Sharp Integral Inequalities Based on a General Four-Point Quadrature Formula via a Generalization of the Montgomery Identity

Abstract

We consider families of general four-point quadrature formulae using a generalization of the Montgomery identity via Taylor’s formula. The results are applied to obtain some sharp inequalities for functions whose derivatives belong to 𝐿_𝑝 spaces. Generalizations of Simpson’s 3/8 formula and the Lobatto four-point formula with related inequalities are considered as special cases.

1. Introduction

The most elementary quadrature rules in four nodes are Simpson’s 3/8 rule based on the following four point formula ^𝑏_𝑎𝑓(𝑡)𝑑𝑡=𝑏−𝑎8𝑓(𝑎)+3𝑓2𝑎+𝑏3+3𝑓𝑎+2𝑏3(𝑏−()+𝑓)𝑏−𝑎⁵𝑓6480⁽⁴⁾(𝜉), (1.1) where 𝜉∈[𝑎,𝑏], and Lobatto rule based on the following four point formula ¹₋₁𝑓(𝑡)1𝑑𝑡=6𝑓()−√−1+5𝑓55√+5𝑓55(1−2+𝑓)𝑓23625⁽⁶⁾𝜂, (1.2) where 𝜂∈[−1,1]. Formula (1.1) is valid for any function 𝑓 with a continuous fourth derivative 𝑓⁽⁴⁾ on [𝑎,𝑏] and formula (1.2) is valid for any function 𝑓 with a continuous sixth derivative 𝑓⁽⁶⁾ on [−1,1].

Let 𝑓∶[𝑎,𝑏]→ℝ be differentiable on [𝑎,𝑏] and 𝑓^′∶[𝑎,𝑏]→ℝ integrable on [𝑎,𝑏].

Then the Montgomery identity holds (see [1]) 𝑓(𝑥)=1𝑏−𝑎^𝑏_𝑎𝑓(𝑡)𝑑𝑡+^𝑏_𝑎𝑃(𝑥,𝑡)𝑓^′(𝑡)𝑑𝑡, (1.3) where the Peano kernel is 𝑃(𝑥,𝑡)=𝑡−𝑎𝑏−𝑎,𝑎≤𝑡≤𝑥,𝑡−𝑏𝑏−𝑎,𝑥<𝑡≤𝑏. (1.4)

In [2], Pečarić proved the following weighted Montgomery identity 𝑓(𝑥)=^𝑏_𝑎𝑤(𝑡)𝑓(𝑡)𝑑𝑡+^𝑏_𝑎𝑃_𝑤(𝑥,𝑡)𝑓^′(𝑡)𝑑𝑡, (1.5) where 𝑤∶[𝑎,𝑏]→[0,∞⟩ is some probability density function, that is, integrable function, satisfying ∫^𝑏_𝑎𝑤(𝑡)𝑑𝑡=1, and 𝑊(𝑡)=∫^𝑡_𝑎𝑤(𝑥)𝑑𝑥 for 𝑡∈[𝑎,𝑏], 𝑊(𝑡)=0 for 𝑡<𝑎 and 𝑊(𝑡)=1 for 𝑡>𝑏 and 𝑃_𝑤(𝑥,𝑡) is the weighted Peano kernel defined by 𝑃_𝑤()=𝑊(𝑡)𝑊(𝑡)𝑥,𝑡,𝑎≤𝑡≤𝑥,−1,𝑥<𝑡≤𝑏. (1.6) Now, let us suppose that 𝐼 is an open interval in ℝ, [𝑎,𝑏]⊂𝐼, 𝑓∶𝐼→ℝ is such that 𝑓^(𝑛−1) is absolutely continuous for some 𝑛≥2, 𝑤∶[𝑎,𝑏]→[0,∞⟩ is a probability density function. Then the following generalization of the weighted Montgomery identity via Taylor’s formula states (given by Aglić Aljinović and Pečarić in [3]) 𝑓(𝑥)=^𝑏_𝑎𝑤(𝑡)𝑓(𝑡)𝑑𝑡−^𝑛−2_𝑖=0𝑓^(𝑖+1)(𝑥)(𝑖+1)!^𝑏_𝑎𝑤(𝑠)(𝑠−𝑥)^𝑖+1+1𝑑𝑠(𝑛−1)!^𝑏_𝑎𝑇_𝑤,𝑛(𝑥,𝑠)𝑓^(𝑛)(𝑠)𝑑𝑠, (1.7) where 𝑥∈[𝑎,𝑏] and 𝑇_𝑤,𝑛(𝑥,𝑠)=^𝑠𝑎𝑤(𝑢)(𝑢−𝑠)^𝑛−1𝑑𝑢,𝑎≤𝑠≤𝑥,−^𝑏_𝑠𝑤(𝑢)(𝑢−𝑠)^𝑛−1𝑑𝑢,𝑥<𝑠≤𝑏. (1.8) If we take 𝑤(𝑡)=1/(𝑏−𝑎), 𝑡∈[𝑎,𝑏], equality (1.7) reduces to 𝑓(𝑥)=1𝑏−𝑎^𝑏_𝑎𝑓(𝑡)𝑑𝑡−^𝑛−2_𝑖=0𝑓^(𝑖+1)(𝑥)(𝑏−𝑥)^𝑖+2−(𝑎−𝑥)^𝑖+2()!()+1𝑖+2𝑏−𝑎(𝑛−1)!^𝑏_𝑎𝑇_𝑛(𝑥,𝑠)𝑓^(𝑛)(𝑠)𝑑𝑠, (1.9) where 𝑥∈[𝑎,𝑏] and 𝑇_𝑛()=−()𝑥,𝑠𝑎−𝑠^𝑛𝑛()−()𝑏−𝑎,𝑎≤𝑠≤𝑥,𝑏−𝑠^𝑛𝑛(𝑏−𝑎),𝑥<𝑠≤𝑏. (1.10) For 𝑛=1, (1.9) reduces to the Montgomery identity (1.3).

