On Hermite-Hadamard Type Inequalities for 𝑠-Convex Functions on the Coordinates via Riemann-Liouville Fractional Integrals

Abstract

We obtain some Hermite-Hadamard type inequalities for 𝑠-convex functions on the coordinates via Riemann-Liouville integrals. Some integral inequalities with the right-hand side of the fractional Hermite-Hadamard type inequality are also established.

1. Introduction

If 𝑓∶𝐼→𝑅 is a convex function on the interval 𝐼, then, for any 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.

Since then, some refinements of the Hermite-Hadamard inequality on convex functions have been extensively investigated by a number of authors (e.g., [1–7]).

Definition 1 (see [8]). Let 𝑠∈(0,1] be a fixed real number. A function 𝑓∶𝐼⊂[0,∞)→[0,∞) is said to be 𝑠-convex in the second sense, or that 𝑓 belongs to the class 𝐾²_𝑠, if the inequality, 𝑓𝛼𝑥+(1−𝛼)𝑦≤𝛼^𝑠𝑓(𝑥)+(1−𝛼)^𝑠𝑓𝑦, (2) holds for all 𝑥,𝑦∈𝐼 and 𝛼∈[0,1].

It can be easily seen that, for 𝑠=1, 𝑠-convexity reduces to ordinary convexity of functions defined on [0,∞).

In [4], Dragomir defined convex functions on the coordinates as follows.

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 (3) 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 convex functions on the coordinates may be stated as follows.

Definition 2. 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 (4) holds for all (𝑥,𝑦), (𝑧,𝑦), (𝑥,𝑤), (𝑧,𝑤)∈Δ, and 𝑡,𝜆∈[0,1].

In [4], Dragomir established the following Hadamard-type inequalities for convex functions on the coordinates in a rectangle from the plane ℝ².

Theorem 3 (see [4]). Suppose that 𝑓∶Δ=[𝑎,𝑏]×[𝑐,𝑑]→ℝ is convex on the coordinates on Δ. Then one has the inequalities: 𝑓𝑎+𝑏2,𝑐+𝑑2≤1(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓≤𝑓()()()()𝑥,𝑦𝑑𝑦𝑑𝑥𝑎,𝑐+𝑓𝑎,𝑑+𝑓𝑏,𝑐+𝑓𝑏,𝑑4. (5)

The concept of 𝑠-convex functions on the coordinates in the second sense was introduced by Alomari and Darus in [9].

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 𝑠-convex on Δ if the inequality, 𝑓𝜆𝑥+(1−𝜆)𝑧,𝜆𝑦+(1−𝜆)𝑤≤𝜆^𝑠𝑓𝑥,𝑦+(1−𝜆)^𝑠𝑓(𝑧,𝑤), (6) holds for all (𝑥,𝑦), (𝑧,𝑤)∈Δ with 𝜆∈[0,1] and for some fixed 𝑠∈(0,1].

A function 𝑓∶Δ→ℝ is said to be 𝑠-convex on the coordinates on Δ in the second sense 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 𝑠-convex in the second sense for all 𝑦∈[𝑐,𝑑] and 𝑥∈[𝑎,𝑏] with some fixed 𝑠∈(0,1].

A formal definition for convex functions on the coordinates in the second sense may be stated as follows.

Definition 4. A function 𝑓∶Δ→ℝ is said to be 𝑠-convex on coordinates in the second sense on Δ if the inequality, 𝑓𝜆𝑥+(1−𝜆)𝑧,𝑡𝑦+(1−𝑡)𝑤≤𝜆^𝑠𝑡^𝑠𝑓𝑥,𝑦+𝜆^𝑠(1−𝑡)^𝑠𝑓()+()𝑥,𝑤1−𝜆^𝑠𝑡^𝑠𝑓𝑧,𝑦+(1−𝑡)^𝑠(1−𝜆)^𝑠𝑓(𝑧,𝑤), (7) holds for all (𝑥,𝑦), (𝑧,𝑦), (𝑥,𝑤), (𝑧,𝑤)∈Δ with 𝑡,𝜆∈[0,1] and some fixed 𝑠∈(0,1].

In [10], Alomari and Darus proved the following inequalities based on the above definition.

Theorem 5 (see [10]). Suppose that 𝑓∶Δ=[𝑎,𝑏]×[𝑐,𝑑]⊆[0,∞)²→[0,∞) is s-convex on the coordinates in the second sense on Δ. Then one has the inequalities: 4^𝑠−1𝑓𝑎+𝑏2,𝑐+𝑑2≤1(𝑏−𝑎)(𝑑−𝑐)^𝑏_𝑎^𝑑_𝑐𝑓≤𝑓()()()()𝑥,𝑦𝑑𝑦𝑑𝑥𝑎,𝑐+𝑓𝑎,𝑑+𝑓𝑏,𝑐+𝑓𝑏,𝑑)(𝑠+1²⁻¹. (8)

It is remarkable that Sarikaya et al. [11] proved the following interesting inequalities of Hermite-Hadamard type involving Riemann-Liouville fractional integrals.

Theorem 6 (see [11]). Let 𝑓∶[𝑎,𝑏]→𝑅 be a positive function with 𝑎<𝑏 and 𝑓∈𝐿₁[𝑎,𝑏]. If 𝑓 is a convex function on [𝑎,𝑏], then the following inequalities for fractional integrals hold 𝑓𝑎+𝑏2≤Γ(𝛼+1)2(𝑏−𝑎)^𝛼𝐽^𝛼_𝑎+𝑓(𝑏)+𝐽^𝛼_𝑏⁻𝑓(𝑎≤𝑓(𝑎)(𝑏))+𝑓2, (9) with 𝛼>0.

