Generalizations of Inequalities for Differentiable Co-Ordinated Convex Functions

Abstract

A generalized lemmas is proved and several new inequalities for differentiable co-ordinated convex and concave functions in two variables are obtained.

1. Introduction

Let 𝑓∶𝐼⊆ℝ→ℝ be a convex function and 𝑎,𝑏∈𝐼 with 𝑎<𝑏; we have the following double inequality: 𝑓𝑎+𝑏2≤1𝑏−𝑎^𝑏_𝑎𝑓(𝑡)𝑓(𝑎)(𝑏)𝑑𝑡≤+𝑓2. (1) This remarkable result is well known in the literature as the Hermite-Hadamard inequality for convex mapping.

Since then, some refinements of the Hermite-Hadamard inequality on convex functions have been extensively investigated by a number of authors (e.g., [1–4]).

A modification for convex functions which is also known as coordinated convex functions was introduced as follows by Dragomir in [5].

Let us consider the bidimensional interval Δ∶=[𝑎,𝑏]×[𝑐,𝑑] in ℝ² with 𝑎<𝑏 and 𝑐<𝑑; a mapping 𝑓∶Δ→ℝ is said to be convex on Δ if the inequality 𝑓()()𝑤+()𝑓()𝜆𝑥+1−𝜆𝑧,𝜆𝑦+1−𝜆≤𝜆𝑓𝑥,𝑦1−𝜆𝑧,𝑤 (2) holds for all (𝑥,𝑦), (𝑧,𝑤)∈Δ, and 𝜆∈[0,1].

A function 𝑓∶Δ→ℝ is said to be coordinated convex on Δ if the partial mappings 𝑓_𝑦∶[𝑎,𝑏]→ℝ, 𝑓_𝑦(𝑢)=𝑓(𝑢,𝑦) and 𝑓_𝑥∶[𝑐,𝑑]→ℝ, 𝑓_𝑥(𝑣)=𝑓(𝑥,𝑣) are convex for all 𝑦∈[𝑐,𝑑] and 𝑥∈[𝑎,𝑏].

A formal definition for coordinated convex functions may be stated as follows.

Definition 1. A function 𝑓∶Δ→ℝ is said to be convex on coordinates on Δ if the inequality 𝑓()()𝑤()𝑓()+()+()𝑓()𝜆𝑥+1−𝜆𝑧,𝑡𝑦+1−𝑡≤𝜆𝑡𝑓𝑥,𝑦+𝜆1−𝑡𝑥,𝑤1−𝜆𝑡𝑓𝑧,𝑦1−𝑡)(1−𝜆𝑧,𝑤 (3) holds for all (𝑥,𝑦), (𝑧,𝑦), (𝑥,𝑤), (𝑧,𝑤)∈Δ, and 𝑡,𝜆∈[0,1].

Dragomir in [5] established the following Hadamard-type inequalities for coordinated convex functions in a rectangle from the plane ℝ².

Theorem 2. Suppose that 𝑓∶Δ=[𝑎,𝑏]×[𝑐,𝑑]→ℝ is convex on the coordinates on Δ. Then one has the inequalities as follows: 𝑓𝑎+𝑏2,𝑐+𝑑2≤1(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓≤𝑓()()()()𝑥,𝑦𝑑𝑦𝑑𝑥𝑎,𝑐+𝑓𝑎,𝑑+𝑓𝑏,𝑐+𝑓𝑏,𝑑4. (4)

Some new integral inequalities that are related to the Hermite-Hadamard type for coordinated convex functions are also established by many authors.

In ([6], 2008), Alomari and Darus defined coordinated 𝑠-convex functions and proved some inequalities based on this definition. In ([7], 2009), analogous results for ℎ-convex functions on the coordinates were proved by Latif and Alomari. In ([8], 2009), Alomari and Darus established some Hadamard-type inequalities for coordinated log-convex functions.

In ([9], 2012), Latif and Dragomir obtained some new Hadamard type inequalities for differentiable coordinated convex and concave functions in two variables which are related to the left-hand side of Hermite-Hadamard type inequality for coordinated convex functions in two variables based on the following lemma.

Lemma 3. Let 𝑓∶Δ⊆ℝ²→ℝ be a partial differentiable mapping on Δ∶=[𝑎,𝑏]×[𝑐,𝑑] in ℝ² with 𝑎<𝑏 and 𝑐<𝑑. If 𝜕²𝑓/𝜕𝑢𝜕𝑣∈𝐿(Δ), then the following equality holds: 1(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓𝑥,𝑦𝑑𝑦𝑑𝑥+𝑓𝑎+𝑏2,𝑐+𝑑2−1𝑏−𝑎^𝑏_𝑎𝑓𝑥,𝑐+𝑑2−1𝑑𝑥𝑑−𝑐^𝑑_𝑐𝑓𝑎+𝑏2=()×,𝑦𝑑𝑦𝑏−𝑎)(𝑑−𝑐₁₀𝐾()𝜕𝑢,𝑣²𝑓(()()𝑑)𝜕𝑢𝜕𝑣𝑢𝑎+1−𝑢𝑏,𝑣𝑐+1−𝑣𝑑𝑢𝑑𝑣, (5) where 𝐾()=()∈1𝑢,𝑣𝑢𝑣,𝑢,𝑣0,2×10,2𝑢(),()∈1𝑣−1𝑢,𝑣0,2×12()()∈1,1𝑢−1𝑣,𝑢,𝑣2×1,10,2(),()∈1𝑢−1)(𝑣−1𝑢,𝑣2×1,12,1. (6)

Theorem 4 (see [9]). Let 𝑓∶Δ⊆ℝ²→ℝ be a partial differentiable mapping on Δ∶=[𝑎,𝑏]×[𝑐,𝑑] in ℝ² with 𝑎<𝑏 and 𝑐<𝑑. If |𝜕²𝑓/𝜕𝑢𝜕𝑣| is convex on the coordinates on Δ, then the following equality holds: 1(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓𝑥,𝑦𝑑𝑦𝑑𝑥+𝑓𝑎+𝑏2,𝑐+𝑑2≤()−𝐴𝑏−𝑎)(𝑑−𝑐×𝜕16₂𝑓(+𝜕𝜕𝑢𝜕𝑣𝑎,𝑐)²𝑓(+𝜕𝜕𝑢𝜕𝑣𝑎,𝑑)²𝑓(+𝜕𝜕𝑢𝜕𝑣𝑏,𝑐)₂𝑓(×(4)𝜕𝑢𝜕𝑣𝑏,𝑑)⁻¹, (7) where 1𝐴=𝑏−𝑎^𝑏_𝑎𝑓𝑥,𝑐+𝑑21𝑑𝑥+𝑑−𝑐^𝑑_𝑐𝑓𝑎+𝑏2,𝑦𝑑𝑦. (8)

Theorem 5 (see [9]). Let 𝑓∶Δ⊆ℝ²→ℝ be a partial differentiable mapping on Δ∶=[𝑎,𝑏]×[𝑐,𝑑] in ℝ² with 𝑎<𝑏 and 𝑐<𝑑. If |𝜕²𝑓/𝜕𝑢𝜕𝑣|^𝑞 is convex on the coordinates on Δ and 𝑝, 𝑞>1, 1/𝑝+1/𝑞=1, then the following equality holds: 1(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓𝑥,𝑦𝑑𝑦𝑑𝑥+𝑓𝑎+𝑏2,𝑐+𝑑2≤()−𝐴𝑏−𝑎)(𝑑−𝑐4𝑝+1^2/𝑝×𝜕²𝑓(𝜕𝑢𝜕𝑣𝑎,𝑐)^𝑞+𝜕²𝑓(𝜕𝑢𝜕𝑣𝑎,𝑑)^𝑞+𝜕²𝑓(𝜕𝑢𝜕𝑣𝑏,𝑐)^𝑞+𝜕²𝑓(𝜕𝑢𝜕𝑣𝑏,𝑑)^𝑞×(4)^−11/𝑞, (9) where 𝐴 is as given in Theorem 4.

