Some Inequalities for Multiple Integrals on the 𝑛-Dimensional Ellipsoid, Spherical Shell, and Ball

Abstract

The authors establish some new inequalities of Pólya type for multiple integrals on the 𝑛-dimensional ellipsoid, spherical shell, and ball, in terms of bounds of the higher order derivatives of the integrands. These results generalize the main result in the paper by Feng Qi, Inequalities for a multiple integral, Acta Mathematica Hungarica (1999).

1. Introduction

In [1], it was obtained that if 𝑓 is differentiable and if 𝑓(𝑎)=𝑓(𝑏)=0, then 𝑓^′(𝜏)>4(𝑏−𝑎)^2𝑏𝑎𝑓(𝑡)d𝑡, (1) for a certain 𝜏 between 𝑎 and 𝑏. This inequality can be found in [2–4] and many other textbooks. It can be reformulated as follows. If 𝑓(𝑥) is differentiable and not identically constant, such that 𝑓(𝑎)=𝑓(𝑏)=0 and |𝑓^′(𝑥)|≤𝑀 on [𝑎,𝑏], then ^𝑏_𝑎𝑓(𝑥)d𝑥≤(𝑏−𝑎)²4𝑀. (2) In the literature, the inequalities (1) or (2) is called the Pólya integral inequality.

In [5], the inequality (1), or say (2), was generalized as ^𝑏_𝑎𝑓(𝑥)d1𝑥−2(𝑓(𝑎)(𝑏≤𝑀()𝑏−𝑎)+𝑓)𝑏−𝑎²4−𝑓(𝑏)−𝑓(𝑎)²,4𝑀 (3) where 𝑓∶[𝑎,𝑏]→ℝ is a differentiable function and |𝑓^′(𝑥)|≤𝑀.

In [6–9], the above inequalities were refined and generalized as follows.

Theorem 1 (see [9, Proposition 1]). Let 𝑓(𝑥) be continuous on [𝑎,𝑏] and differentiable in (𝑎,𝑏). Suppose that 𝑓(𝑎)=𝑓(𝑏)=0, and that 𝑚≤𝑓^′(𝑥)≤𝑀 in (𝑎,𝑏). If 𝑓(𝑥) is not identically zero, then 𝑚<0<𝑀 and ^𝑏_𝑎𝑓(𝑥)()𝑑𝑥≤−𝑏−𝑎²2𝑚𝑀.𝑀−𝑚 (4)

Theorem 2 (see [6, 7, 9]). Let 𝑓(𝑥) be continuous on [𝑎,𝑏] and differentiable in (𝑎,𝑏). Suppose that 𝑓(𝑥) is not identically 𝑎 constant, and that 𝑚≤𝑓^′(𝑥)≤𝑀 in (𝑎,𝑏). Then, 1𝑏−𝑎^𝑏_𝑎𝑓(𝑥)𝑓(𝑎)(𝑏)𝑑𝑥−+𝑓2≤𝑓(𝑏)(𝑎)(×𝑀()(𝑏)(𝑎×(2(−𝑓−𝑚𝑏−𝑎)𝑏−𝑎−𝑓+𝑓)𝑀−𝑚)(𝑏−𝑎))⁻¹=−𝑀−𝑆₀(𝑎,𝑏)𝑚−𝑆₀(𝑎,𝑏)2()(),𝑀−𝑚𝑏−𝑎 (5) where 𝑆₀()=𝑓(𝑏)(𝑎)𝑎,𝑏−𝑓.𝑏−𝑎 (6)

