# Generalizations of the Simpson-Like Type Inequalities for Co-Ordinateds-Convex Mappings in the Second Sense

Authors: Jaekeun Park
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

A generalized identity for some partial differentiable mappings on a bidimensional interval is obtained, and, by using this result, the author establishes generalizations of Simpson-like type inequalities for coordinateds-convex mappings in the second sense.


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## Body

## 1. Introduction

In recent years, a number of authors have considered error estimate inequalities for some known and some new quadrature formulas. Sometimes they have considered generalizations of the Simpson-like type inequality which gives an error bound for the well-known Simpson rule.Theorem 1.1.
Letf:𝕀⊂[0,∞)→ℝ be a four-time continuous differentiable mapping on [a,b] and ∥f(4)∥∞=supx∈[a,b]|f(4)(x)|<∞. Then, the following inequality holds:
(1.1)|16[f(a)+4f(a+b2)+f(b)]-1b-a∫abf(x)dx|≤(b-a)42880‖f(4)‖∞.It is well known that the mappingf is neither four times differentiable nor is the fourth derivative f(4) bounded on (a,b), then we cannot apply the classical Simpson quadrature formula.For recent results on Simpson type inequalities, you may see the papers [1–5].In [2, 6–8], Dragomir et al. and Park considered among others the class of mappings which are s-convex on the coordinates.In the sequel, in this paper letΔ=[a,b]×[c,d] be a bidimensional interval in ℝ2 with a<b and c<d.Definition 1.2.
A mappingf:Δ→ℝ will be called s-convex in the second sense on Δ if the following inequality:
(1.2)f(λx+(1-λ)z,λy+(1-λ)w)≤λsf(x,y)+(1-λ)sf(z,w),
holds, for all (x,y),(z,w)∈Δ, λ∈[0,1] and s∈[0,1].Modification for convex ands-convex mapping on Δ, which are also known as co-ordinated convex, s-convex mapping, and s-r-convex, respectively, were introduced by Dragomir, Sarikaya [5, 9, 10], and Park [4, 8, 11, 12].Definition 1.3.
A mappingf:Δ→ℝ will be called coordinated s-convex in the second sense on Δ if the partial mappings
(1.3)fy:[a,b]⟶R,fy(u)=f(u,y),fx:[c,d]⟶R,fx(v)=f(x,v),
are s-convex in the second sense, for all x∈[a,b], y∈[c,d], and s∈[0,1] [5, 9, 10].A formal definition for coordinateds-convex mappings may be stated as follow [8].Definition 1.4.
A mappingf:Δ→ℝ will be called coordinated s-convex in the second sense on Δ if the following inequality:
(1.4)f(tx+(1-t)z,λy+(1-λ)w)≤tsλsf(x,y)+(1-t)sλsf(z,y)+ts(1-λ)sf(x,w)+(1-t)s(1-λ)sf(z,w),
holds, for all t,λ∈[0,1], (x,y),(z,w)∈Δ, and s∈[0,1].In [2], S.S. Dragomir established the following theorem.Theorem 1.5.
Letf:Δ→ℝ be convex on the coordinates on Δ. Then, one has the inequalities:
(1.5)f(a+b2,c+d2)≤1(b-a)(d-c)∫ab∫cdf(x,y)dydx≤14[f(a,c)+f(b,c)+f(a,d)+f(b,d)].In [13], Hwang et al. gave a refinement of Hadamard’s inequality on the coordinates and they proved some inequalities for coordinated convex mappings.In [1, 6, 14], Alomari and Darus proved inequalities for coordinated s-convex mappings.In [15], Latif and Alomari defined coordinated h-convex mappings, established some inequalities for co-ordinated h-convex mappings and proved inequalities involving product of convex mappings on the coordinates.In [3], Özdemir et al. gave the following theorems:Theorem 1.6.