Theorem 6 (see [9]). Let 𝑓∶Δ⊆ℝ²→ℝ be a partial differentiable mapping on Δ∶=[𝑎,𝑏]×[𝑐,𝑑] in ℝ² with 𝑎<𝑏 and 𝑐<𝑑. If |𝜕²𝑓/𝜕𝑢𝜕𝑣|^𝑞 is convex on the coordinates on Δ and 𝑞≥1, then the following equality holds: 1(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓𝑥,𝑦𝑑𝑦𝑑𝑥+𝑓𝑎+𝑏2,𝑐+𝑑2≤()−𝐴𝑏−𝑎)(𝑑−𝑐×𝜕16₂𝑓(𝜕𝑢𝜕𝑣𝑎,𝑐)^𝑞+𝜕²𝑓(𝜕𝑢𝜕𝑣𝑎,𝑑)^𝑞+𝜕²𝑓(𝜕𝑢𝜕𝑣𝑏,𝑐)^𝑞+𝜕²𝑓(𝜕𝑢𝜕𝑣𝑏,𝑑)^𝑞×(4)^−11/𝑞, (10) where 𝐴 is as given in Theorem 4.

In ([10], 2012), analogous results which are related to the right-hand side of Hermite-Hadamard type inequality for coordinated convex functions in two variables were proved by Sarıkaya et al. based on the following lemma.

Lemma 7. Let 𝑓∶Δ⊆ℝ²→ℝ be a partial differentiable mapping on Δ∶=[𝑎,𝑏]×[𝑐,𝑑] in ℝ² with 𝑎<𝑏 and 𝑐<𝑑. If 𝜕²𝑓/𝜕𝑢𝜕𝑣∈𝐿(Δ), then the following equality holds: 𝑓(𝑎,𝑐)+𝑓(𝑎,𝑑)+𝑓(𝑏,𝑐)+𝑓(𝑏,𝑑)4+1(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓−1𝑥,𝑦𝑑𝑦𝑑𝑥21𝑏−𝑎^𝑏_𝑎𝑓()(+1𝑥,𝑐+𝑓𝑥,𝑑)𝑑𝑥𝑑−𝑐^𝑑_𝑐𝑓=()𝑎,𝑦+𝑓𝑏,𝑦𝑑𝑦𝑏−𝑎)(𝑑−𝑐4×¹₀()𝜕1−2𝑢)(1−2𝑣²𝑓×(()()𝑑)𝜕𝑢𝜕𝑣𝑢𝑎+1−𝑢𝑏,𝑣𝑐+1−𝑣𝑑𝑢𝑑𝑣. (11)

Theorem 8 (see [10]). Let 𝑓∶Δ⊆ℝ²→ℝ be a partial differentiable mapping on Δ∶=[𝑎,𝑏]×[𝑐,𝑑] in ℝ² with 𝑎<𝑏 and 𝑐<𝑑. If |𝜕²𝑓/𝜕𝑢𝜕𝑣| is convex on the coordinates on Δ, then the following equality holds: 𝑓(𝑎,𝑐)+𝑓(𝑎,𝑑)+𝑓(𝑏,𝑐)+𝑓(𝑏,𝑑)4+1(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓≤()𝑥,𝑦𝑑𝑦𝑑𝑥−𝐴𝑏−𝑎)(𝑑−𝑐×𝜕16₂𝑓(+𝜕𝜕𝑢𝜕𝑣𝑎,𝑐)²𝑓(+𝜕𝜕𝑢𝜕𝑣𝑎,𝑑)²𝑓(+𝜕𝜕𝑢𝜕𝑣𝑏,𝑐)₂𝑓(×(4)𝜕𝑢𝜕𝑣𝑏,𝑑)⁻¹, (12) where 1𝐴=21𝑏−𝑎^𝑏_𝑎𝑓()(+1𝑥,𝑐+𝑓𝑥,𝑑)𝑑𝑥𝑑−𝑐^𝑑_𝑐𝑓𝑎,𝑦+𝑓𝑏,𝑦𝑑𝑦. (13)

Theorem 9 (see [10]). Let 𝑓∶Δ⊆ℝ²→ℝ be a partial differentiable mapping on Δ∶=[𝑎,𝑏]×[𝑐,𝑑] in ℝ² with 𝑎<𝑏 and 𝑐<𝑑. If |𝜕²𝑓/𝜕𝑢𝜕𝑣|^𝑞 is convex on the coordinates on Δ and 𝑝, 𝑞>1, 1/𝑝+1/𝑞=1, then the following equality holds: 𝑓(𝑎,𝑐)+𝑓(𝑎,𝑑)+𝑓(𝑏,𝑐)+𝑓(𝑏,𝑑)4+1(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓≤()𝑥,𝑦𝑑𝑦𝑑𝑥−𝐴𝑏−𝑎)(𝑑−𝑐4𝑝+1^2/𝑝×𝜕²𝑓(𝜕𝑢𝜕𝑣𝑎,𝑐)^𝑞+𝜕²𝑓(𝜕𝑢𝜕𝑣𝑎,𝑑)^𝑞+𝜕²𝑓(𝜕𝑢𝜕𝑣𝑏,𝑐)^𝑞+𝜕²𝑓(𝜕𝑢𝜕𝑣𝑏,𝑑)^𝑞×(4)^−11/𝑞, (14) where 𝐴 is as given in Theorem 8.

Theorem 10 (see [10]). Let 𝑓∶Δ⊆ℝ²→ℝ be a partial differentiable mapping on Δ∶=[𝑎,𝑏]×[𝑐,𝑑] in ℝ² with 𝑎<𝑏 and 𝑐<𝑑. If |𝜕²𝑓/𝜕𝑢𝜕𝑣|^𝑞 is convex on the coordinates on Δ and 𝑞≥1, then the following equality holds: 𝑓(𝑎,𝑐)+𝑓(𝑎,𝑑)+𝑓(𝑏,𝑐)+𝑓(𝑏,𝑑)4+1(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓≤()𝑥,𝑦𝑑𝑦𝑑𝑥−𝐴𝑏−𝑎)(𝑑−𝑐×𝜕16₂𝑓(𝜕𝑢𝜕𝑣𝑎,𝑐)^𝑞+𝜕²𝑓(𝜕𝑢𝜕𝑣𝑎,𝑑)^𝑞+𝜕²𝑓(𝜕𝑢𝜕𝑣𝑏,𝑐)^𝑞+𝜕²𝑓(𝜕𝑢𝜕𝑣𝑏,𝑑)^𝑞×(4)^−11/𝑞, (15) where 𝐴 is as given in Theorem 8.