We remark that the symbol 𝐽^𝛼_𝑎+ and 𝐽^𝛼_𝑏⁻𝑓 denote the left-sided and right-sided Riemann-Liouville fractional integrals of the order 𝛼≥0 with 𝑎≥0 which are defined by 𝐽^𝛼_𝑎+𝑓(𝑥)=1Γ(𝛼)^𝑥_𝑎(𝑥−𝑡)^𝛼−1𝑓(𝑡)𝐽𝑑𝑡,𝑥>𝑎,_𝛼𝑏−𝑓(𝑥)=1Γ(𝛼)^𝑏_𝑥(𝑡−𝑥)^𝛼−1𝑓(𝑡)𝑑𝑡,𝑥<𝑏, (10) respectively. Here, Γ(𝛼) is the Gamma function defined by Γ(𝛼)=∫^∞₀𝑒^−𝑡𝑡^𝛼−1𝑑𝑡.

Definition 7 (see [12]). Let 𝑓∈𝐿₁[𝑎,𝑏]×[𝑐,𝑑]. The Riemann-Liouville fractional integrals 𝐽_𝑎^𝛼,𝛽_+,𝑐+, 𝐽_𝑎^𝛼,𝛽_+,𝑑−, 𝐽_𝑏^𝛼,𝛽_−,𝑐+ and 𝐽_𝑏^𝛼,𝛽_−,𝑑− of order 𝛼,𝛽>0 with 𝑎,𝑐≥0 are defined by 𝐽_𝑎^𝛼,𝛽_+,𝑐+𝑓=1𝑥,𝑦Γ(𝛼)Γ𝛽^𝑥_𝑎^𝑦_𝑐(𝑥−𝑡)^𝛼−1𝑦−𝑠^𝛽−1𝑓()𝐽𝑡,𝑠𝑑𝑠𝑑𝑡,𝑥>𝑎,𝑦>𝑐,_{𝑎𝛼,𝛽+,𝑑−}𝑓=1𝑥,𝑦Γ(𝛼)Γ𝛽^𝑥_𝑎^𝑑_𝑦(𝑥−𝑡)^𝛼−1𝑠−𝑦^𝛽−1𝑓()𝐽𝑡,𝑠𝑑𝑠𝑑𝑡,𝑥>𝑎,𝑦<𝑑,_{𝑏𝛼,𝛽−,𝑐+}𝑓=1𝑥,𝑦Γ(𝛼)Γ𝛽^𝑏_𝑥^𝑦_𝑐(𝑡−𝑥)^𝛼−1𝑦−𝑠^𝛽−1𝑓()𝐽𝑡,𝑠𝑑𝑠𝑑𝑡,𝑥<𝑏,𝑦>𝑐,_{𝑏𝛼,𝛽−,𝑑−}𝑓=1𝑥,𝑦Γ(𝛼)Γ𝛽^𝑏_𝑥^𝑑_𝑦(𝑡−𝑥)^𝛼−1𝑠−𝑦^𝛽−1𝑓(𝑡,𝑠)𝑑𝑠𝑑𝑡,𝑥<𝑏,𝑦<𝑑. (11)

In [12], Sarıkaya proposed the following Hermite-Hadamard-type inequalities for Riemann-Liouville fractional integrals by using convex functions of two variables on the coordinates.

Theorem 8 (see [12]). Let 𝑓∶Δ⊆ℝ²→ℝ be convex functions on the coordinates on Δ=[𝑎,𝑏]×[𝑐,𝑑] in ℝ² with 0≤𝑎<𝑏, 0≤𝑐<𝑑, and 𝑓∈𝐿₁(Δ). Then one has the inequalities: 𝑓𝑎+𝑏2,𝑐+𝑑2≤Γ(𝛼+1)Γ𝛽+14(𝑏−𝑎)^𝛼(𝑑−𝑐)^𝛽×𝐽_𝑎^𝛼,𝛽_+,𝑐+𝑓(𝑏,𝑑)+𝐽_𝑎^𝛼,𝛽_+,𝑑−𝑓(𝑏,𝑐)+𝐽_𝑏^𝛼,𝛽_−,𝑐+𝑓(𝑎,𝑑)+𝐽_𝑏^𝛼,𝛽_−,𝑑−𝑓(≤𝑓()()()()𝑎,𝑐)𝑎,𝑐+𝑓𝑎,𝑑+𝑓𝑏,𝑐+𝑓𝑏,𝑑4. (12)

In [12], Sarıkaya established some Hermite-Hadamard inequalities for convex functions on the coordinates in the second sense via fractional integrals based on the following lemma.

