On improvements of the Rozanova's inequality

Chang-Jian Zhao; Wing-Sum Cheung

doi:10.1186/1029-242X-2011-33

Research
Open Access
Published: 18 August 2011

On improvements of the Rozanova's inequality

Chang-Jian Zhao¹ &
Wing-Sum Cheung²

Journal of Inequalities and Applications volume 2011, Article number: 33 (2011) Cite this article

1673 Accesses
1 Citations
Metrics details

Abstract

In the present paper, we establish some new Rozanova's type integral inequalities involving higher-order partial derivatives. The results in special cases yield some of the interrelated results on Rozanova's inequality and provide new estimates on inequalities of this type.

MS (2000) Subject Classifiication: 26D15.

1 Introduction

In the year 1960, Opial [1] established the following integral inequality:

Theorem A Suppose f ∈ C¹[0, h] satisfies f(0) = f(h) = 0 and f(x) > 0 for all x ∈ (0, h). Then

∫ 0 h f (x) f ′ (x) d x ≤ h 4 ∫ 0 h (f ′ (x)) 2 d x .

(1.1)

The first Opial's type inequality was established by Willett [2] as follows:

Theorem B Let x(t) be absolutely continuous in [0, a], and x(0) = 0. Then

∫ 0 a | x (t) x ′ (t) | d t ≤ a 2 ∫ 0 a | x ′ (t) | 2 d t .

(1.2)

A non-trivial generalization of Theorem B was established by Hua [3] as follows:

Theorem C Let x(t) be absolutely continuous in [0, a], and x(0) = 0. Futher, let l be a positive integer. Then

∫ 0 a | x (t) x ′ (t) | d t ≤ a l l + 1 ∫ 0 a | x ′ (t) | l + 1 d t .

(1.3)

A sharper inequality was established by Godunova [4] as follows:

Theorem D Let f(t) be convex and increasing functions on [0, ∞) with f(0) = 0. Further, let x(t) be absolutely continuous on [0, τ], and x(α) = 0. Then, following inequality holds

∫ α τ f ′ (| x (t) |) | x ′ (t) | d t ≤ f ∫ α τ | x ′ (t) | d t .

(1.4)

Rozanova [5] proved an extension of inequality (1.4) is embodied in the following:

Theorem F Let f(t) and g(t) be convex and increasing functions on [0, ∞) with f(0) = 0, and let p(t) ≥ 0, p'(t) > 0, t ∈ [α, a] with p(α) = 0. Further, let x(t) be absolutely continuous on [α, a), and x(α) = 0. Then, following inequality holds

∫ α a p ′ (t) ⋅ g | x ′ (t) | p ′ (t) ⋅ f ′ p (t) ⋅ g | x (t) | p (t) d t ≤ f ∫ α a p ′ (t) ⋅ g | x ′ (t) | p ′ (t) d t .

(1.5)

The inequality (1.5) will be called as Rozanova's inequality in the paper.

Opial's inequality and its generalizations, extensions and discretizations play a fundamental role in establishing the existence and uniqueness of initial and boundary value problems for ordinary and partial differential equations as well as difference equations [6–13]. For Opial-type integral inequalities involving high-order partial derivatives, see [14, 15]. For an extensive survey on these inequalities, see [16].

The first aim of the present paper is to establish the following Opial-type inequality involving higher-order partial derivatives, which is an extension of the Rozanova's inequality (1.5).

Theorem 1.1 Let f and g be convex and increasing functions on [0, ∞) with f(0) = 0, and let p(s, t) ≥ 0, $D 1 D 2 p (s, t) = ∂ 2 ∂ s ∂ t p (s, t)$ , D₁D₂p(s, t) > 0, s ∈ [α, a], t ∈ [β, b] with p(s, β) = p(α, t) = p(α, β) = 0 and D₁D₂p(s, t) |_{t = τ}= 0. Further, let x(s, t) be absolutely continuous on [α, a) × [β, b], and x(s, β) = x(α, t) = x(α, β) = 0. Then following inequality holds

∫ α a ∫ β b D 1 D 2 p (s, t) ⋅ g | D 1 D 2 x (s, t) | D 1 D 2 p (s, t) ⋅ ∂ ∂ t f p (s, t) ⋅ g | x (s, t) | p (s, t) d s d t ≤ f ∫ α a ∫ β b D 1 D 2 p (s, t) ⋅ g | D 1 D 2 x (s, t) | D 1 D 2 p (s, t) d s d t .

