Generalizations of the Simpson-Like Type Inequalities for Co-Ordinated 𝑠-Convex Mappings in the Second Sense

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 coordinated s-convex mappings in the second sense.

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. Let 𝑓∶𝕀⊂[0,∞)→ℝ be a four-time continuous differentiable mapping on [𝑎,𝑏] and ‖𝑓⁽⁴⁾‖_∞=sup_{𝑥∈[𝑎,𝑏]}|𝑓⁽⁴⁾(𝑥)|<∞. Then, the following inequality holds: 16𝑓(𝑎)+4𝑓𝑎+𝑏2(𝑏−1+𝑓)𝑏−𝑎^𝑏_𝑎𝑓(𝑥)≤()𝑑𝑥𝑏−𝑎⁴𝑓2880^(4)∞. (1.1)

It is well known that the mapping 𝑓 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 let Δ=[𝑎,𝑏]×[𝑐,𝑑] be a bidimensional interval in ℝ² with 𝑎<𝑏 and 𝑐<𝑑.

Definition 1.2. A mapping 𝑓∶Δ→ℝ will be called 𝑠-convex in the second sense on Δ if the following inequality: 𝑓𝜆𝑥+(1−𝜆)𝑧,𝜆𝑦+(1−𝜆)𝑤≤𝜆^𝑠𝑓𝑥,𝑦+(1−𝜆)^𝑠𝑓(𝑧,𝑤), (1.2) holds, for all (𝑥,𝑦),(𝑧,𝑤)∈Δ, 𝜆∈[0,1] and 𝑠∈[0,1].

Modification for convex and 𝑠-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 mapping 𝑓∶Δ→ℝ will be called coordinated 𝑠-convex in the second sense on Δ if the partial mappings 𝑓_𝑦∶[𝑎,𝑏]⟶ℝ,𝑓_𝑦(𝑢),𝑓=𝑓𝑢,𝑦_𝑥∶[𝑐,𝑑]⟶ℝ,𝑓_x(𝑣)=𝑓(𝑥,𝑣), (1.3) 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 coordinated 𝑠-convex mappings may be stated as follow [8].

Definition 1.4. A mapping 𝑓∶Δ→ℝ will be called coordinated 𝑠-convex in the second sense on Δ if the following inequality: 𝑓𝑡𝑥+(1−𝑡)𝑧,𝜆𝑦+(1−𝜆)𝑤≤𝑡^𝑠𝜆^𝑠𝑓𝑥,𝑦+(1−𝑡)^𝑠𝜆^𝑠𝑓𝑧,𝑦+𝑡^𝑠(1−𝜆)^𝑠𝑓(𝑥,𝑤)+(1−𝑡)^𝑠(1−𝜆)^𝑠𝑓(𝑧,𝑤), (1.4) 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. Let 𝑓∶Δ→ℝ be convex on the coordinates on Δ. Then, one has the inequalities: 𝑓𝑎+𝑏2,𝑐+𝑑2≤1(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓≤1𝑥,𝑦𝑑𝑦𝑑𝑥4𝑓(𝑎,𝑐)+𝑓(𝑏,𝑐)+𝑓(𝑎,𝑑)+𝑓(𝑏,𝑑). (1.5)

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. Let 𝑓∶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: 19𝑓𝑎,𝑐+𝑑2+𝑓𝑏,𝑐+𝑑2+4𝑓𝑎+𝑏2,𝑐+𝑑2+𝑓𝑎+𝑏2,𝑐+𝑓𝑎+𝑏2+1,𝑑𝑓()()()(+136𝑎,𝑐+𝑓𝑎,𝑑+𝑓𝑏,𝑐+𝑓𝑏,𝑑)(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓𝑥,𝑦𝑑𝑦𝑑𝑥−A≤572²()×𝜕𝑏−𝑎)(𝑑−𝑐₂𝑓(+𝜕𝜕𝑡𝜕𝜆𝑎,𝑐)²𝑓(+𝜕𝜕𝑡𝜕𝜆𝑎,𝑑)²𝑓(+𝜕𝜕𝑡𝜕𝜆𝑏,𝑐)²𝑓(,𝜕𝑡𝜕𝜆𝑏,𝑑) (1.6) where 1𝐴=𝑏−𝑎^𝑏_𝑎𝑓(𝑥,𝑐)+4𝑓(𝑥,(𝑐+𝑑)/2)+𝑓(𝑥,𝑑)6+1𝑑𝑥𝑑−𝑐^𝑑_𝑐𝑓𝑎,𝑦+4𝑓(𝑎+𝑏)/2,𝑦+𝑓𝑏,𝑦6𝑑𝑦. (1.7)

