A Note on Hermite-Hadamard Inequalities for Products of Convex Functions

Abstract

We obtain some new Hermite-Hadamard type inequalities for products of convex functions. We conclude that the results obtained in this work are the refinements of the present results.

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]).

In [8], Pachpatte established two new Hermite-Hadamard type inequalities for products of convex functions as follows.

Theorem 1. Let 𝑓 and 𝑔 be real-valued, nonnegative, and convex functions on [𝑎,𝑏]. Then, 1𝑏−𝑎^𝑏_𝑎𝑓(𝑥)𝑔(𝑥)1𝑑𝑥≤3𝑀()+1𝑎,𝑏6𝑁(𝑎,𝑏),2𝑓𝑎+𝑏2𝑔𝑎+𝑏2≤1𝑏−𝑎^𝑏_𝑎𝑓(𝑥)𝑔(𝑥)+1𝑑𝑥6𝑀()+1𝑎,𝑏3𝑁(𝑎,𝑏), (2)(3) where 𝑀(𝑎,𝑏)=𝑓(𝑎)𝑔(𝑎)+𝑓(𝑏)𝑔(𝑏) and 𝑁(𝑎,𝑏)=𝑓(𝑎)𝑔(𝑏)+𝑓(𝑏)𝑔(𝑎).

Some new integral inequalities involving two nonnegative and integrable functions that are related to the Hermite-Hadamard type are also established by many authors. In [9], Pachpatte established some Hermite-Hadamard type inequalities involving two log-convex functions. An analogous result for 𝑠-convex functions is obtained by Kirmaci et al. in [10]. In [11], Sarikaya presented some integral inequalities for two ℎ-convex functions. Some Hermite-Hadamard type inequalities for two operator convex functions are given by Bacak and Türkmen in [12]. For recent results and generalizations concerning Hermite-Hadamard type inequality for product of two functions, see [13] and the references given therein.

In this paper, we obtain new generalizations of Hermite-Hadamard type inequalities for products of convex functions. This result refines the Hermite-Hadamard type inequalities given in Theorem 1.

2. Lemma

Lemma 2. Let 𝑓 and 𝑔 be real-valued, nonnegative, and convex functions on [𝑎,𝑏]. If [𝑐,𝑑]⊂[𝑎,𝑏], then 1𝑑−𝑐^𝑑_𝑐𝑓(𝑥)𝑔(𝑥)1𝑑𝑥≤3𝑀()+1𝑐,𝑑6𝑁(𝑐,𝑑),2𝑓𝑐+𝑑2𝑔𝑐+𝑑2≤1𝑑−𝑐^𝑑_𝑐𝑓(𝑥)𝑔(𝑥)+1𝑑𝑥6𝑀()+1𝑐,𝑑3𝑁(𝑐,𝑑), (4) where 𝑀(𝑐,𝑑)=𝑓(𝑐)𝑔(𝑐)+𝑓(𝑑)𝑔(𝑑) and 𝑁(𝑐,𝑑)=𝑓(𝑐)𝑔(𝑑)+𝑓(𝑑)𝑔(𝑐).

Proof. We get the conclusions by 𝑓 and 𝑔 are nonnegative and convex functions on [𝑐,𝑑] and Theorem 1.

3. Main Results

Theorem 3. Let 𝑓 and 𝑔 be real-valued, nonnegative, and convex functions on [𝑎,𝑏]. Then, 1𝑏−𝑎^𝑏_𝑎𝑓(𝑥)𝑔(𝑥)(𝑡)≤1𝑑𝑥≤𝐿3𝑀()+1𝑎,𝑏6𝑁(𝑎,𝑏), (5) where 𝑀(𝑎,𝑏)=𝑓(𝑎)𝑔(𝑎)+𝑓(𝑏)𝑔(𝑏), 𝑁(𝑎,𝑏)=𝑓(𝑎)𝑔(𝑏)+𝑓(𝑏)𝑔(𝑎), 𝑡∈[0,1], and 𝐿(𝑡)=𝑡3𝑀(())+𝑡𝑎,1−𝑡𝑎+𝑡𝑏6𝑁(())+𝑎,1−𝑡𝑎+𝑡𝑏1−𝑡3𝑀))+((1−𝑡𝑎+𝑡𝑏,𝑏1−𝑡6𝑁((1−𝑡)𝑎+𝑡𝑏,𝑏). (6)

