C om mun.Fac.Sci.U niv.A nk.Series A 1 Volum e 66, N umb er 2, Pages 53–61 (2017) D O I: 10.1501/C om mua1_ 0000000800 ISSN 1303–5991
http://com munications.science.ankara.edu.tr/index.php?series= A 1
ON WEIGHTED GRÜSS TYPE INEQUALITIES FOR DOUBLE INTEGRALS
HÜSEYIN BUDAK AND MEHMET ZEKI SARIKAYA
Abstract. In this study, we obtain some new weighted inequalities of Grüss type for functions of two independent variables. Special cases of the results presented here reduce the results given in earlier works.
1. Introduction In 1935, G. Grüss [3] proved the following inequality:
1 b a b Z a f (x)g(x)dx 1 b a b Z a f (x)dx 1 b a b Z a g(x)dx 1 4( ')( ); (1.1)
provided that f and g are two integrable function on [a; b] satisfying the condition
' f (x) and g(x) for all x 2 [a; b]: (1.2)
The constant 1
4 is best possible.
In 1882, P. L. µCebyšev [1] gave the following inequality:
jT (f; g)j 121 (b a)2kf0k1kg0k1; (1.3) where f; g : [a; b] ! R are absolutely continuous function, whose …rst derivatives f0
and g0 are bounded,
T (f; g) = 1 b a b Z a f (x)g(x)dx 0 @ 1 b a b Z a f (x)dx 1 A 0 @ 1 b a b Z a g(x)dx 1 A (1.4) and k:k1 denotes the norm in L1[a; b] de…ned as kpk1= ess sup
t2[a;b] jp(t)j :
The following result of weighted Grüss type was proved by Dragomir [2]: Received by the editors: February 9, 2016; Accepted: October 15, 2016.
2010 Mathematics Subject Classi…cation. Primary 26D15; Secondary 26A51.
Key words and phrases. Weighted Grüss type inequality, double integrals, two independent variables.
c 2 0 1 7 A n ka ra U n ive rsity C o m m u n ic a tio n s d e la Fa c u lté d e s S c ie n c e s d e l’U n ive rs ité d ’A n ka ra . S é rie s A 1 . M a th e m a t ic s a n d S t a tis tic s .
Theorem 1. Let f and g be two functions de…ned and integrable on [a; b] : If (1.2) holds, where '; ; and are given real constant, and w : [a; b] ! [0; 1) is integrable and b R a w(x)dx > 0; then b Z a w(x)dx: b Z a f (x)g(x)dx b Z a f (x)w(x)dx: b Z a g(x)w(x)dx (1.5) 1 4( ')( ) 0 @ b Z a w(x)dx 1 A 2
and the constant 14 is the best possible.
In the last years, many authors were interested in the generalization of Grüss type inequalities for mapping of one variable, we can mention the works [2], [4]-[8]. Recently, Sarikaya and Kiris have proved the following Grüss type inequality for double integrals in [9]:
Theorem 2. Let f; g : [a; b] [c; d] ! R be two functions de…ned and integrable on [a; b] [c; d] : Then for
' f (x; y) and g(x; y) for all (x; y) 2 [a; b] [c; d] we have 1 (b a) (d c) b Z a d Z c f (x; y)g(x; y)dydx (1.6) 0 @ 1 (b a) (d c) b Z a d Z c f (x; y)dydx 1 A 0 @ 1 (b a) (d c) b Z a d Z c g(x; y)dydx 1 A 1 4( ')( ):
In this study, we establish some new inequalities of weighted Grüss type involving functions of two independent variables for double integrals.
2. Main Results
Throughout this work, we assume that the weight function h : [a; b] [c; d] ! [0; 1) is integrable, nonnegative and satis…es
m(a; b; c; d) = b Z a d Z c h(x; y)dydx < 1:
De…nition 1. Consider a function f : V ! R de…ned on a subset V of Rn; n 2 N.
Let L = (L1; L2; :::; Ln) where Li 0; i = 1; 2; :::; n: We say that f is L-Lipschitzian
function if jf(x) f (y)j n X i=1 Lijxi yij for all x; y 2 V [9]:
Theorem 3. Let h be as above and let f; g : [a; b] ! R be two functions de…ned and integrable on [a; b] [c; d]. If
' f (x; y) and g(x; y) for all x 2 [a; b] [c; d] ; (2.1) then we have b R a d R c h(x; y)dydx ! b R a d R c
f (x; y)g(x; y)h(x; y)dydx
b R a d R c f (x; y)h(x; y)dydx ! b R a d R c g(x; y)h(x; y)dydx ! 1 4( ')( ) b R a d R c h(x; y)dydx !2 : (2.2) The constant1
4 is the best possible in the sense that it cannot be replaced by smaller
one.