In this paper, we generalize the results from [4]. Namely, we use identities (1.7) and (1.9) to establish for each number 𝑥∈(𝑎,(𝑎+𝑏)/2] a general four-point quadrature formula of the type ^𝑏_𝑎𝑤(𝑡)𝑓(𝑡)1𝑑𝑡=2(𝑥)𝑓(𝑎)(𝑏(𝑥𝑓(𝑥)(,−𝐴+𝑓)+𝐴)+𝑓𝑎+𝑏−𝑥)+𝑅𝑓,𝑤;𝑥 (1.11) where 𝑅(𝑓,𝑤;𝑥) is the remainder and 𝐴∶(𝑎,(𝑎+𝑏)/2]→ℝ is a real function. The obtained formula is used to prove a number of inequalities which give error estimates for the general four-point formula for functions whose derivatives are from 𝐿_𝑝-spaces. These inequalities are generally sharp. As special cases of the general non-weighted four-point quadrature formula, we obtain generalizations of the well-known Simpson’s 3/8 formula and Lobatto four-point formula with related inequalities.

2. General Weighted Four-Point Formula

Let 𝑓∶[𝑎,𝑏]→ℝ be such that 𝑓^(𝑛−1) exists on [𝑎,𝑏] for some 𝑛≥2. We introduce the following notation for each 𝑥∈(𝑎,(𝑎+𝑏)/2]: 𝐷(𝑥)=12(𝑥𝑓(𝑎)(𝑏(𝑥𝑓(𝑥)(,𝑡−𝐴)+𝑓)+𝐴)+𝑓𝑎+𝑏−𝑥)_𝑤,𝑛(𝑥)=𝐴(𝑥)^𝑛−2_𝑖=0𝑓^(𝑖+1)(𝑥)(𝑖+1)!^𝑏_𝑎𝑤(𝑠)(𝑠−𝑥)^𝑖+1+𝑑𝑠_{𝑛−2𝑖=0}𝑓^(𝑖+1)(𝑎+𝑏−𝑥)(𝑖+1)!^𝑏_𝑎𝑤(𝑠)(𝑠−𝑎−𝑏+𝑥)^𝑖+1+1𝑑𝑠2−𝐴(𝑥)^𝑛−2_𝑖=0𝑓^(𝑖+1)(𝑎)(𝑖+1)!^𝑏_𝑎𝑤(𝑠)(𝑠−𝑎)^𝑖+1+𝑑𝑠_{𝑛−2𝑖=0}𝑓^(𝑖+1)(𝑏)(𝑖+1)!^𝑏_𝑎𝑤(𝑠)(𝑠−𝑏)^𝑖+1,𝑇𝑑𝑠_𝑤,𝑛()1𝑥,𝑠=−2−𝐴(𝑥)𝑇_𝑤,𝑛(𝑎,𝑠)+𝑇_𝑤,𝑛(𝑏,𝑠)−𝐴(𝑥)𝑇_𝑤,𝑛(𝑥,𝑠)+𝑇_𝑤,𝑛(=−1𝑎+𝑏−𝑥,𝑠)2+𝐴(𝑥)^𝑠_𝑎𝑤(𝑢)(𝑢−𝑠)^𝑛−1+1𝑑𝑢2−𝐴(𝑥)^𝑏_𝑠𝑤(𝑢)(𝑢−𝑠)^𝑛−1−1𝑑𝑢,𝑎≤𝑠≤𝑥,2^𝑠_𝑎𝑤(𝑢)(𝑢−𝑠)^𝑛−1𝑑𝑢−^𝑏_𝑠𝑤(𝑢)(𝑢−𝑠)^𝑛−1−1𝑑𝑢,𝑥<𝑠≤𝑎+𝑏−𝑥,2−𝐴(𝑥)^𝑠_𝑎𝑤(𝑢)(𝑢−𝑠)^𝑛−1+1𝑑𝑢2+𝐴(𝑥)^𝑏_𝑠𝑤(𝑢)(𝑢−𝑠)^𝑛−1𝑑𝑢,𝑎+𝑏−𝑥<𝑠≤𝑏. (2.1)

