Refinements on the Hermite-Hadamard Inequalities for 𝑟-Convex Functions

Abstract

We give some new generalizations of the well-known Hermite-Hadamard inequality for 𝑟-convex functions.

1. Introduction

Let 𝑓∶𝐼⊂𝑅→𝑅 be 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. Note that some of the classical inequalities for means can be derived from (1) for appropriate particular selections of the mapping 𝑓. Both inequalities hold in the reversed direction if 𝑓 is concave. Some refinements of the Hermite-Hadamard inequality on convex functions have been extensively investigated by a number of authors (e.g., [1–9]). The Hermite-Hadamard inequality was generalized in [10] to an 𝑟-convex positive function which is defined on an interval [𝑎,𝑏]. A positive function 𝑓 is called 𝑟-convex on [𝑎,𝑏], if for each 𝑥, 𝑦∈[𝑎,𝑏] and 𝑡∈[0,1]𝑓𝑡𝑥+(1−𝑡)𝑦≤𝑡𝑓^𝑟(𝑥)+(1−𝑡)𝑓^𝑟𝑦^1/𝑟𝑓(𝑥)^𝑡𝑓𝑦^1−𝑡. (2) It is obvious that 0-convex functions are simply log-convex functions and 1-convex functions are ordinary convex functions. If 𝑓 is a positive 𝑟-concave function, then inequality (2) is reversed. We note that if 𝑓 and 𝑔 are convex and 𝑔 is increasing, then 𝑔∘𝑓 is convex; moreover, since 𝑓=exp(log𝑓), it follows that a log-convex function is convex. This follows directly from (2) because, by the arithmetic-geometric mean inequality, we have 𝑓(𝑥)^𝑡𝑓𝑦^1−𝑡≤𝑡𝑓(𝑥)+(1−𝑡)𝑓𝑦. (3)

In [6], Gill et al. proved the following two theorems.

Theorem 1. Suppose that 𝑓 is a positive 𝑟-convex function on [𝑎,𝑏]. Then 1𝑏−𝑎^𝑏_𝑎𝑓(𝑡)𝑟𝑑𝑡≤𝑓𝑟+1^𝑟+1(𝑏)−𝑓^𝑟+1(𝑎)𝑓^𝑟(𝑏)−𝑓^𝑟(𝑎). (4) If 𝑓 is a positive 𝑟-concave function, then the inequality is reversed.

Theorem 2. Suppose that 𝑓 is a positive log-convex function on [𝑎,𝑏]. Then 1𝑏−𝑎^𝑏_𝑎𝑓(𝑡)𝑓(𝑏)(𝑎)𝑑𝑡≤−𝑓(𝑏)(𝑎).ln𝑓−ln𝑓 (5) If 𝑓 is a positive log-concave function, then the inequality is reversed.

In [11], Sulaiman obtained the following result for 𝑟-convex functions.

Theorem 3. Let 𝑓 be a positive 𝑟-convex function on [𝑎,𝑏], 0<𝑟≤1. Set 𝑤(𝑡)=𝑡𝑓^𝑟(2−𝑡)𝑎+𝑡𝑏2+(1−𝑡)𝑓^𝑟(1−𝑡)𝑎+(1+𝑡)𝑏2^1/𝑟,𝑊(𝑡)=𝑟𝑡𝑓𝑟+1^𝑟+1((1−𝑡)𝑎+𝑡𝑏)−𝑓^𝑟+1(𝑎)𝑓^𝑟((1−𝑡)𝑎+𝑡𝑏)−𝑓^𝑟(𝑎)+()𝑓1−𝑡^𝑟+1(𝑏)−𝑓^𝑟+1((1−𝑡)𝑎+𝑡𝑏)𝑓^𝑟(𝑏)−𝑓^𝑟)).((1−𝑡𝑎+𝑡𝑏 (6) Then, the following inequality holds: 𝑓𝑎+𝑏2(𝑡)≤1≤𝑤𝑏−𝑎^𝑏_𝑎𝑓(𝑥)(𝑡)≤𝑟𝑑𝑥≤𝑊𝑓𝑟+1^𝑟+1(𝑏)−𝑓^𝑟+1(𝑎)𝑓^𝑟(𝑏)−𝑓^𝑟(𝑎)≤𝑓(𝑎)+𝑓(𝑏)2. (7)

In [12], Sulaiman obtained the following result for log-convex functions.

