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volume 5, issue 3, article 72, 2004.

Received 08 March, 2004;

accepted 11 April, 2004.

Communicated by:N. Elezovi´c

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Journal of Inequalities in Pure and Applied Mathematics

ON LANDAU TYPE INEQUALITIES FOR FUNCTIONS WITH HÖLDER CONTINUOUS DERIVATIVES

LJ. MARANGUNI ´C AND J. PE ˇCARI ´C

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

2000c Victoria University ISSN (electronic): 1443-5756 079-04

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On Landau Type Inequalities for Functions with HÖlder Continuous Derivatives Lj. Maranguni´c and J. Peˇcari´c

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J. Ineq. Pure and Appl. Math. 5(3) Art. 72, 2004

http://jipam.vu.edu.au

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

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 onI. Iff ∈L∞(I)and the derivativef⁰ :I →Rsatisfies Hölder’s condition

(1.1) |f⁰(t)−f⁰(s)| ≤H· |t−s|^α for any t, s∈I,

where H > 0 and α ∈ (0,1] are given, then f⁰ ∈ L∞(I) and one has the inequalities:

(1.2) ||f⁰|| ≤











2 1 + _α¹_α+1^α

· ||f||^α+1^α ·H^α+1¹ if m(I)≥2^α+2^α+1_||f||

H

_α+1¹

1 + _α¹_α+1¹

;

4·||f||

m(I) + ₂α(α+1)^H [m(I)]^α

if 0< m(I)≤2^α+2^α+1 _||f_||

H

_α+1¹

(1 + _α¹)^α+1¹ , where|| · ||is the∞-norm on the intervalI, andm(I)is the length ofI.

In our paper we shall give an improvement of this theorem.

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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 TheoremA. Thenf⁰ ∈L∞(I)and the following inequlities hold:

(2.1) ||f⁰|| ≤











2 1 + _α¹_α+1^α

· ||f||^α+1^α ·H^α+1¹ if m(I)≥2^α+1¹ _||f||

H

_α+1¹

1 + _α¹_α+1¹

;

2||f||

m(I) +_α+1^H [m(I)]^α

if 0< m(I)≤2^α+1¹ _||f_||

H

_α+1¹

1 + _α¹_α+1¹ ,

where|| · ||is the∞-norm on the intervalI, andm(I)is the length ofI.

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].

Proof. Consider the functiony: [a, b]→Rgiven by:

y(x) = (b−x)^α+1+ (x−a)^α+1. We observe that the unique solution of the equation

y⁰(x) = (α+ 1) [(x−a)^α−(b−x)^α] = 0

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isx₀ = ^a+b₂ ∈ [a, b]. The functiony⁰(x)is decreasing on(a, x₀)and increasing on (x₀, b). Thus, the maximal values fory(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. LetA, B >0andα∈(0,1]. Consider the functiongα: (0,∞)→ Rgiven by:

(2.3) g_α(λ) = A

λ +B·λ^α. Defineλ₀ := _αB^A _α+1¹

∈(0,∞). Then forλ₁ ∈(0,∞)we have the bound

(2.4) inf

λ∈(0,λ1]g_α(λ) =







A

λ1 +B·λ^α₁ if 0< λ₁ < λ₀ (α+ 1)α⁻^α+1^α ·A^α+1^α ·B^α+1¹ if λ₁ ≥λ₀.

Proof. We have:

g⁰_α(λ) =−A

λ² +α·B·λ^α−1.

The unique solution of the equationg⁰_α(λ) = 0,λ∈(0,∞), isλ0 = _αB^A _α+1¹

∈ (0,∞). The functiong_α(λ)is decreasing on(0, λ₀)and increasing on(λ₀,∞).

The global minimum forg_α(λ)on(0,∞)is:

(2.5) g_α(λ₀) = A αB

A _α+1¹

+B A

αB _α+1^α

= (α+ 1)α⁻^α+1^α ·A^α+1^α ·B^α+1¹ , which proves (2.4).

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Lemma 2.4. Let A, B > 0 and α ∈ (0,1]. Consider the functions g_α : (0,∞)→Randhα : (0,∞)→Rdefined by:

(2.6)







g_α(λ) = ^A_λ +B·λ^α h_α(λ) = ^2A_λ + ₂^Bαλ^α. Defineλ0 := _αB^A _α+1¹

∈(0,∞). Then forλ1 ∈(0,∞)we have:

(2.7)







λ∈(0,λinf1]gα(λ)< inf

λ∈(0,λ1]hα(λ) if 0< λ1 <2λ0

inf

λ∈(0,λ1]g_α(λ) = inf

λ∈(0,λ1]h_α(λ) if λ₁ ≥2λ₀.

Proof. In Lemma2.3, we found that the global minimum forg_α(λ)is obtained forλ=λ₀. Similarly we find that the global minimum forh_α(λ)is obtained for λ = 2λ₀, and its value is equal to the minimal value ofg_α(λ), i.e. h_α(2λ₀) = g_α(λ₀).