In the next theorem we establish the general weighted four-point formula.

Theorem 2.1. Let 𝐼 be an open interval in ℝ, [𝑎,𝑏]⊂𝐼, and let 𝑤∶[𝑎,𝑏]→[0,∞⟩ be some probability density function. Let 𝑓∶𝐼→ℝ be such that 𝑓^(𝑛−1) is absolutely continuous for some 𝑛≥2. Then for each 𝑥∈(𝑎,(𝑎+𝑏)/2] the following identity holds ^𝑏_𝑎𝑤(𝑡)𝑓(𝑡)d𝑡=𝐷(𝑥)+𝑡_𝑤,𝑛(𝑥)+1(𝑛−1)!^𝑏_𝑎𝑇_𝑤,𝑛(𝑥,𝑠)𝑓^(𝑛)(𝑠)𝑑𝑠. (2.2)

Proof. We put 𝑥≡𝑎,𝑥≡𝑥,𝑥≡𝑎+𝑏−𝑥 and 𝑥≡𝑏 in (1.7) to obtain four new formulae. After multiplying these four formulae by 1/2−𝐴(𝑥),𝐴(𝑥),𝐴(𝑥)and1/2−𝐴(𝑥), respectively, and adding, we get (2.2).

Remark 2.2. Identity (2.2) holds true in the case 𝑛=1. It can also be obtained by taking 𝑥≡𝑎,𝑥≡𝑥,𝑥≡𝑎+𝑏−𝑥 and 𝑥≡𝑏 in (1.5), multiplying these four formulae by 1/2−𝐴(𝑥),𝐴(𝑥),𝐴(𝑥)and1/2−𝐴(𝑥), respectively, and adding. In this special case we have ^𝑏_𝑎𝑤(𝑡)𝑓(𝑡)𝑑𝑡=𝐷(𝑥)+^𝑏_𝑎𝑇_𝑤,1(𝑥,𝑠)𝑓^′(𝑠)𝑑𝑠, (2.3) where 𝑇_𝑤,1()1𝑥,𝑠=−2−𝐴(𝑥)𝑇_𝑤,1(𝑎,𝑠)+𝑇_𝑤,1(𝑏,𝑠)−𝐴(𝑥)𝑇_𝑤,1(𝑥,𝑠)+𝑇_𝑤,1(1𝑎+𝑏−𝑥,𝑠)=−2−𝐴(𝑥)𝑃_𝑤(𝑎,𝑠)+𝑃_𝑤(𝑏,𝑠)−𝐴(𝑥)𝑃_𝑤(𝑥,𝑠)+𝑃_𝑤(=1𝑎+𝑏−𝑥,𝑠)2(𝑥)(𝑠)1−𝐴−𝑊,𝑎≤𝑠≤𝑥,2(𝑠)1−𝑊,𝑥<𝑠≤𝑎+𝑏−𝑥,2+𝐴(𝑥)−𝑊(𝑠),𝑎+𝑏−𝑥<𝑠≤𝑏. (2.4)

