New Hermite-Hadamard Type Inequalities for 𝑛-Times Differentiable and 𝑠-Logarithmically Preinvex Functions

Abstract

The concept of s-logarithmically preinvex function is introduced, and by creating an integral identity involving an n-times differentiable function, some new Hermite-Hadamard type inequalities for s-logarithmically preinvex functions are established.

1. Introduction

Throughout this paper, let ℝ=(−∞,∞), ℝ₊=(0,∞), ℕ denote the set of all positive integers, 𝐼 denote the interval in ℝ, and 𝐴 denote the set in ℝ^𝑛,𝑛∈ℕ.

Let us recall some definitions of various convex functions.

Definition 1. A function 𝑓∶𝐼⊆ℝ→ℝ is said to be convex if 𝑓𝜆𝑥+(1−𝜆)𝑦≤𝜆𝑓(𝑥)+(1−𝜆)𝑓𝑦 (1) holds for all 𝑥,𝑦∈𝐼 and 𝜆∈[0,1]. If inequality (1) reverses, then 𝑓 is said to be concave on 𝐼.

Definition 2 (see [1]). A set 𝐴⊆ℝ^𝑛 is said to be invex with respect to the map 𝜂∶𝐴×𝐴→ℝ^𝑛, if for every 𝑥,𝑦∈𝐴 and 𝑡∈[0,1]𝑦+𝑡𝜂𝑥,𝑦∈𝐴. (2) The invex set 𝐴 is also called a 𝜂-connected set.

It is obvious that every convex set is invex with respect to the map 𝜂(𝑥,𝑦)=𝑥−𝑦, but there exist invex sets which are not convex (see [1], e.g.).

Definition 3 (see [1]). Let 𝐴⊆ℝ^𝑛 be an invex set with respect to 𝜂∶𝐴×𝐴→ℝ^𝑛. For every 𝑥,𝑦∈𝐴, the 𝜂-path 𝑃_𝑥𝑣 joining the points 𝑥 and 𝑣=𝑥+𝜂(𝑦,𝑥) is defined by 𝑃_𝑥𝑣=𝑧∣𝑧=𝑥+𝑡𝜂𝑦,𝑥,𝑡∈[0,1]. (3)

Definition 4 (see [2]). Let 𝐴⊆ℝ^𝑛 be an invex set with respect to 𝜂∶𝐴×𝐴→ℝ^𝑛. A function 𝑓∶𝐴→ℝ is said to be preinvex with respect to 𝜂, if for every 𝑥,𝑦∈𝐴 and 𝑡∈[0,1]𝑓𝑦+𝑡𝜂𝑥,𝑦≤𝑡𝑓(𝑥)+(1−𝑡)𝑓𝑦. (4) The function 𝑓 is said to be preincave if and only if −𝑓 is preinvex.

Every convex function is preinvex with respect to the map 𝜂(𝑥,𝑦)=𝑥−𝑦, but not conversely (see [2], e.g.).

Definition 5 (see [3]). Let 𝐴⊆ℝ^𝑛 be an invex set with respect to 𝜂∶𝐴×𝐴→ℝ^𝑛. The function 𝑓∶𝐴→ℝ₊ on the set 𝐴 is said to be logarithmically preinvex with respect to 𝜂, if for every 𝑥,𝑦∈𝐴 and 𝑡∈[0,1]𝑓𝑦+𝑡𝜂𝑥,𝑦≤𝑓(𝑥)^𝑡𝑓(𝑦)^1−𝑡. (5)

For properties and applications of preinvex and logarithmically preinvex functions, please refer to [1–8] and closely related references therein.

The most important inequality in the theory of convex functions, the well known Hermite-Hadamard’s integral inequality, may be stated as follows. Let 𝐼⊆ℝ and 𝑎,𝑏∈𝐼 with 𝑎<𝑏. If 𝑓∶[𝑎,𝑏]⊆𝐼→ℝ is a convex function on [𝑎,𝑏], then 𝑓𝑎+𝑏2≤1𝑏−𝑎^𝑏_𝑎𝑓(𝑥)𝑓(𝑎)(𝑏)𝑑𝑥≤+𝑓2. (6) If 𝑓 is concave on [𝑎,𝑏], then inequality (6) is reversed.

Inequality (6) has been generalized by many mathematicians. Some of them may be recited as follows.

Theorem 6 (see [9, Theorem 2.2]). Let 𝑓∶𝐼⊆ℝ→ℝ be a differentiable mapping on 𝐼 and 𝑎,𝑏∈𝐼 with 𝑎<𝑏. If |𝑓^′(𝑥)| is convex on [𝑎,𝑏], then 𝑓(𝑎)+𝑓(𝑏)2−1𝑏−𝑎^𝑏_𝑎𝑓(𝑥)≤(𝑓𝑑𝑥𝑏−𝑎)^′(𝑎)+𝑓^′(𝑏)8. (7)

Theorem 7 (see [10, Theorem 1]). Let 𝐼⊆ℝ and 𝑎,𝑏∈𝐼 with 𝑎<𝑏. If 𝑓∶[𝑎,𝑏]⊆𝐼→ℝ is differentiable on [𝑎,𝑏] such that |𝑓^′(𝑥)|^𝑞 is a convex function on [𝑎,𝑏] for 𝑞≥1, then 𝑓(𝑎)+𝑓(𝑏)2−1𝑏−𝑎^𝑏_𝑎𝑓(𝑥)≤𝑑𝑥𝑏−𝑎4𝑓^′(𝑎)^𝑞+𝑓^′(𝑏)^𝑞2^1/𝑞. (8)

Theorem 8 (see [11, Theorem 2.3]). Let 𝑓∶𝐼⊆ℝ→ℝ be differentiable on 𝐼, 𝑎,𝑏∈𝐼 with 𝑎<𝑏 and 𝑝>1. If |𝑓^′(𝑥)|^{𝑝/(𝑝−1)} is convex on [𝑎,𝑏], then 𝑓𝑎+𝑏2−1𝑏−𝑎^𝑏_𝑎𝑓(𝑥)≤𝑑𝑥𝑏−𝑎416𝑝+1^1/𝑝×𝑓^′(𝑎)^{𝑝/(𝑝−1)}+3𝑓^′(𝑏)^{𝑝/(𝑝−1)1−1/𝑝}+3𝑓^′(𝑎)^{𝑝/(𝑝−1)}+𝑓^′(𝑏)^{𝑝/(𝑝−1)1−1/𝑝}. (9)

Theorem 9 (see [6]). Let 𝐴⊆ℝ be an open invex set with respect to 𝜂∶𝐴×𝐴→ℝ and 𝑎,𝑏∈𝐴 with 𝜂(𝑎,𝑏)>0 for all 𝑎≠𝑏. If 𝑓∶𝐴→ℝ₊ is a preinvex function on 𝐴, then the following inequality holds: 𝑓2𝑏+𝜂(𝑎,𝑏)2≤1𝜂(𝑎,𝑏)_𝑏^{𝑏+𝜂(𝑎,𝑏)}𝑓(𝑥)≤𝑓(𝑎)(𝑏)𝑑𝑥+𝑓2. (10)

Theorem 10 (see [4, Theorem 4.3]). Let 𝐴⊆ℝ be an open invex set with respect to 𝜂∶𝐴×𝐴→ℝ and 𝑎,𝑏∈𝐴 with 𝜂(𝑎,𝑏)>0 for all 𝑎≠𝑏. Suppose that 𝑓∶𝐴→ℝ is a twice differentiable function on 𝐴 and |𝑓^′′(𝑥)| is preinvex on 𝐴. If 𝑞>1 and 𝑓^′′ is integrable on the 𝜂-path  𝑃_𝑏𝑐  for 𝑐=𝑏+𝜂(𝑎,𝑏), then 𝑓(𝑏)+𝑓𝑏+𝜂(𝑎,𝑏)2−1𝜂(𝑎,𝑏)_𝑏^{𝑏+𝜂(𝑎,𝑏)}𝑓(𝑥)≤𝑑𝑥𝜂(𝑎,𝑏)²1122^1/𝑞𝑓^′′(𝑎)^𝑞+𝑓^′′(𝑏)^𝑞1/𝑞. (11)