Proof. Since 𝑓 and 𝑔 are convex on [𝑎,𝑏] and nonnegative, by (2), we get 1𝑏−𝑎^𝑏_𝑎𝑓(𝑥)𝑔(𝑥)1𝑑𝑥≤3𝑀()+1𝑎,𝑏6𝑁(𝑎,𝑏). (7)
Firstly, 𝑓 and 𝑔 are convex on [𝑎,(1−𝑡)𝑎+𝑡𝑏] and nonnegative; applying (2) on [𝑎,(1−𝑡)𝑎+𝑡𝑏], we get 1𝑡(𝑏−𝑎)_𝑎^{(1−𝑡)𝑎+𝑡𝑏}𝑓(𝑥)𝑔(𝑥)≤1𝑑𝑥3𝑀(())+1𝑎,1−𝑡𝑎+𝑡𝑏6𝑁(𝑎,(1−𝑡)𝑎+𝑡𝑏). (8)
Similarly, we can show that 1(1−𝑡)(𝑏−𝑎)^𝑏_{()1−𝑡𝑎+𝑡𝑏}𝑓(𝑥)𝑔(𝑥)≤1𝑑𝑥3𝑀))+1((1−𝑡𝑎+𝑡𝑏,𝑏6𝑁((1−𝑡)𝑎+𝑡𝑏,𝑏). (9)
Multiplying (8) by 𝑡 and (9) by (1−𝑡) and adding the resulting inequalities, we get 1𝑏−𝑎^𝑏_𝑎𝑓(𝑥)𝑔(𝑥)≤𝑡𝑑𝑥3𝑀(())+𝑡𝑎,1−𝑡𝑎+𝑡𝑏6𝑁(())+𝑎,1−𝑡𝑎+𝑡𝑏1−𝑡3𝑀))+((1−𝑡𝑎+𝑡𝑏,𝑏1−𝑡6𝑁((1−𝑡)𝑎+𝑡𝑏,𝑏)=𝐿(𝑡). (10)
Using the fact that 𝑡3𝑀(𝑎,(1−𝑡)𝑎+𝑡𝑏)+1−𝑡3𝑀))=𝑡((1−𝑡𝑎+𝑡𝑏,𝑏3𝑓(𝑎)𝑔(𝑎))))++𝑓((1−𝑡𝑎+𝑡𝑏×𝑔((1−𝑡𝑎+𝑡𝑏)1−𝑡3𝑓))𝑔))(𝑏)𝑔(𝑏=𝑡((1−𝑡𝑎+𝑡𝑏((1−𝑡𝑎+𝑡𝑏+𝑓)3𝑓(𝑎)𝑔(𝑎)+1−𝑡3𝑓(𝑏)𝑔(𝑏)+13𝑓))𝑔))≤𝑡((1−𝑡𝑎+𝑡𝑏((1−𝑡𝑎+𝑡𝑏3𝑓(𝑎)𝑔(𝑎)+1−𝑡3𝑓(𝑏)𝑔(𝑏)+13)𝑓(𝑎)(𝑏×)𝑔(𝑎)(𝑏=()(1−𝑡+𝑡𝑓)(1−𝑡+𝑡𝑔)𝑡+1−𝑡²3𝑓(𝑎)𝑔(𝑎)+1−𝑡+𝑡²3𝑓(𝑏)𝑔(𝑏)+(1−𝑡)𝑡3𝑁()=𝑎,𝑏1−𝑡+𝑡²3𝑀()+()𝑡𝑎,𝑏1−𝑡3𝑁(),𝑡𝑎,𝑏6𝑁(𝑎,(1−𝑡)𝑎+𝑡𝑏)+1−𝑡6𝑁))=𝑡((1−𝑡𝑎+𝑡𝑏,𝑏6𝑓(𝑎)𝑔))))𝑔(𝑎+((1−𝑡𝑎+𝑡𝑏+𝑓((1−𝑡𝑎+𝑡𝑏)1−𝑡6𝑓))𝑔(𝑏)(𝑏)𝑔)≤𝑡((1−𝑡𝑎+𝑡𝑏+𝑓((1−𝑡𝑎+𝑡𝑏)6𝑓(𝑎)()𝑔(𝑎)(𝑏)+()𝑓(𝑎)(𝑏)𝑔(𝑎+1−𝑡+𝑡𝑔1−𝑡+𝑡𝑓)1−𝑡6)𝑓(𝑎)(𝑏𝑔(𝑏)(𝑏)()𝑔(𝑎)(𝑏)=𝑡(1−𝑡+𝑡𝑓)+𝑓1−𝑡+𝑡𝑔62()𝑓(𝑎)𝑔(𝑎)(+1−𝑡+𝑡𝑁𝑎,𝑏)1−𝑡6(𝑏)𝑔(𝑏)+()𝑁(=𝑡()2𝑡𝑓1−𝑡𝑎,𝑏)1−𝑡3𝑀()+()𝑎,𝑏1−𝑡²+𝑡²6𝑁(𝑎,𝑏), (11) we have 𝐿(𝑡)≤1−𝑡+𝑡²3𝑀()+()𝑡𝑎,𝑏1−𝑡3𝑁()+𝑡()𝑎,𝑏1−𝑡3𝑀()+()𝑎,𝑏1−𝑡²+𝑡²6𝑁()=1𝑎,𝑏3𝑀()+1𝑎,𝑏6𝑁(𝑎,𝑏), (12) which completes the proof.

