# Generalizedn Dimensional Ostrowski Type and Grüss Type Inequalities on Time Scales

Authors: Bin Zheng; Qinghua Feng
Journal: Journal of Applied Mathematics
 (2014)

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## Abstract

Some new generalizedn dimensional Ostrowski type and Grüss type integral inequalities on time scales are established in this paper. The present results unify continuous and discrete analysis, and extend some known results in the literature.


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## Body

## 1. Introduction

In recent years, the research for the Ostrowski type and Grüss type inequalities has been an interesting topic in the literature. The Ostrowski type inequality can be used to estimate the absolute deviation of a function from its integral mean, and it was originally presented by Ostrowski in [1] as follows.Theorem 1.
Letf:I→R be a differentiable mapping in the interior IntI of I, where I⊂R is an interval, and let a,b∈IntI. a<b. If |f′(t)|≤M, for all t∈[a,b], then one has
(1)|f(x)-1b-a∫abf(t)dt|≤[14+(x-((a+b)/2))2(b-a)2](b-a)M,≤[14+(x-((a+b)/2))2(b-a)2]forx∈[a,b].
The Grüss inequality, which can be used to estimate the absolute deviation of the integral of the product of two functions from the product of their respective integral, was originally presented by Grüss in [2] as follows.Theorem 2.
Letf,g be integrable functions on [a,b] and satisfy the condition m≤f(x)≤M, n≤g(x)≤N. Then one has
(2)|1b-a∫abf(x)g(x)dx-(1b-a∫abf(x)dx)×(1b-a∫abg(x)dx)|≤14(M-m)(N-n).
In the last few decades, various generalizations of the Ostrowski inequality and the Grüss inequality including continuous and discrete versions have been established (e.g., see [3–14] and the references therein). On the other hand, Hilger [15] initiated the theory of time scales as a theory capable of treating continuous and discrete analysis in a consistent way, based on which some authors have studied the Ostrowski type and Grüss type inequalities on time scales. For example, in [16], Bohner and Matthews established the following Ostrowski type inequality on time scales for the first time.Theorem 3 (see [16, Theorem 3.5]).
Leta,b,s,t∈𝕋, a<b, and let f:[a,b]→R be differentiable, where 𝕋 is an arbitrary time scale. Then
(3)|f(t)-1b-a∫abfσ(s)Δs|≤Mb-a(h2(t,a)+h2(t,b)),
where M=sup⁡a<t<b⁡|fΔ(t)|. This inequality is sharp in the sense that the right-hand side of it cannot be replaced by a smaller one.In [17], Özkan and Yildirim established the following Grüss type inequality for double integrals on time scales.Theorem 4 (see [17, Theorem 2.2]).
Leta,b∈𝕋1, c,d∈𝕋2 and f,g∈CCrd1([a,b]×[c,d],R), where 𝕋1  and  𝕋2 are two arbitrary time scales. Then
(4)|1k∫∫Rf(x,y)g(x,y)Δ2yΔ1x+12(1k‍∫∫Rf(x,y)Δ2yΔ1x)×(1k∫∫Rg(σ1(x),σ2(y))Δ2yΔ1x)+12(1k∫∫Rf(σ1(x),σ2(y))Δ2yΔ1x)×(1k∫∫Rg(x,y)Δ2yΔ1x)-12k2∫∫R(g(x,y)F(x,y)+f(x,y)G(x,y))Δ2yΔ1x|≤12k2∫∫RM2(x,y)H1(x)H2(y)Δ2yΔ1x,
where  k=(b-a)(d-c),  F(x,y)=(d-c)∫abf(σ1(t),y)Δ1t+(b-a)∫cdf(x,σ2(s))Δ2s, and  G(x,y)=(d-c)∫abg(σ1(t),y)Δ1t+(b-a)∫cdg(x,σ2(s))Δ2s.
In [18], Liu and Ngô established an Ostrowski-Grüss type inequality in one independent variable as follows.Theorem 5 (see [18, Theorem 4]).
