# Inequality of Ostrowski Type for Mappings with Bounded Fourth Order Partial Derivatives

Authors: Waseem Ghazi Alshanti
Journal: Abstract and Applied Analysis
 (2019)

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## Abstract

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


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## Body

## 1. Introduction

In 1938, Ostrowski [1] introduced the following integral inequality.Theorem 1.
Letf: a,b→R be continuous mapping on a,b and differentiable on a,b, whose derivative f′:a,b→R is bounded on a,b, i.e., f′∞=supt∈a,b⁡f′t<∞, then for all x∈a,b(1)fx-1b-a∫abftdt≤14+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 [2] proposed the following generalization of (1) for a function f of several variables whose first order partial derivatives are bounded.Theorem 2.
Letf: Rm→R be a differentiable function defined on D¯ and let ∂f/∂xi≤MiMi>0;i=1,…,m in D. Then, for every X=x1,…,xm∈D¯,(2)fx1,…,xm-1∏i=1mbi-ai∫a1b1⋯∫ambmfy1,…,ymdy1⋯dym≤∑i=1m14+xi-ai+bi/22bi-ai2bi-aiMi.In 1998, Barnett and Dragomir [3] proved the following Ostrowski type inequality for mappings of two variables with bounded second order partial derivatives.Theorem 3.
Letf: a,b×c,d→R continuous on a,b×c,d, fx,y′′=∂2f/∂x∂y exists on a,b×c,d and is bounded, i.e., (3)fs,t′′∞=supx,y∈a,b×c,d∂2fx,y∂x∂y<∞,
Then we have the inequality(4)∫ab∫cdfs,tdsdt-b-a∫cdfx,tdt+d-c∫abfs,yds-d-cb-afx,y≤14b-a2+x-a+b2214d-c2+y-c+d22fs,t′′∞,for all x,y∈a,b×c,d.In [4], Xue et al. derive the following inequality of Ostrowski type.Theorem 4.
Letf: a,b×c,d→R be an absolutely continuous function such that the partial derivatives of order two exist and suppose that there exist constants γ,Γ∈R with γ≤∂2ft,s/∂t∂s≤Γ for all t,s∈a,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-λ∫abft,ydt+λ2∫abft,c+ft,ddt-1d-c1-λ∫cdfx,sds+λ2∫cdfa,s+fb,sds-Γ+γ21-λ2x-a+b2y-c+d2+1b-ad-c∫ab∫cdft,sdsdt≤Γ-γ21b-ad-cλ2+1-λ2b-a24+x-a+b22×λ2+1-λ2d-c24+y-c+d22,
for allx,y∈a+λb-a/2,b-λb-a/2×c+λd-c/2,d-λd-c/2 and λ∈0,1.More recently, Sarikaya et al. [5] establish weighted Ostrowski type inequalities considering function whose second order partial derivatives are bounded as follows.Theorem 5.
Letf: a,b×c,d→R be an absolutely continuous function such that the partial derivatives of order two exist and are bounded, i.e., (6)∂2ft,s∂t∂s∞=supx,y∈a,b×c,d∂2ft,s∂t∂s<∞,
for allt,s∈a,b×c,d.Then we have(7)m∘a,bm∘c,dx-μa,by-μc,dfx,y-m∘c,dy-μc,d∫ax∫atωuduft,ydt+∫xb∫btωuduft,ydt-m∘a,bx-μa,b∫cy∫csωvdvfx,sds+∫yd∫dsωvdvfx,sds-m∘a,bm∘c,d∫ab∫cdft,sdsdt≤m∘a,bm∘c,d4x-μa,b2+σ2a,by-μc,d2+σ2c,d∂2ft,s∂t∂s∞≤m∘a,bm∘c,d4x-a+b2+b-a22y-c+d2+d-c22∂2ft,s∂t∂s∞,
where(8)mia,b=∫abtiωtdt,i=0,1,…,μa,b=m1a,bm∘a,b,σ2a,b=m2a,bm∘a,b-μ2a,b.For other related work, we refer the reader to [6–15].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.