Lemma 9 (see [12]). Let 𝑓∶Δ⊆ℝ²→ℝ be a partial differentiable mapping on Δ∶=[𝑎,𝑏]×[𝑐,𝑑] in ℝ² with 0≤𝑎<𝑏, 0≤𝑐<𝑑. If 𝜕²𝑓/𝜕𝑡𝜕𝑠∈𝐿(Δ), then the following equality holds: 𝑓(𝑎,𝑐)+𝑓(𝑎,𝑑)+𝑓(𝑏,𝑐)+𝑓(𝑏,𝑑)4+Γ(𝛼+1)Γ𝛽+14(𝑏−𝑎)^𝛼(𝑑−𝑐)^𝛽×𝐽_𝑎^𝛼,𝛽_+,𝑐+𝑓(𝑏,𝑑)+𝐽_𝑎^𝛼,𝛽_+,𝑑−𝑓(𝑏,𝑐)+𝐽_𝑏^𝛼,𝛽_−,𝑐+𝑓(𝑎,𝑑)+𝐽_𝑏^𝛼,𝛽_−,𝑑−𝑓(=()𝑎,𝑐)−𝐴𝑏−𝑎)(𝑑−𝑐4×¹₀𝑡^𝛼𝑠^𝛽𝜕²𝑓(()()𝑑)𝜕𝑡𝜕𝑠𝑡𝑎+1−𝑡𝑏,𝑠𝑐+1−𝑠𝑑𝑠𝑑𝑡−¹₀(1−𝑡)^𝛼𝑠^𝛽𝜕²𝑓×(()()𝑑)−𝜕𝑡𝜕𝑠𝑡𝑎+1−𝑡𝑏,𝑠𝑐+1−𝑠𝑑𝑠𝑑𝑡₁₀𝑡^𝛼(1−𝑠)^𝛽𝜕²𝑓×(()()𝑑)+𝜕𝑡𝜕𝑠𝑡𝑎+1−𝑡𝑏,𝑠𝑐+1−𝑠𝑑𝑠𝑑𝑡₁₀(1−𝑡)^𝛼(1−𝑠)^𝛽𝜕²𝑓×(()()𝑑),𝜕𝑡𝜕𝑠𝑡𝑎+1−𝑡𝑏,𝑠𝑐+1−𝑠𝑑𝑠𝑑𝑡 (13) where Γ𝐴=𝛽+14(𝑑−𝑐)^𝛽𝐽^𝛽_𝑐+𝑓(𝑎,𝑑)+𝐽^𝛽_𝑐+𝑓(𝑏,𝑑)+𝐽^𝛽_𝑑−𝑓(𝑎,𝑐)+𝐽^𝛽_𝑑−𝑓(+Γ()𝑏,𝑐)𝛼+14(𝑏−𝑎)^𝛼𝐽^𝛼_𝑎+𝑓(𝑏,𝑐)+𝐽^𝛼_𝑎+𝑓(𝑏,𝑑)+𝐽^𝛼_𝑏⁻𝑓(𝑎,𝑐)+𝐽^𝛼_𝑏⁻𝑓(𝑎,𝑑). (14)

In this paper, we establish some Hermite-Hadamard type inequalities for 𝑠-convex functions on the coordinates functions via Riemann-Liouville integrals. Some integral inequalities with the right-hand side of the fractional Hermite-Hadamard type inequality are also given.

2. Fractional Inequalities for 𝑠-Convex Functions on the Coordinates

Theorem 10. Suppose that 𝑓∶Δ=[𝑎,𝑏]×[𝑐,𝑑]⊆[0,∞)²→[0,∞) is 𝑠-convex function on the coordinates in the second sense on Δ. Then one has the inequalities: 4^𝑠−1𝑓𝑎+𝑏2,𝑐+𝑑2≤Γ(𝛼+1)Γ𝛽+14(𝑏−𝑎)^𝛼(𝑑−𝑐)^𝛽×𝐽_𝑎^𝛼,𝛽_+,𝑐+𝑓(𝑏,𝑑)+𝐽_𝑎^𝛼,𝛽_+,𝑑−𝑓(𝑏,𝑐)+𝐽_𝑏^𝛼,𝛽_−,𝑐+𝑓(𝑎,𝑑)+𝐽_𝑏^𝛼,𝛽_−,𝑑−𝑓(≤𝑓()()()()𝑎,𝑐)𝑎,𝑐+𝑓𝑎,𝑑+𝑓𝑏,𝑐+𝑓𝑏,𝑑4×1(+1𝛼+𝑠)𝛽+𝑠𝐵+1𝛼+1𝛽,𝑠+1()𝐵().𝛽+1×𝐵𝛼,𝑠+1+𝐵𝛽,𝑠+1𝛼,𝑠+1 (15)