(1.6)

We also prove the following Rozanova-type inequality involving higher-order partial derivatives.

Theorem 1.2 Assume that

(i)
f, g and x(s, t) are as in Theorem 1.1,
(ii)
p(s, t) is increasing on [0, a] × [0, b] with p(s, β) = p(α, t) = p(α, β) = 0,
(iii)
h is concave and increasing on [0, ∞),
(iv)
ϕ(t) is increasing on [0, a] with ϕ(0) = 0,
(v)
For $y (s, t) = ∫ 0 s ∫ 0 t D 1 D 2 p (σ, τ) g | D 1 D 2 x (σ, τ) | D 1 D 2 p (σ, τ) d σ d τ$ ,
$D 1 D 2 f y (s, t) D 1 D 2 y (s, t) ⋅ ϕ 1 D 1 D 2 y (s, t) ≤ c (a, b) y (a, b) ⋅ ϕ ′ t y (a, b) .$

Then

∫ 0 a ∫ 0 b D 1 D 2 f p (s, t) g x (s, t) p (s, t) ⋅ v D 1 D 2 p (s, t) g D 1 D 2 x (s, t) D 1 D 2 p (s, t) d s d t ≤ w ∫ 0 a ∫ 0 b D 1 D 2 p (s, t) g x (s, t) D 1 D 2 p (s, t) d s d t,

(1.7)

where

v (z) = z h ϕ 1 z, w (z) = c (a, b) h a ϕ b z,

and

c (a, b) = ∫ 0 a ∫ 0 b D 1 D 2 f y (s, t) D 1 D 2 y (s, t) d s d t .

2 Main results and proofs

Theorem 2.1 Let f and g be convex and increasing functions on [0, ∞) with f(0) = 0, and let p(s, t) ≥ 0, $D 1 D 2 p (s, t) = ∂ 2 ∂ s ∂ t p (s, t)$ , D₁D₂p(s, t) > 0, s ∈ [α, a], t ∈ [β, b] with p(s, β) = p(α, t) = p(α, β) = 0 and D₁D₂p(s, t) |_{t = τ}= 0. Further, let x(s, t) be absolutely continuous on [α, a) × [β, b], and x(s, β) = x(α, t) = x(α, β) = 0. Then, following inequality holds

∫ α a ∫ β b D 1 D 2 p (s, t) ⋅ g | D 1 D 2 x (s, t) | D 1 D 2 p (s, t) ⋅ ∂ ∂ t f p (s, t) ⋅ g | x (s, t) | p (s, t) d s d t ≤ f ∫ α a ∫ β b D 1 D 2 p (s, t) ⋅ g | D 1 D 2 x (s, t) | D 1 D 2 p (s, t) d s d t .

(2.1)

Proof Let $y (s, t) = ∫ α s ∫ β t D 1 D 2 x (σ, τ) d σ d τ$ so that D₁D₂y(s, t) = |D₁D₂x(s, t)| and y(s, t) ≥ |x(s, t)|. Thus, from Jensen's integral inequality, we obtain

g x (s, t) p (s, t) ≤ g y (s, t) p (s, t) ≤ g ∫ α s ∫ β t D 1 D 2 p (σ, τ) D 1 D 2 x (σ, τ) D 1 D 2 p (σ, τ) d σ d τ ∫ α s ∫ β t D 1 D 2 p (σ, τ) d σ d τ ≤ 1 p (s, t) ∫ α s ∫ β t D 1 D 2 p (σ, τ) g | D 1 D 2 x (σ, τ) | D 1 D 2 p (σ, τ) d σ d τ .

(2.2)

By using the inequality (2.2), we have

∫ α a ∫ β b D 1 D 2 p (s, t) ⋅ g | D 1 D 2 x (s, t) | D 1 D 2 p (s, t) ⋅ ∂ ∂ t f p (s, t) ⋅ g | x (s, t) | p (s, t) d s d t ≤ ∫ α a ∫ β b D 1 D 2 p (s, t) ⋅ g D 1 D 2 y (s, t) D 1 D 2 p (s, t) ⋅ ∂ ∂ t f ∫ α s ∫ β t D 1 D 2 p (σ, τ) ⋅ g D 1 D 2 y (σ, τ) D 1 D 2 p (σ, τ) d σ d τ d s d t .