Theorem 1.7. Let 𝑓∶Δ⊂ℝ²→ℝ be a partial differentiable mapping on Δ. If 𝜕²𝑓/𝜕𝑡𝜕𝜆 is bounded, that is, 𝜕²𝑓𝜕𝑡𝜕𝜆(𝑡𝑎+(1−𝑡)𝑏,𝜆𝑐+(1−𝜆)𝑑)_∞=sup_{(𝑥,𝑦)∈(𝑎,𝑏)×(𝑐,𝑑)}𝜕²𝑓(()()𝑑𝜕𝑡𝜕𝜆𝑡𝑎+1−𝑡𝑏,𝜆𝑐+1−𝜆)<∞ (1.8) for all (𝑡,𝜆)∈[0,1]², then the following inequality holds: 19𝑓𝑎,𝑐+𝑑2+𝑓𝑏,𝑐+𝑑2+4𝑓𝑎+𝑏2,𝑐+𝑑2+𝑓𝑎+𝑏2,𝑐+𝑓𝑎+𝑏2+1,𝑑𝑓()()()(+136𝑎,𝑐+𝑓𝑎,𝑑+𝑓𝑏,𝑐+𝑓𝑏,𝑑)(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓≤5𝑥,𝑦𝑑𝑦𝑑𝑥−𝐴72²(𝜕𝑏−𝑎)(𝑑−𝑐)²𝑓(()()𝑑𝜕𝑡𝜕𝜆𝑡𝑎+1−𝑡𝑏,𝜆𝑐+1−𝜆)_∞, (1.9) 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 coordinated 𝑠-convex mappings in the second sense.

2. Main Results

To prove our main results, we need the following lemma.

Lemma 2.1. Let 𝑓∶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: 𝐼𝑓ℎ₁,ℎ₂,𝑟₁,𝑟₂=^let𝑟₁−2𝑟₂−2𝑟₁𝑟₂𝑓ℎ₁𝑎+1−ℎ₁𝑏,ℎ₂𝑐+1−ℎ₂𝑑+𝑟₁−2𝑟₁𝑟₂𝑓ℎ₁𝑎+1−ℎ₁𝑏,𝑐+𝑓ℎ₁𝑎+1−ℎ₁+𝑟𝑏,𝑑₂−2𝑟₁𝑟₂𝑓𝑎,ℎ₂𝑐+1−ℎ₂𝑑+𝑓𝑏,ℎ₂𝑐+1−ℎ₂𝑑+1𝑟₁𝑟₂𝑓()()()(−1𝑎,𝑐+𝑓𝑎,𝑑+𝑓𝑏,𝑐+𝑓𝑏,𝑑)𝑟₂(𝑏−𝑎)^𝑏_𝑎𝑓(𝑥,𝑐)+𝑟₂−2𝑓𝑥,ℎ₂𝑐+1−ℎ₂𝑑(−1+𝑓𝑥,𝑑)𝑑𝑥𝑟₁(𝑑−𝑐)^𝑑_𝑐𝑓𝑎,𝑦+𝑟₁−2𝑓ℎ₁𝑎+1−ℎ₁+1𝑏,𝑦+𝑓𝑏,𝑦𝑑𝑦(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓=()𝑥,𝑦𝑑𝑦𝑑𝑥𝑏−𝑎)(𝑑−𝑐¹₀𝑝ℎ₁,𝑟₁,𝑡𝑞ℎ₂,𝑟₂×𝜕,𝜆₂𝑓(()()𝑑)𝜕𝑡𝜕𝜆𝑡𝑎+1−𝑡𝑏,𝜆𝑐+1−𝜆𝑑𝑡𝑑𝜆, (2.1) where 𝑝ℎ₁,𝑟₁=1,𝑡𝑡−𝑟₁,𝑡∈0,ℎ₁,𝑟𝑡−¹−1𝑟₁,𝑡∈ℎ₁,𝑞ℎ,1₂,𝑟₂=1,𝜆𝜆−𝑟₂,𝜆∈0,ℎ₂,𝑟𝜆−²−1𝑟₂,𝜆∈ℎ₂,1. (2.2)