Theorem 11 (see [12, Theorem 3.1]). For 𝑛∈ℕ and 𝑛≥2, let 𝐴⊆ℝ be an open invex set with respect to 𝜂∶𝐴×𝐴→ℝ and 𝑎,𝑏∈𝐴 with 𝜂(𝑎,𝑏)>0 for all 𝑎≠𝑏. Suppose that 𝑓∶𝐴→ℝ is an 𝑛-times differentiable function on 𝐴 and 𝑓^(𝑛) is integrable on the 𝜂-path 𝑃_𝑏𝑐 for 𝑐=𝑏+𝜂(𝑎,𝑏). If |𝑓^(𝑛)|^𝑞 is preinvex on 𝐴 for 𝑞≥1, then 𝑓(𝑏)+𝑓𝑏+𝜂(𝑎,𝑏)2−1𝜂(𝑎,𝑏)_𝑏^{𝑏+𝜂(𝑎,𝑏)}𝑓(𝑥)+𝑑𝑥_{𝑛−1𝑘=1}𝜂(𝑎,𝑏)^𝑘(1−𝑘)4)!]×𝑓[(𝑘+1_(𝑘)(𝑏)+(−1)^𝑘𝑓^(𝑘)()≤𝑏+𝜂𝑎,𝑏𝜂(𝑎,𝑏)^𝑛(𝑛−1)^1−1/𝑞4[(𝑛+1)!](𝑛+2)^1/𝑞×𝑛𝑓^(𝑛)(𝑎)^𝑞+(𝑛²−2)𝑓^(𝑛)(𝑏)^𝑞1/𝑞+(𝑛²−2)𝑓^(𝑛)(𝑎)^𝑞+𝑛𝑓^(𝑛)(𝑏)^𝑞1/𝑞. (12)

Theorem 12 (see [13, Theorem 5]). For 𝑛∈ℕ and 𝑛≥2, let 𝐴⊆ℝ be an open invex set with respect to 𝜂∶𝐴×𝐴→ℝ and 𝑎,𝑏∈𝐴 with 𝜂(𝑎,𝑏)>0 for all 𝑎≠𝑏. Suppose that 𝑓∶𝐴→ℝ is a function such that 𝑓^(𝑛) exits on 𝐴 and 𝑓^(𝑛) is integrable on [𝑎,𝑎+𝜂(𝑏,𝑎)]. If |𝑓^(𝑛)|^𝑞 is logarithmically preinvex on  𝐴  for 𝑞≥1, then we have the inequality 𝑓(𝑎)+𝑓𝑎+𝜂(𝑏,𝑎)2−1𝜂(𝑏,𝑎)_𝑎^{𝑎+𝜂(𝑏,𝑎)}𝑓(𝑥)−𝑑𝑥_{𝑛−1𝑘=2}(−1)^𝑘(𝑘−1)𝜂(𝑏,𝑎)^𝑘2()!𝑓𝑘+1^(𝑘)()≤𝑎+𝜂𝑏,𝑎𝜂(𝑏,𝑎)^𝑛2𝑛!𝑛−1𝑛+1^1−1/𝑞𝐸₁(𝑛,𝑞)^1/𝑞, (13) where 𝐸₁=()𝑛,𝑞−1^𝑛𝑛!𝑞ln𝑓^(𝑛)(𝑏)−ln𝑓^(𝑛)(𝑎)+2𝑓^(𝑛)(𝑎)^𝑞𝑞^𝑛+1ln𝑓^(𝑛)(𝑏)−ln𝑓^(𝑛)(𝑎)^𝑛+1−2𝑓^(𝑛)(𝑏)^𝑞𝑞ln𝑓^(𝑛)(𝑏)−ln𝑓^(𝑛)(𝑎𝑓)−𝑛!^(𝑛)(𝑏)^𝑞×^𝑛_𝑘=1(−1)^𝑘𝑞ln𝑓^(𝑛)(𝑏)−ln𝑓^(𝑛)(𝑎)+2(𝑛−𝑘)!𝑞^𝑘+1ln𝑓^(𝑛)(𝑏)−ln𝑓^(𝑛)(𝑎)^𝑘+1. (14)

Recently, some related inequalities for preinvex functions were also obtained in [14, 15].

In the paper, the concept of 𝑠-logarithmically preinvex function is introduced, and by creating an integral identity involving an 𝑛-times differentiable function, some new Hermite-Hadamard type inequalities for 𝑠-logarithmically preinvex functions are established which generalize some known results.

2. New Definition and Lemmas

Now we introduce concepts of 𝑠-logarithmically preinvex functions.

Definition 13. Let 𝐴⊆ℝ^𝑛 be an invex set with respect to 𝜂∶𝐴×𝐴→ℝ^𝑛. The function 𝑓∶𝐴→ℝ₊ on the set 𝐴 is said to be 𝑠-logarithmically preinvex with respect to 𝜂, if for every 𝑥,𝑦∈𝐴,𝑡∈[0,1], and some 𝑠∈(0,1]𝑓𝑦+𝑡𝜂𝑥,𝑦≤𝑓(𝑥)^𝑡𝑠𝑓(𝑦)^{(1−𝑡)𝑠}. (15)
Clearly, when taking 𝑠=1 in (15), then 𝑓 becomes the standard logarithmically convex function on 𝐴.
In order to obtain our main results, we need the following lemmas.

Lemma 14. For 𝑛∈ℕ, let 𝐴⊆ℝ be an open invex set with respect to 𝜂∶𝐴×𝐴→ℝ and 𝑎,𝑏∈𝐴 with 𝜂(𝑎,𝑏)>0  for all 𝑎≠𝑏. If 𝑓∶𝐴→ℝ is an 𝑛-times differentiable function on 𝐴 and 𝑓^(𝑛) is integrable on the 𝜂-path 𝑃_𝑏𝑐  for 𝑐=𝑏+𝜂(𝑎,𝑏), then 𝑓(𝑏)+𝑓𝑏+𝜂(𝑎,𝑏)2−1𝜂(𝑎,𝑏)_𝑏^{𝑏+𝜂(𝑎,𝑏)}𝑓(𝑥)+𝑑𝑥_𝑛𝑘=21()×)2𝑘!𝜂𝑎,𝑏(𝑏−𝑥^𝑘−12𝑥−2𝑏−𝑘𝜂(𝑎,𝑏)+𝑏+𝜂(𝑎,𝑏)−𝑥^𝑘−1×2𝑏+2𝜂(𝑎,𝑏)−2𝑥−𝑘𝜂(𝑎,𝑏)𝑓^(𝑘−1)(𝑥)=𝜂(𝑎,𝑏)^𝑛×2𝑛!_{0(𝑥−𝑏)/(𝜂(𝑎,𝑏))}(−𝑡)^𝑛−1(2𝑡−𝑛)𝑓^(𝑛)()+𝑏+𝑡𝜂𝑎,𝑏𝑑𝑡_{1()𝑥−𝑏/(𝜂(𝑎,𝑏))}(1−𝑡)^𝑛−1(2𝑡+𝑛−2)×𝑓^(𝑛)𝑏+𝑡𝜂(𝑎,𝑏)𝑑𝑡, (16) where 𝑥∈[𝑏,𝑏+𝜂(𝑏,𝑎)] and the above summation is zero for 𝑛=1.