(2.3)

On the other hand

∂ 2 ∂ s ∂ t f ∫ α s ∫ β t D 1 D 2 p (σ, τ) ⋅ g D 1 D 2 y (σ, τ) D 1 D 2 p (σ, τ) d σ d τ = ∂ ∂ s ∂ ∂ t f ∫ α s ∫ β t D 1 D 2 p (σ, τ) ⋅ g D 1 D 2 y (σ, τ) D 1 D 2 p (σ, τ) d σ d τ ⋅ ∫ α s p σ t (σ, t) ⋅ g D 1 D 2 y (σ, τ) D 1 D 2 p (σ, t) d σ = ∂ 2 ∂ s ∂ t f ∫ α s ∫ β t D 1 D 2 p (σ, τ) ⋅ g D 1 D 2 y (σ, τ) D 1 D 2 p (σ, τ) d σ d τ ⋅ ∫ α s D 1 D 2 p (σ, t) ⋅ g D 1 D 2 y (σ, τ) p σ t (σ, t) d σ × ∫ β t p s τ (s, τ) ⋅ g D 1 D 2 y (σ, τ) D 1 D 2 p (s, τ) d τ + D 1 D 2 p (s, t) ⋅ g D 1 D 2 y (s, t) D 1 D 2 p (s, t) × ∂ ∂ t f ∫ α s ∫ β t D 1 D 2 p (σ, τ) ⋅ g D 1 D 2 y (σ, τ) D 1 D 2 p (σ, τ) d σ d τ = D 1 D 2 p (s, t) ⋅ g D 1 D 2 y (s, t) D 1 D 2 p (s, t) ⋅ ∂ f ∂ t ∫ α s ∫ β t D 1 D 2 p (σ, τ) ⋅ g D 1 D 2 y (σ, τ) D 1 D 2 p (σ, τ) d σ d τ .

(2.4)

From (2.3) and (2.4), we have

∫ α a ∫ β b D 1 D 2 p (s, t) ⋅ g | D 1 D 2 x (s, t) | D 1 D 2 p (s, t) ⋅ ∂ ∂ t f p (s, t) ⋅ g | x (s, t) | p (s, t) d s d t ≤ ∫ α a ∫ β b ∂ 2 ∂ s ∂ t f ∫ α s ∫ β t D 1 D 2 p (σ, τ) ⋅ g D 1 D 2 y (σ, τ) D 1 D 2 p (σ, τ) d σ d τ d s d t = f ∫ α a ∫ β b D 1 D 2 p (σ, τ) ⋅ g D 1 D 2 y (σ, τ) D 1 D 2 p (σ, τ) d σ d τ = f ∫ α a ∫ β b D 1 D 2 p (s, t) ⋅ g | D 1 D 2 x (s, t) | D 1 D 2 p (s, t) d s d t .

This completes the proof.

Remark 2.2 Let x(s, t) reduce to s(t), and with suitable modifications in the proof of Theorem 2.1, then (2.1) becomes inequality (1.5) stated in Section 1.

Remark 2.3 Taking for g(x) = x in (2.1), then (2.1) becomes the following inequality.

∫ α a ∫ β b D 1 D 2 x (s, t) ⋅ ∂ ∂ t (f (x (s, t))) d s d t ≤ f ∫ α a ∫ β b D 1 D 2 x (s, t) d s d t .

(2.5)

Let x(s, t) reduce to s(t), and with suitable modifications, then (2.5) becomes inequality (1.4) stated in Section 1.

Remark 2.4 For f(t) = t^l⁺¹, l ≥ 0, the inequality (2.5) reduces to

∫ α a ∫ β b x (s, t) l ∂ ∂ t (x (s, t)) d s d t ≤ 1 l + 1 ∫ α a ∫ β b D 1 D 2 x (s, t) d s d t l + 1 .

(2.6)

In the right side of (2.6), by Hölder inequality with indices l + 1 and (l + 1)l, gives

∫ α a ∫ β b x (s, t) l ∂ ∂ t (x (s, t)) d s d t ≤ [(a - α) (b - β)] l l + 1 ∫ α a ∫ β b D 1 D 2 x (s, t) l + 1 d s d t .

(2.7)

Let x(s, t) reduce to s(t) and α = β = 0, then (2.7) becomes Hua's inequality (1.3) stated in Section 1.