Theorem 3 (see [8]). For 𝑎=(𝑎₁,…,𝑎_𝑚)∈ℝ^𝑚 and 𝑏=(𝑏₁,…,𝑏_𝑚)∈ℝ^𝑚 with 𝑎_𝑖<𝑏_𝑖 for 𝑖=1,2,…,𝑚, denote the 𝑚-rectangles by 𝑄_𝑚=^𝑚_𝑖=1𝑎_𝑖,𝑏_𝑖,𝑄_𝑚(𝑡)=^𝑚_𝑖=1𝑎_𝑖,𝑐_𝑖(𝑡)_∘,𝑄_𝑚=^𝑚_𝑖=1𝑎_𝑖,𝑏_𝑖, (7) where 𝑐_𝑖(𝑡)=(1−𝑡)𝑎_𝑖+𝑡𝑏_𝑖for 𝑖=1,2,…,𝑚 and 𝑡∈(0,1). Let 𝜈=(𝜈₁,…,𝜈_𝑚) be a multi-index; that is, 𝜈_𝑖 is a nonnegative integer, with |𝜈|=∑^𝑚_𝑖=1𝜈_𝑖. Let 𝑓∈𝐶^(𝑛+1)(𝑄_𝑚) be a function of 𝑚 variables on 𝑄_𝑚, and let its partial derivatives of (𝑛+1)th order remain between 𝑀_𝑛+1(𝜈) and 𝑁_𝑛+1(𝜈) in 𝑄^∘_𝑚; that is, 𝑁_𝑛+1(𝜈)≤𝐷^𝜈𝑓(𝑥)≤𝑀_𝑛+1(𝜈),𝑥∈𝑄^∘_𝑚, (8) where |𝜈|=𝑛+1 and 𝐷^𝜈𝑓(𝑥)=𝜕^|𝜈|𝑓(𝑥)∏^𝑚_𝑖=1𝜕𝑥^𝜈𝑖_𝑖. (9) Let 𝐴(𝜈)=^𝑚_𝑖=1𝑏_𝑖−𝑎_𝑖^𝜈𝑖+1𝜈_𝑖!𝑀+1^𝑛+1(𝜈),𝐵𝜈,𝑓(𝑥)=^𝑚_𝑖=1𝑏_𝑖−𝑎_𝑖^𝜈𝑖+1𝜈_𝑖!𝜕+1𝜕𝑥_𝑖^𝜈𝑖𝑓(𝑥),𝐶(𝜈)=^𝑚_𝑖=1𝑏_𝑖−𝑎_𝑖^𝜈𝑖+1𝜈_𝑖!𝑁+1^𝑛+1(𝜈),𝑇(𝜈,𝑡)=^𝑚_𝑖=11−(1−𝑡)^𝜈𝑖+1−1, (10) for 𝑡∈(0,1). Then, for any 𝑡∈(0,1), (1) when 𝑛 is even, one has _{|𝜈|=𝑛+1}𝐶(𝜈)𝑡^𝑚+𝑛+1+_{|𝜈|=𝑛+1}𝐴(𝜈)𝑇()≤𝜈,𝑡_𝑄𝑚𝑓(𝑥)d𝑥−^𝑛_{𝑘=0|𝜈|=𝑘}𝐵𝜈,𝑓(𝑎)𝑡^𝑚+𝑘+^𝑛_𝑘=0(−1)^𝑘_|𝜈|=𝑘𝐵𝜈,𝑓(𝑏)𝑇(𝜈,𝑡)≤_{|𝜈|=𝑛+1}𝐴(𝜈)𝑡^𝑚+𝑛+1+_{|𝜈|=𝑛+1}𝐶(𝜈)𝑇(𝜈,𝑡). (11)(2) When 𝑛 is odd, one has _{|𝜈|=𝑛+1}𝐶(𝜈)𝑡^𝑚+𝑛+1(≤+𝑇𝜈,𝑡)_𝑄𝑚𝑓(𝑥)d𝑥−^𝑛_{𝑘=0|𝜈|=𝑘}𝐵𝜈,𝑓(𝑎)𝑡^𝑚+𝑘+^𝑛_𝑘=0(−1)^𝑘_|𝜈|=𝑘𝐵𝜈,𝑓(𝑏)𝑇(𝜈,𝑡)≤_{|𝜈|=𝑛+1}𝐴(𝜈)𝑡^𝑚+𝑛+1+𝑇(𝜈,𝑡). (12)

We remark that Theorem 2 has been applied in [10] to give bounds for the complete elliptic integrals of the first and second kinds.

For more information on this topic, please refer to [11–18] and [19, pp. 558–561], especially to the preprint [20].

In what follows, we will continue to use some notations from Theorem 3. Assume that 𝑏_𝑖, 𝑟_𝑖>0 for 𝑖=1,2,…,𝑛 and 𝜌,𝜌₁,𝜌₂>0 with 𝜌₁<𝜌₂, and adopt the following notations: 𝑥=𝑥₁,𝑥₂,…,𝑥_𝑛,𝑟=𝑟₁,𝑟₂,…,𝑟_𝑛𝑣=𝑣₁,𝑣₂,…,𝑣_𝑛,Ω()=𝑎,𝑏,2𝑟𝑥∶^𝑛_𝑖=1𝑥_𝑖−𝑎_𝑖^2𝑟𝑖𝑏^2𝑟𝑖_𝑖≤1,𝑥₁≥𝑎₁,…,𝑥_𝑛≥𝑎_𝑛=Ω_2𝑟,Ω₁(𝑎,𝑏)=𝑥∶^𝑛_𝑖=1𝑥_𝑖−𝑎_𝑖²𝑏²_𝑖≤1=Ω₁,Ω₂𝜌₁,𝜌₂=𝑥∶𝜌²₁≤^𝑛_𝑖=1𝑥²_𝑖≤𝜌²₂=Ω₂,Ω₃𝑎,𝜌=𝑥∶^𝑛_𝑖=1𝑥_𝑖−𝑎_𝑖²≤𝜌²=Ω₃,Ω₄(𝑡)=𝑥∶^𝑛_𝑖=1𝑥_𝑖−𝑎_𝑖²≤𝜌²(𝑡),𝜌(𝑡)=𝑡𝜌,𝑡∈(0,1]=Ω₄. (13) Moreover, let 𝑓∶𝐼⊆ℝ→ℝ be an (𝑚+1)-times differentiable function, and let 𝑔₁(𝑥)=^𝑛_𝑖=1𝑥_𝑖−𝑎_𝑖²𝑏²_𝑖,𝑥∈Ω₁,𝑔₂(𝑥)=^𝑛_𝑖=1𝑥²_𝑖,𝑥∈Ω₂,𝑔₃(𝑥)=^𝑛_𝑖=1𝑥_𝑖−𝑎_𝑖²,𝑥∈Ω₃. (14)

