;R3
)
and φ ∈W 1, p+
(
ω;R3
)
.
Then let λ := ρ/3, η := 13µρ−8/3, ε := ρ6/3, let Ωε := ω× (−ε,ε), and let Θ : Ωε → R3 and
Φ :Ωε→R3 be the mappings defined by
Θ(x) := θ(y)+x3ν(θ)(y) for all x = (y, x3) ∈Ωε
and
Φ(x) :=φ(y)+x3ν(φ)(y) for almost all x = (y, x3) ∈Ωε.
Note that the above definition of the constants λ, η and ε is justified by the estimates established
in Step (iv) below.
Step (i). There exists a constant C1(p,ρ) > 0 such that
inf
R ∈O3+
‖∇Φ−R∇Θ‖Lp (Ωε)
>C1(p,ρ) inf
R ∈O3+
(∥∥ν(φ)−Rν(θ)∥∥Lp (ω) +∥∥∇φ−R∇θ∥∥Lp (ω) +∥∥∇ν(φ)−R∇ν(θ)∥∥Lp (ω)) .
C. R. Mathématique, 2021, 359, n 2, 105-111
Maria Malin and Cristinel Mardare 109
The proof of this inequality relies on Clarkson’s inequalities in the space Lp (Ωε) (see, e.g.,
Adams [1, Theorem 2.28]) and follows an argument previously used in Ciarlet, Malin & Mar-
dare [4, Proof of Theorem 4.2].
Step (ii). There exists a constant C2(p,ρ) > 0 such that∥∥∥∥∥ infR ∈O3+ |∇Φ−R∇Θ|
∥∥∥∥∥
Lp (Ωε)
6C2(p,ρ)
∥∥∥∥ inf
R ∈O3+
(∣∣ν(φ)−Rν(θ)∣∣+ ∣∣∇φ−R∇θ∣∣+ ∣∣∇ν(φ)−R∇ν(θ)∣∣)∥∥∥∥
Lp (ω)
.
The proof of this inequality uses either Jensen’s inequality if p > 2, or the inequality (a + b
+ c)p/2 6 ap/2 +bp/2 + cp/2 if p 6 2 for some appropriate nonnegative real numbers a,b and c,
followed by an appropriate application of Fubini’s theorem.
Step (iii). The following assertions hold:
A(θ) ∈C 0 (ω;S2>) , A(θ)−1 ∈C 0 (ω;S2>) , B (θ) ∈C 0 (ω;S2) , Θ ∈C 1 (Ωε;R3) ,
and
A(φ)1/2 ∈ Lp (ω;S2>) , A(φ)−1/2B (φ) ∈ Lp (ω;M2) , Φ ∈W 1,p (Ωε;R3) .
These assertions are straightforward generalisations of similar ones established in Ciarlet,
Gratie & Mardare [3] for p = 2, and for this reason their proof is omitted.
Step (iv). The mappingΘ satisfies the following properties:
det∇Θ(x)>λ and |∇Θ(x)|6 1
λ
for all x ∈Ωε,
and
|∇Θ(x)−∇Θ(x̃)|6 η for all x, x̃ ∈Ωε such that |x − x̃|6 δ.
Using the estimates in terms of ρ of the partial derivatives of θ and ν(θ) appearing in the defi-
nition of the set C 1
ρ,δ,µ(ω;R
3) (see the statement of Theorem 1), we first deduce from Weingarten’s
equations that
|∇Θ(x)|6 7
3ρ
and det∇Θ(x)> 11ρ
18
for all x = (y, x3) ∈Ωε,
then we deduce from the definition of the vector field ν(θ) in terms of the partial derivatives of θ
(see relation (3)) that ∣∣ν(y)−ν(ỹ)∣∣6 3
ρ8
∣∣∇θ(y)−∇θ(ỹ)∣∣ for all y, ỹ ∈ω.
Combined with the definition of the mapping Θ in terms of θ and the definition of the
parameter ε in terms of ρ, the last inequality implies that, for each x = (y, x3) ∈ Ωε and each
x̃ = (ỹ , x̃3) ∈Ωε,
|∇Θ(x)−∇Θ(x̃)|6 ∣∣∇θ(y)−∇θ(ỹ)∣∣+ε ∣∣∇ν(y)−∇ν(ỹ)∣∣+ ∣∣ν(y)−ν(ỹ)∣∣
6
(
1+ 3
ρ8
)∣∣∇θ(y)−∇θ(ỹ)∣∣+ ρ6
3
∣∣∇ν(y)−∇ν(ỹ)∣∣ .
Assume next that x and x̃ satisfy |x − x̃|6 δ, so that, in particular, |y − ỹ |6 δ. Then we infer from
the definition of the space C 1
ρ,δ,µ(ω;R
3) and from the previous estimate that
|∇Θ(x)−∇Θ(x̃)|6
(
1+ 3
ρ8
+ ρ
6
3
)
µ6
13µ
3ρ8
.
C. R. Mathématique, 2021, 359, n 2, 105-111
110 Maria Malin and Cristinel Mardare
Step (v). Assume that the given constantµ> 0 satisfiesµ6 ρ11468(1+Cω) . Then there exists a constant
C3(ω, p,ρ,δ) depending only on ω, p,ρ,δ such that
inf
R ∈O3+
‖∇Φ−R∇Θ‖Lp (Ωε)6C3(ω, p,ρ,δ)
∥∥∥∥∥ infR ∈O3+ |∇Φ−R∇Θ|
∥∥∥∥∥
Lp (Ωε)
. (8)
First, the inequalities established in (iv) imply that the mappingΘ belong to the set:
C 1λ,δ,η
(
Ωε;R3
)
:=
Θ ∈C
1
(
Ωε;R3
)
; inf
x∈Ωε
det∇Θ(x)>λ, sup
x∈Ωε
|∇Θ(x)|6 1
λ
, sup{
x, x̃ ∈Ωε,
|x − x̃|6 δ
|∇Θ(x)−∇Θ(x̃)|6 η
 .
Secondly, by (iii),
Φ ∈W 1, p (Ωε;R3) .
Thirdly, the definition of the setΩε in terms of ω, the definition of the geodesic distance inΩε
(see relation (1)), and the definition of the constant Cω (see the statement of Theorem 1), together
show that, for each x = (y, x3) ∈Ωε and each x̃ = (ỹ , x̃3) ∈Ωε,
distΩε (x, x̃) ≤ distω(y, ỹ)+|x3 − x̃3|6Cω
∣∣y − ỹ∣∣+|x3 − x̃3|6 (1+Cω) |x − x̃| .