Proof. From (7), with 𝑥=𝑡₁𝑎+(1−𝑡₁)𝑏, 𝑦=(1−𝑡₁)𝑎+𝑡₁𝑏, 𝑧=𝑡₂𝑐+(1−𝑡₂)𝑑, 𝑤=(1−𝑡₂)𝑐+𝑡₂𝑑, and 𝑡=𝜆=1/2, we get 𝑓𝑎+𝑏2,𝑐+𝑑2≤14^𝑠𝑓𝑡₁𝑎+1−𝑡₁𝑏,𝑡₂𝑐+1−𝑡₂𝑑+𝑓𝑡₁𝑎+1−𝑡₁𝑏,1−𝑡₂𝑐+𝑡₂𝑑+𝑓1−𝑡₁𝑎+𝑡₁𝑏,𝑡₂𝑐+1−𝑡₂𝑑+𝑓1−𝑡₁𝑎+𝑡₁𝑏,1−𝑡₂𝑐+𝑡₂𝑑. (16) Multiplying both sides of above inequality by 𝑡₁^𝛼−1𝑡₂^𝛽−1 then integrating the resulting inequality with respect to 𝑡₁,𝑡₂ over [0,1]×[0,1], we obtain 1𝑓𝛼𝛽𝑎+𝑏2,𝑐+𝑑2≤14^𝑠1₀𝑡₁^𝛼−1𝑡₂^𝛽−1𝑓𝑡₁𝑎+1−𝑡₁𝑏,𝑡₂𝑐+1−𝑡₂𝑑𝑑𝑡₂𝑑𝑡₁+¹₀𝑡₁^𝛼−1𝑡₂^𝛽−1𝑓𝑡₁𝑎+1−𝑡₁𝑏,1−𝑡₂𝑐+𝑡₂𝑑𝑑𝑡₂𝑑𝑡₁+¹₀𝑡₁^𝛼−1𝑡₂^𝛽−1𝑓1−𝑡₁𝑎+𝑡₁𝑏,𝑡₂𝑐+1−𝑡₂𝑑𝑑𝑡₂𝑑𝑡₁+¹₀𝑡₁^𝛼−1𝑡₂^𝛽−1𝑓1−𝑡₁𝑎+𝑡₁𝑏,1−𝑡₂𝑐+𝑡₂𝑑𝑑𝑡₂𝑑𝑡₁. (17) Using the change of the variable, we get 1𝑓𝛼𝛽𝑎+𝑏2,𝑐+𝑑2≤14^𝑠𝑏_𝑎^𝑑_𝑐(𝑏−𝑥)^𝛼−1𝑑−𝑦^𝛽−1𝑓+𝑥,𝑦𝑑𝑦𝑑𝑥_{𝑏𝑎𝑑𝑐}(𝑏−𝑥)^𝛼−1𝑦−𝑐^𝛽−1𝑓𝑥,𝑦𝑑𝑦𝑑𝑥+^𝑏_𝑎^𝑑_𝑐(𝑥−𝑎)^𝛼−1𝑑−𝑦^𝛽−1𝑓+𝑥,𝑦𝑑𝑦𝑑𝑥_{𝑏𝑎𝑑𝑐}(𝑥−𝑎)^𝛼−1𝑦−𝑐^𝛽−1𝑓𝑥,𝑦𝑑𝑦𝑑𝑥, (18) by which the first inequality is proved. For the proof of the second inequality, we note that 𝑓 is 𝑠-convex on coordinates, then 𝑓𝑡₁𝑎+1−𝑡₁𝑏,𝑡₂𝑐+1−𝑡₂𝑑≤𝑡^𝑠₁𝑡^𝑠₂𝑓(𝑎,𝑐)+𝑡^𝑠₁1−𝑡₂^𝑠𝑓()+𝑎,𝑑1−𝑡₁^𝑠𝑡^𝑠₂𝑓(𝑏,𝑐)+1−𝑡₁^𝑠1−𝑡₂^𝑠𝑓(),𝑓𝑡𝑏,𝑑₁𝑎+1−𝑡₁𝑏,1−𝑡₂𝑐+𝑡₂𝑑≤𝑡^𝑠₁1−𝑡₂^𝑠𝑓(𝑎,𝑐)+𝑡^𝑠₁𝑡^𝑠₂𝑓()+𝑎,𝑑1−𝑡₁^𝑠1−𝑡₂^𝑠𝑓(𝑏,𝑐)+1−𝑡₁^𝑠𝑡^𝑠₂𝑓(),𝑓𝑏,𝑑1−𝑡₁𝑎+𝑡₁𝑏,𝑡₂𝑐+1−𝑡₂𝑑≤1−𝑡₁^𝑠𝑡^𝑠₂𝑓(𝑎,𝑐)+1−𝑡₁^𝑠1−𝑡₂^𝑠𝑓(𝑎,𝑑)+𝑡^𝑠₁𝑡^𝑠₂𝑓(𝑏,𝑐)+𝑡^𝑠₁1−𝑡₂^𝑠𝑓(),𝑓𝑏,𝑑1−𝑡₁𝑎+𝑡₁𝑏,1−𝑡₂𝑐+𝑡₂𝑑≤1−𝑡₁^𝑠1−𝑡₂^𝑠𝑓(𝑎,𝑐)+1−𝑡₁^𝑠𝑡^𝑠₂𝑓(𝑎,𝑑)+𝑡^𝑠₁1−𝑡₂^𝑠𝑓(𝑏,𝑐)+𝑡^𝑠₁𝑡^𝑠₂𝑓(𝑏,𝑑). (19) Multiplying both sides of above inequalities by 𝑡₁^𝛼−1𝑡₂^𝛽−1, then integrating the resulting inequality with respect to 𝑡₁,𝑡₂ over [0,1]×[0,1], we obtain ¹₀𝑡₁^𝛼−1𝑡₂^𝛽−1𝑓𝑡₁𝑎+1−𝑡₁𝑏,𝑡₂𝑐+1−𝑡₂𝑑+𝑓𝑡₁𝑎+1−𝑡₁𝑏,1−𝑡₂𝑐+𝑡₂𝑑+𝑓1−𝑡₁𝑎+𝑡₁𝑏,𝑡₂𝑐+1−𝑡₂𝑑+𝑓1−𝑡₁𝑎+𝑡₁𝑏,1−𝑡₂𝑐+𝑡₂𝑑𝑑𝑡₂𝑑𝑡₁≤𝑓(𝑎,𝑐)+𝑓(𝑎,𝑑)+𝑓(𝑏,𝑐)+𝑓(𝑏,𝑑)×¹₀𝑡₁^𝛼−1𝑡₂^𝛽−1𝑡^𝑠₁𝑡^𝑠₂+𝑡^𝑠₁1−𝑡₂^𝑠+1−𝑡₁^𝑠𝑡^𝑠₂+1−𝑡₁^𝑠1−𝑡₂^𝑠𝑑𝑡₂𝑑𝑡₁=𝑓()()()(×1𝑎,𝑐+𝑓𝑎,𝑑+𝑓𝑏,𝑐+𝑓𝑏,𝑑)()+1𝛼+𝑠𝛽+𝑠𝐵+1𝛼+1𝛽,𝑠+1𝐵()𝐵().𝛽+1𝛼,𝑠+1+𝐵𝛽,𝑠+1𝛼,𝑠+1 (20) Here, using the change of the variable, we have Γ(𝛼+1)Γ𝛽+1(𝑏−𝑎)^𝛼(𝑑−𝑐)^𝛽𝐽_𝑎^𝛼,𝛽_+,𝑐+𝑓(𝑏,𝑑)+𝐽_𝑎^𝛼,𝛽_+,𝑑−𝑓(𝑏,𝑐)+𝐽_𝑏^𝛼,𝛽_−,𝑐+𝑓(𝑎,𝑑)+𝐽_𝑏^𝛼,𝛽_−,𝑑−𝑓(≤𝑓()()()(×1𝑎,𝑐)𝑎,𝑐+𝑓𝑎,𝑑+𝑓𝑏,𝑐+𝑓𝑏,𝑑)(+1𝛼+𝑠)𝛽+𝑠𝐵+1𝛼+1𝛽,𝑠+1𝐵()𝐵().𝛽+1𝛼,𝑠+1+𝐵𝛽,𝑠+1𝛼,𝑠+1 (21) The proof is completed.