Proof. By the definitions of 𝑝(ℎ₁,𝑟₁,𝑡) and 𝑞(ℎ₂,𝑟₂,𝜆), we can write 𝐼=¹₀𝑝ℎ₁,𝑟₁,𝑡𝑞ℎ₂,𝑟₂𝜕,𝜆²𝑓(()()𝑑)=𝜕𝑡𝜕𝜆𝑡𝑎+1−𝑡𝑏,𝜆𝑐+1−𝜆𝑑𝑡𝑑𝜆₁₀𝑞ℎ₂,𝑟₂,𝜆¹₀𝑝ℎ₁,𝑟₁𝜕,𝑡²𝑓(()()𝑑)=𝜕𝑡𝜕𝜆𝑡𝑎+1−𝑡𝑏,𝜆𝑐+1−𝜆𝑑𝑡𝑑𝜆₁₀𝑞ℎ₂,𝑟₂,𝜆^ℎ1₀1𝑡−𝑟₁𝜕²𝑓(()()𝑑)+𝜕𝑡𝜕𝜆𝑡𝑎+1−𝑡𝑏,𝜆𝑐+1−𝜆𝑑𝑡_1ℎ1𝑟𝑡−¹−1𝑟₁𝜕²𝑓(()()𝑑)=𝜕𝑡𝜕𝜆𝑡𝑎+1−𝑡𝑏,𝜆𝑐+1−𝜆𝑑𝑡𝑑𝜆₁₀𝑞ℎ₂,𝑟₂,𝜆𝐼₁₁+𝐼₁₂𝑑𝜆, (2.3) where 𝐼₁₁=^ℎ1₀1𝑡−𝑟₁𝜕²𝑓(()()𝑑)𝐼𝜕𝑡𝜕𝜆𝑡𝑎+1−𝑡𝑏,𝜆𝑐+1−𝜆𝑑𝑡,₁₂=¹_ℎ1𝑟𝑡−¹−1𝑟₁𝜕²𝑓(()()𝑑)𝜕𝑡𝜕𝜆𝑡𝑎+1−𝑡𝑏,𝜆𝑐+1−𝜆𝑑𝑡. (2.4)
By integration by parts, we have 𝐼₁₁=^ℎ1₀1𝑡−𝑟₁𝜕²𝑓(()()𝑑)=1𝜕𝑡𝜕𝜆𝑡𝑎+1−𝑡𝑏,𝜆𝑐+1−𝜆𝑑𝑡ℎ𝑎−𝑏¹−1𝑟₁𝜕𝑓ℎ𝜕𝜆¹𝑎+1−ℎ₁()𝑑+1𝑏,𝜆𝑐+1−𝜆𝑟₁𝜕𝑓(()𝑑)−𝜕𝜆𝑏,𝜆𝑐+1−𝜆_ℎ10𝜕𝑓(()()𝑑),𝜕𝜆𝑡𝑎+1−𝑡𝑏,𝜆𝑐+1−𝜆𝑑𝑡𝐼₁₂=¹_ℎ1𝑟𝑡−¹−1𝑟₁𝜕²𝑓(()()𝑑)=1𝜕𝑡𝜕𝜆𝑡𝑎+1−𝑡𝑏,𝜆𝑐+1−𝜆𝑑𝑡1𝑎−𝑏𝑟₁𝜕𝑓(()𝑑)−ℎ𝜕𝜆𝑎,𝜆𝑐+1−𝜆₁−𝑟₁−1𝑟₁𝜕𝑓ℎ𝜕𝜆¹𝑎+1−ℎ₁()𝑑−𝑏,𝜆𝑐+1−𝜆_1ℎ1𝜕𝑓(()()𝑑).𝜕𝜆𝑡𝑎+1−𝑡𝑏,𝜆𝑐+1−𝜆𝑑𝑡 (2.5)(2.6)
By using the equalities (2.5) and (2.6) in (2.3), we have 1𝐼=𝑟𝑎−𝑏¹−2𝑟₁¹₀𝑞ℎ₂,𝑟₂,𝜆𝜕𝑓ℎ𝜕𝜆¹𝑎+1−ℎ₁()𝑑+1𝑏,𝜆𝑐+1−𝜆𝑑𝜆𝑟₁¹₀𝑞ℎ₂,𝑟₂,𝜆𝜕𝑓(()𝑑)+1𝜕𝜆𝑏,𝜆𝑐+1−𝜆𝑑𝜆𝑟₁¹₀𝑞ℎ₂,𝑟₂,𝜆𝜕𝑓(()𝑑)−𝜕𝜆𝑎,𝜆𝑐+1−𝜆𝑑𝜆₁₀𝑞ℎ₂,𝑟₂,𝜆¹₀𝜕𝑓(()()𝑑)=1𝜕𝜆𝑡𝑎+1−𝑡𝑏,𝜆𝑐+1−𝜆𝑑𝜆𝑑𝑡𝑟𝑎−𝑏¹−2𝑟₁𝐼₂₁+1𝑟₁𝐼₂₂+1𝑟₁𝐼₂₃−𝐼₂₄, (2.7) where 𝐼₂₁=¹₀𝑞ℎ₂,𝑟₂,𝜆𝜕𝑓ℎ𝜕𝜆¹𝑎+1−ℎ₁()𝑑𝐼𝑏,𝜆𝑐+1−𝜆𝑑𝜆,₂₂=¹₀𝑞ℎ₂,𝑟₂,𝜆𝜕𝑓(()𝑑)𝐼𝜕𝜆𝑏,𝜆𝑐+1−𝜆𝑑𝜆,₂₃=¹₀𝑞ℎ₂,𝑟₂,𝜆𝜕𝑓(()𝑑)𝐼𝜕𝜆𝑎,𝜆𝑐+1−𝜆𝑑𝜆,₂₄=¹₀𝑞ℎ₂,𝑟₂,𝜆¹₀𝜕𝑓(()()𝑑)𝜕𝜆𝑡𝑎+1−𝑡𝑏,𝜆𝑐+1−𝜆𝑑𝜆𝑑𝑡. (2.8)
Note that (i)^ℎ2₀1𝜆−𝑟₂𝜕𝑓ℎ𝜕𝜆¹𝑎+1−ℎ₁()𝑑=1𝑏,𝜆𝑐+1−𝜆𝑑𝜆ℎ𝑐−𝑑²−1𝑟₂𝑓ℎ₁𝑎+1−ℎ₁𝑏,ℎ₂𝑐+1−ℎ₂𝑑+1𝑟₂𝑓ℎ₁a+1−ℎ₁−𝑏,𝑑_ℎ20𝑓ℎ₁𝑎+1−ℎ₁𝑏,𝜆𝑐+(1−𝜆)𝑑𝑑𝜆,(ii)¹_ℎ2𝑟𝜆−²−1𝑟₂𝜕𝑓ℎ𝜕𝜆¹𝑎+1−ℎ₁()𝑑=1𝑏,𝜆𝑐+1−𝜆𝑑𝜆1𝑐−𝑑𝑟₂𝑓ℎ₁𝑎+1−ℎ₁−ℎ𝑏,𝑐₂−𝑟₂−1𝑟₂𝑓ℎ₁𝑎+1−ℎ₁𝑏,ℎ₂𝑐+1−ℎ₂𝑑−¹_ℎ2𝑓ℎ₁𝑎+1−ℎ₁𝑏,𝜆𝑐+(1−𝜆)𝑑𝑑𝜆. (2.9)(2.10)
By the equalities (2.9) and (2.10), we have 𝐼₂₁=¹₀𝑞ℎ₂,𝑟₂,𝜆𝜕𝑓ℎ𝜕𝜆¹𝑎+1−ℎ₁()𝑑=𝑏,𝜆𝑐+1−𝜆𝑑𝜆_ℎ201𝜆−𝑟₂𝜕𝑓ℎ𝜕𝜆¹𝑎+1−ℎ₁()𝑑+𝑏,𝜆𝑐+1−𝜆𝑑𝜆_1ℎ2𝑟𝜆−²−1𝑟₂𝜕𝑓ℎ𝜕𝜆¹𝑎+1−ℎ₁()𝑑=1𝑏,𝜆𝑐+1−𝜆𝑑𝜆1𝑐−𝑑𝑟₂𝑓ℎ₁𝑎+1−ℎ₁+1𝑏,𝑐𝑟₂𝑓ℎ₁𝑎+1−ℎ₁+𝑟𝑏,𝑑₂−2𝑟₂𝑓ℎ₁𝑎+1−ℎ₁𝑏,ℎ₂𝑐+1−ℎ₂𝑑−¹₀𝑓ℎ₁𝑎+1−ℎ₁𝑏,𝜆𝑐+(1−𝜆)𝑑𝑑𝜆. (2.11)
By the similar way, we get the following: 𝐼₂₂=¹₀𝑞ℎ₂,𝑟₂,𝜆𝜕𝑓(()𝑑)=1𝜕𝜆𝑏,𝜆𝑐+1−𝜆𝑑𝜆1𝑐−𝑑𝑟₂𝑓()+1𝑏,𝑐𝑟₂𝑓()+𝑟𝑏,𝑑²−2𝑟₂𝑓𝑏,ℎ₂𝑐+1−ℎ₂𝑑−¹₀𝑓(𝑏,𝜆𝑐+(1−𝜆)𝑑)𝑑𝜆,𝐼₂₃=¹₀𝑞ℎ₂,𝑟₂,𝜆𝜕𝑓(()𝑑)=1𝜕𝜆𝑎,𝜆𝑐+1−𝜆𝑑𝜆1𝑐−𝑑𝑟₂𝑓()+1𝑎,𝑐𝑟₂𝑓()+𝑟𝑎,𝑑²−2𝑟₂𝑓𝑎,ℎ₂𝑐+1−ℎ₂𝑑−¹₀𝑓(𝑎,𝜆𝑐+(1−𝜆)𝑑)𝑑𝜆,𝐼₂₄=¹₀𝑞ℎ₂,𝑟₂,𝜆¹₀𝜕𝑓(()()𝑑)=1𝜕𝜆𝑡𝑎+1−𝑡𝑏,𝜆𝑐+1−𝜆𝑑𝜆𝑑𝑡1𝑐−𝑑𝑟₂¹₀𝑓(())1𝑡𝑎+1−𝑡𝑏,𝑐𝑑𝑡+𝑟₂¹₀𝑓(())+𝑟𝑡𝑎+1−𝑡𝑏,𝑑𝑑𝑡₂−2𝑟₂¹₀𝑓𝑡𝑎+(1−𝑡)𝑏,ℎ₂𝑐+1−ℎ₂𝑑d𝑡−¹₀𝑓(𝑡𝑎+(1−𝑡)𝑏,𝜆𝑐+(1−𝜆)𝑑)𝑑𝜆𝑑𝑡. (2.12)(2.13)(2.14)
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. Lemma 2.1 is a generalization of the results which proved by Sarikaya, Set, Özdemir, and Dragomir [3, 5, 9, 10].