Proof. For mappings f; g : [a; b] [c; d] ! R, we have the identity
b R a d R c b R a d R c
[f (x; s) f (t; y)] [g(x; s) g(t; y)] h(t; y)h(x; s)dsdtdydx
= b R a d R c b R a d R c
[f (x; s)g(x; s)h(t; y)h(x; s) f (x; s)g(t; y)h(t; y)h(x; s) f (t; y)g(x; s)h(t; y)h(x; s) + f (t; y)g(t; y)h(t; y)h(x; s)] dsdtdydx
= 2 b R a d R c h(t; y)dydt b R a d R c f (x; s)g(x; s)h(x; s)dsdx 2 b R a d R c f (x; s)h(x; s)dsdx ! b R a d R c g(t; y)h(t; y)dydt ! : (2.3)
Appling Cauchy-Buniakowski-Schwarz’s inequality, we have the inequality " 1 2[m(a;b;c;d)]2 b R a d R c b R a d R c
[f (x; s) f (t; y)] [g(x; s) g(t; y)] h(t; y)h(x; s)dsdtdydx #2 1 2[m(a;b;c;d)]2 b R a d R c b R a d R c
[f (x; s) f (t; y)]2h(t; y)h(x; s)dsdtdydx ! 1 2[m(a;b;c;d)]2 b R a d R c b R a d R c
[g(x; s) g(t; y)]2h(t; y)h(x; s)dsdtdydx ! = 2 4 1 m(a;b;c;d) b R a d R c f2(x; s)h(x; s)dsdx 1 m(a;b;c;d) b R a d R c f (x; s)h(x; s)dsdx !23 5 2 4 1 m(a;b;c;d) b R a d R c g2(x; s)h(x; s)dsdx m(a;b;c;d)1 b R a d R c g(x; s)h(x; s)dsdx !23 5 : (2.4) It is easy to observe that
1 m(a;b;c;d) b R a d R c f2(x; s)h(x; s)dsdx 1 m(a;b;c;d) b R a d R c f (x; s)h(x; s)dsdx !2 = m(a;b;c;d)1 b R a d R c f (x; s)h(x; s)dsdx ! 1 m(a;b;c;d) b R a d R c f (x; s)h(x; s)dsdx ' ! 1 m(a;b;c;d) b R a d R c [ f (x; s)] [f (x; s) '] h(x; s)dsdx:
Since [ f (x; s)] [f (x; s) '] 0 for each (x; s) 2 [a; b] [c; d] ; then we get 1 m(a;b;c;d) b R a d R c f2(x; s)h(x; s)dsdx 1 m(a;b;c;d) b R a d R c f (x; s)h(x; s)dsdx !2 1 m(a;b;c;d) b R a d R c f (x; s)h(x; s)dsdx ! 1 m(a;b;c;d) b R a d R c f (x; s)h(x; s)dsdx ' ! : (2.5) Similarly, we obtain 1 m(a;b;c;d) b R a d R c g2(x; s)h(x; s)dsdx 1 m(a;b;c;d) b R a d R c g(x; s)h(x; s)dsdx !2 1 m(a;b;c;d) b R a d R c g(x; s)h(x; s)dsdx ! 1 m(a;b;c;d) b R a d R c g(x; s)h(x; s)dsdx ! : (2.6)
Using (2.5) and (2.6) in (2.4), we get the following inequality " 1 2[m(a;b;c;d)]2 b R a d R c b R a d R c
[f (x; s) f (t; y)] [g(x; s) g(t; y)] dsdtdydx #2 1 m(a;b;c;d) b R a d R c f (x; s)h(x; s)dsdx ! 1 m(a;b;c;d) b R a d R c f (x; s)h(x; s)dsdx ' ! 1 m(a;b;c;d) b R a d R c g(x; s)h(x; s)dsdx ! 1 m(a;b;c;d) b R a d R c g(x; s)h(x; s)dsdx ! : (2.7) Now, using the elementary inequality for real numbers
we get " 1 2[m(a;b;c;d)]2 b R a d R c b R a d R c
[f (x; s) f (t; y)] [g(x; s) g(t; y)] h(t; y)h(x; s)dsdtdydx #2
1
16( ')2( )2
which completes the proof. To prove the sharpness of (2.2), let choose h(x; y) = 1 and f (x; y) = g(x; y) = 8 > > < > > : 1; a x < a+b2 ; c y < c+d2 1; a x < a+b2 ; c+d2 y d 1; a+b2 x b; c y <c+d2 1; a+b2 x b; c+d2 y d then b Z a d Z c h(x; y)dydx = (b a)(d c); b Z a d Z c
f (x; y)g(x; y)h(x; y)dydx = (b a) (d c) ,
b Z a d Z c f (x; y)h(x; y)dydx = b Z a d Z c g(x; y)h(x; y)dydx = 0 and ( ') = ( ) = 2
which the equality (2.2) is realized. This implies the constant 14 is the best possible.