In [11], Ozdemir et al. established some Simpson’s inequalities for coordinated convex functions based on the following lemma.

Lemma 11. Let 𝑓∶Δ⊆ℝ²→ℝ be a partial differentiable mapping on Δ∶=[𝑎,𝑏]×[𝑐,𝑑] in ℝ² with 𝑎<𝑏 and 𝑐<𝑑. If 𝜕²𝑓/𝜕𝑢𝜕𝑣∈𝐿(Δ), then the following equality holds: 𝑓𝑎,𝑐+𝑑2+𝑓𝑏,𝑐+𝑑2+4𝑓𝑎+𝑏2,𝑐+𝑑2+𝑓𝑎+𝑏2,𝑐+𝑓𝑎+𝑏2,𝑑×(9)⁻¹+𝑓(𝑎,𝑐)+𝑓(𝑎,𝑑)+𝑓(𝑏,𝑐)+𝑓(𝑏,𝑑)−1366(𝑏−𝑎)^𝑏_𝑎𝑓(𝑥,𝑐)+4𝑓𝑥,𝑐+𝑑2()−1+𝑓𝑥,𝑑𝑑𝑥6(𝑑−𝑐)^𝑑_𝑐𝑓𝑎,𝑦+4𝑓𝑎+𝑏2+1,𝑦+𝑓𝑏,𝑦𝑑𝑦(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓=()×𝑥,𝑦𝑑𝑦𝑑𝑥𝑏−𝑎)(𝑑−𝑐₁₀𝑝()𝑝𝜕𝑥,𝑢𝑦,𝑣²𝑓×(()()𝑑)𝜕𝑢𝜕𝑣𝑢𝑎+1−𝑢𝑏,𝑣𝑐+1−𝑣𝑑𝑢𝑑𝑣, (16) where 𝑝()=1𝑥,𝑢𝑢−61,𝑡∈0,25𝑢−61,𝑡∈2,𝑝=1,1𝑦,𝑣𝑣−61,𝑠∈0,25𝑣−61,𝑠∈2,1. (17)

Theorem 12 (see [11]). Let 𝑓∶Δ⊆ℝ²→ℝ be a partial differentiable mapping on Δ∶=[𝑎,𝑏]×[𝑐,𝑑]. If |𝜕²𝑓/𝜕𝑢𝜕𝑣| is convex on the coordinates on Δ, then the following equality holds: 𝑓𝑎,𝑐+𝑑2+𝑓𝑏,𝑐+𝑑2+4𝑓𝑎+𝑏2,𝑐+𝑑2+𝑓𝑎+𝑏2,𝑐+𝑓𝑎+𝑏2,𝑑×(9)⁻¹+𝑓(𝑎,𝑐)+𝑓(𝑎,𝑑)+𝑓(𝑏,𝑐)+𝑓(𝑏,𝑑)+136(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓≤5𝑥,𝑦𝑑𝑦𝑑𝑥−𝐴72²()×𝜕𝑏−𝑎)(𝑑−𝑐₂𝑓(+𝜕𝜕𝑢𝜕𝑣𝑎,𝑐)²𝑓(+𝜕𝜕𝑢𝜕𝑣𝑎,𝑑)²𝑓(+𝜕𝜕𝑢𝜕𝑣𝑏,𝑐)₂𝑓(,𝜕𝑢𝜕𝑣𝑏,𝑑) (18) where 1𝐴=6(𝑏−𝑎)^𝑏_𝑎𝑓(𝑥,𝑐)+4𝑓𝑥,𝑐+𝑑2()+1+𝑓𝑥,𝑑𝑑𝑥6(𝑑−𝑐)^𝑑_𝑐𝑓𝑎,𝑦+4𝑓𝑎+𝑏2,𝑦+𝑓𝑏,𝑦𝑑𝑦. (19)

For recent results and generalizations concerning Hermite-Hadamard type inequality for differentiable coordinated convex functions see ([12], 2012) and the references given therein.

In this paper, a generalized lemma is proved and several new inequalities for differentiable coordinated convex and concave functions in two variables are obtained.

2. Lemmas

To establish our results, we need the following lemma.

Lemma 13. Let 𝑓∶Δ⊆ℝ²→ℝ be a partial differentiable mapping on Δ∶=[𝑎,𝑏]×[𝑐,𝑑] in ℝ² with 𝑎<𝑏 and 𝑐<𝑑. If 𝜕²𝑓/𝜕𝑢𝜕𝑣∈𝐿(Δ) and 𝜆∈[0,1], then the following equality holds: 1(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓+()𝑥,𝑦𝑑𝑦𝑑𝑥1−𝜆²𝑓𝑎+𝑏2,𝑐+𝑑2+𝜆(1−𝜆)2×𝑓𝑎,𝑐+𝑑2+𝑓𝑎,𝑐+𝑑2+𝑓𝑎+𝑏2,𝑐+𝑓𝑎+𝑏2+𝜆,𝑑₂4𝑓()()()(−1𝑎,𝑐+𝑓𝑎,𝑑+𝑓𝑏,𝑐+𝑓𝑏,𝑑)2()×𝑏−𝑎_𝑏𝑎𝜆𝑓(𝑥,𝑐)+2(1−𝜆)𝑓𝑥,𝑐+𝑑2()−1+𝜆𝑓𝑥,𝑑𝑑𝑥2()×𝑑−𝑐_𝑑𝑐𝜆𝑓𝑎,𝑦+2(1−𝜆)𝑓𝑎+𝑏2=(),𝑦+𝜆𝑓𝑏,𝑦𝑑𝑦𝑏−𝑎)(𝑑−𝑐×¹₀𝑀()𝜕𝑢,𝑣²𝑓×(()()𝑑)𝜕𝑢𝜕𝑣𝑢𝑎+1−𝑢𝑏,𝑣𝑐+1−𝑣𝑑𝑢𝑑𝑣, (20) where 𝑀()=𝜆𝑢,𝑣𝑢−2𝜆𝑣−2,()∈1𝑢,𝑣0,2×10,2𝜆𝑢−2𝜆𝑣−1−2,()∈1𝑢,𝑣0,2×12𝜆,1𝑢−1−2𝜆𝑣−2,()∈1𝑢,𝑣2×1,10,2𝜆𝑢−1−2𝜆𝑣−1−2,()∈1𝑢,𝑣2×1,12,1. (21)