Remark 4. Applying Theorem 3 for 𝐿(0)=𝐿(1)=(1/3)𝑀(𝑎,𝑏)+(1/6)𝑁(𝑎,𝑏), we get the result in Theorem 1. So, we conclude that our results give an improvement of Theorem 1.

Corollary 5. With notations above, one has the following inequality: 1𝑏−𝑎^𝑏_𝑎𝑓(𝑥)𝑔(𝑥)𝑑𝑥≤inf_0≤𝑡≤1𝐿(𝑡)≤sup_0≤𝑡≤1𝐿(𝑡)≤13𝑀()+1𝑎,𝑏6𝑁(𝑎,𝑏), (13) where 𝐿(𝑡) is defined in Theorem 3.

Theorem 6. Let 𝑓 and 𝑔 be real-valued, nonnegative, and convex functions on [𝑎,𝑏]. Then, 2𝑓𝑎+𝑏2𝑔𝑎+𝑏2≤1𝑏−𝑎^𝑏_𝑎𝑓(𝑥)𝑔(𝑥)+1𝑑𝑥𝑁12𝑎,𝑎+𝑏2+𝑁𝑎+𝑏2+1,𝑎6𝑁𝑎+𝑏2,𝑎+𝑏2(≤1+𝑁𝑎,𝑏)𝑏−𝑎^𝑏_𝑎𝑓(𝑥)𝑔(𝑥)+1𝑑𝑥6𝑀()+1𝑎,𝑏3𝑁(𝑎,𝑏), (14) where 𝑀(𝑎,𝑏)=𝑓(𝑎)𝑔(𝑎)+𝑓(𝑏)𝑔(𝑏)  and  𝑁(𝑎,𝑏)=𝑓(𝑎)𝑔(𝑏)+𝑓(𝑎)𝑔(𝑏).

