IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-3008, p-ISSN:2319-7676. Volume 10, Issue 2 Ver. VII (Mar-Apr. 2014), PP 01-04 www.iosrjournals.org

Dual Results of Opial’s Inequality Y.O. Anthonio1, S.O. Salawu2 and S.O. Sogunro3 1

Department of Mathematics, Lagos State University, Ojo, Nigeria. Department of Mathematics, University of Ilorin, Ilorin, Nigeria. 3 Department of Mathematics, Lagos State Polytechnic, Ikorodu, Nigeria. 2

Abstract: In this paper, we used Jensen’s inequality for the case of convex functions, firstly to obtain a Calvert’s generalization and second, to obtain a Maron’s generalization of Opial’s inequality. The main tool was adaptation of Jensen’s inequality for convex functions. Keywords: Integral inequalities, Maron’s inequality and Calvert’s generalization of Opial’s inequality Jensen’s inequality, convex functions.

I.

Introduction:

Opial [8] established the following interesting integral inequality: Let ( x, y) C [0, b] be such x(0) x(b) 0 and x(t ) 0 in (0, b), then b

x(t ) x (t ) dt

a

Where

b

b ( x (t )) 2 dt 4a

(1)

b in the best possible constant. 4

In 1067 Maroni [5] obtained a generalized Opial’s inequality by using Holder inequality with indices The result obtained is the following: Theorem 1:

and .

Let p(t ) be positive and continuous on

, with p1 (t )dt , where 1, x(t ) be absolutely

function on

, and x(0) 0 . The following inequality holds. 2

x(t ) x (t ) dt

Where

1

1

2

1 1 p (t )dt p(t ) x (t ) dt 2

1. equality holds in (2) in and only if

p

1

(2)

( s)ds

Calvert [2] also established the following result: Theorem 2: [2] assume that (i) x(t ) is absolutely continuous in , and x( ) 0

f (t ) is continuous, complex-valued, defined in the range of x(t ) and for all real for t of the form

(ii) s

t ( s) x (u ) du : f ( t ) for all t and f (t ) is real t 0 and is increasing there,

(iii)

p(t ) is positive, continuous and

p

1

(t )dt , where

1

1

1. then the following inequality

holds. 2

2

f (t ) x (t ) dt F p1 (t )dt p(t ) x (t ) dt www.iosrjournals.org

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Dual Results Of Opial’s Inequality Where F (t )

t

t

0

f (t )ds, t 0. Equality holds in (3) if and only if x(t ) p1 ( s)ds.

The aim of this paper is to generalize Maroni and Calvert results using Jensen’s inequality.

II.

Some Adaptation of Jensen’s inequalities:

be continuous and convex function and let h( s, t ) be a non negative function and be non 1 decreasing function. Let (t ) (t ) and suppose has a continuous inverse (which is Let

necessarily concave). Then,

(t ) (t ) h( s, t )d ( s) ( ) 1 ( h( s, t ) )d ( s ) (t ) 1 ( t ) (4) (t ) (t ) d ( s) d ( s ) (t ) (t ) With the inequality reserved if is concave. The inequality (4) above is known as Jensen’s inequality for convex function. Setting (u) u , (t ) t and (t ) 0 in (4), then we obtain. t h( s, t )d ( s) 0 ( f (t )) f t 0 d (s) III.

1

1 t ( h( s, t ) ) l d ( s ) 0 t d ( s ) 0

(5)

Main Result:

Before stating our main result in this section, we shall need the following useful Lemma: Lemma 1: Let x(t ), (t ) and f (u ) be absolutely continuous and non decreasing functions on a, b for with f (t ) 0. Let l , k , o, and and measurable function on

0ab

be real numbers such that 0, o 0 and also let R(t) be non negative

a, b such that

t x (t ) f x (t ) R(t )d (t ) (t ) l y (t ) R(t ) 1 (t ) 1 y (t ). 0 (6) Then the following inequality holds: b

b

x (t ) f (t ) dt f ( y (t )) y (t )dt

a

a

(7 )

Proof: Setting h( s, t ) x(t ) R(t ) in (5) , we have

t x (t ) R(t )d (t ) 0 ( f (t )) f t 0 d (t ) l By setting f ( (t )) (t ) in (8) yields

1

1 t ( x (t ) R(t ) ) l d (t ) 0 t 0 d (t )

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(8)

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Dual Results Of Opial’s Inequality

t f x (t ) R(t )d (t ) 0 (t ) l

1

t l f x (t ) R(t ) d (t ) 0 (t )