The only solution of equationg_α(λ) =h_α(λ),λ∈(0,∞), is:

λ_S =

A B(1−2^−α)

_α+1¹ ,

and we can easily check that λ0 < λS < 2λ0. Thus, for λ1 < λ0 we have g_α(λ₁)< h_α(λ₁)and inf

λ∈(0,λ1]g_α(λ)< inf

λ∈(0,λ1]h_α(λ), and the rest of the proof is obvious.

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Proof of Theorem2.1. Now we start proving our theorem using the identity:

(2.8) f(x) = f(a) + (x−a)f⁰(a) + Z x

a

[f⁰(s)−f⁰(a)]ds; a, x∈I

or, by changingxwithaandawithx:

(2.9) f(a) =f(x) + (a−x)f⁰(x) + Z a

x

[f⁰(s)−f⁰(x)]ds; a, x∈I.

Analogously, we have forb ∈I:

(2.10) f(b) =f(x) + (b−x)f⁰(x) + Z b

x

[f⁰(s)−f⁰(x)]ds; b, x∈I.

From (2.9) and (2.10) we obtain:

(2.11) f(b)−f(a) = (b−a)f⁰(x) + Z b

x

[f⁰(s)−f⁰(x)]ds +

Z x a

[f⁰(s)−f⁰(x)]ds; a, b, x∈I

and

(2.12) f⁰(x) = f(b)−f(a)

b−a − 1

b−a Z b

x

[f⁰(s)−f⁰(x)]ds

− 1 b−a

Z x a

[f⁰(s)−f⁰(x)]ds.

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Assuming thatb > awe have the inequality:

(2.13) |f⁰(x)| ≤ |f(b)−f(a)|

b−a + 1

b−a

Z b x

|f⁰(s)−f⁰(x)|ds

+ 1

b−a

Z x a

|f⁰(s)−f⁰(x)|ds . Sincef⁰ is ofα−HHölder type, then:

Z b x

≤H·

Z b x

|s−x|^αds (2.14)

=H Z b

x

(s−x)^αds

= H

α+ 1(b−x)^α+1; b, x∈I, b > x

Z x a

≤H·

Z x a

|s−x|^αds (2.15)

=H Z x

a

(x−s)^αds

= H

α+ 1(x−a)^α+1; a, x∈I, a < x.

From (2.13), (2.14) and (2.15) we deduce:

(2.16) |f⁰(x)| ≤ |f(b)−f(a)|

b−a

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+ H

(b−a)(α+ 1)[(b−x)^α+1+ (x−a)^α+1];

a, b, x∈I, a < x < b.

Sincef ∈L∞(I)then|f(b)−f(a)| ≤2· ||f||. Using Lemma2.2we obviously get that:

(2.17) |f⁰(x)| ≤ 2||f||

b−a + H

α+ 1(b−a)^α; a, b, x∈I, a < x < b.

Denoteb−a =λ. Sincea, b ∈ I, b > a, we haveλ ∈ (0, m(I)), and we can analyze the right-hand side of the inequality (2.17) as a function of variableλ.

Thus we obtain:

(2.18) |f⁰(x)| ≤ 2||f||

λ + H

α+ 1λ^α =gα(λ) forx∈Iand for everyλ∈(0, m(I)).

Taking the infimum overλ∈(0, m(I))in (2.18), we get:

(2.19) |f⁰(x)| ≤ inf

λ∈(0,m(I))g_α(λ).

If we take the supremum overx∈Iin (2.19) we conclude that

(2.20) sup

x∈I

|f⁰(x)|=||f⁰|| ≤ inf

λ∈(0,m(I))g_α(λ).

Making use of Lemma2.3we obtain the desired result (2.1).

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Remark 2.1. Denoteλ0 = h

2 1 + _α¹||f||

H

i_α+1¹

. Comparing the results of Theo- remAand Theorem2.1we can see that in the case ofm(I)≥2λ₀the estimated values for ||f⁰|| in both theorems coincide. If0 < m(I) < 2λ₀ the estimated value for||f⁰||given by (2.1) is better than the one given by (1.2). Namely, using Lemma2.4we have:

(2.21) 2||f||

m(I)+ H

α+ 1[m(I)]^α < 4||f||

m(I)+ H

2^α(α+ 1)[m(I)]^α; m(I)∈(0, λ₀] and

(2.22)

2

1 + 1 α

_α+1^α

· ||f||^α+1^α ·H^α+1¹

< 4||f||

m(I) + H

2^α(α+ 1)[m(I)]^α; 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)f⁰(x)−f(b) +f(a)| ≤ H α+ 1

(b−x)^α+1+ (x−a)^α+1

;

a, b, x∈I, a < x < b.

This result is an extension of the result obtained by V.G. Avakumovi´c and S.

Aljanˇci´c 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. AVAKUMOVI ´C AND S. ALJAN ˇCI ´C, 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. MITRINOVI ´C, J.E. PE ˇCARI ´CANDA.M. FINK, Inequalities Involv- ing Functions and Their Integrals and Derivatives, Kluwer Academic Pub- lishers, Dordrecht, Boston, London, 1991.