Proof. Since ¹₀𝑀()𝜕𝑢,𝑣²𝑓×(()()𝑑)𝜕𝑢𝜕𝑣𝑢𝑎+1−𝑢𝑏,𝑣𝑐+1−𝑣𝑑𝑢𝑑𝑣=₀^1/2𝜆𝑢−2𝜆𝑣−2𝜕²𝑓×(()()𝑑)+𝜕𝑢𝜕𝑣𝑢𝑎+1−𝑢𝑏,𝑣𝑐+1−𝑣𝑑𝑢𝑑𝑣_01/211/2𝜆𝑢−2𝜆𝑣−1−2𝜕²𝑓×(()()𝑑)+𝜕𝑢𝜕𝑣𝑢𝑎+1−𝑢𝑏,𝑣𝑐+1−𝑣𝑑𝑢𝑑𝑣_11/201/2𝜆𝑢−1−2𝜆𝑣−2𝜕²𝑓×(()()𝑑)+𝜕𝑢𝜕𝑣𝑢𝑎+1−𝑢𝑏,𝑣𝑐+1−𝑣𝑑𝑢𝑑𝑣_11/2𝜆𝑢−1−2𝜆𝑣−1−2𝜕²𝑓×(()()𝑑)𝜕𝑢𝜕𝑣𝑢𝑎+1−𝑢𝑏,𝑣𝑐+1−𝑣𝑑𝑢𝑑𝑣, (22) thus, by integration by parts, it follows that ₀^1/2𝜆𝑢−2𝜆𝑣−2𝜕²𝑓×(()()𝑑)=𝜕𝑢𝜕𝑣𝑢𝑎+1−𝑢𝑏,𝑣𝑐+1−𝑣𝑑𝑢𝑑𝑣_01/2𝜆𝑢−2×𝜆𝑣−21𝑐−𝑑𝜕𝑓×(()()𝑑𝜕𝑢𝑢𝑎+1−𝑢𝑏,𝑣𝑐+1−𝑣)₀^1/2−1×𝑐−𝑑_01/2𝜕𝑓×(()()𝑑)=𝜕𝑢𝑢𝑎+1−𝑢𝑏,𝑣𝑐+1−𝑣𝑑𝑣𝑑𝑢1−𝜆21𝑐−𝑑₀^1/2𝜆𝑢−2𝜕𝑓()𝜕𝑢𝑢𝑎+1−𝑢𝑏,𝑐+𝑑2+𝜆𝑑𝑢21𝑐−𝑑₀^1/2𝜆𝑢−2𝜕𝑓(())−1𝜕𝑢𝑢𝑎+1−𝑢𝑏,𝑑𝑑𝑢𝑐−𝑑₀^1/2𝜆𝑢−2𝜕𝑓×(()()𝑑)=𝜕𝑢𝑢𝑎+1−𝑢𝑏,𝑣𝑐+1−𝑣𝑑𝑣𝑑𝑢1−𝜆211𝑐−𝑑𝜆𝑎−𝑏𝑢−2×𝑓𝑢𝑎+(1−𝑢)𝑏,𝑐+𝑑2₀^1/2−1−𝜆211𝑐−𝑑𝑎−𝑏₀^1/2𝑓𝑢𝑎+(1−𝑢)𝑏,𝑐+𝑑2+𝜆𝑑𝑢211𝑐−𝑑𝜆𝑎−𝑏𝑢−2𝑓(𝑢𝑎+(1−𝑢)𝑏,𝑑)₀^1/2−𝜆211𝑐−𝑑𝑎−𝑏₀^1/2𝑓(())−1𝑢𝑎+1−𝑢𝑏,𝑑𝑑𝑢1𝑐−𝑑×𝑎−𝑏_01/2𝜆𝑢−2𝑓(𝑢𝑎+(1−𝑢)𝑏,𝑣𝑐+(1−𝑣)𝑑)₀^1/2−₀^1/2𝑓(()()𝑑)=()𝑢𝑎+1−𝑢𝑏,𝑣𝑐+1−𝑣𝑑𝑢𝑑𝑣1−𝜆²4()𝑓𝑐−𝑑)(𝑎−𝑏𝑎+𝑏2,𝑐+𝑑2+𝜆(1−𝜆)4()𝑓𝑐−𝑑)(𝑎−𝑏𝑏,𝑐+𝑑2−1−𝜆2(𝑐−𝑑)(𝑎−𝑏)₀^1/2𝑓𝑢𝑎+(1−𝑢)𝑏,𝑐+𝑑2+𝜆()𝑑𝑢1−𝜆4()𝑓𝑐−𝑑)(𝑎−𝑏𝑎+𝑏2+𝜆,𝑑₂4()𝑓()−𝜆𝑐−𝑑)(𝑎−𝑏𝑏,𝑑2(𝑐−𝑑)(𝑎−𝑏)₀^1/2𝑓(())−𝑢𝑎+1−𝑢𝑏,𝑑𝑑𝑢1−𝜆2(𝑐−𝑑)(𝑎−𝑏)₀^1/2𝑓𝑎+𝑏2()𝑑−𝜆,𝑣𝑐+1−𝑣𝑑𝑣2(𝑐−𝑑)(𝑎−𝑏)₀^1/2𝑓(()𝑑)+1𝑏,𝑣𝑐+1−𝑣𝑑𝑣(𝑐−𝑑)(𝑎−𝑏)×₀^1/2𝑓(𝑢𝑎+(1−𝑢)𝑏,𝑣𝑐+(1−𝑣)𝑑)𝑑𝑢𝑑𝑣. (23)
Similarly, we can get ₀^1/21_1/2𝜆𝑢−2𝜆𝑣−1−2𝜕²𝑓×(()()𝑑)=()𝜕𝑢𝜕𝑣𝑢𝑎+1−𝑢𝑏,𝑣𝑐+1−𝑣𝑑𝑢𝑑𝑣1−𝜆²4()𝑓𝑐−𝑑)(𝑎−𝑏𝑎+𝑏2,𝑐+𝑑2+𝜆(1−𝜆)4()𝑓𝑐−𝑑)(𝑎−𝑏𝑏,𝑐+𝑑2−1−𝜆2()×𝑐−𝑑)(𝑎−𝑏_01/2𝑓𝑢𝑎+(1−𝑢)𝑏,𝑐+𝑑2+𝜆()𝑑𝑢1−𝜆4()𝑓𝑐−𝑑)(𝑎−𝑏𝑎+𝑏2+𝜆,𝑐₂4()𝑓()−𝜆𝑐−𝑑)(𝑎−𝑏𝑏,𝑐2()×𝑐−𝑑)(𝑎−𝑏_01/2𝑓(())−𝑢𝑎+1−𝑢𝑏,𝑐𝑑𝑢1−𝜆2()×𝑐−𝑑)(𝑎−𝑏_11/2𝑓𝑎+𝑏2()𝑑𝜆,𝑣𝑐+1−𝑣𝑑𝑣−2()×𝑐−𝑑)(𝑏−𝑎_11/2𝑓(()𝑑)+1𝑎,𝑣𝑐+1−𝑣𝑑𝑣()×𝑐−𝑑)(𝑎−𝑏_01/211/2𝑓(𝑢𝑎+(1−𝑢)𝑏,𝑣𝑐+(1−𝑣)𝑑)𝑑𝑢𝑑𝑣,_11/201/2𝜆𝑢−1−2𝜆𝑣−2𝜕²𝑓×(()()𝑑)=()𝜕𝑢𝜕𝑣𝑢𝑎+1−𝑢𝑏,𝑣𝑐+1−𝑣𝑑𝑢𝑑𝑣1−𝜆²4()𝑓𝑐−𝑑)(𝑎−𝑏𝑎+𝑏2,𝑐+𝑑2+𝜆(1−𝜆)4()𝑓𝑐−𝑑)(𝑎−𝑏𝑎,𝑐+𝑑2−1−𝜆2(𝑐−𝑑)(𝑎−𝑏)¹_1/2𝑓𝑢𝑎+(1−𝑢)𝑏,𝑐+𝑑2+𝜆()𝑑𝑢1−𝜆4()𝑓𝑐−𝑑)(𝑎−𝑏𝑎+𝑏2+𝜆,𝑑₂4()𝑓()−𝜆𝑑−𝑐)(𝑏−𝑎𝑎,𝑑2()×𝑐−𝑑)(𝑎−𝑏_11/2𝑓(())−𝑢𝑎+1−𝑢𝑏,𝑑𝑑𝑢1−𝜆2()×𝑐−𝑑)(𝑎−𝑏_01/2𝑓𝑎+𝑏2()𝑑−𝜆,𝑣𝑐+1−𝑣𝑑𝑣2()×𝑑−𝑐)(𝑏−𝑎_01/2𝑓(()𝑑)1𝑎,𝑣𝑐+1−𝑣𝑑𝑣+()×𝑐−𝑑)(𝑎−𝑏_11/201/2𝑓(𝑢𝑎+(1−𝑢)𝑏,𝑣𝑐+(1−𝑣)𝑑)𝑑𝑢𝑑𝑣,_11/2𝜆𝑢−1−2𝜆𝑣−1−2𝜕²𝑓×(()()𝑑)=()𝜕𝑢𝜕𝑣𝑢𝑎+1−𝑢𝑏,𝑣𝑐+1−𝑣𝑑𝑢𝑑𝑣,1−𝜆²4()𝑓𝑐−𝑑)(𝑎−𝑏𝑎+𝑏2,𝑐+𝑑2+𝜆(1−𝜆)4()𝑓𝑐−𝑑)(𝑎−𝑏𝑎,𝑐+𝑑2−1−𝜆2(𝑐−𝑑)(𝑎−𝑏)¹_1/2𝑓𝑢𝑎+(1−𝑢)𝑏,𝑐+𝑑2+𝜆()𝑑𝑢1−𝜆4()𝑓𝑐−𝑑)(𝑎−𝑏𝑎+𝑏2+𝜆,𝑐₂4()𝑓()−𝜆𝑐−𝑑)(𝑎−𝑏𝑎,𝑐2()×𝑐−𝑑)(𝑎−𝑏_11/2𝑓(())−𝑢𝑎+1−𝑢𝑏,𝑐𝑑𝑢1−𝜆2(𝑐−𝑑)(𝑎−𝑏)×¹_1/2𝑓𝑎+𝑏2()𝑑−𝜆,𝑣𝑐+1−𝑣𝑑𝑣2(𝑐−𝑑)(𝑎−𝑏)¹_1/2𝑓(()𝑑)+1𝑎,𝑣𝑐+1−𝑣𝑑𝑣()×𝑐−𝑑)(𝑎−𝑏_11/2𝑓(𝑢𝑎+(1−𝑢)𝑏,𝑣𝑐+(1−𝑣)𝑑)𝑑𝑢𝑑𝑣. (24)
Now ¹₀𝑀()𝜕𝑢,𝑣²𝑓(()()𝑑)=()𝜕𝑢𝜕𝑣𝑢𝑎+1−𝑢𝑏,𝑣𝑐+1−𝑣𝑑𝑢𝑑𝑣1−𝜆²()𝑓𝑑−𝑐)(𝑏−𝑎𝑎+𝑏2,𝑐+𝑑2+𝜆(1−𝜆)2()×𝑓𝑑−𝑐)(𝑏−𝑎𝑎,𝑐+𝑑2+𝑓𝑏,𝑐+𝑑2+𝑓𝑎+𝑏2,𝑐+𝑓𝑎+𝑏2+𝜆,𝑑₂4()×𝑓()()()(+1𝑑−𝑐)(𝑏−𝑎𝑎,𝑐+𝑓𝑏,𝑐+𝑓𝑎,𝑑+𝑓𝑏,𝑑)2()×𝑑−𝑐)(𝑏−𝑎₁₀(())()𝑓()𝜆𝑓𝑢𝑎+1−𝑢𝑏,𝑐+21−𝜆𝑢𝑎+1−𝑢𝑏,𝑐+𝑑2(()+1+𝜆𝑓𝑢𝑎+1−𝑢𝑏,𝑑)𝑑𝑢2()×𝑑−𝑐)(𝑏−𝑎₁₀(()𝑑)()𝑓𝜆𝑓𝑎,𝑣𝑐+1−𝑣+21−𝜆𝑎+𝑏2()𝑑(()𝑑+1,𝑣𝑐+1−𝑣+𝜆𝑓𝑏,𝑣𝑐+1−𝑣)𝑑𝑣()×𝑏−𝑎)(𝑑−𝑐₁₀𝜕²𝑓(()()𝑑)𝜕𝑢𝜕𝑣𝑢𝑎+1−𝑢𝑏,𝑣𝑐+1−𝑣𝑑𝑢𝑑𝑣. (25) Multiplying both sides by (𝑏−𝑎)(𝑑−𝑐) and using the change of the variable 𝑥=𝑢𝑎+(1−𝑢)𝑏 and 𝑦=𝑣𝑐+(1−𝑣)𝑑, which completes the proof.