Theorem 4. Assume that 𝑓∶𝐼→𝑅 is an increasing log-convex function. Then for all 𝑡∈[0,1], one has 𝑓𝑎+𝑏2()≤1≤𝑤𝑎,𝑏𝑏−𝑎^𝑏_𝑎𝑓(𝑥)(𝑡)≤𝑓(𝑏)(𝑎)𝑑𝑥≤𝑊−𝑓(𝑏)(𝑎)≤𝑓(𝑎)(𝑏)ln𝑓−ln𝑓+𝑓2, (8) where 𝑤(𝑎,𝑏)=𝑓3𝑎+𝑏4𝑓𝑎+3𝑏4𝑊(𝑡)=()𝑓(()𝑏)(𝑎)1−𝑡𝑡𝑎+1−𝑡−𝑓(()𝑏)(𝑎)𝑓(𝑏)(()𝑏)ln𝑓𝑡𝑎+1−𝑡−ln𝑓+𝑡−𝑓𝑡𝑎+1−𝑡(𝑏)(()𝑏).ln𝑓−ln𝑓𝑡𝑎+1−𝑡 (9)(10)

For recent results and generalizations concerning the Hermite-Hadamard inequality, see [7, 8] and the references given therein.

In this note, we establish some generalizations of the above results for the class of 𝑟-convex functions.

2. Lemmas

Lemma 5. If 𝑎, 𝑏>0, then lim_𝑟→0𝑡𝑎^𝑟+(1−𝑡)𝑏^𝑟1/𝑟=𝑎^𝑡𝑏^1−𝑡. (11)

Remark 6. Applying Lemma 5 and (2), for an 𝑟-convex 𝑓, let 𝑟→0; then 𝑓 is log-convex. Now we let 𝑟→0 in Theorem 1; then we get Theorem 2 from lim_𝑟→0𝑟𝑓𝑟+1^𝑟+1(𝑏)−𝑓^𝑟+1(𝑎)𝑓^𝑟(𝑏)−𝑓^𝑟(𝑎)=𝑓(𝑏)−𝑓(𝑎)(𝑏)(𝑎).ln𝑓−ln𝑓 (12)

Lemma 7. If 0<𝑎<𝑏<𝑐, then 𝑏−𝑎≤ln𝑏−ln𝑎𝑐−𝑎,ln𝑐−ln𝑎𝑐−𝑏≤ln𝑐−ln𝑏𝑐−𝑎.ln𝑐−ln𝑎 (13)

Lemma 8. If 0<𝑎<𝑏<𝑐 and 𝑟>0, then 𝑐^𝑟+1−𝑏^𝑟+1𝑐^𝑟−𝑏^𝑟≤𝑐^𝑟+1−𝑎^𝑟+1𝑐^𝑟−𝑎^𝑟,𝑏^𝑟+1−𝑎^𝑟+1𝑏^𝑟−𝑎^𝑟≤𝑐^𝑟+1−𝑎^𝑟+1𝑐^𝑟−𝑎^𝑟. (14)

Lemma 9. If 𝑎, 𝑏>0, then the following inequality holds: 𝑏−𝑎≤ln𝑏−ln𝑎𝑎+𝑏2. (15)

Lemma 10 (see [11, Theorem  3.1]). Let 𝑓∶[𝑎,𝑏]→(0,∞) be 𝑟-convex on [𝑎,𝑏] and 0<𝑟≤1; then 𝑓𝑎+𝑏2≤1(𝑏−𝑎)^𝑏_𝑎𝑓^𝑟(𝑥)𝑑𝑥^1/𝑟≤1𝑏−𝑎^𝑏_𝑎𝑓(𝑥)≤𝑟𝑑𝑥𝑓𝑟+1^𝑟+1(𝑏)−𝑓^𝑟+1(𝑎)𝑓^𝑟(𝑏)−𝑓^𝑟(𝑎)≤𝑓(𝑎)+𝑓(𝑏)2. (16)