Supposea,b,s,t∈𝕋 and f:[a,b]→R is differentiable, where 𝕋 is an arbitrary time scale, fΔ is rd-continuous, and γ≤fΔ(t)≤Γ,∀t∈[a,b]. Then one has
(5)|f(t)-1b-a∫abfσ(s)Δs-f(b)-f(a)(b-a)2(h2(t,a)-h2(t,b))|≤14(b-a)(Γ-γ),∀t∈[a,b].Other results on the Ostrowski type and Grüss type inequalities on time scales can be found in [19–25]. These inequalities on time scales unify continuous and discrete analysis and can be used to provide explicit error bounds for some known and some new numerical quadrature formulae.Motivated by the above works, in this paper, we establish some new generalized Ostrowski type and Grüss type inequalities on time scales involving functions of n independent variables, which extend some known results in the literature ton dimensional case.We first give the following definition for further use.Definition 6.
h
k
:
𝕋
2
→
ℝ, k=0,1,2,… are defined by
(6)hk+1(t,s)=∫sthk(τ,s)Δτ,∀s,t∈𝕋,
where 𝕋 is an arbitrary time scale and h0(t,s)=1.Throughout this paper,ℝ denotes the set of real numbers and ℝ+=[0,∞), while ℤ denotes the set of integers and ℕ0 denotes the set of nonnegative integers. For a function f and two integers m0  and  m1, one has ∑s=m0m1f=0, provided that m0>m1. 𝕋i denotes an arbitrary time scale, i=1,2,…,n. For an interval [a,b], [a,b]𝕋i:=[a,b]⋂‍𝕋i, i=1,2,…,n.


## 2. Main Results

For the sake of convenience, we present the following notations:(7)L=∏i=1n(bi-ai),Lk=L∏i=1k(bmi-ami),A1(x1,x2,…,xn)=∑k=1n-1(-1)k+1Lk×{∑1≤m1≤m2⋯≤mk≤n∫am1bm1∫am2bm2⋯∫amkbmkf(x1,…,σm1(sm1),…,σm2(sm2),…,σmk(smk),…,xn)×Δmksmk,…,Δm2sm2Δm1sm1∑1≤m1≤m2⋯≤mk≤n∫am1bm1∫am2bm2},A2(x1,x2,…,xn)=∑k=1n-1(-1)k+1Lk×{∑1≤m1≤m2…≤mk≤n∫am1bm1∫am2bm2⋯∑1≤m1≤m2⋯≤mk≤n∫am1bm1∫am2bm2⋯∫amkbmkg(x1,…,σm1(sm1),…,σm2(sm2),…,σmk(smk),…,xn)×Δmksmk,…,Δm2sm2Δm1sm1∑1≤m1≤m2…≤mk≤n∫am1bm1∫am2bm2…},A3(x1,x2,…,xn)=(-1)n+1{∫a1b1∫a2b2⋯∫an-1bn-1∫anbnf(σ1(s1),f(x1,x2,…,xn)×∫a1b1∫a2b2⋯∫an-1bn-1∫anbng(σ1(s1),σ2(s2),…,σn-1(sn-1),σn(sn))×ΔnsnΔn-1sn-1,…,Δ2s2Δ1s1+g(x1,x2,…,xn)×∫a1b1∫a2b2⋯∫an-1bn-1∫anbnf(σ1(s1),σ2(s2),…,σn-1(sn-1),σn(sn))×ΔnsnΔn-1sn-1,…,Δ2s2Δ1s1∫a1b1∫a2b2⋯∫an-1bn-1∫anbnf(σ1(s1),}.Lemma 7 (generalizedn dimensional Montgomery identity).
Let(8)pi(xi,si)={si-ai,si∈[ai,xi],si-bi,si∈(xi,bi],i=1,2,…,n.
Then one has
(9)f(x1,x2,…,xn)L-A1(x1,x2,…,xn)=∫a1b1∫a2b2⋯∫an-1bn-1∫anbn∂nf(s1,s2,…,sn-1,sn)Δ1s1Δ2s2,…,Δn-1xn-1Δnxn×∏i=1npi(xi,si)ΔnsnΔn-1sn-1,…,Δ2s2Δ1s1+(-1)n+1×∫a1b1∫a2b2⋯∫an-1bn-1∫anbnf(σ1(s1),σ2(s2),…,σn-1(sn-1),σn(sn))×ΔnsnΔn-1sn-1,…,Δ2s2Δ1s1.Proof.
Suppose thatn=2. Then we have
(10)∫a1b1∫a2b2∂2f(s1,s2)Δ1s1Δ2s2∏i=12pi(xi,si)Δ2s2Δ1s1=∫a1b1‍(b2-a2)∂f(s1,x2)Δ1s1p1(x1,s1)Δ1s1-∫a1b1‍[p1(x1,s1)∫a2b2‍∂f(s1,σ2(s2))Δ1s1Δ2s2]Δ1s1=∫a1b1‍(b2-a2)∂f(s1,x2)Δ1s1p1(x1,s1)Δ1s1-∫a2b2‍∫a1b1‍[p1(x1,s1)∂f(s1,σ2(s2))Δ1s1Δ1s1]Δ2s2=(b2-a2)(b1-a1)f(x1,x2)-(b2-a2)×∫a1b1‍f(σ1(s1),x2)Δ1s1-(b1-a1)∫a2b2‍f(x1,σ2(s2))Δ2s2+∫a1b1‍∫a2b2‍f(σ1(s1),σ2(s2))Δ2s2Δ1s1.