Letf:Δ⊂ℝ2→ℝ be a partial differentiable mapping on Δ. If ∂2f/∂t∂λ is convex on the coordinates on Δ, then the following inequality holds:
(1.6)|19[f(a,c+d2)+f(b,c+d2)+4f(a+b2,c+d2)+f(a+b2,c)+f(a+b2,d)]+136{f(a,c)+f(a,d)+f(b,c)+f(b,d)}+1(b-a)(d-c)∫ab∫cdf(x,y)dydx-A|≤(572)2(b-a)(d-c)×{|∂2f∂t∂λ(a,c)|+|∂2f∂t∂λ(a,d)|+|∂2f∂t∂λ(b,c)|+|∂2f∂t∂λ(b,d)|},
where
(1.7)A=1b-a∫ab{f(x,c)+4f(x,(c+d)/2)+f(x,d)6}dx+1d-c∫cd{f(a,y)+4f((a+b)/2,y)+f(b,y)6}dy.Theorem 1.7.
Letf:Δ⊂ℝ2→ℝ be a partial differentiable mapping on Δ. If ∂2f/∂t∂λ is bounded, that is,
(1.8)‖∂2f∂t∂λ(ta+(1-t)b,λc+(1-λ)d)‖∞=sup⁡(x,y)∈(a,b)×(c,d)|∂2f∂t∂λ(ta+(1-t)b,λc+(1-λ)d)|<∞
for all (t,λ)∈[0,1]2, then the following inequality holds:
(1.9)|19[f(a,c+d2)+f(b,c+d2)+4f(a+b2,c+d2)+f(a+b2,c)+f(a+b2,d)]+136{f(a,c)+f(a,d)+f(b,c)+f(b,d)}+1(b-a)(d-c)∫ab∫cdf(x,y)dydx-A|≤(572)2(b-a)(d-c)‖∂2f∂t∂λ(ta+(1-t)b,λc+(1-λ)d)‖∞,
where A is defined in Theorem 1.6.In [3], Özdemir et al. proved a new equality and, by using this equality, established some inequalities on coordinated convex mappings.In this paper the author give a generalized identity for some partial differentiable mappings on a bidimensional interval and, by using this result, establish a generalizations of Simpson-like type inequalities for coordinateds-convex mappings in the second sense.


## 2. Main Results

To prove our main results, we need the following lemma.Lemma 2.1.
Letf:Δ→ℝ be a partial differentiable mapping on Δ=[a,b]×[c,d]⊂ℝ2. If (∂2f/∂t∂λ)∈L1(Δ), then, for r1,r2≥2 and h1,h2∈(0,1) with (1/r1)≤h1≤(r1-1/r1) and (1/r2)≤h2≤((r2-1)/r2), the following equality holds:
(2.1)I(f)(h1,h2,r1,r2)=let{(r1-2)(r2-2)r1r2}f(h1a+(1-h1)b,h2c+(1-h2)d)+{(r1-2)r1r2}{f(h1a+(1-h1)b,c)+f(h1a+(1-h1)b,d)}+{(r2-2)r1r2}{f(a,h2c+(1-h2)d)+f(b,h2c+(1-h2)d)}+1r1r2{f(a,c)+f(a,d)+f(b,c)+f(b,d)}-1r2(b-a)∫ab{f(x,c)+(r2-2)f(x,h2c+(1-h2)d)+f(x,d)}dx-1r1(d-c)∫cd{f(a,y)+(r1-2)f(h1a+(1-h1)b,y)+f(b,y)}dy+1(b-a)(d-c)∫ab∫cdf(x,y)dydx=(b-a)(d-c)∬01p(h1,r1,t)q(h2,r2,λ)×∂2f∂t∂λ(ta+(1-t)b,λc+(1-λ)d)dtdλ,
where
(2.2)p(h1,r1,t)={t-1r1,t∈[0,h1],t-r1-1r1,t∈(h1,1],q(h2,r2,λ)={λ-1r2,λ∈[0,h2],λ-r2-1r2,λ∈(h2,1].Proof.