(9)

Hence,

1 t t l f x(t ) R(t )d (t ) (t ) f x(t ) R(t ) l d (t ) (t ) l y(t ) 0 0

(10)

Now let t

y (t ) f x (t ) R(t ) l (t ) 1

(11)

0

y (t ) f x(t ) R(t ) l (t ) 1

then

y (t ) f x(t ) R(t ) (t ) using the fact that f (u ) u to have y (t ) l x(t ) R(t ) l (t ) l l

that is

(12) l

(13) (14)

x(t ) R(t ) (t ) y (t ) Combining both (10) and (15) yields, inequality (6) and the proof is complete. 1

1

(15)

Remarks 1: By setting

f (u) u , R(t ) P(t )

1 k 1

, (t )

1 k 1

, l in lemma 1 yields

1 1 1 1 t x(t ) f x(t ) P(t ) k 1 P(t ) k 1 dt (t ) l 1 y(t ) l P(t ) k 1 P(t ) k 1 y (t ). 0 Integrating both sides of inequality (16) over [a, b] with the respect to t, to get b

a

(16)

t b x (t ) f x (t ) dt y (t ) y (t )dt 0 a

(17)

That is l

b t l a x (t ) 0 x (t ) dt a y(t ) y (t )dt F ( y(b)) F ( y(a)). If y(a) 0, then inequality (18) becomes b

(18)

l

b t l x ( t ) x ( t ) dt a y (t ) y (t )dt F ( y (b)). a 0 By using Holders inequality with o and we obtain b

(19)

1 1 o

1 b b b b y (b) x (t ) dt R o (t ) R x (t ) (t )dt R1o (t )dt R(t ) x (t ) dt a a a a (20) Combing inequality (19) and (20) to obtain inequality (3) if in inequality (20) o and which is 1

our desired result. Furthermore, we need the following Lemma to obtain a generalization of Maroni.

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Dual Results Of Opial’s Inequality Lemma 2: Let x(t ), (t ), f (u), R(t ), l , k , o and

be as in Lemma 1 such that

1 t l x (t ) f x (t ) R(t ) d (t ) (t ) y (t ) l R(t ) 1 (t ) 1 y (t ). 0

(21) Then, the following inequality holds: b

a

b 1 x (t ) f (t ) dt y (t ) y (t )dt y (t ) dt 2a a b

2

(22)

Proof: The proof is similar to the proof of lemma 1. Since

f (u) u l , inequality (10) becomes l

t x (t ) R(t )d (t ) (t ) l y (t ) 0 l t x (t ) R(t )d (t ) (t ) l y (t ) 0 Combining (15) and (24) to obtain the inequality (21)

(23)

(24)

This completes the proof of the Lemma. Consider all conditions of remark 1 1 1 l l l 1 1 t k 1 k 1 l l k 1 k x(t ) x(t ) P(t ) P(t ) dt (t ) y(t ) P(t ) P(t ) 1 y (t ). 0 t x (t ) x (t ) dt y (t ) y (t ) 0

(25)

(26)

t

Putting

x(t ) dt x(t ) and integrate both side of inequality (26) over [a, b] with the respect to t obtain 0 b

a

b 1 x(t ) x(t ) dt y(t ) y (t )dt x(t ) dt 2a a b

1 1 t 1 o x (t ) P(t ) P(t ) o dt 20

2

(27)

2

(28) 2

2

1 b o b 1 P(t )1 dt x(t ) P(t ) o dt 2a a This is the generalization of inequality (2).

(29)

Acknowledgements: The authors gratefully acknowledged with thanks, Professor J.A Oguntuase for his contributions which have improved the final version of this paper.

References [1]. [2]. [3]. [4]. [5].

Adeagbo-Shelkh A.G and Imoru C.O: An integral inequality of the Hardy’s-Type. Kragujevac J Math. 29(2006), 57-61. Calvert J.: Some generalization of Opial’s Inequality, Pron. Amer. Math. Soc. 18(1967) 72-75 Maroni P.M,: Surl’ingelite d’Opial-Bessack, C.R Aca. Sci. Paris A264 (1967), 62-64. C.O Fabulurin, A.G Adeagbo-Shelkh and Y.O Anthonio,: On an inequality relates to Opial, Octogon Mathematics Magazine, 18(2010), No 1, 32-41. Obial Z.: Surune integalite. Ann. Polon. Math. 8(1960), 29-32.

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