On a new ostrowski-type inequality and related results

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KYUNGPOOK Math. J. 54(2014), 545-554 http://dx.doi.Org/10.5666/KMJ.2014.54.4.545

O n a N e w O str o w sk i-T y p e In e q u a lity a n d R e la te d R e su lts

Er h a n Se t*

Department of Mathematics, Faculty of Science and Arts, Ordu University, Ordu, Turkey

e-mail : erliaxiset@yahoo.com

Me h m e t Ze k i Sa r ik a y a

Department of Mathematics, Faculty of Science and Arts, Diizce University, Diizce, Turkey

e-m ail: sarikayamz@gmail.com

Ab s t r a c t. We provide a new Ostrowski-type inequality involving functions of two inde pendent variables, as well as some related results.

1. Introduction

The well-known Ostrowski inequality (see [3]) states that

(

1

)

/(*)

1

b — a

1

4 _{( b - a f} (b - a) M,

where / : [a, 6] —> M. is a differentiable function such that \f'\ < M for all x e [a, b]. This article is greatly motivated and inspired also by the following results.

Theorem l.([l]) Let I C R be an open interval, a,b 6 I ,a < b. f : I -> R is a differentiable function such that there exist constants 7, T G R with 7 < f' {x) <

T , x e [a,b\. Then we have

(

2 )

(x — b) f (6) - (x - a) f (a) 2 [b - a) < (x — a) 2 + (6 — x) 2 _ —

W ^ )

— ( 7) * Corresponding Author.

Received Noverber 4, 2011; accepted May 8, 2013. 2010 Mathematics Subject Classification: 26D15; 26A51.

Key words and phrases: Ostrowski inequality, Functions of two independent variables.

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546 Erhan Set and Mehmet Zeki Sankaya

for all x £ [a, 6].

T h e o r e m 2.([10]) Let f : [a, b] —► R be a twice differentiable mapping on (a,b) and suppose that7 < f " ( t ) < F for all t £ (a, b ). Then we have the double inequality

- f

f ( t ) d t < ^ - J - ( b - a f

a J a 24

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where S = _{b — a}

T h e o r e m 3.([2]) Under the assumptions of Theorem 2, we have

r

(.x - a) 3 + (6 - a; ) 3 1 2 (6 — a) +

8

b — a + a + b (4) < - _{/ (} 2O + (x - a)f{a) + (6 - x)f(b) < 7 b — a 3 , / . \ 3 ‘ b — a

Os-

r)

[ f {t)dt J a (x — a) + (6 — x)" ₁ - + ~ b — a + x —a + b

_(s-

7

) , 12(6- a )

for all x £ [a,

6]

, where S = ^ ^ •

T h e o rem 4.([7]) Let f : [a, 6] x [c, d] -? R be an continuous function such that the partial derivative of order 2 exists and supposes that there exist constants 7, T £ R with7

<

9 Q t Q s s ^

< r

for all (t, s) £ [a,

6]

x [c, d]. Then, we have

b d \ l ( x , , ) +\H(x,y)-

j

-

j

Six, s)ds

2

(

6

-0

,a)^d-c) j ~

C ^ ’

^ + ^ ~

^

dt

(5) 1 2 (b — a) (d — c) a

J

[{x - a )/(a , s) + (6 - x)f(b, s)] ds 6 d t)dsdt < a C [(a? ~ a) 2 + (6 - x) 2] [(y - c) 2 + {d - y)2] 32(6 — a) (d — c)

(r-7)

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O n a N ew O stro w sk i-T y p e In e q u ality a n d R e la te d R esu lts 547

for all (x , y ) € [a, b] x [c, d] where H{x, y)

(x ~ a) [(y - c )/(a , c) + (d - y)f (a, d)] (b — a) (d — c) (b - x) [(y - c)f(b, c) + (d — y)f {b, d)} (b — a ) { d - c) (z - a )/(a , y) + (b - :r)/(fr, y) b — a {y - c )/(x , c) + (d - y)/(:r, d) d — c

T h e o re m 5.([5]) Let f : [a, b] x [c, d] —t R 6e an absolutely continuous function such

that the partial derivative of order 2 exists and suppose that there exist constants

7, T £ K with 7 < d < F for all (t , s) G [a, &] x [c, d]. Then, we have

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(1 - A)2f ( x , y) + - ( 1 - A) [/(a, y ) + f{b, y) + f ( x , c) + f {x, d)] +

_{( 2}

_{) [/(a>c) +}

_{c) + / ( a> + / ( fe> rf)l}

~ ( ! - A ) J f ( t , y ) d t + ^ j [ f { t , c ) + f(t , d)] dt \ '■ a a ' s d d (1 - A ) J f { x , s ) d s + ^ J [ f ( a , s ) + f(b,s)]ds

r + 7

(1 - A)2( z - (y - ~ Y ~ )c + d\ b d < r - 7 2 (b — a)(d — c

)

(Aa + ( 1- + ' (A2 + (1 - A)2)( d - c) 2 X

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548 E rhan Set and Mehmet Zeki Sarikaya

for all (x, y) G [a + A y ^ , f> — A ^ y ] x [c + A^y2, d — A^y2] and A G [0,1]. For the other recent Ostrowski type results, see [4],[6],[8] and [9].