Remark 11. Applying Theorem 10 for 𝑠=1, we get Theorem 8.
We note that the Beta functions is defined by 𝐵𝑥,𝑦=¹₀(1−𝑡)^𝑦−1𝑡^𝑥−1𝑑𝑡,𝑥>0,𝑦>0. (22)

3. Inequalities for Differentiable Functions

Theorem 12. Let 𝑓∶Δ=[𝑎,𝑏]×[𝑐,𝑑]⊆[0,∞)²→[0,∞) be a partial differentiable mapping with 0≤𝑎<𝑏, 0≤𝑐<𝑑. If |𝜕²𝑓/𝜕𝑡₁𝜕𝑡₂| is 𝑠-convex on the coordinates in the second sense on Δ for some fixed 𝑠∈(0,1]. Then one has the inequalities: 𝑓(𝑎,𝑐)+𝑓(𝑎,𝑑)+𝑓(𝑏,𝑐)+𝑓(𝑏,𝑑)4+Γ(𝛼+1)Γ𝛽+14(𝑏−𝑎)^𝛼(𝑑−𝑐)^𝛽×𝐽_𝑎^𝛼,𝛽_+,𝑐+𝑓(𝑏,𝑑)+𝐽_𝑎^𝛼,𝛽_+,𝑑−𝑓(𝑏,𝑐)+𝐽_𝑏^𝛼,𝛽_−,𝑐+𝑓(𝑎,𝑑)+𝐽_𝑏^𝛼,𝛽_−,𝑑−𝑓(=()𝑎,𝑐)−𝐴𝑏−𝑎)(𝑑−𝑐41(×1𝛼+𝑠+1+𝐵𝑠+1,𝛼+1)×𝜕𝛽+𝑠+1+𝐵𝑠+1,𝛽+1₂𝑓𝜕𝑡₁𝜕𝑡₂(+𝜕𝑎,𝑐)²𝑓𝜕𝑡₁𝜕𝑡₂(+𝜕𝑏,𝑐)₂𝑓𝜕𝑡₁𝜕𝑡₂(+𝜕𝑎,𝑑)²𝑓𝜕𝑡₁𝜕𝑡₂(𝑏,𝑑), (23) where 𝐴 is as given in Lemma 9.