Remark 14. Applying Lemma 13 for 𝜆=0, 1, 1/3, we get the results of Lemmas 3, 7, and 11, respectively.

3. Main Results

Theorem 15. Let 𝑓∶Δ⊆ℝ²→ℝ be a partial differentiable mapping on Δ∶=[𝑎,𝑏]×[𝑐,𝑑] in ℝ² with 𝑎<𝑏 and 𝑐<𝑑. If |𝜕²𝑓/𝜕𝑢𝜕𝑣| is convex on the coordinates on Δ and 𝜆∈[0,1], then the following equality holds: 1(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓+()𝑥,𝑦𝑑𝑦𝑑𝑥1−𝜆²𝑓𝑎+𝑏2,𝑐+𝑑2+𝜆²4𝑓()()()(≤()𝑎,𝑐+𝑓𝑎,𝑑+𝑓𝑏,𝑐+𝑓𝑏,𝑑)−𝐴𝑏−𝑎)(𝑑−𝑐2𝜆²−2𝜆+18²×𝜕²𝑓(+𝜕𝜕𝑢𝜕𝑣𝑎,𝑐)²𝑓(+𝜕𝜕𝑢𝜕𝑣𝑎,𝑑)₂𝑓(+𝜕𝜕𝑢𝜕𝑣𝑏,𝑐)²𝑓(,𝜕𝑢𝜕𝑣𝑏,𝑑) (26) where 1𝐴=2()×𝑏−𝑎_𝑏𝑎𝜆𝑓(𝑥,𝑐)+2(1−𝜆)𝑓𝑥,𝑐+𝑑2()+1+𝜆𝑓𝑥,𝑑𝑑𝑥2()×𝑑−𝑐_𝑑𝑐𝜆𝑓𝑎,𝑦+2(1−𝜆)𝑓𝑎+𝑏2−𝜆(),𝑦+𝜆𝑓𝑏,𝑦𝑑𝑦1−𝜆2×𝑓𝑎,𝑐+𝑑2+𝑓𝑎,𝑐+𝑑2+𝑓𝑎+𝑏2,𝑐+𝑓𝑎+𝑏2,𝑑. (27)