That is, (9) holds for n=2.Suppose that (9) holds for n-1. Then for n we have
(11)∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn‍∂nf(s1,2,…,sn-1,sn)Δ1s1Δ2s2,…,Δn-1sn-1Δnsn×∏i=1npi(xi,si)ΔnsnΔn-1sn-1,…,Δ2s2Δ1s1=∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍(bn-an)∂n-1f(s1,s2,…,sn-1,xn)Δ1s1Δ2s2,…,Δn-1sn-1×∏i=1n-1pi(xi,si)Δn-1sn-1,…,Δ2s2Δ1s1-∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍[∏i=1n-1pi(xi,si)×∫anbn∂n-1f(s1,s2,…,sn-1,σn(sn))Δ1s1Δ2s2,…,Δn-1sn-1×Δnsn∏i=1n-1pi(xi,si)]×Δn-1sn-1,…,Δ2s2Δ1s1=(bn-an)×∫a1b1‍∫a2b2‍…∫an-1bn-1‍∂n-1f(s1,s2,…,sn-1,xn)Δ1s1Δ2s2,…,Δn-1sn-1×∏i=1n-1pi(xi,si)Δn-1sn-1,…,Δ2s2Δ1s1-∫anbn‍[∏∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∂n-1f(s1,s2,…,sn-1,σn(sn))Δ1s1Δ2s2,…,Δn-1sn-1×∏i=1n-1pi(xi,si)Δn-1sn-1,…,Δ2s2Δ1s1]Δnsn=(bn-an){∏i=1n-1(bi-ai)+∑‍k=1n-2(-1)kLkbn-anf(x1,x2,…,xn-1,xn)×∏i=1n-1(bi-ai)+∑‍k=1n-2(-1)kLkbn-an×[∑1≤m1≤m2⋯≤mk≤n-1∫am1bm1‍∫am2bm2‍⋯∫amkbmk‍f(x1,…,σm1(sm1),…,σm2(sm2),…,σmk(smk)…,xn-1,xn)×Δmksmk,…,Δm2sm2Δm1sm1∑1≤m1≤m2⋯≤mk≤n-1∫am1bm1‍∫am2bm2‍⋯]+(-1)n-1×∫a1b1‍∫a2b2‍…∫an-1bn-1‍f(σ1(s1),σ2(s2),…,σn-1(sn-1),xn)×Δn-1sn-1,…,Δ2s2Δ1s1∑1≤m1≤m2⋯≤mk≤n-1∫am1bm1‍∫am2bm2‍⋯}-∫anbn‍{Δmksmk,…,Δm2sm2Δm1sm1∑‍1≤m1≤m2⋯≤mk≤n-1∫am1bm1‍∫am2bm2]f(x1,x2,…,xn-1,σn(sn))×∏i=1n-1(bi-ai)+∑‍k=1n-2(-1)kLkbn-an×[∑‍1≤m1≤m2⋯≤mk≤n-1∫am1bm1‍∫am2bm2⋯∫amkbmk‍f(x1,…,σm1(sm1),…,σm2(sm2),…,σmk(smk),…,xn-1,σn(sn))×Δmksmk,…,Δm2sm2Δm1sm1∑‍1≤m1≤m2⋯≤mk≤n-1∫am1bm1‍∫am2bm2]+(-1)n-1∫a1b1‍∫a2b2‍…∫an-1bn-1‍f(σ1(s1),σ2(s2),…,σn-1(sn-1),σn(sn))×Δn-1sn-1,…,Δ2s2Δ1s1Δmksmk,…,Δm2sm2Δm1sm1∑‍1≤m1≤m2⋯≤mk≤n-1∫am1bm1‍∫am2bm2]}Δnsn=f(x1,x2,…,xn)∏i=1n(bi-ai)+∑k=1n-1(-1)kLk×{∑‍1≤m1≤m2⋯≤mk≤n∫am1bm1‍∫am2bm2‍⋯∫amkbmk‍f(x1,…,σm1(sm1),…,σm2(sm2),…,σmk(smk),…,xn)×Δmksmk,…,Δm2sm2Δm1sm1∑‍1≤m1≤m2⋯≤mk≤n∫am1bm1‍∫am2bm2‍}+(-1)n×∫a1b1∫a2b2⋯∫an-1bn-1‍∫anbn‍f(σ1(s1),σ2(s2),…,σn-1(sn-1),σn(sn))×ΔnsnΔn-1sn-1,…,Δ2s2Δ1s1,
which is the desired result.Remark 8.