Proof. Since 𝑓 and 𝑔 are convex on [𝑎,𝑏] and nonnegative, for 𝑡∈[0,1], we observe that 2𝑓𝑎+𝑏2𝑔𝑎+𝑏2𝑡))+()𝑏=2𝑓((𝑎+𝑏/21−𝑡2+(1−𝑡)𝑎+𝑡((𝑎+𝑏)/2)2𝑡))+()𝑏×𝑔((𝑎+𝑏/21−𝑡2+(1−𝑡)𝑎+𝑡((𝑎+𝑏)/2)2≤12𝑓𝑡𝑎+𝑏2+(1−𝑡)𝑏+𝑓(1−𝑡)𝑎+𝑡𝑎+𝑏2×𝑔𝑡𝑎+𝑏2+(1−𝑡)𝑏+𝑔(1−𝑡)𝑎+𝑡𝑎+𝑏2≤12𝑓𝑡𝑎+𝑏2+(1−𝑡)𝑏𝑔𝑡𝑎+𝑏2+()𝑏()1−𝑡+𝑓1−𝑡𝑎+𝑡𝑎+𝑏2𝑔(1−𝑡)𝑎+𝑡𝑎+𝑏2+12𝑡𝑓𝑎+𝑏2+()𝑓(𝑏)×()𝑔(𝑎)1−𝑡1−𝑡+𝑡𝑔𝑎+𝑏2+(1−𝑡)𝑓(𝑎)+𝑡𝑓𝑎+𝑏2×𝑡𝑔𝑎+𝑏2+()𝑔(𝑏)=11−𝑡2𝑓𝑡𝑎+𝑏2+(1−𝑡)𝑏𝑔𝑡𝑎+𝑏2+()𝑏()1−𝑡+𝑓1−𝑡𝑎+𝑡𝑎+𝑏2𝑔(1−𝑡)𝑎+𝑡𝑎+𝑏2+12𝑡(1−𝑡)𝑓𝑎+𝑏2𝑔(𝑎)+𝑡²𝑓𝑎+𝑏2𝑔𝑎+𝑏2+(1−𝑡)²𝑓(𝑏)𝑔(𝑎)+(1−𝑡)𝑡𝑓(𝑏)𝑔𝑎+𝑏2+(1−𝑡)²𝑓(𝑎)𝑔(𝑏)+(1−𝑡)𝑡𝑓(𝑎)𝑔𝑎+𝑏2+𝑡(1−𝑡)𝑓𝑎+𝑏2𝑔(𝑏)+𝑡²𝑓𝑎+𝑏2𝑔𝑎+𝑏2=12𝑓𝑡𝑎+𝑏2+(1−𝑡)𝑏𝑔𝑡𝑎+𝑏2+()𝑏()1−𝑡+𝑓1−𝑡𝑎+𝑡𝑎+𝑏2𝑔(1−𝑡)𝑎+𝑡𝑎+𝑏2+12𝑡(1−𝑡)𝑁𝑎,𝑎+𝑏2+𝑡²𝑁𝑎+𝑏2,𝑎+𝑏2+(1−𝑡)²𝑁(𝑎,𝑏)+𝑡(1−𝑡)𝑁𝑎+𝑏2,𝑏. (15)
Integrating both sides of the previous inequality with respect to 𝑡 over [0,1], we obtain 2𝑓𝑎+𝑏2𝑔𝑎+𝑏2≤12¹₀𝑓𝑡𝑎+𝑏2+()𝑏𝑡1−𝑡×𝑔𝑎+𝑏2+()𝑏+1−𝑡𝑑𝑡₁₀𝑓(1−𝑡)𝑎+𝑡𝑎+𝑏2×𝑔(1−𝑡)𝑎+𝑡𝑎+𝑏2+1𝑑𝑡216𝑁𝑎,𝑎+𝑏2+13𝑁𝑎+𝑏2,𝑎+𝑏2+13𝑁()+1𝑎,𝑏6𝑁𝑎+𝑏2=1,𝑏22𝑏−𝑎^𝑏_{()𝑎+𝑏/2}𝑓(𝑥)𝑔(𝑥)+2𝑑𝑥𝑏−𝑎_𝑎^{(𝑎+𝑏)/2}𝑓(𝑥)𝑔(𝑥)+1𝑑𝑥216𝑁𝑎,𝑎+𝑏2+13𝑁𝑎+𝑏2,𝑎+𝑏2+13𝑁()+1𝑎,𝑏6𝑁𝑎+𝑏2=1,𝑏𝑏−𝑎^𝑏_𝑎𝑓(𝑥)𝑔(𝑥)+1𝑑𝑥𝑁12𝑎,𝑎+𝑏2+16𝑁𝑎+𝑏2,𝑎+𝑏2+16𝑁()+1𝑎,𝑏𝑁12𝑎+𝑏2,𝑏. (16)
On the other hand, we have 𝑁𝑎,𝑎+𝑏2=𝑓(𝑎)𝑔𝑎+𝑏2+𝑓𝑎+𝑏2𝑔(𝑎)≤12𝑓(𝑎)𝑔(𝑎)(𝑏)+𝑓(𝑎)(𝑏)𝑔(𝑎(𝑎)𝑔(𝑎)+1+𝑔+𝑓)=𝑓2𝑁(𝑎,𝑏). (17)
By a similar way, we obtain 𝑁𝑎+𝑏2(𝑏)𝑔,𝑏=𝑓𝑎+𝑏2+𝑓𝑎+𝑏2𝑔(𝑏)≤12𝑓(𝑏)𝑔(𝑎)(𝑏)+𝑓(𝑎)(𝑏)𝑔(𝑏(𝑏)𝑔(𝑏)+1+𝑔+𝑓)=𝑓2𝑁(),𝑁𝑎,𝑏𝑎+𝑏2,𝑎+𝑏2=2𝑓𝑎+𝑏2𝑔𝑎+𝑏2≤12𝑔(𝑎)(𝑏)𝑓(𝑎)(𝑏)=1+𝑔+𝑓2𝑀()+1𝑎,𝑏2𝑁(𝑎,𝑏). (18)
So, 1𝑁12𝑎,𝑎+𝑏2+16𝑁𝑎+𝑏2,𝑎+𝑏2+16𝑁()+1𝑎,𝑏𝑁12𝑎+𝑏2≤1,𝑏𝑓(𝑎)𝑔(𝑎)+1122𝑁()(𝑏)𝑔(𝑏)+1𝑎,𝑏+𝑓2𝑁(+1𝑎,𝑏)612𝑀()+1𝑎,𝑏2𝑁(+1𝑎,𝑏)6𝑁()=1𝑎,𝑏6𝑀()+1𝑎,𝑏3𝑁(𝑎,𝑏), (19) which yields the desired result.

4. Conclusion

In this paper, we establish some new Hermite-Hadamard type inequalities for products of convex functions. The results obtained in this work are the refinements of the present results. An interesting topic is whether we can use the methods in this paper to refine the Hermite-Hadamard inequality for coordinated convex functions.

Acknowledgment

This work is supported by the Youth Project of Chongqing Three Gorges University of China.