Proof. Since 𝑎,𝑏∈𝐴 and 𝐴 is an invex set with respect to 𝜂, for every 𝑡∈[0,1], we have 𝑏+𝑡𝜂(𝑎,𝑏)∈𝐴. When 𝑛=1, by integrating by part in the right-hand side of (16), one gives 𝜂(𝑎,𝑏)2¹₀(2𝑡−1)𝑓^′()=𝑓(𝑏)(𝑏+𝑡𝜂𝑎,𝑏𝑑𝑡+𝑓𝑏+𝜂𝑎,𝑏)2−1𝜂(𝑎,𝑏)_𝑏^{𝑏+𝜂(𝑎,𝑏)}𝑓(𝑥)𝑑𝑥. (17) Hence, the identity (16) holds for 𝑛=1.
When 𝑛=𝑚 and 𝑚≥1, suppose that the identity (16) is valid.
When 𝑛=𝑚+1, by the hypothesis, we have 𝜂(𝑎,𝑏)^𝑚+12()!×𝑚+1_{0(𝑥−𝑏)/(𝜂(𝑎,𝑏))}(−𝑡)^𝑚(2𝑡−𝑚−1)×𝑓^(𝑚+1)()+𝑏+𝑡𝜂𝑎,𝑏𝑑𝑡_{1(𝑥−𝑏)/(𝜂(𝑎,𝑏))}(1−𝑡)^𝑚(2𝑡+𝑚−1)×𝑓^(𝑚+1)()=𝜂(𝑏+𝑡𝜂𝑎,𝑏𝑑𝑡𝑎,𝑏)^𝑚2()!×𝑚+1𝑏−𝑥𝜂(𝑎,𝑏)^𝑚2𝑥−2𝑏𝜂()𝑓𝑎,𝑏−𝑚−1^(𝑚)(𝑥)−₀^{(𝑥−𝑏)/(𝜂(𝑎,𝑏))}(−𝑡)^𝑚−1(𝑚+1)(𝑚−2𝑡)×𝑓^(𝑚)()−()𝑏+𝑡𝜂𝑎,𝑏𝑑𝑡𝑏+𝜂𝑎,𝑏−𝑥𝜂(𝑎,𝑏)^𝑚2𝑥−2𝑏𝜂()𝑓𝑎,𝑏+𝑚−1^(𝑚)(𝑥)−¹_{(𝑥−𝑏)/(𝜂(𝑎,𝑏))}(1−𝑡)^𝑚−1()×()𝑓𝑚+12−2𝑡−𝑚^(𝑚)()=1𝑏+𝑡𝜂𝑎,𝑏𝑑𝑡2()()×)𝑚+1!𝜂𝑎,𝑏(𝑏−𝑥^𝑚()𝜂(−()2𝑥−2𝑏−𝑚+1𝑎,𝑏)𝑏+𝜂𝑎,𝑏−𝑥^𝑚2𝑥−2𝑏+(𝑚−1)𝜂(𝑎,𝑏)×𝑓^(𝑚)(𝑥)+𝜂(𝑎,𝑏)^𝑚×2𝑚!_{0(𝑥−𝑏)/(𝜂(𝑎,𝑏))}(−𝑡)^𝑚−1(2𝑡−𝑚)𝑓^(𝑚)()+𝑏+𝑡𝜂𝑎,𝑏𝑑𝑡_{1()𝑥−𝑏/(𝜂(𝑎,𝑏))}(1−𝑡)^𝑚−1(2𝑡+𝑚−2)×𝑓^(𝑚)(=𝑓(𝑏)()𝑏+𝑡𝜂𝑎,𝑏)𝑑𝑡+𝑓𝑏+𝜂𝑎,𝑏2−1𝜂(𝑎,𝑏)_𝑏^{𝑏+𝜂(𝑎,𝑏)}𝑓(𝑥)+𝑑𝑥_𝑚+1𝑘=21()×)2𝑘!𝜂𝑎,𝑏(𝑏−𝑥^𝑘−1()+()2𝑥−2𝑏−𝑘𝜂𝑎,𝑏𝑏+𝜂𝑎,𝑏−𝑥^𝑘−12𝑏+2𝜂(𝑎,𝑏)−2𝑥−𝑘𝜂(𝑎,𝑏)×𝑓^(𝑘−1)(𝑥). (18)
Therefore, when 𝑛=𝑚+1, the identity (16) holds. By induction, the proof of Lemma 14 is complete.

Remark 15. Under the conditions of Lemma 14, we have 𝑓(𝑏)+𝑓𝑏+𝜂(𝑎,𝑏)2−1𝜂(𝑎,𝑏)_𝑏^{𝑏+𝜂(𝑎,𝑏)}𝑓(𝑥)+𝑑𝑥_𝑛𝑘=2𝜂(𝑎,𝑏)^𝑘−1(2−𝑘)𝑓2𝑘!^(𝑘−1)(𝑏)=𝜂(𝑎,𝑏)^𝑛2𝑛!¹₀(1−𝑡)^𝑛−1(2𝑡+𝑛−2)×𝑓^(𝑛)𝑏+𝑡𝜂(𝑎,𝑏)𝑑𝑡,𝑓(𝑏)+𝑓𝑏+𝜂(𝑎,𝑏)2−1𝜂(𝑎,𝑏)_𝑏^{𝑏+𝜂(𝑎,𝑏)}𝑓(𝑥)+𝑑𝑥_𝑛𝑘=2𝜂(𝑎,𝑏)^𝑘−1(1−𝑘)1+(−1)^𝑘−12^𝑘𝑘!×𝑓^(𝑘−1)𝜂()𝑏+𝑎,𝑏2=𝜂(𝑎,𝑏)^𝑛×2𝑛!_01/2(−𝑡)^𝑛−1(2𝑡−𝑛)𝑓^(𝑛)𝑏+𝑡𝜂(𝑎,𝑏)𝑑𝑡+¹_1/2(1−𝑡)^𝑛−1(2𝑡+𝑛−2)𝑓^(𝑛)𝑏+𝑡𝜂(𝑎,𝑏)𝑑𝑡,𝑓(𝑏)+𝑓𝑏+𝜂(𝑎,𝑏)2−1𝜂(𝑎,𝑏)_𝑏^{𝑏+𝜂(𝑎,𝑏)}𝑓(𝑥)+𝑑𝑥_𝑛𝑘=2−𝜂(𝑎,𝑏)^𝑘−1(2−𝑘)𝑓2𝑘!^(𝑘−1)()=𝜂(𝑏+𝜂𝑎,𝑏𝑎,𝑏)^𝑛2𝑛!¹₀(−𝑡)^𝑛−1(2𝑡−𝑛)𝑓^(𝑛)𝑏+𝑡𝜂(𝑎,𝑏)𝑑𝑡. (19)(20)(21)

Proof. These are special cases of Lemma 14 for 𝑥=𝑏,  𝑏+𝜂(𝑎,𝑏)/2, 𝑏+𝜂(𝑎,𝑏), respectively.

Remark 16. Adding the identities (19) and (21) and then dividing by 2 result in Lemma 14 from [12].

Lemma 17. Let 𝜇>0 and 𝑥≥0. Then 𝐸𝑛;𝜇,𝑥≜^𝑥₀𝑡^𝑛𝜇^𝑡=()𝑑𝑡−1^𝑛+1𝑛!ln𝜇^𝑛+1+𝑛!𝜇^𝑥𝑛_𝑘=0(−1)^𝑘𝑥^𝑛−𝑘(𝑛−𝑘)!ln𝜇^𝑘+1𝑥,𝜇≠1,_𝑛+1𝑛+1,𝜇=1 (22) for 𝑛≥0, 𝑛∈ℕ.

Proof. When 𝜇=1, the proof is straightforward.
When 𝜇≠1, for 𝑛=0, we have ^𝑥₀𝜇^𝑡𝜇𝑑𝑡=^𝑥−1,ln𝜇 (23) which coincides with the right-hand side of (22) for 𝑛=0.
For 𝑛=1, we get ^𝑥₀𝑡𝜇^𝑡𝑑𝑡=𝑥𝜇^𝑥−1ln𝜇ln𝜇^𝑥₀𝜇^𝑡=1𝑑𝑡ln𝜇²+𝜇^𝑥𝑥−1ln𝜇ln𝜇², (24) which coincides with the right-hand side of (22) for 𝑛=1.
Suppose that (22) is true for 𝑛=𝑚,  𝑚≥0, then, for 𝑛=𝑚+1, it follows that ^𝑥₀𝑡^𝑚+1𝜇^𝑡=𝑥𝑑𝑡_𝑚+1𝜇^𝑥−ln𝜇𝑚+1ln𝜇^𝑥₀𝑡^𝑚𝜇^𝑡=𝑥𝑑𝑡_𝑚+1𝜇^𝑥−ln𝜇𝑚+1()ln𝜇−1^𝑚+1𝑚!ln𝜇^𝑚+1+𝑚!𝜇^𝑥𝑚_𝑘=0(−1)^𝑘𝑥^𝑚−𝑘(𝑚−𝑘)!ln𝜇^𝑘+1=(−1)^𝑚+2(𝑚+1)!ln𝜇^𝑚+2+𝑥^𝑚+1𝜇^𝑥+()ln𝜇𝑚+1!𝜇^𝑥𝑚+1_𝑘=1(−1)^𝑘𝑥^{𝑚−𝑘+1}(𝑚−𝑘+1)!ln𝜇^𝑘+1=(−1)^𝑚+2(𝑚+1)!ln𝜇^𝑚+2+(𝑚+1)!𝜇^𝑥𝑚+1_𝑘=0(−1)^𝑘𝑥^{𝑚−𝑘+1}(𝑚−𝑘+1)!ln𝜇^𝑘+1. (25) Therefore, when 𝑛=𝑚+1, the identity (22) holds. By induction, the proof of Lemma 17 is complete.