Lemma7 is the n dimensional extension of [17, Lemma 2.3].
Based on Lemma7, we present two generalized n dimensional Ostrowski type inequalities on time scales.Theorem 9.
Letai,bi∈𝕋i,xi∈[ai,bi]𝕋i, i=1,2,…,n, and f∈Crd([a1,b1]𝕋1×[a2,b2]𝕋2×⋯×[an,bn]𝕋n,ℝ) such that the partial delta derivative of order n exists and there exists a constant K such that sup⁡ai<xi<bi,i=1,2,…,n⁡|∂nf(x1,x2,…,xn)/(Δ1x1Δ2x2,…,Δnxn)| =K. Then one has
(12)|∑k=1n-1(-1)k+1Lkf(x1,x2,…,xn)L-∑k=1n-1(-1)k+1Lk×{∑1≤m1≤m2⋯≤mk≤n∫am1bm1‍∫am2bm2‍⋯∑1≤m1≤m2⋯≤mk≤n∫am1bm1‍∫am2bm2‍…∫amkbmkf(x1,…,σm1(sm1),…,σm2(sm2),…,σmk(smk),…,xn)×Δmksmk,…,Δm2sm2Δm1sm1∑1≤m1≤m2⋯≤mk≤n∫am1bm1‍∫am2bm2‍⋯}+(-1)n×∫a1b1‍∫a2b2‍…∫an-1bn-1∫anbnf(σ1(s1),σ2(s2),‍…,σn-1(sn-1),σn(sn))×ΔnsnΔn-1sn-1,…,Δ2s2Δ1s1∑k=1n-1(-1)k+1Lk|≤K∏i=1n[h2(xi,ai)+h2(xi,bi)].
The inequality (12) is sharp in the sense that the right-hand side of it cannot be replaced by a smaller one.Proof.
From the definition ofpi(xi,si) we can obtain
(13)∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn‍|∏i=1npi(xi,si)|×ΔnsnΔn-1sn-1,…,Δ2s2Δ1s1=∏i=1n[h2(xi,ai)+h2(xi,bi)].
Then by Lemma 7 we have
(14)|∫f(x1,x2,…,xn)L-A1(x1,x2,…,xn)+(-1)n×∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbnf(σ1(s1),σ2(s2),‍…,σn-1(sn-1),σn(sn))×ΔnsnΔn-1sn-1,…,Δ2s2Δ1s1∫|=|∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn‍∂nf(s1,s2,…,sn-1,sn)Δ1s1Δ2s2,…,Δn-1xn-1Δnxn×∏i=1npi(xi,si)ΔnsnΔn-1sn-1,…,Δ2s2Δ1s1|≤∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn|∂nf(s1,s2,…,sn-1,sn)Δ1s1Δ2s2,…,Δn-1xn-1Δnxn|‍×|∏i=1npi(xi,si)|ΔnsnΔn-1sn-1,…,Δ2s2Δ1s1≤K∏i=1n[h2(xi,ai)+h2(xi,bi)],
which is the desired inequality.The proof of the sharpness of (12) can be referred to [16, Theorem 3.5], which is equivalent to the case in Theorem 9 with n=1.Theorem 10.
Under the conditions of Theorem9, if there exist constants K1  and  K2 such that K1≤∂nf(x1,x2,…,xn)/(Δ1x1Δ2x2,…,Δnxn) ≤ K2, then one has
(15)|∏i=1n[h2(xi,ai)-h2(xi,bi)]f(x1,x2,…,xn)L-∑‍k=1n-1(-1)k+1Lk×{∑1≤m1≤m2⋯≤mk≤n∫am1bm1‍∫am2bm2‍⋯∫amkbmkf(x1,…,‍σm1(sm1),…,σm2(sm2),…,σmk(smk),…,xn)×Δmksmk,…,Δm2sm2Δm1sm1∑1≤m1≤m2⋯≤mk≤n∫am1bm1‍∫am2bm2‍}+(-1)n×∫a1b1‍∫a2b2‍⋯∫an-1bn-1∫anbn‍f(σ1(s1),‍σ2(s2),…,σn-1(sn-1),σn(sn))×ΔnsnΔn-1sn-1,…,Δ2s2Δ1s1-K1+K22∏i=1n[h2(xi,ai)-h2(xi,bi)]|≤K2-K12∏i=1n[h2(xi,ai)+h2(xi,bi)].Proof.