Theorem 2.3. Let 𝑓∶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: 1()𝐼𝑓ℎ𝑏−𝑎)(𝑑−𝑐¹,ℎ₂,𝑟₁,𝑟₂≤𝜇₁(𝑟,𝑠)𝜇₂(𝜕𝑟,𝑠)²𝑓(𝜕𝑡𝜕𝜆𝑎,𝑐)+𝜈₂(𝜕𝑟,𝑠)²𝑓(𝜕𝑡𝜕𝜆𝑎,𝑑)+𝜈₁(𝑟,𝑠)𝜇₂(𝜕𝑟,𝑠)²𝑓(𝜕𝑡𝜕𝜆𝑏,𝑐)+𝜈₂(𝜕𝑟,𝑠)²𝑓(,𝜕𝑡𝜕𝜆𝑏,𝑑) (2.15) where 𝜇₁(𝑟,𝑠)=𝑀𝑟₁,𝑠+𝑁ℎ₁,𝜇,𝑠₂(𝑟,𝑠)=𝑀𝑟₂,𝑠+𝑁ℎ₂,𝜈,𝑠₁(𝑟,𝑠)=𝑀𝑟₁,𝑠+𝑁1−ℎ₁,𝜈,𝑠₂(𝑟,𝑠)=𝑀𝑟₂,𝑠−𝑁1−ℎ₂,𝑠 (2.16) for 𝑀()=()𝑟,𝑠2+2𝑟−1^𝑠+2+𝑟^𝑠+1(𝑠−𝑟+2)(𝑠+1)(𝑠+2)𝑟^𝑠+2,𝑁()=ℎℎ,𝑠^𝑠+1((2ℎ−1)𝑠+2(ℎ−1))().𝑠+1)(𝑠+2 (2.17)