Fourthly, the assumption on µ made in (v) implies that
η6
λ3
4(1+Cω)
.
The four observations above together imply that the assumptions of [8, Theorem 1(a)] are
satisfied by the domain Ωε and by the mappings Θ and Φ from Ωε into R3. Thus inequality (8)
holds as a consequence of this theorem.
Step (vi). Combining the three inequalities established in steps (i), (ii) and (v) above yields the
inequality
inf
R ∈O3+
(∥∥ν(φ)−Rν(θ)∥∥Lp (ω) +∥∥∇φ−R∇θ∥∥Lp (ω) +∥∥∇ν(φ)−R∇ν(θ)∥∥Lp (ω))
6C
∥∥∥∥∥ infR ∈O3+
(∣∣∇φ−R∇θ∣∣+ ∣∣ν(φ)−Rν(θ)∣∣+ ∣∣∇ν(φ)−R∇ν(θ)∣∣)∥∥∥∥∥
Lp (ω)
,
where C := (C1(p,ρ))−1C2(p,ρ)C3(ω, p,ρ,δ). This establishes the first inequality of Theorem 1.
Using the polar decomposition of the 3×3 matrix fields
(∇θ|ν(θ)) and (∇φ∣∣ν(φ))
and a method similar to one used in the proof of [7, Theorem 3.1], we next show that the following
inequality holds almost everywhere in ω:
inf
R ∈O3+
(∣∣ν(φ)−Rν(θ)∣∣2 + ∣∣∇φ−R∇θ∣∣2 + ∣∣∇ν(φ)−R∇ν(θ)∣∣2)
6
∣∣A(φ)1/2 − A(θ)1/2∣∣2 + ∣∣A(φ)−1/2B (φ)− A(θ)1/2B (θ)∣∣2 .
C. R. Mathématique, 2021, 359, n 2, 105-111
Maria Malin and Cristinel Mardare 111
Consequently,∥∥∥∥∥ infR ∈O3+
(∣∣ν(φ)−Rν(θ)∣∣+ ∣∣∇φ−R∇θ∣∣+ ∣∣∇ν(φ)−R∇ν(θ)∣∣)∥∥∥∥∥
Lp (ω)
6
p
3
(∥∥A(φ)1/2 − A(θ)1/2∥∥
Lp (ω) +
∥∥A(φ)−1/2B (φ)− A(θ)−1/2B (θ)∥∥
Lp (ω)
)
.
This establishes the second inequality of Theorem 1. 
References
[1] R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, vol. 65, Academic Press Inc., 1975.
[2] S. Anicic, H. Le Dret, A. Raoult, “The infinitesimal rigid displacement lemma in Lipschitz coordinates and application
to shells with minimal regularity”, Math. Methods Appl. Sci. 27 (2004), no. 4, p. 1283-1299.
[3] P. G. Ciarlet, L. Gratie, C. Mardare, “A nonlinear Korn inequality on a surface”, J. Math. Pures Appl. 85 (2006), no. 1,
p. 2-16.
[4] P. G. Ciarlet, M. Malin, C. Mardare, “New nonlinear estimates for surfaces in terms of their fundamental forms”, C. R.
Acad. Sci. Paris 355 (2017), no. 2, p. 226-231.
[5] P. G. Ciarlet, C. Mardare, “Recovery of a manifold with boundary and its continuity as a function of its metric tensor”,
J. Math. Pures Appl. 83 (2004), no. 7, p. 811-843.
[6] M. Malin, C. Mardare, “Nonlinear estimates for hypersurfaces in terms of their fundamental forms”, C. R. Acad. Sci.
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[7] ——— , “Nonlinear Korn inequalities on a hypersurface”, Chin. Ann. Math. 39 (2018), no. 3, p. 513-534.
[8] ——— , “A nonlinear Korn inequality in Rn with an explicitly bounded constant”, C. R., Math., Acad. Sci. Paris 358
(2020), no. 5, p. 621-626.
[9] ——— , “Estimates for the constant in two nonlinear Korn inequalities”, in press, to appear in Mathematics and
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[10] V. G. Maz’ya, Sobolev Spaces, Springer, 1985, translated from the Russian by T. O. Shaposhnikova.
[11] J. Nečas, Les Méthodes Directes en Théorie des Equations Elliptiques, Masson, 1967.
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no. 1, p. 63-89.
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A nonlinear Korn inequality on a surface with an explicit estimate of the constant

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The heredity and bimeromorphic invariance of the \partial \protect \,\protect \overline{\protect \!\partial }-lemma property

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Refinements and Symmetries of the Morris identity for volumes of flow polytopes

Modelling and finite element simulation of multi-sphere swimmers

Some examples of algebraic surfaces with canonical map of degree 20

Combinatorics of Bricard’s octahedra

On the maximum value of a confluent hypergeometric function

A nonlinear Korn inequality on a surface with an explicit estimate of the constant

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