We notice that∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn  ∏i=1npi(xi,si)  ΔnsnΔn-1sn-1,…,Δ2s2Δ1s1=∏i=1n[h2(xi,ai)-h2(xi,bi)].
We also have|∂nf(x1,x2,…,xn)/(Δ1x1Δ2x2,…,Δnxn) − (K1+K2)/2|≤(K2-K1)/2 and
(16)|∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn‍(∂nf(s1,s2,…,sn-1,sn)Δ1s1Δ2s2,…,Δn-1sn-1Δnsn-K1+K22∂nf(s1,s2,…,sn-1,sn)Δ1s1Δ2s2,…,Δn-1sn-1Δnsn)∏i=1npi(xi,si)×ΔnsnΔn-1sn-1,…,Δ2s2Δ1s1∏|≤K2-K12∫a1b1‍∫a2b2‍⋯∫an-1bn-1∫anbn‍|∏i=1npi(xi,si)|‍×ΔnsnΔn-1sn-1,…,Δ2s2Δ1s1=K2-K12∏i=1n[h2(xi,ai)+h2(xi,bi)].
Collecting the information above and Lemma 7 we can get the desired result.Remark 11.
Theorem9 is the n dimensional extension of [16, Theorem 3.5].
In Theorem9, if we take 𝕋i, i=1,2,…,n, for some special time scales, then we immediately obtain the following three corollaries.Corollary 12 (continuous case).
Let𝕋i=ℝ, i=1,2,…,n, in Theorem 9. Then h2(t,s)=(t-s)2/2, and one obtains
(17)|∑∫f(x1,x2,…,xn)L-∑‍k=1n-1(-1)k+1Lk×{∑‍1≤m1≤m2⋯≤mk≤n∫am1bm1‍∫am2bm2‍⋯∫amkbmk‍f(x1,…,sm1,…,sm2,…,smk,…,xn)×dsmk,…,dsm2dsm1∑‍1≤m1≤m2⋯≤mk≤n∫am1bm1‍∫am2bm2‍}+(-1)n×∫a1b1‍∫a2b2‍⋯∫an-1bn-1∫anbn‍f(s1,s2,…,sn-1,‍sn)dsndsn-1,…,ds2ds1∑∫|≤K∏i=1n[(xi-ai)22+(xi-bi)22],
where K  =  sup⁡ai<xi<bi,i=1,2,…,n⁡ |∂nf(x1,x2,…,xn)/(∂x1∂x2,…,∂xn)|.Corollary 13 (discrete case).
Let𝕋i=ℤ, i=1,2,…,n, in Theorem 9. Then h2(t,s)=(t-s)(t-s-1)/2 for ∀t,s∈ℤ, and one has
(18)|∑f(x1,x2,…,xn)L-∑‍k=1n-1(-1)k+1Lk×{∑‍1≤m1≤m2⋯≤mk≤n∑‍sm1=am1+1bm1∑‍sm2=am2+1bm2∑‍1≤m1≤m2⋯≤mk≤n∑‍sm1=am1+1bm1∑‍sm2=am2+1bm2⋯∑‍smk=amk+1bmkf(x1,…,sm1,…,sm2,…,smk,…,xn)∑‍1≤m1≤m2⋯≤mk≤n∑‍sm1=am1+1bm1∑‍sm2=am2+1bm2}+(-1)n∑‍s1=a1+1b1∑‍s2=a2+1b2⋯∑‍sn-1=an-1+1bn-1∑‍sn=an+1bn∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn‍f(s1,s2,…,sn-1,sn)|≤K∏i=1n(xi-ai)(xi-ai-1)+(xi-bi)(xi-bi-1)2,
where K denotes the maximum value of the absolute value of the difference Δ1Δ2,…,Δnf over [a1,b1-1]ℤ×[a2,b2-1]ℤ⋯×[an,bn-1]ℤ.Corollary 14 (quantum calculus case).