Proof. From Lemma 2.1 and by the coordinated 𝑠-convexity in the second sense of 𝜕²𝑓/𝜕𝑡𝜕𝜆, we can write 1()𝐼𝑓ℎ𝑏−𝑎)(𝑑−𝑐¹,ℎ₂,𝑟₁,𝑟₂≤¹₀𝑝ℎ₁,𝑟₁,𝑡𝑞ℎ₂,𝑟₂×𝜕,𝜆₂𝑓(()()𝑑𝑑𝜕𝑡𝜕𝜆𝑡𝑎+1−𝑡𝑏,𝜆𝑐+1−𝜆)t≤𝑑𝜆₁₀𝑝ℎ₁,𝑟₁,𝑡𝑞ℎ₂,𝑟₂×𝑡,𝜆_𝑠𝜆^𝑠𝜕²𝑓(𝜕𝑡𝜕𝜆𝑎,𝑐)+𝑡^𝑠(1−𝜆)^𝑠𝜕²𝑓(+()𝜕𝑡𝜕𝜆𝑎,𝑑)1−𝑡^𝑠𝜆^𝑠𝜕²𝑓(+()𝜕𝑡𝜕𝜆𝑏,𝑐)1−𝑡^𝑠(1−𝜆)^𝑠𝜕²𝑓(=𝜕𝑡𝜕𝜆𝑏,𝑑)𝑑𝑡𝑑𝜆₁₀𝑝ℎ₁,𝑟₁,𝑡𝑡^𝑠𝑑𝑡¹₀𝑞ℎ₂,𝑟₂,𝜆𝜆^𝑠𝜕𝑑𝜆²𝑓(+𝜕𝑡𝜕𝜆𝑎,𝑐)₁₀𝑞ℎ₂,𝑟₂,𝜆(1−𝜆)^𝑠𝜕𝑑𝜆²𝑓(+𝜕𝑡𝜕𝜆𝑎,𝑑)₁₀𝑝ℎ₁,𝑟₁,𝑡(1−𝑡)^𝑠𝑑𝑡¹₀𝑞ℎ₂,𝑟₂,𝜆𝜆^𝑠𝜕𝑑𝜆²𝑓(+𝜕𝑡𝜕𝜆𝑏,𝑐)₁₀𝑞ℎ₂,𝑟₂,𝜆(1−𝜆)^𝑠𝜕𝑑𝜆²𝑓(.𝜕𝑡𝜕𝜆𝑏,𝑑) (2.18)
Note that (i)¹₀𝑝ℎ₁,𝑟₁,𝑡𝑡^𝑠𝑑𝑡=𝜇₁(),(𝑟,𝑠ii)¹₀𝑝ℎ₁,𝑟₁,𝑡(1−𝑡)^𝑠𝑑𝑡=𝜈₁(),(𝑟,𝑠iii)¹₀𝑞ℎ₂,𝑟₂,𝜆𝜆^𝑠𝑑𝜆=𝜇₂(),(𝑟,𝑠iv)¹₀𝑞ℎ₂,𝑟₂,𝜆(1−𝜆)^𝑠𝑑𝜆=𝜈₂(𝑟,𝑠). (2.19)
By (2.18) and (2.19), we get the inequality (2.15) by the simple calculations.