3. Main Results

Theorem 11. Let 𝑓∶[𝑎,𝑏]→(0,∞) be 𝑟-convex and non-decreasing on [𝑎,𝑏] and 0<𝑟≤1; for 𝑛∈𝑁, 𝜆₀=0, 𝜆_𝑛+1=1, and arbitrary 0≤𝜆₁≤𝜆₂≤⋯𝜆_𝑛≤1, the following inequality holds: 𝑓𝑎+𝑏2≤𝑤𝜆₁,…,𝜆_𝑛≤1𝑏−𝑎^𝑏_𝑎𝑓(𝑥)𝜆𝑑𝑥≤𝑊₁,…,𝜆_𝑛≤𝑟𝑓𝑟+1^𝑟+1(𝑏)−𝑓^𝑟+1(𝑎)𝑓^𝑟(𝑏)−𝑓^𝑟(𝑎)≤𝑓(𝑎)+𝑓(𝑏)2, (17) where 𝑤𝜆₁,…,𝜆_𝑛=^𝑛_𝑘=0𝜆_𝑘+1−𝜆_𝑘𝑓^𝑟×2−𝜆_𝑘+1−𝜆_𝑘𝑎+𝜆_𝑘+1+𝜆_𝑘𝑏×(2)^−11/𝑟,𝑊𝜆₁,…,𝜆_𝑛=𝑟×𝑟+1_𝑛𝑘=0𝜆_𝑘+1−𝜆_𝑘×𝑓^𝑟+11−𝜆_𝑘+1𝑎+𝜆_𝑘+1𝑏−𝑓^𝑟+11−𝜆_𝑘𝑎+𝜆_𝑘𝑏×𝑓^𝑟1−𝜆_𝑘+1𝑎+𝜆_𝑘+1𝑏−𝑓^𝑟1−𝜆_𝑘𝑎+𝜆_𝑘𝑏⁻¹. (18)

Proof. Observing that ∑^𝑛_𝑘=0(𝜆_𝑘+1−𝜆_𝑘)=1, ∑^𝑛_𝑘=0(𝜆²_𝑘+1−𝜆²_𝑘)=1 and Jensen^’s inequality for 𝑓^𝑟(𝑥), we have 𝑓^𝑟𝑎+𝑏2=𝑓^𝑟𝑛_𝑘=0𝜆_𝑘+1−𝜆_𝑘×2−𝜆_𝑘+1−𝜆_𝑘𝑎+𝜆_𝑘+1+𝜆_𝑘𝑏2≤^𝑛_𝑘=0𝜆_𝑘+1−𝜆_𝑘×𝑓^𝑟2−𝜆_𝑘+1−𝜆_𝑘𝑎+𝜆_𝑘+1+𝜆_𝑘𝑏2=𝑤^𝑟𝜆₁,…,𝜆_𝑛≤^𝑛_𝑘=0𝜆_𝑘+1−𝜆_𝑘×1𝜆_𝑘+1−𝜆_𝑘()×𝑏−𝑎_{(1−𝜆𝑘+1)𝑎+𝜆𝑘+1𝑏1−𝜆𝑘𝑎+𝜆𝑘𝑏}𝑓^𝑟(𝑥)=𝑑𝑥_𝑛𝑘=01(𝑏−𝑎)^{(1−𝜆𝑘+1)𝑎+𝜆𝑘+1𝑏}_{1−𝜆𝑘𝑎+𝜆𝑘𝑏}𝑓^𝑟(𝑥)=1𝑑𝑥(𝑏−𝑎)^𝑏_𝑎𝑓^𝑟(𝑥)𝑑𝑥, (19) where the second inequality follows from replacing 𝑎 and 𝑏 by (1−𝜆_𝑘)𝑎+𝜆_𝑘𝑏 and (1−𝜆_𝑘+1)𝑎+𝜆_𝑘+1𝑏, respectively, and (1) for 𝑓^𝑟(𝑥). Therefore 𝑓𝑎+𝑏2≤1(𝑏−𝑎)^𝑏_𝑎𝑓^𝑟(𝑥)𝑑𝑥^1/𝑟≤1𝑏−𝑎^𝑏_𝑎𝑓(𝑥)=1𝑑𝑥𝑏−𝑎^𝑛_𝑘=0^{(1−𝜆𝑘+1)𝑎+𝜆𝑘+1𝑏}_{1−𝜆𝑘𝑎+𝜆𝑘𝑏}𝑓(𝑥)=𝑑𝑥_𝑛𝑘=0𝜆_𝑘+1−𝜆_𝑘1𝜆_𝑘+1−𝜆_𝑘)×(𝑏−𝑎_{(1−𝜆𝑘+1)𝑎+𝜆𝑘+1𝑏(1−𝜆𝑘)𝑎+𝜆𝑘𝑏}𝑓(𝑥)≤𝑟𝑑𝑥×𝑟+1_𝑛𝑘=0𝜆_𝑘+1−𝜆_𝑘×𝑓^𝑟+11−𝜆_𝑘+1𝑎+𝜆_𝑘+1𝑏−𝑓^𝑟+11−𝜆_𝑘𝑎+𝜆_𝑘𝑏×𝑓^𝑟1−𝜆_𝑘+1𝑎+𝜆_𝑘+1𝑏−𝑓^𝑟1−𝜆_𝑘𝑎+𝜆_𝑘𝑏)⁻¹=𝑊𝜆₁,…,𝜆_𝑛≤𝑟𝑟+1^𝑛_𝑘=0𝜆_𝑘+1−𝜆_𝑘×𝑓^𝑟+1(𝑏)−𝑓^𝑟+1(𝑎)𝑓^𝑟(𝑏)−𝑓^𝑟(𝑎)=𝑟𝑓𝑟+1^𝑟+1(𝑏)−𝑓^𝑟+1(𝑎)𝑓^𝑟(𝑏)−𝑓^𝑟(𝑎)≤𝑓(𝑎)+𝑓(𝑏)2, (20) where the third inequality follows from replacing 𝑎 and 𝑏 by (1−𝜆_𝑘)𝑎+𝜆_𝑘𝑏 and (1−𝜆_𝑘+1)𝑎+𝜆_𝑘+1𝑏, respectively, and (4). The proof is completed.