Let𝕋i=qiℕ0, i=1,2,…,n, in Theorem 9, where qi>1, i=1,2,…,n. Then one has
(19)|∑f(x1,x2,…,xn)L-∑‍k=1n-1(-1)k+1Lk×{∑‍1≤m1≤m2⋯≤mk≤n[∏i=1kami(qmi-1)]×∑sm1=0log⁡qm1[bm1/(qm1am1)]∑sm2=0log⁡qm2[bm2/(qm2am2)]⋯∑smk=0log⁡qmk[bmk/(qmkamk)](∏i=1kqmismi)×f(x1,…,am1qm1sm1+1,…,am2qm2sm2+1,…,amkqmksmk+1,…,xn)∑‍1≤m1≤m2⋯≤mk≤n[∏i=1kami(qmi-1)]}+(-1)n[∏i=1nai(qi-1)]×∑s1=0log⁡q1[b1/(q1a1)]∑s2=0log⁡q2[b2/(q2a2)]⋯∑sn-1=0log⁡qn-1[(bn-1)/(qn-1an-1)]∑sn-2=0log⁡qn[bn/(qnan)](∏i=1nqisi)×f(q1s1+1a1,q2s2+1a2,…,qn-1sn-1+1an-1,qnsn+1an)∑|≤K∏i=1n[(xi-ai)(xi-qiai)1+qi+(xi-bi)(xi-qibi)1+qi],
where K denotes the maximum value of the absolute value of the q1q2,…,qn-difference Dq1q2,…,qnf(t1,t2,…,tn) over [a1,b1/q1]𝕋1×[a2,b2/q2]𝕋2×⋯×[an,bn/qn]𝕋n.Proof.
Sincehk(t,s)=∏n=0k-1(t-qins)/∑μ=0nqiμ for ∀s,t∈qiℕ0, i=1,2,…,n, then we have
(20)h2(t,s)=(t-s)(t-qis)1+qi,i=1,2⋯,n.
Substituting (20) into (12) we get the desired result.Next we propose then dimensional Grüss inequalities in the following two theorems.Theorem 15.
Letai,bi∈𝕋i,xi∈[ai,bi]𝕋i, i=1,2,…,n,  and f,g∈Crd([a1,b1]𝕋1×[a2,b2]𝕋2×⋯×[an,bn]𝕋n,ℝ) such that the partial delta derivative of order n exists and there exist constants K1 and K2 such that sup⁡ai<xi<bi,i=1,2,…,n⁡|∂nf(x1,x2,…,xn)/(Δ1x1Δ2x2,…,Δnxn)| =  K1,sup⁡ai<xi<bi,i=1,2,…,n⁡|∂ng(x1,x2,…,xn)/(Δ1x1Δ2x2,…,Δnxn)| = K2. Then one has
(21)|1L∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn‍f(x1,x2,…,xn)×g(x1,x2,…,xn)×ΔnxnΔn-1xn-1,…,Δ2x2Δ1x1-(-1)n+12L2×∫a1b1‍∫a2b2‍⋯∫an-1bn-1∫anbn‍f(x1,x2,…,xn)‍×ΔnxnΔn-1xn-1,…,Δ2x2Δ1x1×∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn‍g(σ1(s1),σ2(s2),…,σn-1(sn-1),σn(sn))×ΔnsnΔn-1sn-1,…,Δ2s2Δ1s1-(-1)n+12L2×∫a1b1‍∫a2b2‍⋯∫an-1bn-1∫anbn‍g(x1,x2,…,xn)‍×ΔnxnΔn-1xn-1,…,Δ2x2Δ1x1×∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn‍f(σ1(s1),σ2(s2),…,σn-1(sn-1),σn(sn))×ΔnsnΔn-1sn-1,…,Δ2s2Δ1s1-12L2×∫a1b1‍∫a2b2‍⋯∫an-1bn-1∫anbn‍[g(x1,x2,…,xn)‍×A1(x1,x2,…,xn)+f(x1,x2,…,xn)×A2(x1,x2,…,xn)]×ΔnxnΔn-1xn-1,…,Δ2x2Δ1x1∑|≤K12L2×∫a1b1‍∫a2b2‍⋯∫an-1bn-1∫anbn[×∏i=1nh2(xi,ai)+h2(xi,bi)|g(x1,x2,…,xn)|‍‍×∏i=1nh2(xi,ai)+h2(xi,bi)]×ΔnxnΔn-1xn-1,…,Δ2s2Δ1s1+K22L2×∫a1b1‍∫a2b2‍⋯∫an-1bn-1∫anbn‍[×∏i=1nh2(xi,ai)+h2(xi,bi)|f(x1,x2,…,xn)|‍×∏i=1nh2(xi,ai)+h2(xi,bi)]×ΔnxnΔn-1xn-1,…,Δ2s2Δ1s1.Proof.
For a functiong(x1,x2,…,xn), from Lemma 7, we have
(22)g(x1,x2,…,xn)L-A2(x1,x2,…,xn)=∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn‍∂ng(s1,s2,…,sn-1,sn)Δ1s1Δ2s2,…,Δn-1xn-1Δnxn×∏i=1npi(xi,si)ΔnsnΔn-1sn-1,…,Δ2s2Δ1s1+(-1)n+1∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn‍g(σ1(s1),σ2(s2),…,σn-1(sn-1),σn(sn))×ΔnsnΔn-1sn-1,…,Δ2s2Δ1s1.