By the definitions ofp(h1,r1,t) and q(h2,r2,λ), we can write
(2.3)I=∬01p(h1,r1,t)q(h2,r2,λ)∂2f∂t∂λ(ta+(1-t)b,λc+(1-λ)d)dtdλ=∫01q(h2,r2,λ)[∫01p(h1,r1,t)∂2f∂t∂λ(ta+(1-t)b,λc+(1-λ)d)dt]dλ=∫01q(h2,r2,λ)[∫0h1(t-1r1)∂2f∂t∂λ(ta+(1-t)b,λc+(1-λ)d)dt+∫h11(t-r1-1r1)∂2f∂t∂λ(ta+(1-t)b,λc+(1-λ)d)dt]dλ=∫01q(h2,r2,λ)[I11+I12]dλ,
where
(2.4)I11=∫0h1(t-1r1)∂2f∂t∂λ(ta+(1-t)b,λc+(1-λ)d)dt,I12=∫h11(t-r1-1r1)∂2f∂t∂λ(ta+(1-t)b,λc+(1-λ)d)dt.
By integration by parts, we have(2.5)I11=∫0h1(t-1r1)∂2f∂t∂λ(ta+(1-t)b,λc+(1-λ)d)dt=1a-b[(h1-1r1)∂f∂λ(h1a+(1-h1)b,λc+(1-λ)d)+1r1∂f∂λ(b,λc+(1-λ)d)-∫0h1∂f∂λ(ta+(1-t)b,λc+(1-λ)d)dt],(2.6)I12=∫h11(t-r1-1r1)∂2f∂t∂λ(ta+(1-t)b,λc+(1-λ)d)dt=1a-b[(1r1)∂f∂λ(a,λc+(1-λ)d)-(h1-r1-1r1)∂f∂λ(h1a+(1-h1)b,λc+(1-λ)d)-∫h11∂f∂λ(ta+(1-t)b,λc+(1-λ)d)dt].
By using the equalities (2.5) and (2.6) in (2.3), we have
(2.7)I=(1a-b){(r1-2r1)∫01q(h2,r2,λ)∂f∂λ(h1a+(1-h1)b,λc+(1-λ)d)dλ+(1r1)∫01q(h2,r2,λ)∂f∂λ(b,λc+(1-λ)d)dλ+(1r1)∫01q(h2,r2,λ)∂f∂λ(a,λc+(1-λ)d)dλ-∫01q(h2,r2,λ)∫01∂f∂λ(ta+(1-t)b,λc+(1-λ)d)dλdt}=(1a-b){(r1-2r1)I21+(1r1)I22+(1r1)I23-I24},
where
(2.8)I21=∫01q(h2,r2,λ)∂f∂λ(h1a+(1-h1)b,λc+(1-λ)d)dλ,I22=∫01q(h2,r2,λ)∂f∂λ(b,λc+(1-λ)d)dλ,I23=∫01q(h2,r2,λ)∂f∂λ(a,λc+(1-λ)d)dλ,I24=∫01q(h2,r2,λ)∫01∂f∂λ(ta+(1-t)b,λc+(1-λ)d)dλdt.
Note that(2.9)(i)∫0h2(λ-1r2)∂f∂λ(h1a+(1-h1)b,λc+(1-λ)d)dλ=1c-d{(h2-1r2)f(h1a+(1-h1)b,h2c+(1-h2)d)+1r2f(h1a+(1-h1)b,d)-∫0h2f(h1a+(1-h1)b,λc+(1-λ)d)dλ},(2.10)(ii)∫h21(λ-r2-1r2)∂f∂λ(h1a+(1-h1)b,λc+(1-λ)d)dλ=1c-d{(1r2)f(h1a+(1-h1)b,c)-(h2-r2-1r2)f(h1a+(1-h1)b,h2c+(1-h2)d)-∫h21f(h1a+(1-h1)b,λc+(1-λ)d)dλ}.