Proof. From Lemma 13, we obtain 1(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓+()𝑥,𝑦𝑑𝑦𝑑𝑥1−𝜆²𝑓𝑎+𝑏2,𝑐+𝑑2+𝜆²4𝑓()()()(≤()×𝑎,𝑐+𝑓𝑎,𝑑+𝑓𝑏,𝑐+𝑓𝑏,𝑑)−𝐴𝑏−𝑎)(𝑑−𝑐₁₀|𝑀(×𝜕𝑢,𝑣)|₂𝑓(()()𝑑𝜕𝑢𝜕𝑣𝑢𝑎+1−𝑢𝑏,𝑣𝑐+1−𝑣)𝑑𝑢𝑑𝑣. (28) Because |𝜕²𝑓/𝜕𝑢𝜕𝑣| is a convex function on the coordinates on Δ, then one has 1(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓+()𝑥,𝑦𝑑𝑦𝑑𝑥1−𝜆²𝑓𝑎+𝑏2,𝑐+𝑑2+𝜆²4𝑓()()()(≤()×𝑎,𝑐+𝑓𝑎,𝑑+𝑓𝑏,𝑐+𝑓𝑏,𝑑)−𝐴𝑏−𝑎)(𝑑−𝑐₁₀|𝑀(×𝜕𝑢,𝑣)|𝑢𝑣²𝑓((𝜕𝜕𝑢𝜕𝑣𝑎,𝑐)+𝑢1−𝑣)²𝑓(+()𝑣𝜕𝜕𝑢𝜕𝑣𝑎,𝑑)1−𝑢²𝑓(+(𝜕𝜕𝑢𝜕𝑣𝑏,𝑐)1−𝑢)(1−𝑣)²𝑓(=()×𝜕𝜕𝑢𝜕𝑣𝑏,𝑑)𝑑𝑢𝑑𝑣𝑏−𝑎)(𝑑−𝑐₂𝑓(𝜕𝑢𝜕𝑣𝑎,𝑐)¹₀|𝑀(+𝜕𝑢,𝑣)|𝑢𝑣𝑑𝑢𝑑𝑣₂𝑓(𝜕𝑢𝜕𝑣𝑎,𝑑)¹₀|𝑀(𝑢()+𝜕𝑢,𝑣)|1−𝑣𝑑𝑢𝑑𝑣₂𝑓(𝜕𝑢𝜕𝑣𝑏,𝑐)¹₀|𝑀()+𝜕𝑢,𝑣)|(1−𝑢𝑣𝑑𝑢𝑑𝑣₂𝑓(𝜕𝑢𝜕𝑣𝑏,𝑑)¹₀|𝑀()=()𝑢,𝑣)|(1−𝑢)(1−𝑣𝑑𝑢𝑑𝑣𝑏−𝑎)(𝑑−𝑐2𝜆²−2𝜆+18²×𝜕²𝑓(+𝜕𝜕𝑢𝜕𝑣𝑎,𝑐)²𝑓(+𝜕𝜕𝑢𝜕𝑣𝑎,𝑑)²𝑓(+𝜕𝜕𝑢𝜕𝑣𝑏,𝑐)₂𝑓(.𝜕𝑢𝜕𝑣𝑏,𝑑) (29) On the other hand, we have ¹₀|𝑀(=𝑢,𝑣)|𝑢𝑣𝑑𝑢𝑑𝑣₁₀|𝑀(𝑢()=𝑢,𝑣)|1−𝑣𝑑𝑢𝑑𝑣₁₀|𝑀()=𝑢,𝑣)|(1−𝑢𝑣𝑑𝑢𝑑𝑣₁₀|𝑀(𝑢,𝑣)|(1−𝑢)(1−𝑣)𝑑𝑢𝑑𝑣,₁₀|𝑀(=𝑢,𝑣)|𝑢𝑣𝑑𝑢𝑑𝑣_01/2𝜆𝑢−2𝜆𝑣−2+𝑢𝑣𝑑𝑢𝑑𝑣_01/211/2𝜆𝑢−2𝜆𝑣−1−2+𝑢𝑣𝑑𝑢𝑑𝑣_11/201/2𝜆𝑢−1−2𝜆𝑣−2𝑢𝑣𝑑𝑢𝑑𝑣+¹_1/2𝜆𝑢−1−2𝜆𝑣−1−2=𝑢𝑣𝑑𝑢𝑑𝑣2𝜆²−2𝜆+18², (30) which completes the proof.

Remark 16. Applying Theorem 15 for 𝜆=0, 1, 1/3, we get the results of Theorems 4, 8, and 12, respectively.

Theorem 17. Let 𝑓∶Δ⊆ℝ²→ℝ be a partial differentiable mapping on Δ∶=[𝑎,𝑏]×[𝑐,𝑑] in ℝ² with 𝑎<𝑏 and 𝑐<𝑑. If |𝜕²𝑓/𝜕𝑢𝜕𝑣|^𝑞 is convex on the coordinates on Δ and 𝑞>1, one gets the following inequality: 1(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓+()𝑥,𝑦𝑑𝑦𝑑𝑥1−𝜆²𝑓𝑎+𝑏2,𝑐+𝑑2+𝜆²4𝑓()()()(≤()𝑎,𝑐+𝑓𝑎,𝑑+𝑓𝑏,𝑐+𝑓𝑏,𝑑)−𝐴𝑏−𝑎)(𝑑−𝑐4𝜆^𝑝+1+(1−𝜆)^𝑝+1𝑝+1^2/𝑝×𝜕²𝑓(𝜕𝑢𝜕𝑣𝑎,𝑐)^𝑞+𝜕²𝑓(𝜕𝑢𝜕𝑣𝑎,𝑑)^𝑞+𝜕²𝑓(𝜕𝑢𝜕𝑣𝑏,𝑐)^𝑞+𝜕²𝑓(𝜕𝑢𝜕𝑣𝑏,𝑑)^𝑞×(4)^−11/𝑞, (31) where 𝜆∈[0,1] and 𝐴 is as given in Theorem 15 and 1/𝑝+1/𝑞=1.

