Journal of Inequalities in Pure and
Applied Mathematics
ON LANDAU TYPE INEQUALITIES FOR FUNCTIONS WITH
HÖLDER CONTINUOUS DERIVATIVES
volume 5, issue 3, article 72,
2004.
LJ. MARANGUNIĆ AND J. PEČARIĆ
Department of Applied Mathematics
Faculty of Electrical Engineering and Computing
University of Zagreb
Unska 3, Zagreb, Croatia.
EMail: ljubo.marangunic@fer.hr
Faculty of Textile Technology
University of Zagreb
Pierottijeva 6, Zagreb
Croatia.
EMail: pecaric@element.hr
Received 08 March, 2004;
accepted 11 April, 2004.
Communicated by: N. Elezović
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
079-04
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Abstract
An inequality of Landau type for functions whose derivatives satisfy Hölder’s
condition is studied.
2000 Mathematics Subject Classification: 26D15
Key words: Landau inequality, Hölder continuity
On Landau Type Inequalities for
Functions with HÖlder
Continuous Derivatives
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
4
Lj. Marangunić and J. Pečarić
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1.
Introduction
S.S. Dragomir and C.I. Preda have proved the following theorem (see [1]):
Theorem A. Let I be an interval in R and f : I → R locally absolutely
continuous function on I. If f ∈ L∞ (I) and the derivative f 0 : I → R satisfies
Hölder’s condition
(1.1)
|f 0 (t) − f 0 (s)| ≤ H · |t − s|α
for any
t, s ∈ I,
where H > 0 and α ∈ (0, 1] are given, then f 0 ∈ L∞ (I) and one has the
inequalities:
 α
α+1
α
1
1
2
1
+
· ||f || α+1 · H α+1


α

1
α+1

1

α+2

||f ||
1 α+1

α+1
if m(I) ≥ 2
1
+
;
H
α
(1.2)
||f 0 || ≤
4·||f ||
H

+ 2α (α+1)
[m(I)]α

m(I)


1
α+1


α+2
1

if 0 < m(I) ≤ 2 α+1 ||f ||
(1 + 1 ) α+1 ,
H
α
where || · || is the ∞-norm on the interval I, and m(I) is the length of I.
In our paper we shall give an improvement of this theorem.
On Landau Type Inequalities for
Functions with HÖlder
Continuous Derivatives
Lj. Marangunić and J. Pečarić
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2.
Main Results
Theorem 2.1. Let I be an interval and f : I → R function on I satisfying
conditions of Theorem A. Then f 0 ∈ L∞ (I) and the following inequlities hold:
 α
α
1
2 1 + α1 α+1 · ||f || α+1 · H α+1




1

α+1

1
1

||f ||
1 α+1

α+1
if m(I) ≥ 2
1
+
;

H
α
(2.1)
||f 0 || ≤

2||f ||
H

+ α+1
[m(I)]α

m(I)



1
α+1

1

1
||f ||

1 α+1
α+1
1
+
,
if 0 < m(I) ≤ 2
H
α
On Landau Type Inequalities for
Functions with HÖlder
Continuous Derivatives
Lj. Marangunić and J. Pečarić
where || · || is the ∞-norm on the interval I, and m(I) is the length of I.
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In our proof and in the subsequent discussion we use three lemmas.
Lemma 2.2. Let a, b ∈ R, a < b, α ∈ (0, 1]. Then the following inequality
holds:
(2.2)
(b − x)α+1 + (x − a)α+1 ≤ (b − a)α+1 ,
∀x ∈ [a, b].
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Proof. Consider the function y : [a, b] → R given by:
α+1
y(x) = (b − x)
α+1
+ (x − a)
Close
.
We observe that the unique solution of the equation
y 0 (x) = (α + 1) [(x − a)α − (b − x)α ] = 0
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is x0 = a+b
∈ [a, b]. The function y 0 (x) is decreasing on (a, x0 ) and increasing
2
on (x0 , b). Thus, the maximal values for y(x) are attained on the boundary of
[a, b] : y(a) = y(b) = (b − a)α+1 , which proves the lemma.
A generalization of the following lemma is proved in [1]:
Lemma 2.3. Let A, B > 0 and α ∈ (0, 1]. Consider the function gα : (0, ∞) →
R given by:
gα (λ) =
(2.3)
1
α+1
∈ (0, ∞). Then for λ1 ∈ (0, ∞) we have the bound
 A
if 0 < λ1 < λ0
 λ1 + B · λα1
inf gα (λ) =
α
α
1