Remark 2.4. In Theorem 2.3,(i)if we choose ℎ₁=ℎ₂=1/2, 𝑟₁=𝑟₂=6, and 𝑠=1 in (2.15), then we get 𝐼𝑓12,12≤5,6,672²𝑀(𝑏−𝑎)(𝑑−𝑐), (2.20)(ii)if we choose ℎ₁=ℎ₂=1/2, 𝑟₁=𝑟₂=2, and 𝑠=1 in (2.15), then we get 𝐼𝑓12,12≤1,2,28²𝑀(𝑏−𝑎)(𝑑−𝑐), (2.21)where 𝜕𝑀=²𝑓(+𝜕𝜕𝑡𝜕𝜆𝑎,𝑐)²𝑓(+𝜕𝜕𝑡𝜕𝜆𝑎,𝑑)²𝑓(+𝜕𝜕𝑡𝜕𝜆𝑏,𝑐)²𝑓(,𝜕𝑡𝜕𝜆𝑏,𝑑) (2.22) which implies that Theorem 2.3 is a generalization of Theorem 1.6.

Theorem 2.5. Let 𝑓∶Δ→ℝ² be a partial differentiable mapping on Δ=[𝑎,𝑏]×[𝑐,𝑑]⊂ℝ². If 𝜕²𝑓/𝜕𝑡𝜕𝜆 is bounded, that is, 𝜕²𝑓𝜕𝑡𝜕𝜆_∞=^let𝜕²𝑓(()()𝑑𝜕𝑡𝜕𝜆𝑡𝑎+1−𝑡𝑏,𝜆𝑐+1−𝜆)_∞=sup_{(𝑥,𝑦)∈(𝑎,𝑏)×(𝑐,𝑑)}𝜕²𝑓(()()𝑑𝜕𝑡𝜕𝜆𝑡𝑎+1−𝑡𝑏,𝜆𝑐+1−𝜆)<∞ (2.23) 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: 1()𝐼𝑓ℎ𝑏−𝑎)(𝑑−𝑐¹,ℎ₂,𝑟₁,𝑟₂≤12−ℎ₁+ℎ²₁+2−𝑟₁𝑟²₁12−ℎ₂+ℎ²₂+2−𝑟₂𝑟²₂𝜕²𝑓𝜕𝑡𝜕𝜆_∞. (2.24)

Proof. From Lemma 2.1, using the property of modulus and the boundedness of 𝜕²𝑓/𝜕𝑡𝜕𝑠, we get 1()𝐼𝑓ℎ𝑏−𝑎)(𝑑−𝑐¹,ℎ₂,𝑟₁,𝑟₂≤𝜕²𝑓𝜕𝑡𝜕𝜆_∞¹₀𝑝ℎ₁,𝑟₁,t𝑞ℎ₂,𝑟₂,𝜆𝑑𝑡𝑑𝜆. (2.25)
By the simple calculations, we have (i)¹₀𝑝ℎ₁,𝑟₁1,𝑡𝑑𝑡=2−ℎ₁+ℎ²₁+2−𝑟₁𝑟²₁, (2.26)(ii)¹₀𝑞ℎ₂,𝑟₂1,𝜆𝑑𝑡=2−ℎ₂+ℎ²₂+2−𝑟₂𝑟²₂. (2.27)
By using the inequality (2.25) and the equalities (2.26)-(2.27), the assertion (2.24) holds.

Remark 2.6. In Theorem 2.5,(i)if we choose ℎ₁=ℎ₂=12 and 𝑟₁=𝑟₂=6, then we get 𝐼𝑓12,12≤5,6,636²𝜕²𝑓𝜕𝑡𝜕𝜆_∞(𝑏−𝑎)(𝑑−𝑐), (2.28)(ii)if we choose ℎ₁=ℎ₂=1/2 and 𝑟₁=𝑟₂=2, then we get 𝐼𝑓12,12≤1,2,24²𝜕²𝑓𝜕𝑡𝜕𝜆_∞(𝑏−𝑎)(𝑑−𝑐), (2.29)which implies that Theorem 2.5 is a generalization of Theorem 1.7.

The following theorem is a generalization of Theorem 1.6.