The m ain purpose of this article is to establish a new inequality similar to the inequalities (2)-(6) for higher-order derivatives of / involving functions of two independent variables.

2. M a in R e s u lts

T h e o r e m 6. Let f : [a, b] x [c, d] —> R be an continuous function such that the partial derivative of order 4 exists and supposes that there exist constants 7, F G R

with 7 < < r for all (t, s) G [a, b] x [c, d]. Then, we have

E( x , y ) (x - a)3 + (b - x) 3 (y - c)3 + (d - y)c 288

1 (r + 7)

< (x - a)3 + ( b - x) 3 (y - c f + ( d - y f 288

( r - 7 )

for all (x,y) G [a, b] x [c, d],where {x - a ) ( y - c)

E( x , y ) = [/(:r, y) + f ( x , c) + / ( a , y) + / ( a , c)]

+ —— — — [/(^ , y) + f ( x , d) + / ( a , y) + / ( a , e0]

+ —— ^ r —— [/O, 2/) + / ( a:>c) + /(&> v ) + /(&> C)1

+ —— — — [f(x, y) + f {x, d) + /(&, y) + /(&, d)} y - c x — a 2~ rb fd

J

[f{t,

y)

+ f{t, c)] dt -

J

[f(t,d) + f { t , y ) ] d t J [f(x,s) + f { a , s ) ] d s - h—^ - j [f(b,s) + f ( x , s ) ] ds + n a f ( t, s)dsdt.

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O n a N ew O stro w sk i-T y p e In e q u a lity a n d R e la te d R esu lts 549 q : [c,d] x [c,rf] — given by and p(x,t) = ( x —t ) ( t —a) 2 ’ a < t < x ( x —t ) ( t —b) 2 ’ x < t <b (y - s ) ( s - c) 2 » c < s < y fy - s ) ( s - d ) 2 ’ y < s < d

By the definitions of these kernel functions, we can write b d. j Jp( x, t ) q{y, s) ^ dsdt a c x y = \ j j ( X ~ *)(< - a)(V - S)(S - C^9Qt2Qs2 dsdt a c x d (1.1) + ^ J j { x - t ) { t - a)(y - s ) ( s - d) ° *2 dsdt a y b y +

\f

J { x - t ) { t - b)(y - s ) ( s - c) d/ t} tds2] dsdt X C b d + \ j J ( x - t ) ( t - b)(y - s ) ( s - d) d-Jt ^ 2 dsdt x y

Integrating by parts in the right hand side of (1.1), we have

x y \ J J ( x - t)(.t - <*)(» - s)(s - c^dJ t2Qs2 dsdt a C = ^ - a^ y .. ^ [f(x, y) + f(x, c) + /(a, y) + f(a, c)] J [/(*, y) + f(t, c)]dt - £ [f(x,s) + f{a,s)]ds X y

II

f(t, s)dsdt. a c (1.2)

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550 Erhan Set and Mehmet Zeki Sarikaya

x d

\ J / ( * - *)(* - a)(y - s)(s - d^~dt^ds* dsdt

a y

(1-3) = — — — — [fix, y ) + f ( x , d) + /(a , y ) + /(a , d)}

+ f ( ^ y ) \ d t -

J

[ f ( x >s ) + f ( a >s )]d s x d + J J f{t,s)dsdt. a y (1.4) b y \ J J i x - * ) ( * - b)(y - s ) ( s - c) y ^ ^ - dsdt (y - c ) ( b - x ) [/(z, 2/) + f i x , c) + /(b, y) + /(&, c)] y ~ c 2 b y J [f(t,y) + f ( t , c ) \ d t - ^ - ^ - J [f(b,s) + f (x,s)]ds

+

J J f (t , s)dsdt. (1.5) b d J J \ x ~ t)(t - b)(y - s)(s - d)d4/(M ) dt2ds2 dsdt x y (id - y ) { b - x ) [f{x, y) + f i x , d) + f{b, y ) + fib, d)] d - y 2 b d J [/(M ) + f i t , y ) ] d t - h—^ - J [fib,s) + f i x, s)]ds + J J f i t , s)dsdt. x y

