AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi 10.1155/2019/5648095 5648095 Research Article Inequality of Ostrowski Type for Mappings with Bounded Fourth Order Partial Derivatives http://orcid.org/0000-0002-6403-3448 Alshanti Waseem Ghazi 1 Wong Patricia J. Y. Department of General Studies Jubail University College Saudi Arabia ucj.edu.sa 2019 332019 2019 19 10 2018 07 01 2019 15 01 2019 332019 2019 Copyright © 2019 Waseem Ghazi Alshanti. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A general Ostrowski’s type inequality for double integrals is given. We utilize function whose partial derivative of order four exists and is bounded.

1. Introduction

In 1938, Ostrowski  introduced the following integral inequality.

Theorem 1.

Let f: a,bR be continuous mapping on a,b and differentiable on a,b, whose derivative f:a,bR is bounded on a,b, i.e., f=supta,bft<, then for all xa,b(1)fx-1b-aabftdt14+x-a+b/22b-a2b-af.

The constant 1 / 4 is sharp in the sense that it cannot be replaced by a smaller one.

In 1975, Milovanović  proposed the following generalization of (1) for a function f of several variables whose first order partial derivatives are bounded.

Theorem 2.

Let f: RmR be a differentiable function defined on D¯ and let f/xiMiMi>0;i=1,,m in D. Then, for every X=x1,,xmD¯,(2)fx1,,xm-1i=1mbi-aia1b1ambmfy1,,ymdy1dymi=1m14+xi-ai+bi/22bi-ai2bi-aiMi.

In 1998, Barnett and Dragomir  proved the following Ostrowski type inequality for mappings of two variables with bounded second order partial derivatives.

Theorem 3.

Let f: a,b×c,dR continuous on a,b×c,d, fx,y=2f/xy exists on a,b×c,d and is bounded, i.e., (3)fs,t=supx,ya,b×c,d2fx,yxy<,

Then we have the inequality(4)abcdfs,tdsdt-b-acdfx,tdt+d-cabfs,yds-d-cb-afx,y14b-a2+x-a+b2214d-c2+y-c+d22fs,t,for all x,ya,b×c,d.

In , Xue et al. derive the following inequality of Ostrowski type.

Theorem 4.

Let f: a,b×c,dR be an absolutely continuous function such that the partial derivatives of order two exist and suppose that there exist constants γ,ΓR with γ2ft,s/tsΓ for all t,sa,b×c,d. Then we have(5)1-λ2fx,y+λ21-λfa,y+fb,y+fx,c+fx,d+λ22fa,c+fb,c+fa,d+fb,d-1b-a1-λabft,ydt+λ2abft,c+ft,ddt-1d-c1-λcdfx,sds+λ2cdfa,s+fb,sds-Γ+γ21-λ2x-a+b2y-c+d2+1b-ad-cabcdft,sdsdtΓ-γ21b-ad-cλ2+1-λ2b-a24+x-a+b22×λ2+1-λ2d-c24+y-c+d22,

for all x,ya+λb-a/2,b-λb-a/2×c+λd-c/2,d-λd-c/2 and λ0,1.

More recently, Sarikaya et al.  establish weighted Ostrowski type inequalities considering function whose second order partial derivatives are bounded as follows.

Theorem 5.

Let f: a,b×c,dR be an absolutely continuous function such that the partial derivatives of order two exist and are bounded, i.e., (6)2ft,sts=supx,ya,b×c,d2ft,sts<,

for all t,sa,b×c,d.Then we have(7)ma,bmc,dx-μa,by-μc,dfx,y-mc,dy-μc,daxatωuduft,ydt+xbbtωuduft,ydt-ma,bx-μa,bcycsωvdvfx,sds+yddsωvdvfx,sds-ma,bmc,dabcdft,sdsdtma,bmc,d4x-μa,b2+σ2a,by-μc,d2+σ2c,d2ft,stsma,bmc,d4x-a+b2+b-a22y-c+d2+d-c222ft,sts,

where (8)mia,b=abtiωtdt,i=0,1,,μa,b=m1a,bma,b,σ2a,b=m2a,bma,b-μ2a,b.

For other related work, we refer the reader to .

In this paper, motivated by the ideas in both [4, 5], we shall derive a new inequality of Ostrowski’s type similar to the inequalities (5) and (7), involving functions of two independent variables.

2. Main Results

In order to introduce our main results, we commence with the following lemma.

Lemma 6.

Let f: a,b×c,dR be an absolutely continuous function such that the partial derivative of order 4 exists for all x,ya+hb-a/2,b-hb-a/2×c+hd-c/2,d-hd-c/2 and h0,1. Then for any two mappings K1(t;x):a,b×a,bR and K2(s;y):c,d×c,dR, where(9)K1t;x12t-a+hb-a22,ta,x12t-b-hb-a22,tx,band(10)K2s;y12s-c+hd-c22,tc,y12s-d-hd-c22,ty,d,the identity (11)Ef;h=abcdK1t;xK2s;y4ft,st2s2dsdt=1-h2b-ad-cfx,y+a+b2-xc+d2-yftsx,y+a+b2-xftx,y+c+d2-yfsx,y+h21-hb-ad-c8d-ca+b2-xftsx,c-ftsx,d+b-ac+d2-yftsa,y-ftsb,y+h4b-a2d-c264ftsa,c-ftsa,d-ftsb,c+ftsb,d+h2b-a281-hd-cfta,y-ftb,y+hd-c2fta,c+ftb,c+fta,d-ftb,d-cdfta,s-ftb,sds+h2d-c281-hb-afsx,c-fsx,d+hb-a2fsa,c+fsb,c+fsa,d-fsb,d-abfst,c-fst,ddt+1-hd-cc+d2-yhb-a2fsa,y+fsb,y-abfst,ydt+1-hb-aa+b2-xhd-c2ftx,c+ftx,d-cdftx,sds+h1-hb-ad-c2fa,y+fb,y+fx,c+fx,d+h2b-ad-c4fa,c+fb,c+fa,d+fb,d-1-hd-cabft,ydt+b-acdfx,sds-h2d-cabft,c+ft,ddt+b-acdfa,s+fb,sds+abcdft,sdsdt.

holds.