In this paper, we will establish some new inequalities of Pólya type for multiple integrals of the composition function 𝑓∘𝑔₁ on the 𝑛-dimensional ellipsoid Ω₁, of the composition function 𝑓∘𝑔₂ on the spherical shell Ω₂, and of the composition function 𝑓∘𝑔₃ on the 𝑛-dimensional ball Ω₃. We also obtain a general inequality for the multiple integral ∫_Ω2𝑟𝑓(𝑥)d𝑥.

2. A Lemma

In order to establish some new inequalities of Pólya type for multiple integrals, we need the following lemma.

Lemma 4. For 𝑏_𝑖, 𝑟_𝑖>0, and 𝑣_𝑖>−1, one has _Ω2𝑟^𝑛_𝑖=1𝑥_𝑖−𝑎_𝑖^𝑣𝑖=∏𝑑𝑥_𝑛𝑖=1𝑏^𝑣𝑖_𝑖⁺¹/𝑟_𝑖2^𝑛−1∑^𝑛_𝑖=1𝑣_𝑖+1/𝑟_𝑖∏^𝑛_𝑖=1Γ𝑣_𝑖+1/2𝑟_𝑖Γ∑^𝑛_𝑖=1𝑣_𝑖+1/2𝑟_𝑖, (15) where Γ(𝑧)=^∞₀𝑡^𝑧−1𝑒^−𝑡𝑑𝑡,ℜ(𝑧)>0 (16) is the classical Euler gamma function.

Proof. Using the spherical coordinates on the region Ω_2𝑟 yields 𝑥₁=𝑏₁𝑠^1/𝑟1cos^1/𝑟1𝜑₁+𝑎₁,𝑥_𝑖=𝑏_𝑖𝑠cos𝜑_𝑖^𝑖−1_𝑘=1sin𝜑_𝑘^1/𝑟𝑖+𝑎_𝑖𝑥,2≤𝑖≤𝑛−1,_𝑛=𝑏_𝑛𝑠^𝑛−1_𝑘=1sin𝜑_𝑘^1/𝑟𝑛+𝑎_𝑛, (17) where 0≤𝑠≤1 and 0≤𝜑₁,𝜑₂,…,𝜑_𝑛−1≤𝜋/2, and 𝐹_𝑖≡𝑠^2𝑖−1_𝑘=1sin²𝜑_𝑘−^𝑛_𝑘=𝑖𝑥_𝑘−𝑎_𝑘𝑏_𝑘^2𝑟𝑘=0,1≤𝑖≤𝑛. (18) We note that when 𝑖=1, the empty product in (18) is understood to be 1. It is clear that the expressions in (17) are solutions of (18), and that 𝐽=𝐷𝑥𝐷𝑠,𝜑₁,𝜑₂,…,𝜑_𝑛−1=(−1)^𝑛𝐷𝐹₁,𝐹₂,…,𝐹_𝑛/𝐷𝑠,𝜑₁,𝜑₂,…,𝜑_𝑛−1𝐷𝐹₁,𝐹₂,…,𝐹_𝑛./𝐷𝑥 (19)
A straightforward computation gives 𝐽=^𝑛_𝑘=1𝑏_𝑘𝑟_𝑘𝑠^{∑𝑛𝑖=1(1/𝑟𝑖)−1𝑛−1}_𝑘=1sin^{∑𝑛𝑖=𝑘+1(1/𝑟𝑖)−1}𝜑_𝑘cos^{(1/𝑟𝑘)−1}𝜑_𝑘. (20) Since ₀^𝜋/2cos^𝑚𝜑sin^𝑛𝜑dΓ))Γ))𝜑=((𝑚+1/2((𝑛+1/2)),2Γ((𝑚+𝑛+2/2 (21) we obtain _Ω2𝑟^𝑛_𝑖=1𝑥_𝑖−𝑎_𝑖^𝑣𝑖d𝑥=∏^𝑛_𝑖=1𝑏^𝑣𝑖_𝑖⁺¹∏^𝑛_𝑖=1𝑟_𝑖¹₀𝑠^{∑𝑛𝑘=1((𝑣𝑘+1)/𝑟𝑘)−1}d𝑠×^𝑛−1_𝑘=10^𝜋/2sin^{∑𝑛𝑖=𝑘+1((𝑣𝑖+1)/𝑟𝑖)−1}𝜑_𝑘cos^{((𝑣𝑘+1)/𝑟𝑘)−1}𝜑_𝑘d𝜑_𝑘=∏^𝑛_𝑖=1𝑏^𝑣𝑖_𝑖⁺¹/𝑟_𝑖2^𝑛−1∑^𝑛_𝑖=1𝑣_𝑖+1/𝑟_𝑖∏^𝑛_𝑖=1Γ𝑣_𝑖+1/2𝑟_𝑖Γ∑^𝑛_𝑖=1𝑣_𝑖+1/2𝑟_𝑖. (22) The proof of Lemma 4 is complete.