Proof. From Lemma 9, we obtain 𝑓(𝑎,𝑐)+𝑓(𝑎,𝑑)+𝑓(𝑏,𝑐)+𝑓(𝑏,𝑑)4+Γ(𝛼+1)Γ𝛽+14(𝑏−𝑎)^𝛼(𝑑−𝑐)^𝛽𝐽_𝑎^𝛼,𝛽_+,𝑐+𝑓(𝑏,𝑑)+𝐽_𝑎^𝛼,𝛽_+,𝑑−𝑓(𝑏,𝑐)+𝐽_𝑏^𝛼,𝛽_−,𝑐+𝑓(𝑎,𝑑)+𝐽_𝑏^𝛼,𝛽_−,𝑑−𝑓(≤()𝑎,𝑐)−𝐴𝑏−𝑎)(𝑑−𝑐4×¹₀𝑡^𝛼₁𝑡^𝛽₂𝜕²𝑓𝜕𝑡₁𝜕𝑡₂𝑡₁𝑎+1−𝑡₁𝑏,𝑡₂𝑐+1−𝑡₂𝑑𝑑𝑡₂𝑑𝑡₁+¹₀1−𝑡₁^𝛼𝑡^𝛽₂×𝜕²𝑓𝜕𝑡₁𝜕𝑡₂×𝑡₁𝑎+1−𝑡₁𝑏,𝑡₂𝑐+1−𝑡₂𝑑𝑑𝑡₂𝑑𝑡₁+¹₀𝑡^𝛼₁1−𝑡₂^𝛽×𝜕²𝑓𝜕𝑡₁𝜕𝑡₂×𝑡₁𝑎+1−𝑡₁𝑏,𝑡₂𝑐+1−𝑡₂𝑑𝑑𝑡₂𝑑𝑡₁+¹₀1−𝑡₁^𝛼1−𝑡₂^𝛽×𝜕²𝑓𝜕𝑡₁𝜕𝑡₂𝑡₁𝑎+1−𝑡₁𝑏,𝑡₂𝑐+1−𝑡₂𝑑𝑑𝑡₂𝑑𝑡₁. (24) Because |𝜕²𝑓/𝜕𝑢𝜕𝑣| is 𝑠-convex function on the coordinates on Δ, we obtain 𝑓(𝑎,𝑐)+𝑓(𝑎,𝑑)+𝑓(𝑏,𝑐)+𝑓(𝑏,𝑑)4+Γ(𝛼+1)Γ𝛽+14(𝑏−𝑎)^𝛼(𝑑−𝑐)^𝛽𝐽_𝑎^𝛼,𝛽_+,𝑐+𝑓(𝑏,𝑑)+𝐽_𝑎^𝛼,𝛽_+,𝑑−𝑓(𝑏,𝑐)+𝐽_𝑏^𝛼,𝛽_−,𝑐+𝑓(𝑎,𝑑)+𝐽_𝑏^𝛼,𝛽_−,𝑑−𝑓(≤()𝑎,𝑐)−𝐴𝑏−𝑎)(𝑑−𝑐4×¹₀𝑡^𝛼₁𝑡^𝛽₂+1−𝑡₁^𝛼𝑡^𝛽₂+𝑡^𝛼₁1−𝑡₂^𝛽+1−𝑡₁^𝛼1−𝑡₂^𝛽×𝜕²𝑓𝜕𝑡₁𝜕𝑡₂𝑡₁𝑎+1−𝑡₁𝑏,𝑡₂𝑐+1−𝑡₂𝑑𝑑𝑡₂𝑑𝑡₁≤(𝑏−𝑎)(𝑑−𝑐)4×¹₀𝑡^𝛼₁𝑡^𝛽₂+1−𝑡₁^𝛼𝑡^𝛽₂+𝑡^𝛼₁1−𝑡₂^𝛽+1−𝑡₁^𝛼1−𝑡₂^𝛽×𝑡^𝑠₁𝑡^𝑠₂𝜕²𝑓𝜕𝑡₁𝜕𝑡₂(𝑎,𝑐)+1−𝑡₁^𝑠𝑡^𝑠₂𝜕²𝑓𝜕𝑡₁𝜕𝑡₂(𝑏,𝑐)+𝑡^𝑠₁1−𝑡₂^𝑠𝜕²𝑓𝜕𝑡₁𝜕𝑡₂(+𝑎,𝑑)1−𝑡₁^𝑠1−𝑡₂^𝑠𝜕²𝑓𝜕𝑡₁𝜕𝑡₂(𝑏,𝑑)𝑑𝑡₂𝑑𝑡₁=(𝑏−𝑎)(𝑑−𝑐)41(×1𝛼+𝑠+1+𝐵𝑠+1,𝛼+1)×𝜕𝛽+𝑠+1+𝐵𝑠+1,𝛽+1₂𝑓𝜕𝑡₁𝜕𝑡₂(+𝜕𝑎,𝑐)²𝑓𝜕𝑡₁𝜕𝑡₂(+𝜕𝑏,𝑐)₂𝑓𝜕𝑡₁𝜕𝑡₂(+𝜕𝑎,𝑑)²𝑓𝜕𝑡₁𝜕𝑡₂(𝑏,𝑑), (25)which completes the proof.

Theorem 13. Let 𝑓∶Δ=[𝑎,𝑏]×[𝑐,𝑑]⊆[0,∞)²→[0,∞) be a partial differentiable mapping with 0≤𝑎<𝑏, 0≤𝑐<𝑑. If |𝜕²𝑓/𝜕𝑢𝜕𝑣|^𝑞 is 𝑠-convex on the coordinates in the second sense on Δ for some fixed 𝑠∈(0,1] and 𝑞>1. Then, one has the inequalities: 𝑓(𝑎,𝑐)+𝑓(𝑎,𝑑)+𝑓(𝑏,𝑐)+𝑓(𝑏,𝑑)4+Γ(𝛼+1)Γ𝛽+14(𝑏−𝑎)^𝛼(𝑑−𝑐)^𝛽×𝐽_𝑎^𝛼,𝛽_+,𝑐+𝑓(𝑏,𝑑)+𝐽_𝑎^𝛼,𝛽_+,𝑑−𝑓(𝑏,𝑐)+𝐽_𝑏^𝛼,𝛽_−,𝑐+𝑓(𝑎,𝑑)+𝐽_𝑏^𝛼,𝛽_−,𝑑−𝑓(≤()𝑎,𝑐)−𝐴𝑏−𝑎)(𝑑−𝑐𝛼𝑝+1𝛽𝑝+1^1/𝑝×𝜕²𝑓/𝜕𝑡₁𝜕𝑡₂(𝑎,𝑐)^𝑞+𝜕²𝑓/𝜕𝑡₁𝜕𝑡₂(𝑏,𝑐)^𝑞+𝜕²𝑓/𝜕𝑡₁𝜕𝑡₂(𝑎,𝑑)^𝑞+𝜕²𝑓/𝜕𝑡₁𝜕𝑡₂(𝑏,𝑑)^𝑞(𝑠+1)^21/𝑞, (26) where (1/𝑝)+(1/𝑞)=1 and 𝐴 are as given in Lemma 9.