Proof.

By definitions of K1(t;x) and K2(s;y) in both (9) and (10), we have(12)Ef;h=abcdK1t;xK2s;y4ft,st2s2dsdt=14axcyt-a+hb-a22s-c+hd-c224ft,st2s2dsdt+14axydt-a+hb-a22s-d-hd-c224ft,st2s2dsdt+14xbcyt-b-hb-a22s-c+hd-c224ft,st2s2dsdt+14xbydt-b-hb-a22s-d-hd-c224ft,st2s2dsdt=I1+I2+I3+I4.

For I1, integration by parts yields(13)I1=14axcyt-a+hb-a22s-c+hd-c224ft,st2s2dsdt=14t-a+hb-a22s-c+hd-c22ftsx,y-h216d-c2x-a+hb-a22ftsx,c+b-a2y-c+hd-c22ftsa,y+h4b-a2d-c264ftsa,c+h2d-c28xbt-b-hb-a2ftst,cdt-h2b-a28cys-c+hd-c2ftsb,sds+12cys-c+hd-c2x-b-hb-a22ftsx,sds-12xbt-b-hb-a2y-c+hd-c22ftst,ydt+axcyt-a+hb-a2s-c+hd-c2ftst,sdsdt.

Similarly, I2, I3, and I4 can be obtained.

Thus, by adding I1, I2, I3, and I4, we easily deduce(14)Ef;h=abcdK1t;xK2s;y4ft,st2s2dsdt=14h-1b-aa+b-2xh-1d-cc+d-2yftsx,y-h216d-c2h-1b-aa+b-2xftsx,c-ftsx,d-h216b-a2h-1d-cc+d-2yftsa,y-ftsb,y+h4b-a2d-c264ftsa,c-ftsa,d-ftsb,c+fb,d+h2b-a28cys-c+hd-c2ftsa,s-ftsb,sds+yds-d-hd-c2ftsa,s-ftsb,sds+axcyt-a+hb-a2s-c+hd-c2ftst,sdsdt+axydt-a+hb-a2s-d-hd-c2ftst,sdsdt+xbcyt-b-hb-a2s-c+hd-c2ftst,sdsdt+xbydt-b-hb-a2s-d-hd-c2ftst,sdsdt.

By further algebraic manipulations and assuming result by , the proof of Lemma 6 is completed.

Theorem 7.

Let f: a,b×c,dR such that fC4a,b×c,d be an absolutely continuous function such that the partial derivative of order 4 exists and is bounded; i.e., (15)4ft,st2s2=supx,ya,b×c,d4ft,st2s2<,for all t,sa,b×c,d. Then for all x,ya+hb-a/2,b-hb-a/2×c+hd-c/2,d-hd-c/2 and h0,1, we have(16)Ef;hh3b-a324+1-hb-a21-h2b-a212+x-a+b22×h3d-c324+1-hd-c21-h2d-c212+y-c+d224ft,st2s2,where the functional Ef;h is given by (11).

Proof.

By considering (11), we have(17)Ef;h=abcdK1t;xK2s;y4ft,st2s2dsdtabcdK1t;xK2s;y4ft,st2s2dsdt4ft,st2s2abK1t;xdt·cdK2s;yds.

But,(18)abK1t;xdt=h324b-a3+1-hb-a21-h2b-a212+x-a+b22,

and(19)cdK2s;yds=h324d-c3+1-hd-c21-h2d-c212+y-c+d22.

Now, substituting (18), (19) into (17) gives (16) and, hence, completes the proof.

Corollary 8.

Under the assumption of Theorem 7 with h=0, we have(20)fx,y+a+b2-xc+d2-yftsx,y+a+b2-xftx,y+c+d2-yfsx,y-1b-ac+d2-yabfst,ydt+1d-ca+b2-xcdftx,sds-1b-aabft,ydt+1d-ccdfx,sds+1b-ad-cabcdft,sdsdt14b-a212+x-a+b22d-c212+y-c+d224ft,st2s2.

Corollary 9.

Under the assumption of Theorem 7 with h=0, x=a+b/2, and y=c+d/2  we have(21)fa+b2,c+d2-1b-aabft,c+d2dt+1d-ccdfa+b2,sds+1b-ad-cabcdft,sdsdtb-a2d-c25764ft,st2s2.

Corollary 10.

Under the assumption of Theorem 7 with h=0, x=a+b/4, and y=c+d/4  we have(22)fa+b4,c+d4+a+bc+d16ftsa+b4,c+d4+a+b4fta+b4,c+d4+c+d4fsa+b4,c+d4-14d-cb-aabfst,c+d4dt+b-ad-ccdfta+b4,sds-1b-aabft,c+d4dt+1d-ccdfa+b4,sds+1b-ad-cabcdft,sdsdt14b-a212+a+b216d-c212+c+d2164ft,st2s2.

Remark 11.

In Corollaries 8, 9, and 10 we assume that the involved integrals can more easily be computed than the original double integral.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares no conflicts of interest.

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