Theorem 2.7. Let 𝑓∶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: 1()𝐼𝑓ℎ𝑏−𝑎)(𝑑−𝑐¹,ℎ₂,𝑟₁,𝑟₂≤𝜇₃^1/𝑝𝜈₃^1/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)^21/𝑝, (2.30) where 𝜇₃=2+𝑟₁−𝑟₁ℎ₁−1^𝑝+1+𝑟₁ℎ₁−1^𝑝+1𝑟₁^𝑝+1,𝜈𝑝+1₃=2+𝑟₂−𝑟₂ℎ₂−1^𝑝+1+𝑟₂ℎ₂−1^𝑝+1𝑟₂^𝑝+1.𝑝+1 (2.31)

Proof. From Lemma 2.1, we can write 1()𝐼𝑓ℎ𝑏−𝑎)(𝑑−𝑐¹,ℎ₂,𝑟₁,𝑟₂≤¹₀𝑝ℎ₁,𝑟₁,𝑡𝑞ℎ₂,𝑟₂×𝜕,𝜆₂𝑓(()()𝑑≤𝜕𝑡𝜕𝜆𝑡𝑎+1−𝑡𝑏,𝜆𝑐+1−𝜆)𝑑𝑡𝑑𝜆₁₀𝑝ℎ₁,𝑟₁,𝑡𝑞ℎ₂,𝑟₂,𝜆^𝑝𝑑𝑡𝑑𝜆^1/𝑝×¹₀𝜕²𝑓(()()𝑑𝜕𝑡𝜕𝜆𝑡𝑎+1−𝑡𝑏,𝜆𝑐+1−𝜆)^𝑞𝑑𝑡𝑑𝜆^1/𝑞. (2.32)
Hence, by the inequality (2.32) and the coordinated 𝑠-convexity in the second sense of |𝜕²𝑓/𝜕𝑡𝜕𝜆|^𝑞, it follows that 1()𝐼𝑓ℎ𝑏−𝑎)(𝑑−𝑐¹,ℎ₂,𝑟₁,𝑟₂≤¹₀𝑝ℎ₁,𝑟₁,𝑡𝑞ℎ₂,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)^21/𝑞. (2.33)
Note that (i)¹₀𝑝ℎ₁,𝑟₁,𝑡^𝑝𝑟𝑑𝑡=2+₁−𝑟₁ℎ₁−1^𝑝+1+𝑟₁ℎ₁−1^𝑝+1𝑟₁^𝑝+1,𝑝+1(ii)¹₀𝑞ℎ₂,𝑟₂,𝜆^𝑝𝑟𝑑𝜆=2+₂−𝑟₂ℎ₂−1^𝑝+1+𝑟₂ℎ₂−1^𝑝+1𝑟₂^𝑝+1.𝑝+1 (2.34)(2.35)
By the inequality (2.33) and the equalities (2.34) and (2.35), the assertion (2.30) holds.

Remark 2.8. In Theorem 2.7,(i)if we choose ℎ₁=ℎ₂=1/2, 𝑟₁=𝑟₂=6, and 𝑠=1, then we get 𝐼𝑓12,12≤2,6,61+2^𝑝+16^𝑝+1𝑝+1^2/𝑝(𝑏−𝑎)(𝑑−𝑐)𝑀_𝑞^1/𝑞, (2.36)(ii)if we choose ℎ₁=ℎ₂=1/2, 𝑟₁=𝑟₂=2, and 𝑠=1, then we get 𝐼𝑓12,12≤1,2,22^𝑝𝑝+1^2/𝑝(𝑏−𝑎)(𝑑−𝑐)𝑀_𝑞^1/𝑞, (2.37)(iii)if we choose ℎ₁=ℎ₂=1/2, 𝑟₁=𝑟₂=6, 𝑠=1, and 𝑞=1, then we get 𝐼𝑓12,12≤5,6,636²(𝑏−𝑎)(𝑑−𝑐)𝑀₁, (2.38)where 𝑀_𝑞=𝜕²𝑓/𝜕𝑡𝜕𝜆(𝑎,𝑐)^𝑞+𝜕²𝑓/𝜕𝑡𝜕𝜆(𝑎,𝑑)^𝑞+𝜕²𝑓/𝜕𝑡𝜕𝜆(𝑏,𝑐)^𝑞+𝜕²𝑓/𝜕𝑡𝜕𝜆(𝑏,𝑑)^𝑞4. (2.39)

Theorem 2.9. Let 𝑓∶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: 1()𝐼𝑓ℎ𝑏−𝑎)(𝑑−𝑐¹,ℎ₂,𝑟₁,𝑟₂≤12−ℎ₁+ℎ²₁+2−𝑟₁𝑟²₁12−ℎ₂+ℎ²₂+2−𝑟₂𝑟²₂^1−(1/𝑞)×𝜇₁𝜇₂𝜕²𝑓(𝜕𝑡𝜕𝜆𝑎,𝑐)^𝑞+𝜈₂𝜕²𝑓(𝜕𝑡𝜕𝜆𝑎,𝑑)^𝑞+𝜈₁𝜇₂𝜕²𝑓(𝜕𝑡𝜕𝜆𝑏,𝑐)^𝑞+𝜈₂𝜕²𝑓(𝜕𝑡𝜕𝜆𝑏,𝑑)^𝑞1/𝑞, (2.40) where 𝜇_𝑖 and 𝜈_𝑖(𝑖=1,2) are as given in Theorem 2.3.