Remark 12. Applying Theorem 11 for 𝑛=1, we get Theorem 3.

Corollary 13. With the above notations, if 𝑓∶[𝑎,𝑏]→(0,∞) is 𝑟-convex and nondecreasing on [𝑎,𝑏] and 0<𝑟≤1, one has the following inequality: sup_{0≤𝜆1≤⋯≤𝜆𝑛≤1}𝑤𝜆₁,…,𝜆_𝑛≤1𝑏−𝑎^𝑏_𝑎𝑓(𝑥)𝑑𝑥≤inf_{0≤𝜆1≤⋯≤𝜆𝑛≤1}𝑊𝜆₁,…,𝜆_𝑛, (21) where 𝑊(𝜆₁,…,𝜆_𝑛) and 𝑤(𝜆₁,…,𝜆_𝑛) are defined in Theorem 11.

Theorem 14. Let 𝑓∶[𝑎,𝑏]→(0,∞) be log-convex and nondecreasing on [𝑎,𝑏]; for 𝑛∈𝑁, 𝜆₀=0, 𝜆_𝑛+1=1, and arbitrary 0≤𝜆₁≤𝜆₂≤⋯𝜆_𝑛≤1, the following inequality holds: 𝑓𝑎+𝑏2(𝑡)≤1≤𝑚𝑏−𝑎^𝑏_𝑎𝑓(𝑥)𝑑𝑥≤𝑀𝜆₁,…,𝜆_𝑛≤𝑓(𝑏)−𝑓(𝑎)(𝑏)(𝑎)≤𝑓(𝑎)(𝑏)ln𝑓−ln𝑓+𝑓2, (22) where 𝑚(𝑡)=𝑓(2−𝑡)𝑎+𝑡𝑏2^𝑡𝑓(1−𝑡)𝑎+(1+𝑡)𝑏2^1−𝑡,𝑀𝜆₁,…,𝜆_𝑛=^𝑛_𝑘=0𝜆_𝑘+1−𝜆_𝑘×𝑓1−𝜆_𝑘+1𝑎+𝜆_𝑘+1𝑏−𝑓1−𝜆_𝑘𝑎+𝜆_𝑘𝑏×ln𝑓1−𝜆_𝑘+1𝑎+𝜆_𝑘+1𝑏−ln𝑓1−𝜆_𝑘𝑎+𝜆_𝑘𝑏⁻¹. (23)(24)