3. Main Results

Now, we start out to state and prove our main results.

Theorem 5. Let 𝑓∶[0,1]→ℝ be an (𝑚+1)-times differentiable function satisfying 𝑁(𝑚)≤𝑓^(𝑚+1)(𝑢)≤𝑀(𝑚). (23) Then, one has 2𝜋^𝑛/2(𝑛−1)!∏^𝑛_𝑖=1𝑏_𝑖Γ()!)𝑛/2)(𝑛+𝑚+1min(−1^𝑚+1𝑀(𝑚),(−1)^𝑚+1𝑁(𝑚≤_Ω1)𝑓𝑔₁(𝑥)−𝑑𝑥_𝑚𝑘=0(−1)^𝑘2𝜋^𝑛/2(𝑛−1)!∏^𝑛_𝑖=1𝑏_𝑖()()𝑓𝑛+𝑘!Γ𝑛/2^(𝑘)(1)≤2𝜋^𝑛/2(𝑛−1)!∏^𝑛_𝑖=1𝑏_𝑖Γ()!)𝑛/2)(𝑛+𝑚+1×max(−1^𝑚+1𝑀(𝑚),(−1)^𝑚+1𝑁(𝑚). (24)

Proof. Using the transformation in (17) on Ω₁ and letting 𝑟_𝑖=1 for 𝑖=1,2,…,𝑛 yield the Jacobian determinant 𝐽=𝑠^𝑛𝑛−1_𝑘=1𝑏_𝑘^𝑛−2_𝑘=1sin^{𝑛−𝑘−1}𝜑_𝑘,0≤𝑠≤1,0≤𝜑₁,𝜑₂,…,𝜑_𝑛−2≤𝜋,0≤𝜑_𝑛−1≤2𝜋. (25)(26) Because ^𝜋₀sin^𝑛𝑡d𝑡=2₀^𝜋/2cos^𝑛𝑡d√𝑡=𝜋Γ((𝑛+1)/2)Γ)),((𝑛+2/2 (27) we have ^𝑛−2_𝑘=1^𝜋₀sin^{𝑛−𝑘−1}𝜑_𝑘d𝜑_𝑘0^2𝜋d𝜑_𝑛−1=2𝜋^𝑛/2Γ().𝑛/2 (28) By integration by parts, one has ^𝛽_𝛼𝑠^𝑛−1𝑓(𝑠)d𝑠=^𝑚_𝑘=0(−1)^𝑘(𝑛−1)!𝛽^𝑛+𝑘𝑓^(𝑘)𝛽−𝛼^𝑛+𝑘𝑓^(𝑘)(𝛼)()!+()𝑛+𝑘−1^𝑚+1(𝑛−1)!(𝑛+𝑚)!^𝛽_𝛼𝑓^(𝑚+1)(𝑠)𝑠^𝑛+𝑚d𝑠. (29) Choosing 𝛼=0 and 𝛽=1 in the above equality shows that _Ω1𝑓𝑔₁(𝑥)d𝑥=^𝑛_𝑘=1𝑏_𝑘¹₀𝑠^𝑛−1𝑓(𝑠)d𝑠^𝑛−2_𝑘=1^𝜋₀sin^{𝑛−𝑘−1}𝜑_𝑘d𝜑_𝑘0^2𝜋d𝜑_𝑛−1=2𝜋^𝑛/2∏^𝑛_𝑖=1𝑏_𝑖Γ(𝑛/2)^𝑚_𝑘=0(−1)^𝑘(𝑛−1)!𝑓^(𝑘)(1)()!+()𝑛+𝑘−1^𝑚+12𝜋^𝑛/2∏^𝑛_𝑖=1𝑏_𝑖Γ()()!𝑛/2𝑛−1(𝑛+𝑚)!¹₀𝑓^(𝑚+1)(𝑠)𝑠^𝑛+𝑚d𝑠. (30) Further utilizing the condition (23) leads to the inequality (24). The proof of Theorem 5 is completed.