Remark 1. If we choose h(x; y) = 1 in (2.2), then the inequality (2.2) reduces the inequality (1.6).
The following inequality of weighted Gruss type for Lipschitzian mappings holds: Theorem 4. Let f; g : R2 ! R satis…es L-Lipschitzian conditions. That is,
for (x; s) and (t; y) belong to := [a; b] [c; d] ; then we have jf(x; s) f (t; y)j L1jx tj + L2js yj
where L1; L2; L3 and L4 are nonnegative constants. Then, we have the following inequality: b R a d R c h(x; y)dydx ! b R a d R c f (x; y)g(x; y)dydx b R a d R c f (x; y)h(x; y)dydx ! b R a d R c g(x; y)h(x; y)dydx ! L1L3 2 4m(a; b; c; d)Rb a d R c x2h(x; y)dydx Rb a d R c xh(x; y)dydx !23 5 +L2L4 2 4m(a; b; c; d)Rb a d R c y2h(x; y)dydx Rb a d R c yh(x; y)dydx !23 5 +L1L4+L2L3 2 b R a d R c A(x; y)dxdy (2.8) where A(x; y) = b Z a d Z c jx tj jy sj h(t; y)h(x; s)dsdt:
Proof. Since f; g are L-Lipschitzian, we have
j[f(x; s) f (t; y)] [g(x; s) g(t; y)]j
for all (x; s), (t; y) 2 := [a; b] [c; d] : Then, we have 1 2 b R a d R c b R a d R c
[f (x; s) f (t; y)] [g(x; s) g(t; y)] h(t; y)h(x; s)dsdtdydx
1 2 b R a d R c b R a d R c j[f(x; s)
f (t; y)] [g(x; s) g(t; y)]j h(t; y)h(x; s)dsdtdydx
1 2 b R a d R c b R a d R c L1L3(x t)2+ L2L4(s y)2
+(L1L4+ L2L3) jx tj js yj] h(t; y)h(x; s)dsdtdydx
= L1L3 2 4m(a; b; c; d)Rb a d R c x2h(x; y)dydx Rb a d R c xh(x; y)dydx !23 5 +L2L4 2 4m(a; b; c; d)Rb a d R c y2h(x; y)dydx Rb a d R c yh(x; y)dydx !23 5 +L1L4+L2L3 2 b R a d R c A(x; y)dxdy which completes the proof.
Remark 2. If we choose h(x; y) = 1 in the inequality (2.8), then we have the inequality 1 (b a)(d c) b R a d R c f (x; y)g(x; y)dydx 1 (b a)(d c) b R a d R c f (x; y)dydx ! 1 (b a)(d c) b R a d R c g(x; y)dydx ! (b a)2 12 L1L3+ (d c)2 12 L2L4+ (b a)(d c) 18 (L1L4+ L2L3)
which was given by Sarikaya and Kiri¸s in [9]. References
[1] P. L. µCebyšev, Sur less expressions approximatives des integrales de…nies par les autres prises entre les memes limites, Proc. Math. Soc. Charkov, 2, 93-98, 1882.
[2] S. S. Dragomir , Some integral inequalities of Grüss type, Indian J. Pure Appl. Math., 31(4), 397-415, April 2000.
[3] G. Grüss, Über das maximum des absoluten Betrages von 1 b a b R a f (x)g(x)dx 1 (b a)2 b R a f (x)dx b R a g(x)dx;Math. Z., 39, 215-226, 1935.
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[5] B. G. Pachpatte, On µCebyšev-Grüss type inequalities via Pecaric’s extention of the Mont-gomery identity, J. Inequal. Pure and Appl. Math. 7(1), Art 108, 2006.
[6] J. E. Pecaric and B. Tebes, On Grüss type inequalities Dragomir and Fedotov, J. Inequal. Pure and Appl. Math. 4(5), Art 91, 2003.
[7] M. Z. Sarikaya, N. Aktan, H. Y¬ld¬r¬m, Weighted µCebyšev-Grüss type inequalities on time scales, J. Math. Inequal. 2, 2 (2008), 185–195.
[8] M. Z. Sarikaya, A Note on Grüss type inequalities on time scales, Dynamic Systems and Applications, 17 (2008), 663-666.
[9] M. Z. Sarikaya and M. E. Kiris, On µCebysev-Grüss type inequalities for double integrals, TJMM, 7(2015), No.1, 75-83.
Current address : H. Budak: Düzce University, Faculty of Science and Arts, Department of Mathematics, Konuralp Campus, Düzce, Turkey.
E-mail address : hsyn.budak@gmail.com
Current address : M. Z. Sar¬kaya: Düzce University, Faculty of Science and Arts, Department of Mathematics, Konuralp Campus, Düzce, Turkey.