λ∈(0,λ1 ]
(α + 1)α− α+1 · A α+1 · B α+1 if λ1 ≥ λ0 .
Define λ0 :=
(2.4)
A
αB
A
+ B · λα .
λ
Proof. We have:
gα0 (λ) = −
A
+ α · B · λα−1 .
λ2
1
A α+1
The unique solution of the equation gα0 (λ) = 0, λ ∈ (0, ∞), is λ0 = αB
∈
(0, ∞). The function gα (λ) is decreasing on (0, λ0 ) and increasing on (λ0 , ∞).
The global minimum for gα (λ) on (0, ∞) is:
1
α
α
α
1
αB α+1
A α+1
(2.5) gα (λ0 ) = A
+B
= (α + 1)α− α+1 · A α+1 · B α+1 ,
A
αB
which proves (2.4).
On Landau Type Inequalities for
Functions with HÖlder
Continuous Derivatives
Lj. Marangunić and J. Pečarić
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Lemma 2.4. Let A, B > 0 and α ∈ (0, 1]. Consider the functions gα :
(0, ∞) → R and hα : (0, ∞) → R defined by:

 gα (λ) = Aλ + B · λα
(2.6)
 h (λ) = 2A + B λα .
α
λ
2α
Define λ0 :=
(2.7)
A
αB



1
α+1
∈ (0, ∞). Then for λ1 ∈ (0, ∞) we have:
inf gα (λ) <
λ∈(0,λ1 ]
inf gα (λ) =
λ∈(0,λ1 ]
inf hα (λ) if
0 < λ1 < 2λ0
inf hα (λ) if
λ1 ≥ 2λ0 .
λ∈(0,λ1 ]
λ∈(0,λ1 ]
Proof. In Lemma 2.3, we found that the global minimum for gα (λ) is obtained
for λ = λ0 . Similarly we find that the global minimum for hα (λ) is obtained for
λ = 2λ0 , and its value is equal to the minimal value of gα (λ), i.e. hα (2λ0 ) =
gα (λ0 ).
The only solution of equation gα (λ) = hα (λ), λ ∈ (0, ∞), is:
A
λS =
B(1 − 2−α )
1
α+1
,
and we can easily check that λ0 < λS < 2λ0 . Thus, for λ1 < λ0 we have
gα (λ1 ) < hα (λ1 ) and inf gα (λ) < inf hα (λ), and the rest of the proof is
λ∈(0,λ1 ]
λ∈(0,λ1 ]
On Landau Type Inequalities for
Functions with HÖlder
Continuous Derivatives
Lj. Marangunić and J. Pečarić
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obvious.
J. Ineq. Pure and Appl. Math. 5(3) Art. 72, 2004
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Proof of Theorem 2.1. Now we start proving our theorem using the identity:
Z x
0
(2.8)
f (x) = f (a) + (x − a)f (a) +
[f 0 (s) − f 0 (a)]ds;
a, x ∈ I
a
or, by changing x with a and a with x:
(2.9)
a
Z
0
[f 0 (s) − f 0 (x)]ds;
f (a) = f (x) + (a − x)f (x) +
a, x ∈ I.
x
On Landau Type Inequalities for
Functions with HÖlder
Continuous Derivatives
Analogously, we have for b ∈ I:
(2.10)
b
Z
0
[f 0 (s) − f 0 (x)]ds;
f (b) = f (x) + (b − x)f (x) +
b, x ∈ I.
Lj. Marangunić and J. Pečarić
x
From (2.9) and (2.10) we obtain:
(2.11)
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Z
0
Contents
b
0
0
f (b) − f (a) = (b − a)f (x) +
[f (s) − f (x)]ds
Z xx
+
[f 0 (s) − f 0 (x)]ds; a, b, x ∈ I
a
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and
(2.12) f 0 (x) =
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f (b) − f (a)
1
−
b−a
b−a
Z
x
b
[f 0 (s) − f 0 (x)]ds
Z x
1
[f 0 (s) − f 0 (x)]ds.
−
b−a a
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Assuming that b > a we have the inequality:
Z
1 b 0
|f (b) − f (a)|
0
0
(2.13)
|f (x)| ≤
+
|f
(s)
−
f
(x)|ds
b−a
b−a x
Z x
1 0
0
.
+
|f
(s)
−
f
(x)|ds
b−a a
Since f 0 is of α − H Hölder type, then:
Z b
Z b
0
0
α
|f
(s)
−
f
(x)|ds
≤
H
·
(2.14)
|s
−
x|
ds
x
x
Z b
(s − x)α ds
=H
On Landau Type Inequalities for
Functions with HÖlder
Continuous Derivatives
Lj. Marangunić and J. Pečarić
x
H
(b − x)α+1 ;
=
α+1
(2.15)
Z
a
x
b, x ∈ I, b > x
Contents
Z x
0
0
α
|f (s) − f (x)|ds ≤ H · |s − x| ds