Proof. From Lemma 13, we obtain 1(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓+()𝑥,𝑦𝑑𝑦𝑑𝑥1−𝜆²𝑓𝑎+𝑏2,𝑐+𝑑2+𝜆²4𝑓()()()(≤()×𝑎,𝑐+𝑓𝑎,𝑑+𝑓𝑏,𝑐+𝑓𝑏,𝑑)−𝐴𝑏−𝑎)(𝑑−𝑐₁₀|𝑀(×𝜕𝑢,𝑣)|₂𝑓(()()𝑑𝜕𝑢𝜕𝑣𝑢𝑎+1−𝑢𝑏,𝑣𝑐+1−𝑣)𝑑𝑢𝑑𝑣. (32) By using the well-known Hölder inequality for double integrals, then one has 1(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓+()𝑥,𝑦𝑑𝑦𝑑𝑥1−𝜆²𝑓𝑎+𝑏2,𝑐+𝑑2+𝜆²4𝑓()()()(≤()𝑎,𝑐+𝑓𝑎,𝑑+𝑓𝑏,𝑐+𝑓𝑏,𝑑)−𝐴𝑏−𝑎)(𝑑−𝑐¹₀|𝑀(𝑢,𝑣)|^𝑝𝑑𝑢𝑑𝑣^1/𝑝×¹₀𝜕²𝑓()+()𝑑𝜕𝑢𝜕𝑣𝑢𝑎+1−𝑢𝑏,𝑣𝑐1−𝑣)^𝑞𝑑𝑢𝑑𝑣^1/𝑞. (33) Because |𝜕²𝑓/𝜕𝑢𝜕𝑣|^𝑞 is a convex function on the coordinates on Δ, by (4), then one has ¹₀𝜕²𝑓(()()𝑑𝜕𝑢𝜕𝑣𝑢𝑎+1−𝑢𝑏,𝑣𝑐+1−𝑣)^𝑞≤𝜕𝑑𝑢𝑑𝑣₂𝑓(𝜕𝑢𝜕𝑣𝑎,𝑐)^𝑞+𝜕²𝑓(𝜕𝑢𝜕𝑣𝑎,𝑑)^𝑞+𝜕²𝑓(𝜕𝑢𝜕𝑣𝑏,𝑐)^𝑞+𝜕²𝑓(𝜕𝑢𝜕𝑣𝑏,𝑑)^𝑞×(4)⁻¹. (34) We note that ¹₀|𝑀(𝑢,𝑣)|^𝑝=𝑑𝑢𝑑𝑣_01/2𝜆𝑢−2𝜆𝑣−2^𝑝+𝑑𝑢𝑑𝑣_01/211/2𝜆𝑢−2𝜆𝑣−1−2^𝑝+𝑑𝑢𝑑𝑣_11/201/2𝜆𝑢−1−2𝜆𝑣−2^𝑝+𝑑𝑢𝑑𝑣_11/2𝜆𝑢−1−2𝜆𝑣−1−2^𝑝=𝜆𝑑𝑢𝑑𝑣_𝑝+1+(1−𝜆)^𝑝+1𝑝+12^𝑝2. (35) Hence, it follows that 1(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓+()𝑥,𝑦𝑑𝑦𝑑𝑥1−𝜆²𝑓𝑎+𝑏2,𝑐+𝑑2+𝜆²4𝑓()()()(≤()𝑎,𝑐+𝑓𝑎,𝑑+𝑓𝑏,𝑐+𝑓𝑏,𝑑)−𝐴𝑏−𝑎)(𝑑−𝑐4𝜆^𝑝+1+(1−𝜆)^𝑝+1𝑝+1^2/𝑝×𝜕²𝑓(𝜕𝑢𝜕𝑣𝑎,𝑐)^𝑞+𝜕²𝑓(𝜕𝑢𝜕𝑣𝑎,𝑑)^𝑞+𝜕²𝑓(𝜕𝑢𝜕𝑣𝑏,𝑐)^𝑞+𝜕²𝑓(𝜕𝑢𝜕𝑣𝑏,𝑑)^𝑞×(4)^−11/𝑞. (36)

Remark 18. Applying Theorem 17 for 𝜆=0, 1, we get the results of Theorems 5 and 9, respectively.

Theorem 19. Let 𝑓∶Δ⊆ℝ²→ℝ be a partial differentiable mapping on Δ∶=[𝑎,𝑏]×[𝑐,𝑑] in ℝ² with 𝑎<𝑏 and 𝑐<𝑑. If |𝜕²𝑓/𝜕𝑢𝜕𝑣|^𝑞 is convex on the coordinates on Δ and 𝑞≥1, then 1(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓+()𝑥,𝑦𝑑𝑦𝑑𝑥1−𝜆²𝑓𝑎+𝑏2,𝑐+𝑑2+𝜆²4𝑓()()()(≤()𝑎,𝑐+𝑓𝑎,𝑑+𝑓𝑏,𝑐+𝑓𝑏,𝑑)−𝐴𝑏−𝑎)(𝑑−𝑐2𝜆²−2𝜆+14²×𝜕²𝑓(𝜕𝑢𝜕𝑣𝑎,𝑐)^𝑞+𝜕²𝑓(𝜕𝑢𝜕𝑣𝑎,𝑑)^𝑞+𝜕²𝑓(𝜕𝑢𝜕𝑣𝑏,𝑐)^𝑞+𝜕²𝑓(𝜕𝑢𝜕𝑣𝑏,𝑑)^𝑞×(4)^−11/𝑞, (37) where 𝜆∈[0,1] and 𝐴 is as given in Theorem 15.