By the equalities (2.9) and (2.10), we have
(2.11)I21=∫01q(h2,r2,λ)∂f∂λ(h1a+(1-h1)b,λc+(1-λ)d)dλ=∫0h2(λ-1r2)∂f∂λ(h1a+(1-h1)b,λc+(1-λ)d)dλ+∫h21(λ-r2-1r2)∂f∂λ(h1a+(1-h1)b,λc+(1-λ)d)dλ=1c-d{1r2f(h1a+(1-h1)b,c)+1r2f(h1a+(1-h1)b,d)+r2-2r2f(h1a+(1-h1)b,h2c+(1-h2)d)-∫01f(h1a+(1-h1)b,λc+(1-λ)d)dλ}.
By the similar way, we get the following:(2.12)I22=∫01q(h2,r2,λ)∂f∂λ(b,λc+(1-λ)d)dλ=1c-d{1r2f(b,c)+1r2f(b,d)+(r2-2r2)f(b,h2c+(1-h2)d)-∫01f(b,λc+(1-λ)d)dλ},(2.13)I23=∫01q(h2,r2,λ)∂f∂λ(a,λc+(1-λ)d)dλ=1c-d{1r2f(a,c)+1r2f(a,d)+(r2-2r2)f(a,h2c+(1-h2)d)-∫01f(a,λc+(1-λ)d)dλ},(2.14)I24=∫01q(h2,r2,λ)∫01∂f∂λ(ta+(1-t)b,λc+(1-λ)d)dλdt=1c-d{1r2∫01f(ta+(1-t)b,c)dt+1r2∫01f(ta+(1-t)b,d)dt+(r2-2r2)∫01f(ta+(1-t)b,h2c+(1-h2)d)dt-∬01f(ta+(1-t)b,λc+(1-λ)d)dλdt}.
By the equalities (2.7) and (2.11)–(2.14) and using the change of the variables x=ta+(1-t)b and y=λc+(1-λ)d for (t,λ)∈[0,1]2, then multiplying both sides with (b-a)(d-c), we have the required result (2.1), which completes the proof.Remark 2.2.
Lemma2.1 is a generalization of the results which proved by Sarikaya, Set, Özdemir, and Dragomir [3, 5, 9, 10].Theorem 2.3.
Letf:Δ→ℝ2 be a partial differentiable mapping on Δ=[a,b]×[c,d]⊂ℝ2. If ∂2f/∂t∂λ is in L1(Δ) and is a coordinated s-convex mapping in the second sense on Δ, then, for r1,r2≥2 and h1,h2∈(0,1) with (1/r1)≤h1≤((r1-1)/r1), and (1/r2)≤h2≤((r2-1)/r2) the following inequality holds:
(2.15)1(b-a)(d-c)|I(f)(h1,h2,r1,r2)|≤μ1(r,s){μ2(r,s)|∂2f∂t∂λ(a,c)|+ν2(r,s)|∂2f∂t∂λ(a,d)|}+ν1(r,s){μ2(r,s)|∂2f∂t∂λ(b,c)|+ν2(r,s)|∂2f∂t∂λ(b,d)|},
where
(2.16)μ1(r,s)=M(r1,s)+N(h1,s),μ2(r,s)=M(r2,s)+N(h2,s),ν1(r,s)=M(r1,s)+N(1-h1,s),ν2(r,s)=M(r2,s)-N(1-h2,s)
for
(2.17)M(r,s)=2+2(r-1)s+2+rs+1(s-r+2)(s+1)(s+2)rs+2,N(h,s)=hs+1((2h-1)s+2(h-1))(s+1)(s+2).Proof.