Theorem 6. Let 𝑓∶[𝜌₁,𝜌₂]→ℝ be an (𝑚+1)-times differentiable function satisfying the inequality (23). Then, one has 2𝜋^𝑛/2𝜌₂^𝑛+𝑚+1−𝜌₁^𝑛+𝑚+1(𝑛−1)!Γ()!)𝑛/2)(𝑛+𝑚+1×min(−1^𝑚+1𝑀(𝑚),(−1)^𝑚+1𝑁(𝑚≤_Ω2)𝑓𝑔₂(𝑥)𝑑𝑥−2𝜋^𝑛/2Γ()×𝑛/2_𝑚𝑘=0(−1)^𝑘(𝑛−1)!𝜌₂^𝑛+𝑘𝑓^(𝑘)𝜌₂−𝜌₁^𝑛+𝑘𝑓^(𝑘)𝜌₁()!≤𝑛+𝑘2𝜋^𝑛/2𝜌₂^𝑛+𝑚+1−𝜌₁^𝑛+𝑚+1(𝑛−1)!Γ()!)𝑛/2)(𝑛+𝑚+1×max(−1^𝑚+1𝑀(𝑚),(−1)^𝑚+1𝑁(𝑚). (31)

Proof. Using the transformation in (17) on Ω₂ and choosing 𝑟_𝑖=1, 𝑎_𝑖=0, and 𝑏_𝑖=1 for 𝑖=1,2,…,𝑛 yield 𝐽=𝑠^{𝑛−1𝑛−2}_𝑘=1sin^{𝑛−𝑘−1}𝜑_𝑘,𝜌₁≤𝑠≤𝜌₂,0≤𝜑₁,𝜑₂,…,𝜑_𝑛−2≤𝜋,0≤𝜑_𝑛−1≤2𝜋. (32) Further letting 𝛼=𝜌₁ and 𝛽=𝜌₂ in (29) gives _Ω2𝑓𝑔₂(𝑥)d𝑥=^𝜌2_𝜌1𝑠^𝑛−1𝑓(𝑠)d𝑠^𝑛−2_𝑘=1^𝜋₀sin^{𝑛−𝑘−1}𝜑_𝑘d𝜑_𝑘0^2𝜋d𝜑_𝑛−1=2𝜋^𝑛/2Γ(𝑛/2)^𝑚_𝑘=0(−1)^𝑘()!×𝜌𝑛−1_2𝑛+𝑘𝑓^(𝑘)𝜌₂−𝜌₁^𝑛+𝑘𝑓^(𝑘)𝜌₁×((𝑛+𝑘)!)⁻¹+(−1)^𝑚+12𝜋^𝑛/2(𝑛−1)!Γ(𝑛/2)(𝑛+𝑚)!^𝜌2_𝜌1𝑓^(𝑚+1)(𝑠)𝑠^𝑛+𝑚d𝑠. (33) Hence, by virtue of the condition (23), the inequality (31) follows immediately. The proof of Theorem 6 is completed.

Theorem 7. Let 𝑓∶[0,𝜌]→ℝ be an (𝑚+1)-times differentiable function satisfying (23). Then, one has 2𝜋^𝑛/2(𝑛−1)!𝜌^𝑛+𝑚+1Γ()!)𝑛/2)(𝑛+𝑚+1min(−1^𝑚+1𝑀(𝑚),(−1)^𝑚+1𝑁(𝑚≤_Ω3)𝑓𝑔₃(𝑥−)𝑑𝑥_𝑚𝑘=0(−1)^𝑘2𝜋^𝑛/2(𝑛−1)!𝜌^𝑛+𝑘()()𝑓𝑛+𝑘!Γ𝑛/2^(𝑘)𝜌≤2𝜋^𝑛/2(𝑛−1)!𝜌^𝑛+𝑚+1Γ()!)𝑛/2)(𝑛+𝑚+1×max(−1^𝑚+1𝑀(𝑚),(−1)^𝑚+1𝑁(𝑚). (34)

Proof. Similar to the proof of Theorem 5, by choosing 𝑏₁=𝑏₂=⋯=𝑏_𝑛=𝜌 and 0≤𝑠≤𝜌, we obtain the inequality (34). The proof is complete.