Lemma 18 (see [16]). Let 𝛾>0, 𝜇>0, and 𝑥≥0. Then 𝐺𝛾;𝜇,𝑥≜^𝑥₀𝑡^𝛾−1𝜇^𝑡𝑑𝑡=𝑥^𝛾𝜇^𝑥∞_𝑘=1−𝑥ln𝜇^𝑘−1𝛾_𝑘<∞, (26) where (𝛾)_𝑘=𝛾(𝛾+1)(𝛾+2)⋯(𝛾+𝑘−1).

By Lemmas 17 and 18, a straightforward computation gives the following lemmas.

Lemma 19. Let 𝜇>0 and 𝑥≥0. Then 𝐹=𝑛;𝜇,𝑥≜𝑛𝐸𝑛−1;𝜇,𝑥−2𝐸𝑛;𝜇,𝑥2𝜇^𝑥𝑥^𝑛×()ln𝜇+𝑛!ln𝜇+2−1^𝑛ln𝜇^𝑛+1+𝜇^𝑥𝑛_𝑘=1(−1)^𝑘−1𝑥^𝑛−𝑘(𝑛−𝑘)!ln𝜇^𝑘+1𝑥,𝜇≠1,_𝑛(𝑛+1−2𝑥)𝑛+1,𝜇=1 (27) for 𝑛≥0, 𝑛∈ℕ.

Lemma 20. Let 𝛾>0, 𝜇>0, and 𝑥≥0. Then 𝐻𝛾;𝜇,𝑥≜𝑛𝐺𝛾;𝜇,𝑥−2𝐺𝛾+1;𝜇,𝑥=𝑥^𝛾𝜇^𝑥∞_𝑘=1−𝑥ln𝜇^𝑘−1𝑛𝛾+𝑘−2𝑥𝛾𝛾_𝑘+1, (28) where (𝛾)_𝑘+1=𝛾(𝛾+1)(𝛾+2)⋯(𝛾+𝑘).

Lemma 21 (see [17]). Let 0<𝜙≤1≤𝜓 and 0<𝑡, 𝑠≤1. Then 𝜙^𝑡𝑠≤𝜙^𝑠𝑡,𝜓^𝑡𝑠≤𝜓^{𝑠𝑡+1−𝑠}. (29)

3. Hermite-Hadamard Type Inequalities

Now we start out to establish some new Hermite-Hadamard type inequalities for 𝑛-times differentiable and 𝑠-logarithmically preinvex functions.

Theorem 22. For 𝑛∈ℕ and 𝑛≥2, let 𝐴⊆ℝ be an open invex set with respect to 𝜂∶𝐴×𝐴→ℝ and 𝑎,𝑏∈𝐴 with 𝜂(𝑎,𝑏)>0 for all 𝑎≠𝑏. Suppose that 𝑓∶𝐴→ℝ is an 𝑛-times differentiable function on 𝐴 and 𝑓^(𝑛) is integrable on the 𝜂-path 𝑃_𝑏𝑐 for 𝑐=𝑏+𝜂(𝑎,𝑏). If  |𝑓^(𝑛)|^𝑞 is 𝑠-logarithmically preinvex on 𝐴 for 𝑞≥1, then for 𝑥∈[𝑏,𝑏+𝜂(𝑎,𝑏)] and some 𝑠∈(0,1]𝑓(𝑏)+𝑓𝑏+𝜂(𝑎,𝑏)2−1𝜂(𝑎,𝑏)_𝑏^{𝑏+𝜂(𝑎,𝑏)}𝑓(𝑥)+𝑑𝑥_𝑛𝑘=21()×)2𝑘!𝜂𝑎,𝑏(𝑏−𝑥^𝑘−1(+2𝑥−2𝑏−𝑘𝜂𝑎,𝑏)𝑏+𝜂(𝑎,𝑏)−𝑥^𝑘−12𝑏+2𝜂(𝑎,𝑏)−2𝑥−𝑘𝜂(𝑎,𝑏)×𝑓^(𝑘−1)(𝑥≤)𝜂(𝑎,𝑏)^{(𝑛+1−𝑞)/𝑞}2𝑛!(𝑛+1)^1−1/𝑞×𝑓^(𝑛)(𝑎)^𝛼𝑓^(𝑛)(𝑏)^𝛽×(𝑥−𝑏)^𝑛(𝑛+1)𝜂(𝑎,𝑏)−2𝑥+2𝑏^1−1/𝑞×𝐹𝑛;𝜉^𝑠𝑞,𝑥−𝑏𝜂(𝑎,𝑏)^1/𝑞+𝑓^(𝑛)(𝑎)^𝛿𝑓^(𝑛)(𝑏)^𝜃×𝑏+𝜂(𝑎,𝑏)−𝑥^𝑛(𝑛−1)𝜂(𝑎,𝑏)+2𝑥−2𝑏^1−1/𝑞×𝐹𝑛;𝜉^−𝑠𝑞,𝑏+𝜂(𝑎,𝑏)−𝑥𝜂(𝑎,𝑏)^1/𝑞, (30) where 𝜉=|𝑓^(𝑛)(𝑎)/𝑓^(𝑛)(𝑏)|, 𝐹(𝑛;𝜇,𝑥) is defined in Lemma 19, and ={},𝛼,𝛽0,𝑠if0<𝑓^(𝑛)(𝑎),𝑓^(𝑛)(𝑏{},)≤1,0,1if0<𝑓^(𝑛)(𝑎)<1<𝑓^(𝑛)(𝑏,{},)1−𝑠,𝑠if0<𝑓^(𝑛)(𝑏)<1<𝑓^(𝑛)(𝑎,{},)1−𝑠,1if𝑓^(𝑛)(𝑎),𝑓^(𝑛)(𝑏{}={},)>1,𝛿,𝜃𝑠,0if0<𝑓^(𝑛)(𝑎),𝑓^(𝑛)(𝑏{},)≤1,𝑠,1−𝑠if0<𝑓^(𝑛)(𝑎)<1<𝑓^(𝑛)(𝑏,{},)1,0if0<𝑓^(𝑛)(𝑏)<1<𝑓^(𝑛)(𝑎,{},)1,1−𝑠if𝑓^(𝑛)(𝑎),𝑓^(𝑛)(𝑏)>1. (31)