Theorem 2.3. Suppose that all assumptions of Theorem 2.1 hold. Additionally, assume that (𝑝,𝑞) is a pair of conjugate exponents, that is, 1≤𝑝,𝑞≤∞, 1/𝑝+1/𝑞=1, let 𝑓^(𝑛)∈𝐿^𝑝[𝑎,𝑏] for some 𝑛≥1. Then for each 𝑥∈(𝑎,(𝑎+𝑏)/2] we have ^𝑏_𝑎𝑤(𝑡)𝑓(𝑡)d𝑡−𝐷(𝑥)−𝑡_𝑤,𝑛(𝑥≤1)()!𝑇𝑛−1^𝑤,𝑛(𝑥,⋅)_𝑞𝑓^(𝑛)_𝑝. (2.5) Inequality (2.5) is sharp for 1<𝑝≤∞.

Proof. By applying the Hölder inequality we have 1(𝑛−1)!^𝑏_𝑎𝑇_𝑤,𝑛(𝑥,𝑠)𝑓^(𝑛)(𝑠)≤1𝑑𝑠()!𝑇𝑛−1^𝑤,𝑛(𝑥,⋅)_𝑞𝑓^(𝑛)_𝑝. (2.6) By using the above inequality from (2.2) we obtain estimate (2.5). Let us denote 𝑈^𝑥_𝑛(𝑠)=𝑇_𝑤,𝑛(𝑥,𝑠). For the proof of sharpness, we will find a function 𝑓 such that ^𝑏_𝑎𝑈^𝑥_𝑛(𝑠)𝑓^(𝑛)(𝑠)𝑑𝑠=𝑈^𝑥_𝑛𝑞𝑓^(𝑛)_𝑝. (2.7) For 1<𝑝<∞, take 𝑓 to be such that 𝑓^(𝑛)(𝑠)=sign𝑈^𝑥_𝑛(𝑠)⋅𝑈^𝑥_𝑛(𝑠)^1/(𝑝−1), (2.8) where for 𝑝=∞ we put 𝑓^(𝑛)(𝑠)=sign𝑈^𝑥_𝑛(𝑠). (2.9)

Remark 2.4. Inequality (2.5) for 𝐴(𝑥)=1/4 was proved by Aglić Aljinović et al. in [4].

3. Non-Weighted Four-Point Formula and Applications

Here we define 𝑡_𝑛̂(𝑥)=𝐴(𝑥)^𝑛−2_𝑖=0𝑓^(𝑖+1)(𝑥)+(−1)^𝑖+1𝑓^(𝑖+1)(𝑎+𝑏−𝑥)()𝑏−𝑥^𝑖+2−(𝑎−𝑥)^𝑖+2()!()+1𝑖+2𝑏−𝑎2−𝐴(𝑥)^𝑛−2_𝑖=0𝑓^(𝑖+1)(𝑎)+(−1)^𝑖+1𝑓^(𝑖+1)(𝑏())𝑏−𝑎^𝑖+1()!,𝑇𝑖+2_𝑛()1𝑥,𝑠=−𝑛2−𝐴(𝑥)𝑇_𝑛(𝑎,𝑠)+𝑇_𝑛(𝑏,𝑠)+𝐴(𝑥)𝑇_𝑛(𝑥,𝑠)+𝑇_𝑛(=1𝑎+𝑏−𝑥,𝑠)2(𝑥)()+𝐴𝑎−𝑠^𝑛()+1𝑏−𝑎2(𝑥)()−𝐴𝑏−𝑠^𝑛()()𝑏−𝑎,𝑎≤𝑠≤𝑥,𝑎−𝑠^𝑛+(𝑏−𝑠)^𝑛2()1𝑏−𝑎,𝑥<𝑠≤𝑎+𝑏−𝑥,2(𝑥)()−𝐴𝑎−𝑠^𝑛()+1𝑏−𝑎2(𝑥)()+𝐴𝑏−𝑠^𝑛(𝑏−𝑎),𝑎+𝑏−𝑥<𝑠≤𝑏. (3.1)(3.2)

Theorem 3.1. Let 𝐼 be an open interval in ℝ, [𝑎,b]⊂𝐼, and let 𝑓∶𝐼→ℝ be such that 𝑓^(𝑛−1) is absolutely continuous for some 𝑛≥1. Then for each 𝑥∈(𝑎,(𝑎+𝑏)/2] the following identity holds 1𝑏−𝑎^𝑏_𝑎𝑓(𝑡)d𝑡=𝐷(𝑥)+𝑡_𝑛̂(𝑥)+1𝑛!^𝑏_𝑎𝑇_𝑛(𝑥,𝑠)𝑓^(𝑛)(𝑠)𝑑𝑠. (3.3)

Proof. We take 𝑤(𝑡)=1/(𝑏−𝑎), 𝑡∈[𝑎,𝑏] in (2.2).