Theorem 2.5 Assume that

(i)
f, g and x(s, t) are as in Theorem 2.1,
(ii)
p(s, t) is increasing on [0, a] × [0, b] with p(s, β) = p(α, t) = p(α, β) = 0,
(iii)
h is concave and increasing on [0, ∞),
(iv)
ϕ(t) is increasing on [0, a] with ϕ(0) = 0,
(v)
For $y (s, t) = ∫ 0 s ∫ 0 t D 1 D 2 p (σ, τ) g | D 1 D 2 x (σ, τ) | D 1 D 2 p (σ, τ) d σ d τ$ ,
$D 1 D 2 f y (s, t) D 1 D 2 y (s, t) ⋅ ϕ 1 D 1 D 2 y (s, t) ≤ c (a, b) y (a, b) ⋅ ϕ ′ t y (a, b) .$
(2.8)

Then

∫ 0 a ∫ 0 b D 1 D 2 f p (s, t) g x (s, t) p (s, t) ⋅ v D 1 D 2 p (s, t) g D 1 D 2 x (s, t) D 1 D 2 p (s, t) d s d t ≤ w ∫ 0 a ∫ 0 b D 1 D 2 p (s, t) g D 1 D 2 x (s, t) D 1 D 2 p (s, t) d s d t,

(2.9)

where

v (z) = z h ϕ 1 z,

(2.10)

w (z) = c (a, b) h a ϕ b z .

(2.11)

and

c (a, b) = ∫ 0 a ∫ 0 b D 1 D 2 f y (s, t) D 1 D 2 y (s, t) d s d t .

Proof From (2.2), we easily obtain

p (s, t) g x (s, t) p (s, t) ≤ ∫ 0 s ∫ 0 t D 1 D 2 p (σ, τ) g | D 1 D 2 x (σ, τ) | D 1 D 2 p (σ, τ) d σ d τ = y (s, t) .

(2.12)

From (2.8), (2.10-2.12) and Jensen's inequality(for concave function), hence

∫ 0 a ∫ 0 b D 1 D 2 f p (s, t) g x (s, t) p (s, t) ⋅ v D 1 D 2 p (s, t) g D 1 D 2 x (s, t) D 1 D 2 p (s, t) d s d t ≤ ∫ 0 a ∫ 0 b D 1 D 2 f y (s, t) ⋅ v D 1 D 2 y (s, t) d s d t = ∫ 0 a ∫ 0 b D 1 D 2 f y (s, t) D 1 D 2 y (s, t) ⋅ h ϕ 1 D 1 D 2 y (s, t) d s d t = ∫ 0 a ∫ 0 b D 1 D 2 f y (s, t) D 1 D 2 y (s, t) ⋅ h ϕ 1 D 1 D 2 y (s, t) d s d t ∫ 0 a ∫ 0 b D 1 D 2 f y (s, t) D 1 D 2 y (s, t) d s d t × ∫ 0 a ∫ 0 b D 1 D 2 f y (s, t) D 1 D 2 y (s, t) d s d t ≤ h ∫ 0 a ∫ 0 b D 1 D 2 f y (s, t) D 1 D 2 y (s, t) ⋅ ϕ 1 D 1 D 2 y (s, t) d s d t ∫ 0 a ∫ 0 b D 1 D 2 f y (s, t) D 1 D 2 y (s, t) d s d t ⋅ c (a, b) ≤ h ∫ 0 a ∫ 0 b c (a, b) y (a, b) ⋅ ϕ ′ t y (a, b) d s d t c (a, b) ⋅ c (a, b) = h 1 y (a, b) ∫ 0 a y (a, b) ϕ t y (a, b) | t = 0 t = b d s ⋅ c (a, b) = h a ϕ b y (a, b) ⋅ c (a, b) = w ∫ 0 a ∫ 0 b D 1 D 2 p (s, t) g | D 1 D 2 x (s, t) | D 1 D 2 p (s, t) d s d t .

This completes the proof.

Remark 2.6 Let x(s, t) reduce to s(t), and with suitable modifications in the proof of Theorem 2.5, then (2.9) becomes the following inequality:

∫ 0 a f ′ p (t) g x (t) p (t) ⋅ v p ′ (t) g x ′ (t) p ′ (t) d t ≤ w ∫ 0 a p ′ (t) g | x ′ (t) | p ′ (t) d t .

(2.13)

The inequality has been obtained by Rozanova in [17]. For $x (t) = x 1 (t), x 1 ′ (t) > 0, x 1 (0) = 0, x (a) = b, g (t) = t, f (t) = ϕ (t) = t 2$ and $h (t) = 1 + t$ , the inequality (2.13) reduces to Polya's inequality (see [17]).

Remark 2.7 Taking for g(x) = x in (2.9), then (2.9) becomes the following interesting inequality.