Proof. Since 𝑎,𝑏∈𝐴 and 𝐴 is an invex set with respect to 𝜂, for every 𝑡∈[0,1], we have 𝑏+𝑡𝜂(𝑎,𝑏)∈𝐴. Using Lemma 14, Hölder’s inequality, and 𝑠-logarithmically preinvexity of |𝑓^(𝑛)|^𝑞, it yields that 𝑓(𝑏)+𝑓𝑏+𝜂(𝑎,𝑏)2−1𝜂(𝑎,𝑏)_𝑏^{𝑏+𝜂(𝑎,𝑏)}𝑓(𝑥)+𝑑𝑥_𝑛𝑘=21()×)2𝑘!𝜂𝑎,𝑏(𝑏−𝑥^𝑘−1()+()2𝑥−2𝑏−𝑘𝜂𝑎,𝑏𝑏+𝜂𝑎,𝑏−𝑥^𝑘−12𝑏+2𝜂(𝑎,𝑏)−2𝑥−𝑘𝜂(𝑎,𝑏)×𝑓^(𝑘−1)(𝑥≤)𝜂(𝑎,𝑏)^𝑛×2𝑛!_{0(𝑥−𝑏)/(𝜂(𝑎,𝑏))}𝑡^𝑛−1(𝑛−2𝑡)𝑓^(𝑛)𝑏+𝑡𝜂(𝑎,𝑏)𝑑𝑡+¹_{()())𝑥−𝑏/(𝜂𝑎,𝑏}(1−𝑡)^𝑛−1()×𝑓2𝑡+𝑛−2_(𝑛)()≤𝑏+𝑡𝜂𝑎,𝑏𝑑𝑡𝜂(𝑎,𝑏)^𝑛×2𝑛!_{0(𝑥−𝑏)/(𝜂(𝑎,𝑏))}𝑡^𝑛−1(𝑛−2𝑡)𝑑𝑡^1−1/𝑞×₀^{(𝑥−𝑏)/(𝜂(𝑎,𝑏))}𝑡^𝑛−1(𝑛−2𝑡)𝑓^(𝑛)(𝑏)^{𝑞(1−𝑡)𝑠}×𝑓^(𝑛)(𝑎)^𝑞𝑡𝑠𝑑𝑡^1/𝑞+¹_{())(𝑥−𝑏)/(𝜂𝑎,𝑏}(1−𝑡)^𝑛−1(2𝑡+𝑛−2)𝑑𝑡^1−1/𝑞×¹_{()())𝑥−𝑏/(𝜂𝑎,𝑏}(1−𝑡)^𝑛−1()×𝑓2𝑡+𝑛−2_(𝑛)(𝑏)^{𝑞(1−𝑡)𝑠}𝑓^(𝑛)(𝑎)^𝑞𝑡𝑠𝑑𝑡^1/𝑞. (32) By Lemmas 17 and 21, for 0<|𝑓^(𝑛)(𝑎)|,|𝑓^(𝑛)(𝑏)|≤1, we give ₀^{(𝑥−𝑏)/(𝜂(𝑎,𝑏))}𝑡^𝑛−1(𝑛−2𝑡)𝑓^(𝑛)(𝑏)^{𝑞(1−𝑡)𝑠}𝑓^(𝑛)(𝑎)^𝑞𝑡𝑠≤𝑑𝑡_{0(𝑥−𝑏)/(𝜂(𝑎,𝑏))}𝑡^𝑛−1(𝑛−2𝑡)𝑓^(𝑛)(𝑏)^{𝑠𝑞(1−𝑡)}𝑓^(𝑛)(𝑎)^𝑠𝑞𝑡=𝑓𝑑𝑡_(𝑛)(𝑏)^𝑠𝑞𝑛𝐸𝑛−1;𝜉^𝑠𝑞,𝑥−𝑏𝜂(𝑎,𝑏)−2𝐸𝑛;𝜉^𝑠𝑞,𝑥−𝑏𝜂(),𝑎,𝑏_{1()())𝑥−𝑏/(𝜂𝑎,𝑏}(1−𝑡)^𝑛−1(2𝑡+𝑛−2)𝑓^(𝑛)(𝑏)^{𝑞(1−𝑡)𝑠}×𝑓^(𝑛)(𝑎)^𝑞𝑡𝑠≤𝑑𝑡_{1()())𝑥−𝑏/(𝜂𝑎,𝑏}(1−𝑡)^𝑛−1(2𝑡+𝑛−2)𝑓^(𝑛)(𝑏)^{𝑠𝑞(1−𝑡)}×𝑓^(𝑛)(𝑎)^𝑠𝑞𝑡=𝑑𝑡_{0(𝑏+𝜂(𝑎,𝑏)−𝑥)/(𝜂(𝑎,𝑏))}𝑡^𝑛−1(𝑛−2𝑡)𝑓^(𝑛)(𝑏)^𝑠𝑞𝑡𝑓^(𝑛)(𝑎)^{𝑠𝑞(1−𝑡)}=𝑓𝑑𝑡_(𝑛)(𝑎)^𝑠𝑞𝑛𝐸𝑛−1;𝜉^−𝑠𝑞,𝑏+𝜂(𝑎,𝑏)−𝑥𝜂(𝑎,𝑏)−2𝐸𝑛;𝜉^−𝑠𝑞,𝑏+𝜂(𝑎,𝑏)−𝑥𝜂().𝑎,𝑏 (33)(34) For 0<|𝑓^(𝑛)(𝑎)|<1<|𝑓^(𝑛)(𝑏)|, we get ₀^{(𝑥−𝑏)/(𝜂(𝑎,𝑏))}𝑡^𝑛−1(𝑛−2𝑡)𝑓^(𝑛)(𝑏)^{𝑞(1−𝑡)𝑠}𝑓^(𝑛)(𝑎)^𝑞𝑡𝑠≤𝑑𝑡_{0(𝑥−𝑏)/(𝜂(𝑎,𝑏))}𝑡^𝑛−1(𝑛−2𝑡)𝑓^(𝑛)(𝑏)^{𝑞[𝑠(1−𝑡)+1−𝑠]}𝑓^(𝑛)(𝑎)^𝑠𝑞𝑡=𝑓𝑑𝑡_(𝑛)(𝑏)^𝑞𝑛𝐸𝑛−1;𝜉^𝑠𝑞,𝑥−𝑏𝜂(𝑎,𝑏)−2𝐸𝑛;𝜉^𝑠𝑞,𝑥−𝑏𝜂(),𝑎,𝑏_{1()())𝑥−𝑏/(𝜂𝑎,𝑏}(1−𝑡)^𝑛−1(2𝑡+𝑛−2)𝑓^(𝑛)(𝑏)^{𝑞(1−𝑡)𝑠}𝑓^(𝑛)(𝑎)^𝑞𝑡𝑠𝑑𝑡≤¹_{()())𝑥−𝑏/(𝜂𝑎,𝑏}(1−𝑡)^𝑛−1(2𝑡+𝑛−2)𝑓^(𝑛)(𝑏)^{𝑞[𝑠(1−𝑡)+1−𝑠]}×𝑓^(𝑛)(𝑎)^𝑠𝑞𝑡=𝑑𝑡_{0(𝑏+𝜂(𝑎,𝑏)−𝑥)/(𝜂(𝑎,𝑏))}𝑡^𝑛−1(𝑛−2𝑡)𝑓^(𝑛)(𝑏)^{𝑞(𝑠𝑡+1−𝑠)}×𝑓^(𝑛)(𝑎)^{𝑠𝑞(1−𝑡)}=𝑓𝑑𝑡_(𝑛)(𝑎)^𝑠𝑞𝑓^(𝑛)(𝑏)^{𝑞(1−𝑠)}𝑛𝐸𝑛−1;𝜉^−𝑠𝑞,𝑏+𝜂(𝑎,𝑏)−𝑥𝜂(𝑎,𝑏)−2𝐸𝑛;𝜉^−𝑠𝑞,𝑏+𝜂(𝑎,𝑏)−𝑥𝜂().𝑎,𝑏 (35)(36) For 0<|𝑓^(𝑛)(𝑏)|<1<|𝑓^(𝑛)(𝑎)|, we obtain ₀^{(𝑥−𝑏)/(𝜂(𝑎,𝑏))}𝑡^𝑛−1(𝑛−2𝑡)𝑓^(𝑛)(𝑏)^{𝑞(1−𝑡)𝑠}𝑓^(𝑛)(𝑎)^𝑞𝑡𝑠≤𝑑𝑡_{0(𝑥−𝑏)/(𝜂(𝑎,𝑏))}𝑡^𝑛−1(𝑛−2𝑡)𝑓^(𝑛)(𝑏)^{𝑞𝑠(1−𝑡)}𝑓^(𝑛)(𝑎)^{𝑞(𝑠𝑡+1−𝑠)}=𝑓𝑑𝑡_(𝑛)(𝑏)^𝑞𝑠𝑓^(𝑛)(𝑎)^{𝑞(1−𝑠)}𝑛𝐸𝑛−1;𝜉^𝑠𝑞,𝑥−𝑏𝜂(𝑎,𝑏)−2𝐸𝑛;𝜉^𝑠𝑞,𝑥−𝑏𝜂(),𝑎,𝑏_{1()())𝑥−𝑏/(𝜂𝑎,𝑏}(1−𝑡)^𝑛−1(2𝑡+𝑛−2)𝑓^(𝑛)(𝑏)^{𝑞(1−𝑡)𝑠}𝑓^(𝑛)(𝑎)^𝑞𝑡𝑠𝑑𝑡≤¹_{()())𝑥−𝑏/(𝜂𝑎,𝑏}(1−𝑡)^𝑛−1(2𝑡+𝑛−2)𝑓^(𝑛)(𝑏)^{𝑞𝑠(1−𝑡)}×𝑓^(𝑛)(𝑎)^{𝑞(𝑠𝑡+1−𝑠)}𝑑𝑡=₀^{(𝑏+𝜂(𝑎,𝑏)−𝑥)/(𝜂(𝑎,𝑏))}𝑡^𝑛−1(𝑛−2𝑡)𝑓^(𝑛)(𝑏)^𝑞𝑠𝑡×𝑓^(𝑛)(𝑎)^{𝑞[𝑠(1−𝑡)+1−𝑠]}=𝑓𝑑𝑡_(𝑛)(𝑎)^𝑞𝑛𝐸𝑛−1;𝜉^−𝑠𝑞,𝑏+𝜂(𝑎,𝑏)−𝑥𝜂(𝑎,𝑏)−2𝐸𝑛;𝜉^−𝑠𝑞,𝑏+𝜂(𝑎,𝑏)−𝑥𝜂().𝑎,𝑏 (37) For |𝑓^(𝑛)(𝑎)|,|𝑓^(𝑛)(𝑏)|>1, we have ₀^{(𝑥−𝑏)/(𝜂(𝑎,𝑏))}𝑡^𝑛−1(𝑛−2𝑡)𝑓^(𝑛)(𝑏)^{𝑞(1−𝑡)𝑠}𝑓^(𝑛)(𝑎)^𝑞𝑡𝑠≤𝑑𝑡_{0(𝑥−𝑏)/(𝜂(𝑎,𝑏))}𝑡^𝑛−1(𝑛−2𝑡)𝑓^(𝑛)(𝑏)^{𝑞[𝑠(1−𝑡)+1−𝑠]}×𝑓^(𝑛)(𝑎)^{𝑞(𝑠𝑡+1−𝑠)}=𝑓𝑑𝑡_(𝑛)(𝑏)^𝑞𝑓^(𝑛)(𝑎)^{𝑞(1−𝑠)}𝑛𝐸𝑛−1;𝜉^𝑠𝑞,𝑥−𝑏𝜂(𝑎,𝑏)−2𝐸𝑛;𝜉^𝑠𝑞,𝑥−𝑏𝜂(),𝑎,𝑏_{1()())𝑥−𝑏/(𝜂𝑎,𝑏}(1−𝑡)^𝑛−1(2𝑡+𝑛−2)𝑓^(𝑛)(𝑏)^{𝑞(1−𝑡)𝑠}𝑓^(𝑛)(𝑎)^𝑞𝑡𝑠≤𝑑𝑡_{1()())𝑥−𝑏/(𝜂𝑎,𝑏}(1−𝑡)^𝑛−1()×𝑓2𝑡+𝑛−2_(𝑛)(𝑏)^{𝑞[𝑠(1−𝑡)+1−𝑠]}𝑓^(𝑛)(𝑎)^{𝑞(𝑠𝑡+1−𝑠)}=𝑑𝑡_{0(𝑏+𝜂(𝑎,𝑏)−𝑥)/(𝜂(𝑎,𝑏))}𝑡^𝑛−1()×𝑓𝑛−2𝑡_(𝑛)(𝑏)^{𝑞(𝑠𝑡+1−𝑠)}𝑓^(𝑛)(𝑎)^{𝑞[𝑠(1−𝑡)+1−𝑠]}=𝑓𝑑𝑡_(𝑛)(𝑎)^𝑞𝑓^(𝑛)(𝑏)^{𝑞(1−𝑠)}𝑛𝐸𝑛−1;𝜉^−𝑠𝑞,𝑏+𝜂(𝑎,𝑏)−𝑥𝜂(𝑎,𝑏)−2𝐸𝑛;𝜉^−𝑠𝑞,𝑏+𝜂(𝑎,𝑏)−𝑥𝜂().𝑎,𝑏 (38)(39) Using Lemma 19 and substituting (33) to (39) into (32), we get inequality (30).
Theorem 22 is thus proved.