Letf: a,b×c,d→R be an absolutely continuous function such that the partial derivative of order 4 exists for all x,y∈a+hb-a/2,b-hb-a/2×c+hd-c/2,d-hd-c/2 and h∈0,1. Then for any two mappings K1(t;x):a,b×a,b→R and K2(s;y):c,d×c,d→R, where(9)K1t;x≔12t-a+hb-a22,t∈a,x12t-b-hb-a22,t∈x,band(10)K2s;y≔12s-c+hd-c22,t∈c,y12s-d-hd-c22,t∈y,d,the identity (11)Ef;h=∫ab∫cdK1t;xK2s;y∂4ft,s∂t2∂s2dsdt=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-c∫abft,ydt+b-a∫cdfx,sds-h2d-c∫abft,c+ft,ddt+b-a∫cdfa,s+fb,sds+∫ab∫cdft,sdsdt.
holds.Proof.
By definitions ofK1(t;x) and K2(s;y) in both (9) and (10), we have(12)Ef;h=∫ab∫cdK1t;xK2s;y∂4ft,s∂t2∂s2dsdt=14∫ax∫cyt-a+hb-a22s-c+hd-c22∂4ft,s∂t2∂s2dsdt+14∫ax∫ydt-a+hb-a22s-d-hd-c22∂4ft,s∂t2∂s2dsdt+14∫xb∫cyt-b-hb-a22s-c+hd-c22∂4ft,s∂t2∂s2dsdt+14∫xb∫ydt-b-hb-a22s-d-hd-c22∂4ft,s∂t2∂s2dsdt=I1+I2+I3+I4.
ForI1, integration by parts yields(13)I1=14∫ax∫cyt-a+hb-a22s-c+hd-c22∂4ft,s∂t2∂s2dsdt=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-c28∫xbt-b-hb-a2ftst,cdt-h2b-a28∫cys-c+hd-c2ftsb,sds+12∫cys-c+hd-c2x-b-hb-a22ftsx,sds-12∫xbt-b-hb-a2y-c+hd-c22ftst,ydt+∫ax∫cyt-a+hb-a2s-c+hd-c2ftst,sdsdt.
Similarly,I2, I3, and I4 can be obtained.
Thus, by addingI1, I2, I3, and I4, we easily deduce(14)Ef;h=∫ab∫cdK1t;xK2s;y∂4ft,s∂t2∂s2dsdt=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-a28∫cys-c+hd-c2ftsa,s-ftsb,sds+∫yds-d-hd-c2ftsa,s-ftsb,sds+∫ax∫cyt-a+hb-a2s-c+hd-c2ftst,sdsdt+∫ax∫ydt-a+hb-a2s-d-hd-c2ftst,sdsdt+∫xb∫cyt-b-hb-a2s-c+hd-c2ftst,sdsdt+∫xb∫ydt-b-hb-a2s-d-hd-c2ftst,sdsdt.
By further algebraic manipulations and assuming result by [4], the proof of Lemma 6 is completed.Theorem 7.
Letf: a,b×c,d→R such that f∈C4a,b×c,d be an absolutely continuous function such that the partial derivative of order 4 exists and is bounded; i.e., (15)∂4ft,s∂t2∂s2∞=supx,y∈a,b×c,d∂4ft,s∂t2∂s2<∞,for all t,s∈a,b×c,d. Then for all x,y∈a+hb-a/2,b-hb-a/2×c+hd-c/2,d-hd-c/2 and h∈0,1, we have(16)Ef;h≤h3b-a324+1-hb-a21-h2b-a212+x-a+b22×h3d-c324+1-hd-c21-h2d-c212+y-c+d22∂4ft,s∂t2∂s2∞,where the functional Ef;h is given by (11).Proof.
By considering (11), we have(17)Ef;h=∫ab∫cdK1t;xK2s;y∂4ft,s∂t2∂s2dsdt≤∫ab∫cdK1t;xK2s;y∂4ft,s∂t2∂s2dsdt≤∂4ft,s∂t2∂s2∞∫abK1t;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 Theorem7 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-y∫abfst,ydt+1d-ca+b2-x∫cdftx,sds-1b-a∫abft,ydt+1d-c∫cdfx,sds+1b-ad-c∫ab∫cdft,sdsdt≤14b-a212+x-a+b22d-c212+y-c+d22∂4ft,s∂t2∂s2∞.Corollary 9.
Under the assumption of Theorem7 with h=0, x=a+b/2, and y=c+d/2  we have(21)fa+b2,c+d2-1b-a∫abft,c+d2dt+1d-c∫cdfa+b2,sds+1b-ad-c∫ab∫cdft,sdsdt≤b-a2d-c2576∂4ft,s∂t2∂s2∞.Corollary 10.
Under the assumption of Theorem7 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-a∫abfst,c+d4dt+b-ad-c∫cdfta+b4,sds-1b-a∫abft,c+d4dt+1d-c∫cdfa+b4,sds+1b-ad-c∫ab∫cdft,sdsdt≤14b-a212+a+b216d-c212+c+d216∂4ft,s∂t2∂s2∞.Remark 11.
In Corollaries8, 9, and 10 we assume that the involved integrals can more easily be computed than the original double integral.



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*Source: 5648095-2019-03-03.xml*