Proof. From Lemma 13, we obtain 1(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓+()𝑥,𝑦𝑑𝑦𝑑𝑥1−𝜆²𝑓𝑎+𝑏2,𝑐+𝑑2+𝜆²4𝑓()()()(≤()×𝑎,𝑐+𝑓𝑎,𝑑+𝑓𝑏,𝑐+𝑓𝑏,𝑑)−𝐴𝑏−𝑎)(𝑑−𝑐₁₀|𝑀(×𝜕𝑢,𝑣)|₂𝑓(()()𝑑𝜕𝑢𝜕𝑣𝑢𝑎+1−𝑢𝑏,𝑣𝑐+1−𝑣)𝑑𝑢𝑑𝑣. (38) By using the well-known power mean inequality for double integrals, then one has 1(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓+()𝑥,𝑦𝑑𝑦𝑑𝑥1−𝜆²𝑓𝑎+𝑏2,𝑐+𝑑2+𝜆²4𝑓()()()(≤()𝑎,𝑐+𝑓𝑎,𝑑+𝑓𝑏,𝑐+𝑓𝑏,𝑑)−𝐴𝑏−𝑎)(𝑑−𝑐¹₀|𝑀(𝑢,𝑣)|𝑑𝑢𝑑𝑣^1−(1/𝑞)×¹₀|𝑀(×𝜕𝑢,𝑣)|₂𝑓()+()𝑑𝜕𝑢𝜕𝑣𝑢𝑎+1−𝑢𝑏,𝑣𝑐1−𝑣)^𝑞𝑑𝑢𝑑𝑣^1/𝑞. (39) Because |𝜕²𝑓/𝜕𝑢𝜕𝑣|^𝑞 is a convex function on the coordinates on Δ, then one has 𝜕²𝑓(()()𝑑𝜕𝑢𝜕𝑣𝑢𝑎+1−𝑢𝑏,𝑣𝑐+1−𝑣)^𝑞𝜕≤𝑢𝑣²𝑓(𝜕𝑢𝜕𝑣𝑎,𝑐)^𝑞(𝜕+𝑢1−𝑣)²𝑓(𝜕𝑢𝜕𝑣𝑎,𝑑)^𝑞+()𝑣𝜕1−𝑢²𝑓(𝜕𝑢𝜕𝑣𝑏,𝑐)^𝑞+(𝜕1−𝑢)(1−𝑣)²𝑓(𝜕𝑢𝜕𝑣𝑏,𝑑)^𝑞. (40) Thus, it follows that 1(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓+()𝑥,𝑦𝑑𝑦𝑑𝑥1−𝜆²𝑓𝑎+𝑏2,𝑐+𝑑2+𝜆²4𝑓()()()(≤()𝑎,𝑐+𝑓𝑎,𝑑+𝑓𝑏,𝑐+𝑓𝑏,𝑑)−𝐴𝑏−𝑎)(𝑑−𝑐2𝜆²−2𝜆+14^2−(2/𝑞)×¹₀|𝑀(×𝜕𝑢,𝑣)|𝑢𝑣²𝑓(𝜕𝑢𝜕𝑣𝑎,𝑐)^𝑞(𝜕+𝑢1−𝑣)²𝑓(𝜕𝑢𝜕𝑣𝑎,𝑑)^𝑞+()𝑣𝜕1−𝑢²𝑓(𝜕𝑢𝜕𝑣𝑏,𝑐)^𝑞𝜕+(1−𝑢)(1−𝑣)²𝑓(𝜕𝑢𝜕𝑣𝑏,𝑑)^𝑞𝑑𝑢𝑑𝑣^1/𝑞. (41) On the other hand, we obtain ¹₀|𝑀(×𝜕𝑢,𝑣)|𝑢𝑣²𝑓(𝜕𝑢𝜕𝑣𝑎,𝑐)^𝑞(𝜕+𝑢1−𝑣)²𝑓(𝜕𝑢𝜕𝑣𝑎,𝑑)^𝑞+()𝑣𝜕1−𝑢²𝑓(𝜕𝑢𝜕𝑣𝑏,𝑐)^𝑞+(𝜕1−𝑢)(1−𝑣)²𝑓(𝜕𝑢𝜕𝑣𝑏,𝑑)^𝑞=𝜕𝑑𝑢𝑑𝑣₂𝑓(𝜕𝑢𝜕𝑣𝑎,𝑐)^𝑞1₀|𝑀(+𝜕𝑢,𝑣)|𝑢𝑣𝑑𝑢𝑑𝑣₂𝑓(𝜕𝑢𝜕𝑣𝑎,𝑑)^𝑞1₀|𝑀(𝑢()+𝜕𝑢,𝑣)|1−𝑣𝑑𝑢𝑑𝑣₂𝑓(𝜕𝑢𝜕𝑣𝑏,𝑐)^𝑞1₀|𝑀()+𝜕𝑢,𝑣)|(1−𝑢𝑣𝑑𝑢𝑑𝑣₂𝑓(𝜕𝑢𝜕𝑣𝑏,𝑑)^𝑞1₀|𝑀()=𝑢,𝑣)|(1−𝑢)(1−𝑣𝑑𝑢𝑑𝑣2𝜆²−2𝜆+18²×𝜕²𝑓(𝜕𝑢𝜕𝑣𝑎,𝑐)^𝑞+𝜕²𝑓(𝜕𝑢𝜕𝑣𝑎,𝑑)^𝑞+𝜕²𝑓(𝜕𝑢𝜕𝑣𝑏,𝑐)^𝑞+𝜕²𝑓(𝜕𝑢𝜕𝑣𝑏,𝑑)^𝑞. (42) Thus, we get the following inequality: 1(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓+()𝑥,𝑦𝑑𝑦𝑑𝑥1−𝜆²𝑓𝑎+𝑏2,𝑐+𝑑2+𝜆²4𝑓()()()(≤()𝑎,𝑐+𝑓𝑎,𝑑+𝑓𝑏,𝑐+𝑓𝑏,𝑑)−𝐴𝑏−𝑎)(𝑑−𝑐2𝜆²−2𝜆+14²×𝜕²𝑓(𝜕𝑢𝜕𝑣𝑎,𝑐)^𝑞+𝜕²𝑓(𝜕𝑢𝜕𝑣𝑎,𝑑)^𝑞+𝜕²𝑓(𝜕𝑢𝜕𝑣𝑏,𝑐)^𝑞+𝜕²𝑓(𝜕𝑢𝜕𝑣𝑏,𝑑)^𝑞×(4)^−11/𝑞, (43) which completes the proof.

Remark 20. Applying Theorem 19 for 𝜆=0, 1, we get the result of Theorems 6 and 10, respectively.

Theorem 21. Let 𝑓∶Δ⊆ℝ²→ℝ be a partial differentiable mapping on Δ∶=[𝑎,𝑏]×[𝑐,𝑑] in ℝ² with 𝑎<𝑏 and 𝑐<𝑑. If |𝜕²𝑓/𝜕𝑢𝜕𝑣|^𝑞 is concave on the coordinates on Δ and 𝑞>1, then 1(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓+()𝑥,𝑦𝑑𝑦𝑑𝑥1−𝜆²𝑓𝑎+𝑏2,𝑐+𝑑2+𝜆²4×𝑓()()()(≤()𝑎,𝑐+𝑓𝑎,𝑑+𝑓𝑏,𝑐+𝑓𝑏,𝑑)−𝐴𝑏−𝑎)(𝑑−𝑐4𝜆^𝑝+1+(1−𝜆)^𝑝+1𝑝+1^2/𝑝×𝜕²𝑓𝜕𝑢𝜕𝑣𝑎+𝑏2,𝑐+𝑑2, (44) where 𝜆∈[0,1] and 𝐴 is as given in Theorem 15 and 1/𝑝+1/𝑞=1.

Proof. Similarly as in Theorem 17, because |𝜕²𝑓/𝜕𝑢𝜕𝑣|^𝑞 is a concave function on the coordinates on Δ, by the reversed direction of (4), we get ¹₀𝜕²𝑓(()()𝑑𝜕𝑢𝜕𝑣𝑢𝑎+1−𝑢𝑏,𝑣𝑐+1−𝑣)^𝑞≤𝜕𝑑𝑢𝑑𝑣₂𝑓𝜕𝑢𝜕𝑣𝑎+𝑏2,𝑐+𝑑2^𝑞. (45) Hence, it follows that 1(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓+()𝑥,𝑦𝑑𝑦𝑑𝑥1−𝜆²𝑓𝑎+𝑏2,𝑐+𝑑2+𝜆²4×𝑓()()()(≤()𝑎,𝑐+𝑓𝑎,𝑑+𝑓𝑏,𝑐+𝑓𝑏,𝑑)−𝐴𝑏−𝑎)(𝑑−𝑐4𝜆^𝑝+1+(1−𝜆)^𝑝+1𝑝+1^2/𝑝×𝜕²𝑓𝜕𝑢𝜕𝑣𝑎+𝑏2,𝑐+𝑑2, (46) which yields the desired result.

Conflict of Interests

The author has declared that no conflict of interests exists.

Acknowledgment

This work is supported by Youth Project of Chongqing Three Gorges University of China (No. 13QN11).