Corollary 8. Under the conditions of Theorem 7, if 𝑓^(𝑘)(𝜌)=0 for 𝑘=0,1,2,…,𝑚, then 2𝜋^𝑛/2(𝑛−1)!𝜌^𝑛+𝑚+1Γ()!𝑁(𝑚)≤()𝑛/2)(𝑛+𝑚+1−1^𝑚−1_Ω3𝑓𝑔₃(𝑥≤)𝑑𝑥2𝜋^𝑛/2(𝑛−1)!𝜌^𝑛+𝑚+1Γ()!𝑀(𝑚).𝑛/2)(𝑛+𝑚+1 (35)

4. A More General Inequality

Let 𝜈=(𝜈₁,𝜈₂,…,𝜈_𝑛) be an 𝑛-tuple index; that is, the numbers 𝜈₁,𝜈₂,…,𝜈_𝑛 are nonnegative and denote |𝜈|=∑^𝑛_𝑖=1𝜈_𝑖. Let 𝑓∶Ω_2𝑟→ℝ be a function which has an 𝑚+1 times continuous derivative on Ω_2𝑟, and let 𝐷^𝜈𝑓(𝑥)=𝜕^|𝜈|𝑓(𝑥)∏^𝑛_𝑖=1𝜕𝑥^𝜈𝑖_𝑖,𝐻()=∏𝜈,𝑏,𝑟^𝑛𝑖=1𝑏^𝜈𝑖_𝑖⁺¹/𝑟_𝑖2^𝑛−1∑^𝑛_𝑖=1𝜈_𝑖+1/𝑟_𝑖∏^𝑛_𝑖=1Γ𝜈_𝑖+1/2𝑟_𝑖Γ∑^𝑛_𝑖=1𝜈_𝑖+1/2𝑟_𝑖, (36) and |𝜈|=𝑚+1.

Theorem 9. Let 𝑓∈𝐶^𝑚+1(Ω_2𝑟) satisfy 𝑁_𝑚+1(𝜈)≤𝐷^𝜈𝑓(𝑥)≤𝑀_𝑚+1(𝜈),𝑥∈Ω_2𝑟. (37) Then 𝐻(𝜈,𝑏,𝑟)_{|𝜈|=𝑚+1}𝑁_𝑚+1(𝜈)∏^𝑛_𝑖=1𝜈_𝑖!≤_Ω2𝑟𝑓(𝑥)𝑑𝑥−𝐻(𝜈,𝑏,𝑟)^𝑚_{𝑗=0|𝜈|=𝑗}𝐷^𝜈𝑓(𝑎)∏^𝑛_𝑖=1𝜈_𝑖!≤𝐻(𝜈,𝑏,𝑟)_{|𝜈|=𝑚+1}𝑀_𝑚+1(𝜈)∏^𝑛_𝑖=1𝜈_𝑖!. (38)