Theorem 3.2. Suppose that all assumptions of Theorem 3.1 hold. Additionally, assume that (𝑝,𝑞) is a pair of conjugate exponents, that is, 1≤𝑝,𝑞≤∞, 1/𝑝+1/𝑞=1 and 𝑓^(𝑛)∈𝐿^𝑝[𝑎,𝑏] for some 𝑛≥1. Then for each 𝑥∈(𝑎,(𝑎+𝑏)/2] we have 1𝑏−𝑎^𝑏_𝑎𝑓(𝑡)𝑑𝑡−𝐷(𝑥)−𝑡_𝑛̂(𝑥≤1)𝑇𝑛!^𝑛(𝑥,⋅)_𝑞𝑓^(𝑛)_𝑝. (3.4) Inequality (3.4) is sharp for 1<𝑝≤∞.

Proof. We take 𝑤(𝑡)=1/(𝑏−𝑎), 𝑡∈[𝑎,𝑏] in (2.5).

Now, we set 𝐴(𝑥)=(𝑏−𝑎)²12(𝑥−𝑎)(𝑏−𝑥),𝑥∈𝑎,𝑎+𝑏2. (3.5) This special choice of the function 𝐴 enables us to consider generalizations of the well-known Simpson’s 3/8 formula (1.1) and Lobatto formula (1.2)

3.1. 𝑥=(2𝑎+𝑏)/3

Suppose that all assumptions of Theorem 3.1 hold. Then the following generalization of Simpson’s 3/8 formula reads 1𝑏−𝑎^𝑏_𝑎𝑓(𝑡)𝑑𝑡=𝐷2𝑎+𝑏3+𝑡_𝑛̂2𝑎+𝑏3+1𝑛!^𝑏_𝑎𝑇_𝑛2𝑎+𝑏3,𝑠𝑓^(𝑛)(𝑠)𝑑𝑠, (3.6) where 𝐷2𝑎+𝑏3=18𝑓(𝑎)+3𝑓2𝑎+𝑏3+3𝑓𝑎+2𝑏3(𝑏),𝑡̂+𝑓_𝑛2𝑎+𝑏3=18^𝑛−2_𝑖=0𝑓^(𝑖+1)2𝑎+𝑏3+(−1)^𝑖+1𝑓^(𝑖+1)𝑎+2𝑏32^𝑖+2+(−1)^𝑖+1(𝑏−𝑎)^𝑖+13^𝑖+1()!+1𝑖+28^𝑛−2_𝑖=0𝑓^(𝑖+1)(𝑎)+(−1)^𝑖+1𝑓^(𝑖+1)(𝑏())𝑏−𝑎^𝑖+1()!,𝑇𝑖+2_𝑛2𝑎+𝑏3𝑛,𝑠=−8𝑇_𝑛(𝑎,𝑠)+3𝑇_𝑛2𝑎+𝑏3,𝑠+3𝑇_𝑛𝑎+2𝑏3,𝑠+𝑇_𝑛(=7()𝑏,𝑠)𝑎−𝑠^𝑛+(𝑏−𝑠)^𝑛8(𝑏−𝑎)𝑎≤𝑠≤2𝑎+𝑏3,(𝑎−𝑠)^𝑛+(𝑏−𝑠)^𝑛2(),𝑏−𝑎2𝑎+𝑏3<𝑠≤𝑎+2𝑏3,(𝑎−𝑠)^𝑛+7(𝑏−𝑠)^𝑛8(𝑏−𝑎)𝑎+2𝑏3<𝑠≤𝑏. (3.7)