∫ 0 a ∫ 0 b D 1 D 2 f (x (s, t)) ⋅ v (D 1 D 2 x (s, t)) d s d t ≤ w ∫ 0 a ∫ 0 b D 1 D 2 x (s, t) d s d t .

References

1.
Opial Z: Sur une inégalité. Ann Polon Math 1960, 8: 29–32.
MathSciNet Google Scholar
2.
Willett D: The existence-uniqueness theorem for an n -th order linear ordinary differential equation. Am Math Monthly 1968, 75: 174–178. 10.2307/2315901
MathSciNet Article Google Scholar
3.
Hua LK: On an inequality of Opial. Sci China 1965, 14: 789–790.
Google Scholar
4.
Godunova EK: Integral'nye neravenstva s proizvodnysi i proizvol'nymi vypuklymi funkcijami. Uc Zap Mosk Gos Ped In-ta im Lenina 1972, 460: 58–65.
Google Scholar
5.
Rozanova GI: On an inequality of Maroni (Russian). Math Zametki 1967, 2: 221–224.
Google Scholar
6.
Das KM: An inequality similar to Opial's inequality. Proc Am Math Soc 1969, 22: 258–261.
Google Scholar
7.
Agarwal RP, Thandapani E: On some new integrodifferential inequalities. Anal sti Univ "Al I Cuza" din Iasi 1982, 28: 123–126.
MathSciNet Google Scholar
8.
Yang GS: A note on inequality similar to Opial inequality. Tamkang J Math 1987, 18: 101–104.
MathSciNet Google Scholar
9.
Agarwal RP, Lakshmikantham V: Uniqueness and Nonuniqueness Criteria for Ordinary Differential Equations. World Scientific, Singapore; 1993.
Google Scholar
10.
Bainov D, Simeonov P: Integral Inequalities and Applications. Kluwer Academic Publishers, Dordrecht; 1992.
Google Scholar
11.
Li JD: Opial-type integral inequalities involving several higher order derivatives. J Math Anal Appl 1992, 167: 98–100. 10.1016/0022-247X(92)90238-9
MathSciNet Article Google Scholar
12.
Cheung WS: On Opial-type inequalities in two variables. Aequationes Math 1989, 38: 236–244. 10.1007/BF01840008
MathSciNet Article Google Scholar
13.
Cheung WS: Some generalized Opial-type inequalities. J Math Anal Appl 1991, 162: 317–321. 10.1016/0022-247X(91)90152-P
MathSciNet Article Google Scholar
14.
Zhao CJ, Cheung WS: Sharp integral inequalities involving high-order partial derivatives. J Inequal Appl 2008. Article ID 571417
Google Scholar
15.
Agarwal RP, Pang PYH: Sharp opial-type inequalities in two variables. Appl Anal 1996,56(3):227–242. 10.1080/00036819508840324
MathSciNet Google Scholar
16.
Agarwal RP, Pang PYH: Opial Inequalities with Applications in Differential and Difference Equations. Kluwer Academic Publishers, Dordrecht; 1995.
Google Scholar
17.
Rozanova GI: Ob odnom integral'nom neravenstve, svjazannom s neravenstvom Polia. Izvestija Vyss Ucebn, Zaved Mat 1975, 125: 75–80.
MathSciNet Google Scholar

Download references

Acknowledgements

The authors express their deep gratitude to the referees for their many very valuable suggestions and comments. The research of Chang-Jian Zhao was supported by National Natural Science Foundation of China (10971205), and the research of Wing-Sum Cheung was partially supported by a HKU URC grant.

Author information

Affiliations

Department of Mathematics, China Jiliang University, Hangzhou, 310018, China
Chang-Jian Zhao
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong
Wing-Sum Cheung

Authors

Chang-Jian Zhao
View author publications
You can also search for this author in PubMed Google Scholar
Wing-Sum Cheung
View author publications
You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Chang-Jian Zhao.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

C-JZ and W-SC jointly contributed to the main results Theorems 2.1 and 2.5. Both authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Zhao, C., Cheung, W. On improvements of the Rozanova's inequality. J Inequal Appl 2011, 33 (2011). https://doi.org/10.1186/1029-242X-2011-33

Download citation

Received: 14 March 2011
Accepted: 18 August 2011
Published: 18 August 2011
DOI: https://doi.org/10.1186/1029-242X-2011-33

Keywords

Opial's inequality
Hölder's inequality
Rozanova's inequality