Proof. By Taylor’s formula, we obtain 𝑓(𝑥)=^𝑚_𝑗=01𝑗!^𝑛_𝑖=1(𝑥_𝑖−𝑎_𝑖)𝜕𝜕𝑥_𝑖^𝑗𝑓(𝑎)+𝑅_𝑚(𝑥), (39) where 𝑅_𝑚(𝑥)=1(𝑚+1)!^𝑛_𝑖=1(𝑥_𝑖−𝑎_𝑖)𝜕𝜕𝑥_𝑖^|𝜈|𝑓((,().𝑎+𝜃𝑥−𝑎))𝜃∈0,1 (40) Using ^𝑛_𝑖=1𝑞_𝑖^𝑗=𝑗!_|𝜈|^𝑛_=𝑗𝑖=1𝑞^𝜈𝑖_𝑖𝜈_𝑖!, (41) we have 𝑓(𝑥)=^𝑚_{𝑗=0|𝜈|=𝑗}1∏^𝑛_𝑖=1𝜈_𝑖!^𝑛_𝑖=1𝑥_𝑖−𝑎_𝑖𝜕𝜕𝑥_𝑖^𝜈𝑖𝑓(𝑎)+_{|𝜈|=𝑚+1}1∏^𝑛_𝑖=1𝜈_𝑖!×^𝑛_𝑖=1𝑥_𝑖−𝑎_𝑖𝜕𝜕𝑥_𝑖^𝜈𝑖𝑓(𝑎+𝜃(𝑥−𝑎)). (42) Integrating on both sides of the above equality leads to _Ω2𝑟𝑓(𝑥)d𝑥=^𝑚_{𝑗=0|𝜈|=𝑗}1∏^𝑛_𝑖=1𝜈_𝑖!×_Ω2𝑟^𝑛_𝑖=1(𝑥_𝑖−𝑎_𝑖)𝜕𝜕𝑥_𝑖^𝜈𝑖𝑓(𝑎)d𝑥+_{|𝜈|=𝑚+1}1∏^𝑛_𝑖=1𝜈_𝑖!×_Ω2𝑟^𝑛_𝑖=1𝑥_𝑖−𝑎_𝑖𝜕𝜕𝑥_𝑖^𝜈𝑖×𝑓(𝑎+𝜃(𝑥−𝑎))d𝑥=^𝑚_{𝑗=0|𝜈|=𝑗}1∏^𝑛_𝑖=1𝜈_𝑖!𝜕^|𝜈|𝑓(𝑎)∏^𝑛_𝑖=1𝜕𝑥^𝜈𝑖_𝑖×_Ω2𝑟^𝑛_𝑖=1𝑥_𝑖−𝑎_𝑖^𝜈𝑖d𝑥+_{|𝜈|=𝑚+1}1∏^𝑛_𝑖=1𝜈_𝑖!×_Ω2𝑟^𝑛_𝑖=1𝑥_𝑖−𝑎_𝑖^𝜈𝑖𝜕^|𝜈|𝑓(𝑎+𝜃(𝑥−𝑎))∏^𝑛_𝑖=1𝜕𝑥^𝜈𝑖_𝑖d𝑥=𝐼₁+𝐼₂, (43) where 𝐼₁=^𝑚_{𝑗=0|𝜈|=𝑗}1∏^𝑛_𝑖=1𝜈_𝑖!𝜕^|𝜈|𝑓(𝑎)∏^𝑛_𝑖=1𝜕𝑥^𝜈𝑖_𝑖Ω2𝑟^𝑛_𝑖=1𝑥_𝑖−𝑎_𝑖^𝜈𝑖d𝑥,𝐼₂=_{|𝜈|=𝑚+1}1∏^𝑛_𝑖=1𝜈_𝑖!_Ω2𝑟^𝑛_𝑖=1𝑥_𝑖−𝑎_𝑖^𝜈𝑖𝜕^|𝜈|𝑓(𝑎+𝜃(𝑥−𝑎))∏^𝑛_𝑖=1𝜕𝑥^𝜈𝑖_𝑖d𝑥. (44)(45) By Lemma 4 and (44), one has 𝐼₁=^𝑚_{𝑗=0|𝜈|=𝑗}1∏^𝑛_𝑖=1𝜈_𝑖!𝜕^|𝜈|𝑓(𝑎)∏^𝑛_𝑖=1𝜕𝑥^𝜈𝑖_𝑖×∏^𝑛_𝑖=1𝑏^𝜈𝑖_𝑖⁺¹/𝑟_𝑖2^𝑛−1∑^𝑛_𝑖=1𝜈_𝑖+1/𝑟_𝑖∏^𝑛_𝑖=1Γ𝜈_𝑖+1/2𝑟_𝑖Γ∑^𝑛_𝑖=1𝜈_𝑖+1/2𝑟_𝑖=^𝑚_{𝑗=0|𝜈|=𝑗}𝐷^𝜈𝑓(𝑎)∏^𝑛_𝑖=1𝜈_𝑖!𝐻(𝜈,𝑏,𝑟). (46) From (37) and 𝐼₂=_{|𝜈|=𝑚+1}1∏^𝑛_𝑖=1𝜈_𝑖!_Ω2𝑟^𝑛_𝑖=1𝑥_𝑖−𝑎_𝑖^𝜈𝑖𝐷^𝜈𝑓(𝑎+𝜃(𝑥−𝑎))d𝑥, (47) we have _{|𝜈|=𝑚+1}𝑁_𝑚+1(𝜈)∏^𝑛_𝑖=1𝜈_𝑖!𝐻(𝜈,𝑏,𝑟)≤𝐼₂≤_{|𝜈|=𝑚+1}𝑀_𝑚+1(𝜈)∏^𝑛_𝑖=1𝜈_𝑖!𝐻(𝜈,𝑏,𝑟). (48) Consequently, the proof of Theorem 9 is complete.

Corollary 10. Let |𝜈|=𝑚+1, and let 𝑓∈𝐶^𝑚+1(Ω₄) with (37). Then, for 𝑡∈(0,1] one has 𝐻(𝜈,𝑡)_{|𝜈|=𝑚+1}𝑁_𝑚+1(𝜈)∏^𝑛_𝑖=1𝜈_𝑖!≤_Ω4(𝑡)𝑓(𝑥)𝑑𝑥−𝐻(𝜈,𝑡)^𝑚_{𝑗=0|𝜈|=𝑗}𝐷^𝜈𝑓(𝑎)∏^𝑛_𝑖=1𝜈_𝑖!≤𝐻(𝜈,𝑡)_{|𝜈|=𝑚+1}𝑀_𝑚+1(𝜈)∏^𝑛_𝑖=1𝜈_𝑖!, (49) where 𝐻()=𝜌𝜈,𝑡^𝑛+𝑚+1∏𝑛+𝑚+1^𝑛𝑖=11+(−1)^𝜈𝑖2^𝑛−1×∏^𝑛_𝑖=1Γ𝜈_𝑖+1/2Γ∑^𝑛_𝑖=1𝜈_𝑖𝑡+1/2^𝑛+𝑚+1. (50)