In the next corollaries we will use the beta function and the incomplete beta function of Euler type defined by 𝐵𝑥,𝑦=¹₀𝑡^𝑥−1(1−𝑡)^𝑦−1𝑑𝑡,𝐵_𝑟𝑥,𝑦=^𝑟₀𝑡^𝑥−1(1−𝑡)^𝑦−1𝑑𝑡,𝑥,𝑦>0. (3.8)

Corollary 3.3. Suppose that all assumptions of Theorem 3.1 hold. Additionally, assume that (𝑝,𝑞) is a pair of conjugate exponents and 𝑛∈ℕ. (a)If 𝑓^(𝑛)∈𝐿^∞[𝑎,𝑏], then 1𝑏−𝑎^𝑏_𝑎𝑓(𝑡)d𝑡−𝐷2𝑎+𝑏3≤25(𝑓288𝑏−𝑎)^′_∞,1𝑏−𝑎^𝑏_𝑎𝑓(𝑡)d𝑡−𝐷2𝑎+𝑏3−𝑡_𝑛̂2𝑎+𝑏3≤1()!3𝑛+1^𝑛+1+3⋅2^𝑛+1+3(−1)^𝑛(𝑏−𝑎)^𝑛4⋅3^𝑛+1−𝑏−𝑎2^𝑛(−1)^𝑛+1+12𝑓^(𝑛)_∞,𝑛≥2. (3.3)(b) If 𝑓^(𝑛)∈𝐿²[𝑎,𝑏], then 1𝑏−𝑎^𝑏_𝑎𝑓(𝑡)d𝑡−𝐷2𝑎+𝑏3−𝑡_𝑛̂2𝑎+𝑏3≤13𝑛!^2𝑛+5⋅2^2𝑛+1+11(𝑏−𝑎)^2𝑛−132⋅3^2𝑛()+()2𝑛+1−1^𝑛(𝑏−𝑎)^2𝑛−1×()327𝐵𝑛+1,𝑛+1+9𝐵_2/3(𝑛+1,𝑛+1)−9𝐵_1/3(𝑛+1,𝑛+1)^1/2𝑓^(𝑛)₂. (3.10)(c) If 𝑓^(𝑛)∈𝐿¹[𝑎,𝑏], then 1𝑏−𝑎^𝑏_𝑎𝑓(𝑡)𝑑𝑡−𝐷2𝑎+𝑏3−𝑡_𝑛̂2𝑎+𝑏3≤1𝐾𝑛!^𝑛2𝑎+𝑏3𝑓^(𝑛)₁, (3.11)where 𝐾₁((2𝑎+𝑏)/3)=5/24, 𝐾₂((2𝑎+𝑏)/3)=(5/18)(𝑏−𝑎), 𝐾₃((2𝑎+𝑏)/3)=(7/54)(𝑏−𝑎)² and 𝐾_𝑛((2𝑎+𝑏)/3)=(1/8)(𝑏−𝑎)^𝑛−1, for 𝑛≥4.
The first and the second inequality are sharp.