Z xa
=H
(x − s)α ds
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H
(x − a)α+1 ;
α+1
From (2.13), (2.14) and (2.15) we deduce:
(2.16) |f 0 (x)| ≤
|f (b) − f (a)|
b−a
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a
=
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a, x ∈ I, a < x.
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+
H
[(b − x)α+1 + (x − a)α+1 ];
(b − a)(α + 1)
a, b, x ∈ I, a < x < b.
Since f ∈ L∞ (I) then |f (b) − f (a)| ≤ 2 · ||f ||. Using Lemma 2.2 we obviously
get that:
(2.17)
|f 0 (x)| ≤
2||f ||
H
+
(b − a)α ;
b−a α+1
a, b, x ∈ I, a < x < b.
Denote b − a = λ. Since a, b ∈ I, b > a, we have λ ∈ (0, m(I)), and we can
analyze the right-hand side of the inequality (2.17) as a function of variable λ.
Thus we obtain:
(2.18)
|f 0 (x)| ≤
2||f ||
H α
+
λ = gα (λ)
λ
α+1
for x ∈ I and for every λ ∈ (0, m(I)).
Taking the infimum over λ ∈ (0, m(I)) in (2.18), we get:
|f 0 (x)| ≤
(2.19)
inf
λ∈(0,m(I))
gα (λ).
If we take the supremum over x ∈ I in (2.19) we conclude that
(2.20)
sup |f 0 (x)| = ||f 0 || ≤
x∈I
On Landau Type Inequalities for
Functions with HÖlder
Continuous Derivatives
Lj. Marangunić and J. Pečarić
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inf
λ∈(0,m(I))
gα (λ).
Making use of Lemma 2.3 we obtain the desired result (2.1).
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1
i α+1
h
. Comparing the results of TheoRemark 2.1. Denote λ0 = 2 1 + α1 ||fH||
rem A and Theorem 2.1 we can see that in the case of m(I) ≥ 2λ0 the estimated
values for ||f 0 || in both theorems coincide. If 0 < m(I) < 2λ0 the estimated
value for ||f 0 || given by (2.1) is better than the one given by (1.2). Namely, using
Lemma 2.4 we have:
(2.21)
2||f ||
H
4||f ||
H
+
[m(I)]α <
+ α
[m(I)]α ;
m(I) α + 1
m(I) 2 (α + 1)
m(I) ∈ (0, λ0 ]
On Landau Type Inequalities for
Functions with HÖlder
Continuous Derivatives
and
(2.22)
α
α+1
α
1
1
· ||f || α+1 · H α+1
2 1+
α
4||f ||
H
[m(I)]α ;
<
+ α
m(I) 2 (α + 1)
Lj. Marangunić and J. Pečarić
m(I) ∈ [λ0 , 2λ0 ).
Remark 2.2. Let the conditions of Theorem 2.1 be fulfilled. Then a simple
consequence of (2.11) is the following inequality:
|(b − a)f 0 (x) − f (b) + f (a)| ≤
H (b − x)α+1 + (x − a)α+1 ;
α+1
a, b, x ∈ I, a < x < b.
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This result is an extension of the result obtained by V.G. Avakumović and S.
Aljančić in [2] (see also [3]).
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References
[1] S.S. DRAGOMIR AND C.J. PREDA, Some Landau type inequalities for
functions whose derivatives are Hölder continuous, RGMIA Res. Rep. Coll.,
6(2) (2003), Article 3. ONLINE [http://rgmia.vu.edu.au/v6n2.
html].
[2] V.G. AVAKUMOVIĆ AND S. ALJANČIĆ, Sur la meilleure limite de la
dérivée d’une function assujetie à des conditions supplementaires, Acad.
Serbe Sci. Publ. Inst. Math., 3 (1950), 235–242.
[3] D.S. MITRINOVIĆ, J.E. PEČARIĆ AND A.M. FINK, Inequalities Involving Functions and Their Integrals and Derivatives, Kluwer Academic Publishers, Dordrecht, Boston, London, 1991.
On Landau Type Inequalities for
Functions with HÖlder
Continuous Derivatives
Lj. Marangunić and J. Pečarić
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