5. An Application

Now, we list some special cases of Ω_2𝑟 as follows. (1)If we take 𝑟₁=𝑟₂=⋯=𝑟_𝑛=1/2, the body Ω_2𝑟 becomes a closed region between the 𝑛-dimensional pyramid and the rectangle 𝑥_𝑖=𝑎_𝑖 for 𝑖=1,2,…,𝑛. (2)If we take 𝑟₁=𝑟₂=⋯=𝑟_𝑛=1, the body Ω_2𝑟 is a closed region between the 𝑛-dimensional ellipsoid Ω₁(𝑎,𝑏) and the rectangle 𝑥_𝑖=𝑎_𝑖 for 𝑖=1,2,…,𝑛. (3)If we take 𝑟₁=𝑟₂=⋯=𝑟_𝑛=1 and 𝑏₁=𝑏₂=⋯=𝑏_𝑛=𝜌, the body Ω_2𝑟 is a closed region between the 𝑛-dimensional ball Ω₃(𝑎,𝜌) and the rectangle 𝑥_𝑖=𝑎_𝑖 for 𝑖=1,2,…,𝑛.

In the calculation of the uniform 𝑛-dimensional volume, static moment, the moment of inertia, the centrifugal moment, and so on, have important applications. See [21, 22].

To show the applicability of the above main results, we now estimate the value of a triple integral 𝐼=_𝑉𝑥sin²𝑎²+𝑦²𝑏²+𝑧²𝑐^25/2d𝑥d𝑦d𝑧, (51) where 𝑉 is the ellipsoid 𝑥²𝑎²+𝑦²𝑏²+𝑧²𝑐²≤1. (52) Choosing 𝑛=3, 𝑏₁=𝑎, 𝑏₂=𝑏, and 𝑏₃=𝑐 in (25), the Jacobian determinant is 𝐽=𝑎𝑏𝑐𝑠²sin𝜑₁,𝐼=₀^2𝜋d𝜑₂^𝜋₀d𝜑₁¹₀𝑎𝑏𝑐𝑠²sin𝜑₁sin𝑠⁵d𝑠=4𝜋𝑎𝑏𝑐¹₀𝑠²sin𝑠⁵d𝑠. (53)(54) Using Taylor’s formula, it follows that sin𝑥=^𝑚_𝑘=1(−1)^𝑘−1𝑥^2𝑘−1()!+()2𝑘−1−1^𝑚𝑥^2𝑚+1()!(),2𝑚+1cos𝜃𝑥0<𝜃<1. (55) Specially, we have sin𝑥=³_𝑘=1(−1)^𝑘−1𝑥^2𝑘−1()!−𝑥2𝑘−17⁷!cos𝜃₁𝑥,sin𝑥=⁶_𝑘=1(−1)^𝑘−1𝑥^2𝑘−1()!+𝑥2𝑘−113¹³!cos𝜃₂𝑥, (56) where 0<𝜃₁,𝜃₂<1 and 0<𝑥<1. Therefore, ⁶_𝑘=1(−1)^𝑘−1𝑥^2𝑘−1(2𝑘−1)!≤sin𝑥≤³_𝑘=1(−1)^𝑘−1𝑥^2𝑘−1()!.2𝑘−1 (57) By (54) and the above inequality, we have ⁶_𝑘=1(−1)^𝑘−1(2𝑘−1)!¹₀𝑠^10𝑘−3d𝑠≤¹₀𝑠²sin𝑠⁵d𝑠≤³_𝑘=1(−1)^𝑘−1(2𝑘−1)!¹₀𝑠^10𝑘−3d𝑠,_6𝑘=1(−1)^𝑘−1()!()≤2𝑘−110𝑘−2₁₀𝑠²sin𝑠⁵d𝑠≤³_𝑘=1(−1)^𝑘−1()!(),2𝑘−110𝑘−261249255037131964940800𝜋𝑎𝑏𝑐≤𝐼≤35097560𝜋𝑎𝑏𝑐. (58)

Acknowledgments

The authors appreciate the anonymous referees for their very careful suggestions and their greatly valuable comments on the original version of this paper. This work was partially supported by the Foundation of the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region under Grant no. NJZY13159, China.