Proof. We apply (3.4) with 𝑥=(2𝑎+𝑏)/3 and 𝑝=∞^𝑏_𝑎𝑇_𝑛2𝑎+𝑏3,𝑠𝑑𝑠=_𝑎^{(2𝑎+𝑏)/3}7(𝑎−𝑠)^𝑛+(𝑏−𝑠)^𝑛8()+𝑏−𝑎𝑑𝑠_{(𝑎+2𝑏)/3(2𝑎+𝑏)/3}(𝑎−𝑠)^𝑛+(𝑏−𝑠)^𝑛2(𝑏−𝑎)𝑑𝑠+^𝑏_{(𝑎+2𝑏)/3}(𝑎−𝑠)^𝑛+7(𝑏−𝑠)^𝑛8()3𝑏−𝑎𝑑𝑠=2^𝑛+1−2^𝑛+1+7⋅(−1)^𝑛(𝑏−𝑎)^𝑛8⋅3^𝑛+1()+2𝑛+1_𝑛+1+(−1)^𝑛+1(𝑏−𝑎)^𝑛3^𝑛+1()−()𝑛+11+−1^𝑛+1(𝑏−𝑎)^𝑛2^𝑛+1()=3𝑛+1_𝑛+1+3⋅2^𝑛+1+3(−1)^𝑛(𝑏−𝑎)^𝑛4⋅3^𝑛+1()−𝑛+1𝑏−𝑎2^𝑛(−1)^𝑛+1+12(),𝑛+1 (3.12) for 𝑛≥2 and ^𝑏_𝑎𝑇₁2𝑎+𝑏3,𝑠𝑑𝑠=25().288𝑏−𝑎 (3.13) To obtain the second inequality we take 𝑝=2^𝑏_𝑎𝑇_𝑛2𝑎+𝑏3,𝑠²𝑑𝑠=_𝑎^{(2𝑎+𝑏)/3}7(𝑎−𝑠)^𝑛+(𝑏−𝑠)^𝑛8(𝑏−𝑎)²+𝑑𝑠_{(𝑎+2𝑏)/3(2𝑎+𝑏)/3}(𝑎−𝑠)^𝑛+(𝑏−𝑠)^𝑛2(𝑏−𝑎)²𝑑𝑠+^𝑏_{(𝑎+2𝑏)/3}(𝑎−𝑠)^𝑛+7(𝑏−𝑠)^𝑛8(𝑏−𝑎)²=3𝑑𝑠_2𝑛+5⋅2^2𝑛+1+11(𝑏−𝑎)^2𝑛−132⋅3^2𝑛()+()2𝑛+1−1^𝑛(𝑏−𝑎)^2𝑛−1×()327𝐵𝑛+1,𝑛+1+9𝐵_2/3(𝑛+1,𝑛+1)−9𝐵_1/3(𝑛+1,𝑛+1). (3.14) If 𝑝=1, we have sup_{[]𝑠∈𝑎,𝑏}𝑇_𝑛2𝑎+𝑏3,𝑠=maxsup_{[()]𝑠∈𝑎,2𝑎+𝑏/3}7(𝑎−𝑠)^𝑛+(𝑏−𝑠)^𝑛8(),𝑏−𝑎sup_{)()]𝑠∈[(2𝑎+𝑏/3,𝑎+2𝑏/3}(𝑎−𝑠)^𝑛+(𝑏−𝑠)^𝑛2(𝑏−𝑎)sup_{)]𝑠∈[(𝑎+2𝑏/3,𝑏}(𝑎−𝑠)^𝑛+7(𝑏−𝑠)^𝑛8().𝑏−𝑎 (3.15)
By an elementary calculation we get sup_{[()]𝑠∈𝑎,2𝑎+𝑏/3}7(𝑎−𝑠)+(𝑏−𝑠)8(𝑏−𝑎)=sup_{[]𝑠∈(𝑎+2𝑏)/3,𝑏}(𝑎−𝑠)+7(𝑏−𝑠)8()=5𝑏−𝑎(),24𝑏−𝑎sup_{[()]𝑠∈𝑎,2𝑎+𝑏/3}7(𝑎−𝑠)²+(𝑏−𝑠)²8(𝑏−𝑎)=sup_{[]𝑠∈(𝑎+2𝑏)/3,𝑏}(𝑎−𝑠)²+7(𝑏−𝑠)²8()=𝑏−𝑎11(),72𝑏−𝑎sup_{[]𝑠∈𝑎,(2𝑎+𝑏)/3}7(𝑎−𝑠)^𝑛+(𝑏−𝑠)^𝑛8(𝑏−𝑎)=sup_{[]𝑠∈(𝑎+2𝑏)/3,𝑏}(𝑎−𝑠)^𝑛+7(𝑏−𝑠)^𝑛8()=()𝑏−𝑎𝑏−𝑎^𝑛−18, (3.16) for 𝑛≥3. The function 𝑦∶[𝑎,𝑏]→ℝ, 𝑦(𝑥)=(𝑎−𝑥)^𝑛+(𝑏−𝑥)^𝑛, is decreasing on ⟨𝑎,(𝑎+𝑏)/2⟩ and increasing on ⟨(𝑎+𝑏)/2,𝑏⟩ if 𝑛 is even, and decreasing on ⟨𝑎,𝑏⟩ if 𝑛 is odd. Thus sup_{)()]𝑠∈[(2𝑎+𝑏/3,𝑎+2𝑏/3}(𝑎−𝑠)^𝑛+(𝑏−𝑠)^𝑛2()=()𝑏−𝑎−1^𝑛+2^𝑛(𝑏−𝑎)^𝑛−12⋅3^𝑛. (3.17) Finally, sup_{[]𝑠∈𝑎,𝑏}𝑇₁2𝑎+𝑏3=5,𝑠24 (3.18) and for 𝑛≥2sup_{[]𝑠∈𝑎,𝑏}𝑇_𝑛2𝑎+𝑏3,𝑠=(𝑏−𝑎)^𝑛−11max8,2^𝑛+(−1)^𝑛2⋅3^𝑛. (3.19)