Corollary 23. Under the assumptions of Theorem 22,
(1) if 𝑞=1, then 𝑓(𝑏)+𝑓𝑏+𝜂(𝑎,𝑏)2−1𝜂(𝑎,𝑏)_𝑏^{𝑏+𝜂(𝑎,𝑏)}𝑓(𝑥)+𝑑𝑥_𝑛𝑘=21()×)2𝑘!𝜂𝑎,𝑏(𝑏−𝑥^𝑘−12𝑥−2𝑏−𝑘𝜂(𝑎,𝑏)+𝑏+𝜂(𝑎,𝑏)−𝑥^𝑘−1×2𝑏+2𝜂(𝑎,𝑏)−2𝑥−𝑘𝜂(𝑎,𝑏)𝑓^(𝑘−1)(𝑥≤𝜂()𝑎,𝑏)^𝑛𝑓2𝑛!^(𝑛)(𝑎)^𝛼𝑓^(𝑛)(𝑏)^𝛽𝐹𝑛;𝜉^𝑠,𝑥−𝑏𝜂()+𝑓𝑎,𝑏_(𝑛)(𝑎)^𝛿𝑓^(𝑛)(𝑏)^𝜃×𝐹𝑛;𝜉^−𝑠,𝑏+𝜂(𝑎,𝑏)−𝑥𝜂();𝑎,𝑏 (40)
(2) if 𝑠=1, then 𝑓(𝑏)+𝑓𝑏+𝜂(𝑎,𝑏)2−1𝜂(𝑎,𝑏)_𝑏^{𝑏+𝜂(𝑎,𝑏)}𝑓(𝑥)+𝑑𝑥_𝑛𝑘=21()×)2𝑘!𝜂𝑎,𝑏(𝑏−𝑥^𝑘−12𝑥−2𝑏−𝑘𝜂(𝑎,𝑏)+𝑏+𝜂(𝑎,𝑏)−𝑥^𝑘−1×2𝑏+2𝜂(𝑎,𝑏)−2𝑥−𝑘𝜂(𝑎,𝑏)𝑓^(𝑘−1)(𝑥≤)𝜂(𝑎,𝑏)^{(𝑛+1−𝑞)/𝑞}2𝑛!(𝑛+1)^1−1/𝑞×𝑓^(𝑛)(𝑏)(𝑥−𝑏)^𝑛(𝑛+1)𝜂(𝑎,𝑏)−2𝑥+2𝑏^1−1/𝑞×𝐹𝑛;𝜉^𝑞,𝑥−𝑏𝜂(𝑎,𝑏)^1/𝑞+𝑓^(𝑛)(𝑎×)𝑏+𝜂(𝑎,𝑏)−𝑥^𝑛(𝑛−1)𝜂(𝑎,𝑏)+2𝑥−2𝑏^1−1/𝑞×𝐹𝑛;𝜉^−𝑞,𝑏+𝜂(𝑎,𝑏)−𝑥𝜂(𝑎,𝑏)^1/𝑞. (41)