Proof. From Lemma 2.1, we can write 1()𝐼𝑓ℎ𝑏−𝑎)(𝑑−𝑐¹,ℎ₂,𝑟₁,𝑟₂≤¹₀𝑝ℎ₁,𝑟₁,𝑡𝑞ℎ₂,𝑟₂×𝜕,𝜆₂𝑓(()()𝑑≤𝜕𝑡𝜕𝜆𝑡𝑎+1−𝑡𝑏,𝜆𝑐+1−𝜆)𝑑𝑡𝑑𝜆₁₀𝑝ℎ₁,𝑟₁,𝑡𝑞ℎ₂,𝑟₂,𝜆𝑑𝑡𝑑𝜆^1−(1/𝑞)×¹₀𝑝ℎ₁,𝑟₁,𝑡𝑞ℎ₂,𝑟₂×𝜕,𝜆₂𝑓(()()𝑑𝜕𝑡𝜕𝜆𝑡𝑎+1−𝑡𝑏,𝜆𝑐+1−𝜆)^𝑞𝑑𝑡𝑑𝜆^1/𝑞. (2.41)
By the simple calculations, we have (i)¹₀𝑝ℎ₁,𝑟₁1,𝑡𝑑𝑡=2−ℎ₁+ℎ²₁+2−𝑟₁𝑟²₁, (2.42)(ii)¹₀𝑞ℎ₂,𝑟₂1,𝜆𝑑𝜆=2−ℎ₂+ℎ²₂+2−𝑟₂𝑟²₂. (2.43)
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], ¹₀𝑝ℎ₁,𝑟₁,𝑡𝑞ℎ₂,𝑟₂𝜕,𝜆²𝑓(()()𝑑𝜕𝑡𝜕𝜆𝑡𝑎+1−𝑡𝑏,𝜆𝑐+1−𝜆)^𝑞≤𝑑𝑡𝑑𝜆₁₀𝑝ℎ₁,𝑟₁,𝑡𝑞ℎ₂,𝑟₂×𝑡,𝜆_s𝜆^𝑠𝜕²𝑓(𝜕𝑡𝜕𝜆𝑎,𝑐)^𝑞+𝑡^𝑠(1−𝜆)^𝑠𝜕²𝑓(𝜕𝑡𝜕𝜆𝑎,𝑑)^𝑞+(1−𝑡)^𝑠𝜆^𝑠𝜕²𝑓(𝜕𝑡𝜕𝜆𝑏,𝑐)^𝑞+(1−𝑡)^𝑠(1−𝜆)^𝑠𝜕²𝑓(𝜕𝑡𝜕𝜆𝑏,𝑑)^𝑞=𝑑𝑡𝑑𝜆₁₀𝑝ℎ₁,𝑟₁,𝑡𝑡^𝑠𝑑𝑡¹₀𝑞ℎ₂,𝑟₂,𝜆𝜆^𝑠𝜕𝑑𝜆²𝑓(𝜕𝑡𝜕𝜆𝑎,𝑐)^𝑞+¹₀𝑞ℎ₂,𝑟₂,𝜆(1−𝜆)^𝑠𝜕𝑑𝜆²𝑓(𝜕𝑡𝜕𝜆𝑎,𝑑)^𝑞+¹₀𝑝ℎ₁,𝑟₁,𝑡(1−𝑡)^𝑠𝑑𝑡¹₀𝑞ℎ₂,𝑟₂,𝑠𝜆^𝑠𝜕𝑑𝜆²𝑓(𝜕𝑡𝜕𝑠𝑏,𝑐)^𝑞+¹₀𝑞ℎ₂,𝑟₂,𝜆(1−𝜆)^𝑠𝜕𝑑𝜆²𝑓(𝜕𝑡𝜕𝜆𝑏,𝑑)^𝑞=𝜇₁𝜇₂𝜕²𝑓(𝜕𝑡𝜕𝜆𝑎,𝑐)^𝑞+𝜈₂𝜕²𝑓𝜕t(𝜕𝜆𝑎,𝑑)^𝑞+𝜈₁𝜇₂𝜕²𝑓(𝜕𝑡𝜕𝜆𝑏,𝑐)^𝑞+𝜈₂𝜕²𝑓(𝜕𝑡𝜕𝜆𝑏,𝑑)^𝑞. (2.44)
By (2.41)–(2.44), the assertion (2.40) holds.