Proof. Let 𝑓 be a positive 𝑟-convex function on [𝑎,𝑏], 0<𝑟≤1, by Theorem 3: then 𝑤(𝑡)=𝑡𝑓^𝑟(2−𝑡)𝑎+𝑡𝑏2+(1−𝑡)𝑓^𝑟(1−𝑡)𝑎+(1+𝑡)𝑏2^1/𝑟≤1𝑏−𝑎^𝑏_𝑎𝑓(𝑥)𝑑𝑥. (25) Applying Lemma 5, let 𝑟→0; then 𝑓 is log-convex and lim_𝑟→0𝑤(𝑡)(𝑡)≤1=𝑚𝑏−𝑎^𝑏_𝑎𝑓(𝑥)𝑓𝑑𝑥.𝑎+𝑏2𝑡()=𝑓2−𝑡𝑎+𝑡𝑏2+()×()()𝑏1−𝑡1−𝑡𝑎+1+𝑡2≤𝑓(2−𝑡)𝑎+𝑡𝑏2^𝑡×𝑓(1−𝑡)𝑎+(1+𝑡)𝑏2^1−𝑡=𝑚(𝑡). (26) Observing that ∑^𝑛_𝑘=0(𝜆_𝑘+1−𝜆_𝑘)=1, 1(𝑏−𝑎)^𝑏_𝑎𝑓(𝑥)=1𝑑𝑥𝑏−𝑎^𝑛_𝑘=0^{(1−𝜆𝑘+1)𝑎+𝜆𝑘+1𝑏}_{1−𝜆𝑘𝑎+𝜆𝑘𝑏}𝑓(𝑥)=𝑑𝑥_𝑛𝑘=0𝜆_𝑘+1−𝜆_𝑘1𝜆_𝑘+1−𝜆_𝑘()×𝑏−𝑎_{(1−𝜆𝑘+1)𝑎+𝜆𝑘+1𝑏(1−𝜆𝑘)𝑎+𝜆𝑘𝑏}𝑓(𝑥)≤𝑑𝑥_𝑛𝑘=0𝜆_𝑘+1−𝜆_𝑘×𝑓1−𝜆_𝑘+1𝑎+𝜆_𝑘+1𝑏−𝑓1−𝜆_𝑘𝑎+𝜆_𝑘𝑏×ln𝑓1−𝜆_𝑘+1𝑎+𝜆_𝑘+1𝑏−ln𝑓1−𝜆_𝑘𝑎+𝜆_𝑘𝑏)⁻¹=𝑀𝜆₁,…,𝜆_𝑛≤^𝑛_𝑘=0𝜆_𝑘+1−𝜆_𝑘𝑓(𝑏)−𝑓(𝑎)(𝑏)(𝑎)=𝑓(𝑏)(𝑎)ln𝑓−ln𝑓−𝑓(𝑏)(𝑎)≤𝑓(𝑎)(𝑏)ln𝑓−ln𝑓+𝑓2, (27) where the first inequality follows from replacing 𝑎 and 𝑏 by (1−𝜆_𝑘)𝑎+𝜆_𝑘𝑏 and (1−𝜆_𝑘+1)𝑎+𝜆_𝑘+1𝑏, respectively, and (5). The proof is completed.

Remark 15. Applying Theorem 14 for 𝑡=1/2 and 𝑛=1, 𝑚(1/2)=𝑤(𝑎,𝑏) = 𝑓((3𝑎+𝑏)/4)𝑓((𝑎+3𝑏)/4), we get Theorem 4.

Corollary 16. With the above notations, suppose that 𝑓∶[𝑎,𝑏]→(0,∞) is log-convex and nondecreasing on [𝑎,𝑏]; one has the following inequality: sup_0≤𝑡≤1𝑚(𝑡)≤1𝑏−𝑎^𝑏_𝑎𝑓(𝑥)𝑑𝑥≤inf_{0≤𝜆1≤⋯≤𝜆𝑛≤1}𝑀𝜆₁,…,𝜆_𝑛, (28) where 𝑚(𝑡) and 𝑀(𝜆₁,…,𝜆_𝑛) are defined in Theorem 14.

Acknowledgments

This work is supported by the Natural Science Foundation Project of “CQ CSTC” (no. cstc2012jjA00013) and Youth Project of Chongqing Three Gorges University of China.