Theorem 24. For 𝑛∈ℕ and 𝑛≥2, let 𝐴⊆ℝ be an open invex set with respect to 𝜂∶𝐴×𝐴→ℝ and 𝑎,𝑏∈𝐴 with 𝜂(𝑎,𝑏)>0 for all 𝑎≠𝑏. Suppose that 𝑓∶𝐴→ℝ is an 𝑛-times differentiable function on 𝐴 and 𝑓^(𝑛) is integrable on the 𝜂-path 𝑃_𝑏𝑐 for 𝑐=𝑏+𝜂(𝑎,𝑏). If |𝑓^(𝑛)|^𝑞 is 𝑠-logarithmically preinvex on 𝐴 for 𝑞>1, then for 𝑥∈[𝑏,𝑏+𝜂(𝑎,𝑏)] and some 𝑠∈(0,1]𝑓(𝑏)+𝑓𝑏+𝜂(𝑎,𝑏)2−1𝜂(𝑎,𝑏)_𝑏^{𝑏+𝜂(𝑎,𝑏)}𝑓(𝑥)+𝑑𝑥_𝑛𝑘=21()×)2𝑘!𝜂𝑎,𝑏(𝑏−𝑥^𝑘−12𝑥−2𝑏−𝑘𝜂(𝑎,𝑏)+𝑏+𝜂(𝑎,𝑏)−𝑥^𝑘−1×2𝑏+2𝜂(𝑎,𝑏)−2𝑥−𝑘𝜂(𝑎,𝑏)𝑓^(𝑘−1)(𝑥≤)𝜂(𝑎,𝑏)^{𝑛−2+1/𝑞}2^2−1/𝑞𝑛!1−𝑞2𝑞−1^1−1/𝑞×𝑛𝜂(𝑎,𝑏)−2𝑥+2𝑏^{(2𝑞−1)/(𝑞−1)}−𝑛𝜂(𝑎,𝑏)^{(2𝑞−1)/(𝑞−1)1−1/𝑞}𝑓^(𝑛)(𝑎)^𝛼𝑓^(𝑛)(𝑏)^𝛽×𝐺𝑛𝑞−𝑞+1;𝜉^𝑠𝑞,𝑥−𝑏𝜂(𝑎,𝑏)^1/𝑞+(𝑛−2)𝜂(𝑎,𝑏)+2𝑥−2𝑏^{(2𝑞−1)/(𝑞−1)}−𝑛𝜂(𝑎,𝑏)^{(2𝑞−1)/(𝑞−1)1−1/𝑞}×𝑓^(𝑛)(𝑎)^𝛿𝑓^(𝑛)(𝑏)^𝜃×𝐺𝑛𝑞−𝑞+1;𝜉^−𝑠𝑞,𝑏+𝜂(𝑎,𝑏)−𝑥𝜂(𝑎,𝑏)^1/𝑞, (42) where 𝜉, {𝛼,𝛽} and {𝛿,𝜃} are given in Theorem 22 and 𝐺(𝛾;𝜇,𝑥) is defined by (26).

Proof. Since 𝑎,𝑏∈𝐴 and 𝐴 is an invex set with respect to 𝜂, for every 𝑡∈[0,1], we have 𝑏+𝑡𝜂(𝑎,𝑏)∈𝐴. Using Lemma 14, Hölder’s inequality, and 𝑠-logarithmically preinvexity of |𝑓^(𝑛)|^𝑞, it follows that 𝑓(𝑏)+𝑓𝑏+𝜂(𝑎,𝑏)2−1𝜂(𝑎,𝑏)_𝑏^{𝑏+𝜂(𝑎,𝑏)}𝑓(𝑥)+𝑑𝑥_𝑛𝑘=21()×)2𝑘!𝜂𝑎,𝑏(𝑏−𝑥^𝑘−12𝑥−2𝑏−𝑘𝜂(𝑎,𝑏)+𝑏+𝜂(𝑎,𝑏)−𝑥^𝑘−1×2𝑏+2𝜂(𝑎,𝑏)−2𝑥−𝑘𝜂(𝑎,𝑏)𝑓^(𝑘−1)(𝑥≤)𝜂(𝑎,𝑏)^𝑛×2𝑛!_{0(𝑥−𝑏)/(𝜂(𝑎,𝑏))}(𝑛−2𝑡)^{𝑞/(𝑞−1)}𝑑𝑡^1−1/𝑞×₀^{(𝑥−𝑏)/(𝜂(𝑎,𝑏))}𝑡^{(𝑛−1)𝑞}𝑓^(𝑛)(𝑏)^{𝑞(1−𝑡)𝑠}𝑓^(𝑛)(𝑎)^𝑞𝑡𝑠𝑑𝑡^1/𝑞+¹_{())(𝑥−𝑏)/(𝜂𝑎,𝑏}(2𝑡+𝑛−2)^{𝑞/(𝑞−1)}𝑑𝑡^1−1/𝑞×¹_{()/𝜂()𝑥−𝑏𝑎,𝑏}(1−𝑡)^{(𝑛−1)𝑞}𝑓^(𝑛)(𝑏)^{𝑞(1−𝑡)𝑠}×𝑓^(𝑛)(𝑎)^𝑞𝑡𝑠𝑑𝑡^1/𝑞. (43)
The rest is the same as the proof of Theorem 22.

Corollary 25. Under the assumptions of Theorem 24, if 𝑠=1, then 𝑓(𝑏)+𝑓𝑏+𝜂(𝑎,𝑏)2−1𝜂(𝑎,𝑏)_𝑏^{𝑏+𝜂(𝑎,𝑏)}𝑓(𝑥)+𝑑𝑥_𝑛𝑘=21()×)2𝑘!𝜂𝑎,𝑏(𝑏−𝑥^𝑘−12𝑥−2𝑏−𝑘𝜂(𝑎,𝑏)+𝑏+𝜂(𝑎,𝑏)−𝑥^𝑘−1×2𝑏+2𝜂(𝑎,𝑏)−2𝑥−𝑘𝜂(𝑎,𝑏)𝑓^(𝑘−1)(𝑥≤)𝜂(𝑎,𝑏)^{𝑛−2+1/𝑞}2^2−1/𝑞𝑛!1−𝑞2𝑞−1^1−1/𝑞×𝑛𝜂(𝑎,𝑏)−2𝑥+2𝑏^{(2𝑞−1)/(𝑞−1)}−𝑛𝜂(𝑎,𝑏)^{(2𝑞−1)/(𝑞−1)1−1/𝑞}𝑓^(𝑛)(𝑏×𝐺)𝑛𝑞−𝑞+1;𝜉^𝑞,𝑥−𝑏𝜂(𝑎,𝑏)^1/𝑞+(𝑛−2)𝜂(𝑎,𝑏)+2𝑥−2𝑏^{(2𝑞−1)/(𝑞−1)}−𝑛𝜂(𝑎,𝑏)^{(2𝑞−1)/(𝑞−1)1−1/𝑞}𝑓^(𝑛)(𝑎×𝐺)𝑛𝑞−𝑞+1;𝜉^−𝑞,𝑏+𝜂(𝑎,𝑏)−𝑥𝜂(𝑎,𝑏)^1/𝑞. (44)

Theorem 26. For 𝑛∈ℕ and 𝑛≥2, let 𝐴⊆ℝ be an open invex set with respect to 𝜂∶𝐴×𝐴→ℝ and 𝑎,𝑏∈𝐴 with 𝜂(𝑎,𝑏)>0 for all 𝑎≠𝑏. Suppose that 𝑓∶𝐴→ℝ is an 𝑛-times differentiable function on 𝐴 and 𝑓^(𝑛) is integrable on the 𝜂-path 𝑃_𝑏𝑐 for 𝑐=𝑏+𝜂(𝑎,𝑏). If |𝑓^(𝑛)|^𝑞 is 𝑠-logarithmically preinvex on 𝐴 for 𝑞>1, then for 𝑥∈[𝑏,𝑏+𝜂(𝑎,𝑏)], 0≤𝑟≤(𝑛−1)𝑞, and some 𝑠∈(0,1]𝑓(𝑏)+𝑓𝑏+𝜂(𝑎,𝑏)2−1𝜂(𝑎,𝑏)_𝑏^{𝑏+𝜂(𝑎,𝑏)}𝑓(𝑥)+𝑑𝑥_𝑛𝑘=21()×)2𝑘!𝜂𝑎,𝑏(𝑏−𝑥^𝑘−12𝑥−2𝑏−𝑘𝜂(𝑎,𝑏)+𝑏+𝜂(𝑎,𝑏)−𝑥^𝑘−1×2𝑏+2𝜂(𝑎,𝑏)−2𝑥−𝑘𝜂(𝑎,𝑏)𝑓^(𝑘−1)(𝑥≤)𝜂(𝑎,𝑏)^{(𝑟+2−𝑞)/𝑞}2𝑛!𝑞−1(𝑛𝑞−𝑟−1)(𝑛𝑞+𝑞−𝑟−2)^1−1/𝑞×𝑓^(𝑛)(𝑎)^𝛼𝑓^(𝑛)(𝑏)^𝛽(𝑥−𝑏)^{(𝑛𝑞−𝑟−1)/𝑞}×𝑛𝑛𝑞+𝑞−𝑟−2𝜂(𝑎,𝑏)−2(𝑛𝑞−𝑟−1)(𝑥−𝑏)^1−1/𝑞×𝐻𝑟+1;𝜉^𝑠𝑞,𝑥−𝑏𝜂(𝑎,𝑏)^1/𝑞+𝑓^(𝑛)(𝑎)^𝛿𝑓^(𝑛)(𝑏)^𝜃×𝑛𝑛𝑞+𝑞−𝑟−2𝜂(𝑎,𝑏)−2(𝑛𝑞−𝑟−1)(𝑏+𝜂(𝑎,𝑏)−𝑥)^1−1/𝑞×𝑏+𝜂(𝑎,𝑏)−𝑥^{(𝑛𝑞−𝑟−1)/𝑞}×𝐻𝑟+1;𝜉^−𝑠𝑞,𝑏+𝜂(𝑎,𝑏)−𝑥𝜂(𝑎,𝑏)^1/𝑞, (45) where 𝜉, {𝛼,𝛽} and {𝛿,𝜃} are given in Theorem 22 and 𝐻(𝛾;𝜇,𝑥) is defined by (28).