From Lemma2.1 and by the coordinated s-convexity in the second sense of ∂2f/∂t∂λ, we can write
(2.18)1(b-a)(d-c)|I(f)(h1,h2,r1,r2)|≤∬01|p(h1,r1,t)q(h2,r2,λ)|×|∂2f∂t∂λ(ta+(1-t)b,λc+(1-λ)d)|dtdλ≤∬01|p(h1,r1,t)q(h2,r2,λ)|×{tsλs|∂2f∂t∂λ(a,c)|+ts(1-λ)s|∂2f∂t∂λ(a,d)|+(1-t)sλs|∂2f∂t∂λ(b,c)|+(1-t)s(1-λ)s|∂2f∂t∂λ(b,d)|}dtdλ={∫01|p(h1,r1,t)|tsdt}{∫01|q(h2,r2,λ)|λsdλ|∂2f∂t∂λ(a,c)|+∫01|q(h2,r2,λ)|(1-λ)sdλ|∂2f∂t∂λ(a,d)|}+{∫01|p(h1,r1,t)|(1-t)sdt}{∫01|q(h2,r2,λ)|λsdλ|∂2f∂t∂λ(b,c)|+∫01|q(h2,r2,λ)|(1-λ)sdλ|∂2f∂t∂λ(b,d)|}.
Note that(2.19)(i)∫01|p(h1,r1,t)|tsdt=μ1(r,s),(ii)∫01|p(h1,r1,t)|(1-t)sdt=ν1(r,s),(iii)∫01|q(h2,r2,λ)|λsdλ=μ2(r,s),(iv)∫01|q(h2,r2,λ)|(1-λ)sdλ=ν2(r,s).
By (2.18) and (2.19), we get the inequality (2.15) by the simple calculations.Remark 2.4.
In Theorem2.3,(i)
if we chooseh1=h2=1/2, r1=r2=6, and s=1 in (2.15), then we get
(2.20)|I(f)(12,12,6,6)|≤(572)2M(b-a)(d-c),(ii)
if we chooseh1=h2=1/2, r1=r2=2, and s=1 in (2.15), then we get
(2.21)|I(f)(12,12,2,2)|≤(18)2M(b-a)(d-c),where
(2.22)M=|∂2f∂t∂λ(a,c)|+|∂2f∂t∂λ(a,d)|+|∂2f∂t∂λ(b,c)|+|∂2f∂t∂λ(b,d)|,
which implies that Theorem 2.3 is a generalization of Theorem 1.6.Theorem 2.5.
Letf:Δ→ℝ2 be a partial differentiable mapping on Δ=[a,b]×[c,d]⊂ℝ2. If ∂2f/∂t∂λ is bounded, that is,
(2.23)‖∂2f∂t∂λ‖∞=let‖∂2f∂t∂λ(ta+(1-t)b,λc+(1-λ)d)‖∞=sup⁡(x,y)∈(a,b)×(c,d)|∂2f∂t∂λ(ta+(1-t)b,λc+(1-λ)d)|<∞
for all (t,λ)∈[0,1]2, then, for r1,r2≥2 and h1,h2∈(0,1) with (1/r1)≤h1≤((r1-1)/r1) and (1/r2)≤h2≤((r2-1)/r2), the following inequality holds:
(2.24)1(b-a)(d-c)|I(f)(h1,h2,r1,r2)|≤{12-h1+h12+(2-r1)r12}{12-h2+h22+(2-r2)r22}‖∂2f∂t∂λ‖∞.Proof.
From Lemma2.1, using the property of modulus and the boundedness of ∂2f/∂t∂s, we get
(2.25)1(b-a)(d-c)|I(f)(h1,h2,r1,r2)|≤‖∂2f∂t∂λ‖∞∬01|p(h1,r1,t)q(h2,r2,λ))|dtdλ.
By the simple calculations, we have(2.26)(i)∫01|p(h1,r1,t)|dt=12-h1+h12+(2-r1)r12,(2.27)(ii)∫01|q(h2,r2,λ)|dt=12-h2+h22+(2-r2)r22.