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O n a N ew O stro w sk i-T y p e In e q u a lity a n d R e la te d R esu lts 551 b d

j J

P{ x , t ) q ( y , s ) a c (■x - a ) ( y - c) d4f ( t , s ) dt 2ds2 dsdt = (1.6) + We also have b d [f(x, y ) + f ( x , c) + f (a, y) + /( a , c)] + —— — — l f ( x, y) + f ( x , d) + f {a, y) + f ( a , d )] + —— — ~ [/(*> y ) + / ( x >c) + / ( 6> v) + / ( b> c)l + — — — — t/(x’ 2/) + Z(x’d) + / ( &> y) + / ( fe> d)l

^~2~ fa + f(^c)}dt - Ja

L f M + /(*»»)]*

Jc

[f(x, s) + f { a , s ) ] d s - if(b,s) + f ( x , s ) ] ds r b p d / j f (t, s)dsdt. J a J c (!-7)

J J

p (x, t) q (y, s) dsdt = a c

Let M = Prom (1.6) and (1.7), we can write

J J

p (x, t) q (y, s) ^ / ( M ) (x - a)3 + (b - x f (y - c)3 + (d - y f 144 dt 2ds 2 M dsdt a c b d (1.8)

J J

_{p ( x , t ) q (y , s) dd ft}gsf dsdt}

r + 7

{x - a) + (b - x f (y - c)3 + ( d - y)c 144

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552 E rhan Set and M ehm et Zeki Sarikaya

On the other hand, we get b d (1.9)

J J

P ( x , t ) q ( y , i d4f (t , s ) dt2ds2 M dsdt < max (t,s)< E [a,6] x [c,d] d4f { t, s ) b d dt2ds2 M

J J

\P {x,t)q(y,s)\ dsdt Since (1.10) and max (£ ,s )€ [a ,6 ] x [c,d\ d4f ( t , s ) dt2ds2 M <

r-7

b d

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/ /

' \P (X' t ') q (y , S ') \dsdt = (x - a)3 + ( b - x f (y - c)3 + { d - y)“ 144 By using (1.10) and (1-11) in (1.9), we get

b d (1.12)

J J

P ( x , t ) q ( y , s ) d4f ( t , s) dt2ds2 M dsdt < {x - a)3 + (6 - x)3 (y - c)3 + {d - y)" 288

(r-7)-□

From (1.8) and (1.12), we see th at the required result holds.

Corollaty 7. By taking x = a and y — c or x = b and y = d in Theorem 6, we have (d — c) (b — a) d — c [/(a, c) + / ( a , d) + f(b, c) + f(b, d)]

J

[f(t,c) + f { t , d ) ] d t - h—^ - [f(a,s) + f{b,s)} ds

n

df ( t , s)dsdt — + (■b — a)3 (d — c)3 288 (F + 7) < ( b - a f ( d - c ) 3 _ 288 v '

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O n a New O stro w sk i-T y p e In e q u a lity a n d R e la te d R esu lts 553 {b —a ) { d - c ) a + b c + d +2 16 2 ’ 2 a H~ b , d ) + f [a, c + d + f ( b, + [/(a , c) + f(a, d) + f(b, c) + /(£>, d)} d — c b — a + ^ ) + / ( ( . cj

/

Jr

| / K s ) + 2/ 2

n

d f(t, s)dsdt (b — a)3(d — c)3 2 a + b +/(6,s) ds (& — a )3(d — c)3 16 x 288

(r + 7)

< 16 x 288

( r - 7).

c + d

R e fe r e n c e s

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[2] W-J. Liu, Q-L. Xue and S-F. Wang, Several new Perturbed Ostrowski-like type in equalities, J. Inequal. Pure and Appl. Math. (JIPAM), 8(4)(2007), Article: 110. [3] A. Ostrowski, Uber die Absolutabweichung einer differentierbaren Funktion von ihren

Integralmittelwert, Comment. Math. Helv., 10, 226-227, (1938).

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[5] Q-L. Xue, J. Zhu and W-J. Liu, A new generalization of Ostrowski-type inequality involving functions of two independent variables, Comp. Math. Appl., 60(2010), 2219- 2224.

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554 Erhan Set and Mehmet Zeki Sarikaya

[9] M. Z. Sarikaya and H. Ogunmez, On the weighted Ostrowski type integral inequal ity fo r double integrals, The Arabian Journal for Science and Engineering (AJSE)- Mathematics, 36(2011), 1153-1160.

[10] N. Ujevic, Some double integral inequalities and applications, Acta Math. Univ. Come- nianae, LXXI, 2(2002), 189-199.

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