Then multiplying (9) by g(x1,x2,…,xn) and (22) by f(x1,x2,…,xn), and adding the resulting identities, we obtain
(23)2f(x1,x2,…,xn)g(x1,x2,…,xn)L=g(x1,x2,…,xn)×∫a1b1‍∫a2b2‍⋯∫an-1bn-1∫anbn‍∂nf(s1,s2,…,sn-1,sn)Δ1s1Δ2s2,…,Δn-1xn-1Δnxn‍×∏i=1npi(xi,si)ΔnsnΔn-1sn-1,…,Δ2s2Δ1s1+f(x1,x2,…,xn)×∫a1b1‍∫a2b2‍⋯∫an-1bn-1∫anbn‍∂ng(s1,s2,…,sn-1,sn)Δ1s1Δ2s2,…,Δn-1xn-1Δnxn‍×∏i=1npi(xi,si)ΔnsnΔn-1sn-1,…,Δ2s2Δ1s1+g(x1,x2,…,xn)A1(x1,x2,…,xn)+f(x1,x2,…,xn)A2(x1,x2,…,xn)+A3(x1,x2,…,xn).
An integration for (23) on [a1,b1]𝕋1×[a2,b2]𝕋2×⋯×[an,bn]𝕋n yields that
(24)∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn‍f(x1,x2,…,xn)g(x1,x2,…,xn)×ΔnxnΔn-1xn-1,…,Δ2x2Δ1x1=12L×∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn[∏i=1npi(xi,si)g(x1,x2,…,xn)‍×∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn‍∂nf(s1,s2,…,sn-1,sn)Δ1s1Δ2s2,…,Δn-1xn-1Δnxn×∏i=1npi(xi,si)×ΔnsnΔn-1sn-1,…,Δ2s2Δ1s1∏i=1npi(xi,si)]×ΔnxnΔn-1xn-1,…,Δ2x2Δ1x1+12L×∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn‍[∏i=1npi(xi,si)f(x1,x2,…,xn)×∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn‍∂ng(s1,s2,…,sn-1,sn)Δ1s1Δ2s2,…,Δn-1xn-1Δnxn×∏i=1npi(xi,si)×ΔnsnΔn-1sn-1,…,Δ2s2Δ1s1×∏i=1npi(xi,si)]×ΔnxnΔn-1xn-1,…,Δ2x2Δ1x1+12L×∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn‍[g(x1,x2,…,xn)×A1(x1,x2,…,xn)+f(x1,x2,…,xn)×A2(x1,x2,…,xn)]×ΔnxnΔn-1xn-1,…,Δ2x2Δ1x1+12L∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn‍A3(x1,x2,…,xn)×ΔnxnΔn-1xn-1,…,Δ2x2Δ1x1.
So we have(25)|∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn‍f(x1,x2,…,xn)g(x1,x2,…,xn)×ΔnxnΔn-1xn-1,…,Δ2x2Δ1x1-12L∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn‍[g(x1,x2,…,xn)×A1(x1,x2,…,xn)+f(x1,x2,…,xn)×A2(x1,x2,…,xn)]×ΔnxnΔn-1xn-1,…,Δ2x2Δ1x1-12L∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn‍A3(x1,x2,…,xn)×ΔnxnΔn-1xn-1,…,Δ2x2Δ1x1∫a1b1‍∫a2b2‍|≤K12L∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn‍[∏|g(x1,x2,…,xn)|×∏i=1n(h2(xi,ai)+h2(xi,bi))∏]×ΔnxnΔn-1xn-1,…,Δ2x2Δ1x1+K22L∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn‍[∏|f(x1,x2,…,xn)|×∏i=1n(h2(xi,ai)+h2(xi,bi))∏]×ΔnxnΔn-1xn-1,…,Δ2x2Δ1x1.
After simple computation we can get the desired result.Remark 16.
Theorem15 is the n dimensional extension of [17, Theorem 2.2].Theorem 17.
Letf,g∈Crd([a1,b1]𝕋1×[a2,b2]𝕋2×⋯×[an,bn]𝕋n,ℝ) such that ϕ≤f(x1,x2,n)≤Φ and γ≤g(x1,x2,…,xn)≤Γ for all xi∈[ai,bi]𝕋i, i=1,2,…,n, where ϕ,Φ,γ,   and  Γ are constants. Then one has
(26)|1L∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn‍f(x1,x2,…,xn)g(x1,x2,…,xn)×ΔnsnΔn-1sn-1,…,Δ2s2Δ1s1-1L∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn‍f(x1,x2,…,xn)×ΔnsnΔn-1sn-1,…,Δ2s2Δ1s1×1L∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn‍g(x1,x2,…,xn)×ΔnsnΔn-1sn-1,…,Δ2s2Δ1s11L∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn‍f(x1,x2,…,xn)g(x1,x2,…,xn)|≤14(Φ-ϕ)(Γ-γ).