By using the inequality (2.25) and the equalities (2.26)-(2.27), the assertion (2.24) holds.Remark 2.6.
In Theorem2.5,(i)
if we chooseh1=h2=12 and r1=r2=6, then we get
(2.28)|I(f)(12,12,6,6)|≤(536)2‖∂2f∂t∂λ‖∞(b-a)(d-c),(ii)
if we chooseh1=h2=1/2 and r1=r2=2, then we get
(2.29)|I(f)(12,12,2,2)|≤(14)2‖∂2f∂t∂λ‖∞(b-a)(d-c),which implies that Theorem 2.5 is a generalization of Theorem 1.7.The following theorem is a generalization of Theorem1.6.Theorem 2.7.
Letf:Δ=[a,b]×[c,d]⊂ℝ2→ℝ2 be a partial differentiable mapping on Δ=[a,b]×[c,d]. If |∂2f/∂t∂λ|q(q>1) is in L1(Δ) and is a coordinated s-convex mapping in the second sense on Δ, then, for r1,r2≥2 and h1,h2∈(0,1) with (1/r1)≤h1≤((r1-1)/r1) and (1/r2)≤h2≤((r2-1)/r2), the following inequality holds:
(2.30)1(b-a)(d-c)|I(f)(h1,h2,r1,r2)|≤μ31/pν31/p×{|∂2f/∂t∂λ(a,c)|q+|∂2f/∂t∂λ(a,d)|q+|∂2f/∂t∂λ(b,c)|q+|∂2f/∂t∂λ(b,d)|q(s+1)2}1/p,
where
(2.31)μ3=2+(r1-r1h1-1)p+1+(r1h1-1)p+1r1p+1(p+1),ν3=2+(r2-r2h2-1)p+1+(r2h2-1)p+1r2p+1(p+1).Proof.
From Lemma2.1, we can write
(2.32)1(b-a)(d-c)|I(f)(h1,h2,r1,r2)|≤∬01|p(h1,r1,t)q(h2,r2,λ)|×|∂2f∂t∂λ(ta+(1-t)b,λc+(1-λ)d)|dtdλ≤{∬01|p(h1,r1,t)q(h2,r2,λ)|pdtdλ}1/p×{∬01|∂2f∂t∂λ(ta+(1-t)b,λc+(1-λ)d)|qdtdλ}1/q.
Hence, by the inequality (2.32) and the coordinated s-convexity in the second sense of |∂2f/∂t∂λ|q, it follows that
(2.33)1(b-a)(d-c)|I(f)(h1,h2,r1,r2)|≤{∬01|p(h1,r1,t)q(h2,r2,λ)|pdtdλ}1/p×{|∂2f/∂t∂λ(a,c)|q+|∂2f/∂t∂λ(a,d)|q+|∂2f/∂t∂λ(b,c)|q+|∂2f/∂t∂λ(b,d)|q(s+1)2}1/q.
Note that(2.34)(i)∫01|p(h1,r1,t)|pdt=2+(r1-r1h1-1)p+1+(r1h1-1)p+1r1p+1(p+1),(2.35)(ii)∫01|q(h2,r2,λ)|pdλ=2+(r2-r2h2-1)p+1+(r2h2-1)p+1r2p+1(p+1).
By the inequality (2.33) and the equalities (2.34) and (2.35), the assertion (2.30) holds.Remark 2.8.
In Theorem2.7,(i)
if we chooseh1=h2=1/2, r1=r2=6, and s=1, then we get
(2.36)|I(f)(12,12,6,6)|≤{2(1+2p+1)6p+1(p+1)}2/p(b-a)(d-c)Mq1/q,(ii)
if we chooseh1=h2=1/2, r1=r2=2, and s=1, then we get
(2.37)|I(f)(12,12,2,2)|≤{12p(p+1)}2/p(b-a)(d-c)Mq1/q,(iii)
if we chooseh1=h2=1/2, r1=r2=6, s=1, and q=1, then we get
(2.38)|I(f)(12,12,6,6)|≤(536)2(b-a)(d-c)M1,where
(2.39)Mq=|∂2f/∂t∂λ(a,c)|q+|∂2f/∂t∂λ(a,d)|q+|∂2f/∂t∂λ(b,c)|q+|∂2f/∂t∂λ(b,d)|q4.Theorem 2.9.