Proof. From Lemma 9, we obtain 𝑓(𝑎,𝑐)+𝑓(𝑎,𝑑)+𝑓(𝑏,𝑐)+𝑓(𝑏,𝑑)4+Γ(𝛼+1)Γ𝛽+14(𝑏−𝑎)^𝛼(𝑑−𝑐)^𝛽×𝐽_𝑎^𝛼,𝛽_+,𝑐+𝑓(𝑏,𝑑)+𝐽_𝑎^𝛼,𝛽_+,𝑑−𝑓(𝑏,𝑐)+𝐽_𝑏^𝛼,𝛽_−,𝑐+𝑓(𝑎,𝑑)+𝐽_𝑏^𝛼,𝛽_−,𝑑−𝑓(≤()𝑎,𝑐)−𝐴𝑏−𝑎)(𝑑−𝑐4×¹₀𝑡^𝛼₁𝑡^𝛽₂×𝜕²𝑓𝜕𝑡₁𝜕𝑡₂𝑡₁𝑎+1−𝑡₁𝑏,𝑡₂𝑐+1−𝑡₂𝑑𝑑𝑡₂𝑑𝑡₁+¹₀1−𝑡₁^𝛼𝑡^𝛽₂×𝜕²𝑓𝜕𝑡₁𝜕𝑡₂𝑐𝑡₁𝑎+1−𝑡₁𝑏,𝑡₂𝑐+1−𝑡₂𝑑𝑑𝑡₂𝑑𝑡₁+¹₀𝑡^𝛼₁1−𝑡₂^𝛽×𝜕²𝑓𝜕𝑡₁𝜕𝑡₂𝑡₁𝑎+1−𝑡₁𝑏,𝑡₂𝑐1−𝑡₂𝑑𝑑𝑡₂𝑑𝑡₁+¹₀1−𝑡₁^𝛼1−𝑡₂^𝛽×𝜕²𝑓𝜕𝑡₁𝜕𝑡₂×𝑡₁𝑎+1−𝑡₁𝑏,𝑡₂𝑐+1−𝑡₂𝑑𝑑𝑡₂𝑑𝑡₁. (27)
By using the well-known Hölder's inequality for double integrals, then one has 𝑓(𝑎,𝑐)+𝑓(𝑎,𝑑)+𝑓(𝑏,𝑐)+𝑓(𝑏,𝑑)4+Γ(𝛼+1)Γ𝛽+14(𝑏−𝑎)^𝛼(𝑑−𝑐)^𝛽×𝐽_𝑎^𝛼,𝛽_+,𝑐+𝑓(𝑏,𝑑)+𝐽_𝑎^𝛼,𝛽_+,𝑑−𝑓(𝑏,𝑐)+𝐽_𝑏^𝛼,𝛽_−,𝑐+𝑓(𝑎,𝑑)+𝐽_𝑏^𝛼,𝛽_−,𝑑−𝑓(≤()𝑎,𝑐)−𝐴𝑏−𝑎)(𝑑−𝑐4¹₀𝑡₁^𝑝𝛼𝑡₂^𝑝𝛽𝑑𝑡₂𝑑𝑡₁^1/𝑝+¹₀1−𝑡₁^𝑝𝛼𝑡₂^𝑝𝛽𝑑𝑡₂𝑑𝑡₁^1/𝑝+¹₀𝑡₁^𝑝𝛼1−𝑡₂^𝑝𝛽𝑑𝑡₂𝑑𝑡₁^1/𝑝+¹₀(1−𝑡)^𝑝𝛼(1−𝑠)^𝑝𝛽𝑑𝑡₂𝑑𝑡₁^1/𝑝×¹₀𝜕²𝑓𝜕𝑡₁𝜕𝑡₂𝑡₁𝑎+1−𝑡₁𝑏,𝑡₂𝑐+1−𝑡₂𝑑^𝑞𝑑𝑡₂𝑑𝑡₁^1/𝑞. (28) Because |𝜕²𝑓/𝜕𝑡₁𝜕𝑡₂| is 𝑠-convex function on the coordinates on Δ, by (8), we have¹₀𝜕²𝑓𝜕𝑡₁𝜕𝑡₂𝑡₁𝑎+1−𝑡₁𝑏,𝑡₂𝑐+1−𝑡₂𝑑^𝑞𝑑𝑡₂𝑑𝑡₁≤𝜕²𝑓/𝜕𝑡₁𝜕𝑡₂(𝑎,𝑐)^𝑞+𝜕²𝑓/𝜕𝑡₁𝜕𝑡₂(𝑏,𝑐)^𝑞+𝜕²𝑓/𝜕𝑡₁𝜕𝑡₂(𝑎,𝑑)^𝑞+𝜕²𝑓/𝜕𝑡₁𝜕𝑡₂(𝑏,𝑑)^𝑞(𝑠+1)². (29) So 𝑓(𝑎,𝑐)+𝑓(𝑎,𝑑)+𝑓(𝑏,𝑐)+𝑓(𝑏,𝑑)4+Γ(𝛼+1)Γ𝛽+14(𝑏−𝑎)^𝛼(𝑑−𝑐)^𝛽×𝐽_𝑎^𝛼,𝛽_+,𝑐+𝑓(𝑏,𝑑)+𝐽_𝑎^𝛼,𝛽_+,𝑑−𝑓(𝑏,𝑐)+𝐽_𝑏^𝛼,𝛽_−,𝑐+𝑓(𝑎,𝑑)+𝐽_𝑏^𝛼,𝛽_−,𝑑−𝑓(≤()𝑎,𝑐)−𝐴𝑏−𝑎)(𝑑−𝑐𝛼𝑝+1𝛽𝑝+1^1/𝑝×𝜕²𝑓/𝜕𝑡₁𝜕𝑡₂(𝑎,𝑐)^𝑞+𝜕²𝑓/𝜕𝑡₁𝜕𝑡₂(𝑏,𝑐)^𝑞+𝜕²𝑓/𝜕𝑡₁𝜕𝑡₂(𝑎,𝑑)^𝑞+𝜕²𝑓/𝜕𝑡₁𝜕𝑡₂(𝑏,𝑑)^𝑞(𝑠+1)^21/𝑞, (30) which completes the proof.