The proof for Theorem 17 is similar to [26, pp: 295–296], which is omitted here.Finally, we present one Ostrowski-Grüss type inequality as follows.Theorem 18.
Suppose thatf∈Crd([a1,b1]𝕋1×[a2,b2]𝕋2×⋯×[an,bn]𝕋n,ℝ) such that the partial delta derivative of order n exists and there exist constants K1  and  K2 such that K1≤∂nf(x1,x2,…,xn)/(Δ1x1Δ2x2,…,Δnxn)≤K2. Then one has
(27)|f(x1,x2,…,xn)-1L∑‍k=1n-1(-1)k+1Lk×{∑‍1≤m1≤m2⋯≤mk≤n∫am1bm1‍∫am2bm2‍⋯∫amkbmk‍f(x1,…,σm1(sm1),…,σm2(sm2),…,σmk(smk),…,xn)×Δmksmk,…,Δm2sm2Δm1sm1∑‍1≤m1≤m2⋯≤mk≤n∫am1bm1‍∫am2bm2‍}+1L(-1)n×∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbnf(σ1(s1),σ2(s2),…,‍σn-1(sn-1),σn(sn))×ΔnsnΔn-1sn-1,…,Δ2s2Δ1s1-[1L∑‍j(1)=12∑‍j(2)=12⋯∑‍j(n)=12(-1)k1+k2+⋯+kn×f(λ1,j(1),λ2,j(2),…,λn,j(n))1L∑‍j(1)=12∑‍j(2)=12⋯∑‍j(n)=12(-1)k1+k2+⋯+kn]×1L∏i=1n[h2(xi,ai)-h2(xi,bi)]|≤L(K2-K1)4.Proof.
From the definition ofpi(xi,si) in (8) we obtain that max⁡(∏i=1npi(xi,si))-min⁡(∏i=1npi(xi,si))≤∏i=1n(bi-ai)=L. On the other hand, we have
(28)∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn‍∂nf(s1,s2,…,sn-1,sn)Δ1s1Δ2s2,…,Δn-1sn-1Δnsn×ΔnsnΔn-1sn-1,…,Δ2s2Δ1s1=∑‍j(1)=12∑‍j(2)=12⋯∑‍j(n)=12(-1)k1+k2+⋯+kn×f(λ1,j(1),λ2,j(2),…,λn,j(n)),∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn‍∏i=1npi(xi,si)×ΔnsnΔn-1sn-1,…,Δ2s2Δ1s1=∏i=1n[h2(xi,ai)-h2(xi,bi)],
where λi,1=ai, λi,2=bi, ki={1,λi,j(i)=ai0,λi,j(i)=bi,i=1,2,…,n. So by Theorem 17 we deduce that
(29)1L∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn‍∂nf(s1,s2,…,sn-1,sn)Δ1s1Δ2s2,…,Δn-1sn-1Δnsn×∏i=1npi(xi,si)×ΔnsnΔn-1sn-1,…,Δ2s2Δ1s1-1L∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn‍∂nf(s1,s2,…,sn-1,sn)Δ1s1Δ2s2,…,Δn-1sn-1Δnsn×ΔnsnΔn-1sn-1,…,Δ2s2Δ1s1×1L∫a1b1‍∫a2b2‍⋯∫an-1bn-1‍∫anbn‍∏i=1npi(xi,si)×ΔnsnΔn-1sn-1,…,Δ2s2Δ1s1≤[max⁡⁡(∏i=1npi(xi,si))-min⁡⁡(∏i=1npi(xi,si))]4(K2-K1)≤L4(K2-K1).
Then combining (9), (28), and (29) we get the desired result.Remark 19.
Theorem18 is the n dimensional extension of [18, Theorem 4] and is the n dimensional extension on time scales of [12, Theorem 2.1].Remark 20.
For Theorems10, 15, 17, and 18, we can also obtain similar results as shown in Corollaries 12–14, which are omitted here.


## 3. Conclusions

In this paper, we have presented some new generalizedn dimensional Ostrowski type and Grüss type integral inequalities on time scales. These inequalities are of new forms compared with the existing results in the literature and are n dimensional extension on time scales of some known continuous and discrete inequalities. Based on these inequalities, some new bounds as well as some sharp bounds for unknown functions are derived. These inequalities can also be used in the estimate of explicit error bounds for some numerical quadrature formulae.



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