Letf:Δ→ℝ2 be a partial differentiable mapping on Δ=[a,b]×[c,d]⊂ℝ2. If |∂2f/∂t∂λ|q(q≥1) is in L1(Δ) and is a coordinated s-convex mapping in the second sense on Δ, then, for r1,r2≥2 and h1,h2∈(0,1) with (1/r1)≤h1≤((r1-1)/r1) and (1/r2)≤h2≤((r2-1)/r2), the following inequality holds:
(2.40)1(b-a)(d-c)|I(f)(h1,h2,r1,r2)|≤{(12-h1+h12+(2-r1)r12)(12-h2+h22+(2-r2)r22)}1-(1/q)×{μ1(μ2|∂2f∂t∂λ(a,c)|q+ν2|∂2f∂t∂λ(a,d)|q)+ν1(μ2|∂2f∂t∂λ(b,c)|q+ν2|∂2f∂t∂λ(b,d)|q)}1/q,
where μi and νi(i=1,2) are as given in Theorem 2.3.Proof.
From Lemma2.1, we can write
(2.41)1(b-a)(d-c)|I(f)(h1,h2,r1,r2)|≤∬01|p(h1,r1,t)q(h2,r2,λ)|×|∂2f∂t∂λ(ta+(1-t)b,λc+(1-λ)d)|dtdλ≤{∬01|p(h1,r1,t)q(h2,r2,λ)|dtdλ}1-(1/q)×{∬01|p(h1,r1,t)q(h2,r2,λ)|×|∂2f∂t∂λ(ta+(1-t)b,λc+(1-λ)d)|qdtdλ}1/q.
By the simple calculations, we have(2.42)(i)∫01|p(h1,r1,t)|dt=12-h1+h12+(2-r1)r12,(2.43)(ii)∫01|q(h2,r2,λ)|dλ=12-h2+h22+(2-r2)r22.
Since|∂2f/∂t∂λ|q is a coordinated s-convex mapping in the second sense on Δ=[a,b]×[c,d], we have that, for t∈[0,1],
(2.44)∬01|p(h1,r1,t)q(h2,r2,λ)||∂2f∂t∂λ(ta+(1-t)b,λc+(1-λ)d)|qdtdλ≤∬01|p(h1,r1,t)q(h2,r2,λ)|×{tsλs|∂2f∂t∂λ(a,c)|q+ts(1-λ)s|∂2f∂t∂λ(a,d)|q+(1-t)sλs|∂2f∂t∂λ(b,c)|q+(1-t)s(1-λ)s|∂2f∂t∂λ(b,d)|q}dtdλ={∫01|p(h1,r1,t)|tsdt}{(∫01|q(h2,r2,λ)|λsdλ)|∂2f∂t∂λ(a,c)|q+(∫01|q(h2,r2,λ)|(1-λ)sdλ)|∂2f∂t∂λ(a,d)|q}+{∫01|p(h1,r1,t)|(1-t)sdt}{(∫01|q(h2,r2,s)|λsdλ)|∂2f∂t∂s(b,c)|q+(∫01|q(h2,r2,λ)|(1-λ)sdλ)|∂2f∂t∂λ(b,d)|q}=μ1{μ2|∂2f∂t∂λ(a,c)|q+ν2|∂2f∂t∂λ(a,d)|q}+ν1{μ2|∂2f∂t∂λ(b,c)|q+ν2|∂2f∂t∂λ(b,d)|q}.
By (2.41)–(2.44), the assertion (2.40) holds.



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*Source: 715751-2012-02-28.xml*