Theorem 14. Let 𝑓∶Δ=[𝑎,𝑏]×[𝑐,𝑑]⊆[0,∞)²→[0,∞) be a partial differentiable mapping with 0≤𝑎<𝑏, 0≤𝑐<𝑑. If |𝜕²𝑓/𝜕𝑢𝜕𝑣|^𝑞 is 𝑠-concave on the coordinates in the second sense on Δ for some fixed 𝑠∈(0,1] and 𝑞>1, then one has the inequalities: 𝑓(𝑎,𝑐)+𝑓(𝑎,𝑑)+𝑓(𝑏,𝑐)+𝑓(𝑏,𝑑)4+Γ(𝛼+1)Γ𝛽+14(𝑏−𝑎)^𝛼(𝑑−𝑐)^𝛽×𝐽_𝑎^𝛼,𝛽_+,𝑐+𝑓(𝑏,𝑑)+𝐽_𝑎^𝛼,𝛽_+,𝑑−𝑓(𝑏,𝑐)+𝐽_𝑏^𝛼,𝛽_−,𝑐+𝑓(𝑎,𝑑)+𝐽_𝑏^𝛼,𝛽_−,𝑑−𝑓(≤()𝑎,𝑐)−𝐴𝑏−𝑎)(𝑑−𝑐𝛼𝑝+1𝛽𝑝+1^1/𝑝×4^{(𝑠−1)/𝑞}𝜕²𝑓𝜕𝑡₁𝜕𝑡₂𝑎+𝑏2,𝑐+𝑑2, (31) where (1/𝑝)+(1/𝑞)=1 and 𝐴 is as given in Lemma 9.

Proof. Similarly as in Theorem 13, we obtain 𝑓(𝑎,𝑐)+𝑓(𝑎,𝑑)+𝑓(𝑏,𝑐)+𝑓(𝑏,𝑑)4+Γ(𝛼+1)Γ𝛽+14(𝑏−𝑎)^𝛼(𝑑−𝑐)^𝛽×𝐽_𝑎^𝛼,𝛽_+,𝑐+𝑓(𝑏,𝑑)+𝐽_𝑎^𝛼,𝛽_+,𝑑−𝑓(𝑏,𝑐)+𝐽_𝑏^𝛼,𝛽_−,𝑐+𝑓(𝑎,𝑑)+𝐽_𝑏^𝛼,𝛽_−,𝑑−𝑓(≤()𝑎,𝑐)−𝐴𝑏−𝑎)(𝑑−𝑐4×¹₀𝑡₁^𝑝𝛼𝑡₂^𝑝𝛽𝑑𝑡₂𝑑𝑡₁^1/𝑝+¹₀1−𝑡₁^𝑝𝛼𝑡₂^𝑝𝛽𝑑𝑡₂𝑑𝑡₁^1/𝑝+¹₀𝑡₁^𝑝𝛼1−𝑡₂^𝑝𝛽𝑑𝑡₂𝑑𝑡₁^1/𝑝+¹₀(1−𝑡)^𝑝𝛼(1−𝑠)^𝑝𝛽𝑑𝑡₂𝑑𝑡₁^1/𝑝×¹₀𝜕²𝑓𝜕𝑡₁𝜕𝑡₂𝑡₁𝑎+1−𝑡₁𝑏,𝑡₂𝑐+1−𝑡₂𝑑^𝑞𝑑𝑡₂𝑑𝑡₁^1/𝑞. (32) Because |𝜕²𝑓/𝜕𝑡₁𝜕𝑡₂| is 𝑠-concave function on the coordinates on Δ, by the reversed direction of (8), then one has ¹₀𝜕²𝑓𝜕𝑡₁𝜕𝑡₂𝑡₁𝑎+1−𝑡₁𝑏,𝑡₂𝑐+1−𝑡₂𝑑^𝑞𝑑𝑡₂𝑑𝑡₁≤4^𝑠−1𝜕²𝑓𝜕𝑡₁𝜕𝑡₂𝑎+𝑏2,𝑐+𝑑2^𝑞. (33) Then 𝑓(𝑎,𝑐)+𝑓(𝑎,𝑑)+𝑓(𝑏,𝑐)+𝑓(𝑏,𝑑)4+Γ(𝛼+1)Γ𝛽+14(𝑏−𝑎)^𝛼(𝑑−𝑐)^𝛽×𝐽_𝑎^𝛼,𝛽_+,𝑐+𝑓(𝑏,𝑑)+𝐽_𝑎^𝛼,𝛽_+,𝑑−𝑓(𝑏,𝑐)+𝐽_𝑏^𝛼,𝛽_−,𝑐+𝑓(𝑎,𝑑)+𝐽_𝑏^𝛼,𝛽_−,𝑑−𝑓(≤()𝑎,𝑐)−𝐴𝑏−𝑎)(𝑑−𝑐𝛼𝑝+1𝛽𝑝+1^1/𝑝×4^{(𝑠−1)/𝑞}𝜕²𝑓𝜕𝑡₁𝜕𝑡₂𝑎+𝑏2,𝑐+𝑑2. (34) We get the desired results.

4. Conclusion

In this paper, we obtain some Hermite-Hadamard type inequalities for coordinated 𝑠-convex functions via Riemann-Liouville integrals. An interesting topic is whether we can use the methods in this paper to establish the Hermite-Hadamard inequalities for other kinds of convex functions on the coordinates via Riemann-Liouville integrals.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

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