3.2. [𝑎,𝑏]=[−1,1],  𝑥=−√5/5

Suppose that all assumptions of Theorem 3.1 hold. Then the following generalization of Lobatto formula reads 12¹₋₁𝑓(𝑡)−√𝑑𝑡=𝐷55+𝑡_𝑛̂−√55+1𝑛!¹₋₁𝑇_𝑛−√55,𝑠𝑓^(𝑛)(𝑠)𝑑𝑠, (3.20) where 𝐷−√55=1𝑓()−√12−1+5𝑓55√+5𝑓55(1),𝑡̂+𝑓_𝑛−√55=512^𝑛−2_𝑖=0𝑓^(𝑖+1)−√55+(−1)^𝑖+1𝑓^(𝑖+1)√55×5+√5^𝑖+2+(−1)^𝑖+15−√5^𝑖+22⋅5^𝑖+2()!+1𝑖+212^𝑛−2_𝑖=0𝑓^(𝑖+1)(−1)+(−1)^𝑖+1𝑓^(𝑖+1)(12)^𝑖+1()!,𝑇𝑖+2_𝑛−√55𝑛,𝑠=−𝑇12^𝑛(−1,𝑠)+5𝑇_𝑛−√55,𝑠+5𝑇_𝑛√55,𝑠+𝑇_𝑛(=()1,𝑠)11−1−𝑠^𝑛+(1−𝑠)^𝑛√24−1≤𝑠≤−55,(−1−𝑠)^𝑛+(1−𝑠)^𝑛4−√55√<𝑠≤55,(−1−𝑠)^𝑛+11(1−𝑠)^𝑛√2455<𝑠≤1. (3.21)

Corollary 3.4. Suppose that all assumptions of Theorem 3.1 hold. Additionally, assume that (𝑝,𝑞) is a pair of conjugate exponents and 𝑛∈ℕ. (a) if 𝑓^(𝑛)∈𝐿^∞[−1,1], then 12¹₋₁𝑓(𝑡)−√𝑑𝑡−𝐷55≤101−√18056𝑓^′_∞,12¹₋₁𝑓(𝑡)−√𝑑𝑡−𝐷55−𝑡_𝑛̂−√55≤1()!2𝑛+1^𝑛+1⋅5^𝑛+5+√5^𝑛+1−−5+√5^𝑛+112⋅5^𝑛−1+(−1)^𝑛+12𝑓^(𝑛)_∞,𝑛≥2. (3.22)(b) if 𝑓^(𝑛)∈𝐿²[−1,1], then 12¹₋₁𝑓(𝑡)−√𝑑𝑡−𝐷55−𝑡_𝑛̂−√55≤1⋅2𝑛!3^𝑛−2355+√5^2𝑛+1+855−√5^2𝑛+1+10^2𝑛+110^2𝑛+1()+()2𝑛+1−1^𝑛11𝐵(𝑛+1,𝑛+1)+25𝐵_√(5+5)/10(𝑛+1,𝑛+1)−25𝐵_{√(5−5)/10}(𝑛+1,𝑛+1)^1/2𝑓^(𝑛)₂. (3.23)(c) if 𝑓^(𝑛)∈𝐿¹[−1,1], then 12¹₋₁𝑓(𝑡)−√𝑑𝑡−𝐷55−𝑡_𝑛̂−√55≤1𝐾𝑛!^𝑛−√55𝑓^(𝑛)₁, (3.24)where 𝐾₁(−√5/5)=1/(2√5), 𝐾₂(−√5/5)=3/5, 𝐾₃(−√5/5)=8/(5√5), 𝐾₄(−√5/5)=28/25, 𝐾₅(−√5/5)=88/(25√5), 𝐾_𝑛(−√5/5)=2^𝑛−3/3, for 𝑛≥6.
The first and the second inequality are sharp.

Proof. Applying (3.4) with [𝑎,𝑏]=[−1,1], 𝑥=−√5/5 and 𝑝=∞,𝑝=2,𝑝=1 and carrying out the same analysis as in Corollay 3.3 we obtain the above inequalities.