Proof. Since 𝑎,𝑏∈𝐴 and 𝐴 is an invex set with respect to 𝜂, for every 𝑡∈[0,1], we have 𝑏+𝑡𝜂(𝑎,𝑏)∈𝐴. Using Lemma 14, Hölder’s inequality, and 𝑠-logarithmically preinvexity of |𝑓^(𝑛)|^𝑞, it turns out that 𝑓(𝑏)+𝑓𝑏+𝜂(𝑎,𝑏)2−1𝜂(𝑎,𝑏)_𝑏^{𝑏+𝜂(𝑎,𝑏)}𝑓(𝑥)+𝑑𝑥_𝑛𝑘=21()×)2𝑘!𝜂𝑎,𝑏(𝑏−𝑥^𝑘−12𝑥−2𝑏−𝑘𝜂(𝑎,𝑏)+𝑏+𝜂(𝑎,𝑏)−𝑥^𝑘−1×2𝑏+2𝜂(𝑎,𝑏)−2𝑥−𝑘𝜂(𝑎,𝑏)𝑓^(𝑘−1)(𝑥≤)𝜂(𝑎,𝑏)^𝑛×2𝑛!_{0(𝑥−𝑏)/(𝜂(𝑎,𝑏))}𝑡^{(𝑛𝑞−𝑞−𝑟)/(𝑞−1)}(𝑛−2𝑡)𝑑𝑡^1−1/𝑞×₀^{(𝑥−𝑏)/(𝜂(𝑎,𝑏))}𝑡^𝑟(𝑛−2𝑡)𝑓^(𝑛)(𝑏)^{𝑞(1−𝑡)𝑠}×𝑓^(𝑛)(𝑎)^𝑞𝑡𝑠𝑑𝑡^1/𝑞+¹_{())(𝑥−𝑏)/(𝜂𝑎,𝑏}(1−𝑡)^{(𝑛𝑞−𝑞−𝑟)/(𝑞−1)}(2𝑡+𝑛−2)𝑑𝑡^1−1/𝑞×¹_{())(𝑥−𝑏)/(𝜂𝑎,𝑏}(1−𝑡)^𝑟()×𝑓2𝑡+𝑛−2_(𝑛)(𝑏)^{𝑞(1−𝑡)𝑠}𝑓^(𝑛)(𝑎)^𝑞𝑡𝑠𝑑𝑡^1/𝑞. (46)
The rest is also similar to the proof of Theorem 22.

Corollary 27. Under the assumptions of Theorem 26,
(1) if 𝑟=0, then 𝑓(𝑏)+𝑓𝑏+𝜂(𝑎,𝑏)2−1𝜂(𝑎,𝑏)_𝑏^{𝑏+𝜂(𝑎,𝑏)}𝑓(𝑥)+𝑑𝑥_𝑛𝑘=21()×)2𝑘!𝜂𝑎,𝑏(𝑏−𝑥^𝑘−12𝑥−2𝑏−𝑘𝜂(𝑎,𝑏)+𝑏+𝜂(𝑎,𝑏)−𝑥^𝑘−1×2𝑏+2𝜂(𝑎,𝑏)−2𝑥−𝑘𝜂(𝑎,𝑏)𝑓^(𝑘−1)(𝑥≤)𝜂(𝑎,𝑏)^{(2−𝑞)/𝑞}2𝑛!𝑞−1(𝑛𝑞−1)(𝑛𝑞+𝑞−2)^1−1/𝑞×𝑓^(𝑛)(𝑎)^𝛼𝑓^(𝑛)(𝑏)^𝛽(𝑥−𝑏)^{(𝑛𝑞−1)/𝑞}×𝑛(𝑛𝑞+𝑞−2)𝜂(𝑎,𝑏)−2(𝑛𝑞−1)(𝑥−𝑏)^1−1/𝑞×𝐻1;𝜉^𝑠𝑞,𝑥−𝑏𝜂(𝑎,𝑏)^1/𝑞+𝑓^(𝑛)(𝑎)^𝛿𝑓^(𝑛)(𝑏)^𝜃×(𝑛−1)(𝑛𝑞−2)𝜂(𝑎,𝑏)−2(𝑛𝑞−1)(𝑏−𝑥)^1−1/𝑞×𝑏+𝜂(𝑎,𝑏)−𝑥^{𝑛𝑞−1/𝑞}×𝐻1;𝜉^−𝑠𝑞,𝑏+𝜂(𝑎,𝑏)−𝑥𝜂(𝑎,𝑏)^1/𝑞; (47)
(2) if 𝑟=(𝑛−1)𝑞, then 𝑓(𝑏)+𝑓𝑏+𝜂(𝑎,𝑏)2−1𝜂(𝑎,𝑏)_𝑏^{𝑏+𝜂(𝑎,𝑏)}𝑓(𝑥)+𝑑𝑥_𝑛𝑘=21()×)2𝑘!𝜂𝑎,𝑏(𝑏−𝑥^𝑘−12𝑥−2𝑏−𝑘𝜂(𝑎,𝑏)+𝑏+𝜂(𝑎,𝑏)−𝑥^𝑘−1×2𝑏+2𝜂(𝑎,𝑏)−2𝑥−𝑘𝜂(𝑎,𝑏)𝑓^(𝑘−1)(𝑥≤)𝜂(𝑎,𝑏)^{𝑛−2+2/𝑞}×𝑓2𝑛!_(𝑛)(𝑎)^𝛼𝑓^(𝑛)(𝑏)^𝛽(𝑥−𝑏)(𝑛𝜂(𝑎,𝑏)−𝑥+𝑏)^1−1/𝑞×𝐻(𝑛−1)𝑞+1;𝜉^𝑠𝑞,𝑥−𝑏𝜂(𝑎,𝑏)^1/𝑞+𝑓^(𝑛)(𝑎)^𝛿×𝑓^(𝑛)(𝑏)^𝜃()×()𝜂()𝑏+𝜂𝑎,𝑏−𝑥𝑛−1𝑎,𝑏+𝑥−𝑏^1−1/𝑞×𝐻(𝑛−1)𝑞+1;𝜉^−𝑠𝑞,𝑏+𝜂(𝑎,𝑏)−𝑥𝜂(𝑎,𝑏)^1/𝑞; (48)
(3) if 𝑠=1, then 𝑓(𝑏)+𝑓𝑏+𝜂(𝑎,𝑏)2−1𝜂(𝑎,𝑏)_𝑏^{𝑏+𝜂(𝑎,𝑏)}𝑓(𝑥)+𝑑𝑥_𝑛𝑘=21()×)2𝑘!𝜂𝑎,𝑏(𝑏−𝑥^𝑘−12𝑥−2𝑏−𝑘𝜂(𝑎,𝑏)+𝑏+𝜂(𝑎,𝑏)−𝑥^𝑘−1×2𝑏+2𝜂(𝑎,𝑏)−2𝑥−𝑘𝜂(𝑎,𝑏)𝑓^(𝑘−1)(𝑥≤)𝜂(𝑎,𝑏)^{(𝑟+2−𝑞)/𝑞}2𝑛!𝑞−1(𝑛𝑞−𝑟−1)(𝑛𝑞+𝑞−𝑟−2)^1−1/𝑞×𝑓^(𝑛)(𝑏)(𝑥−𝑏)^{(𝑛𝑞−𝑟−1)/𝑞}×𝑛𝜂()(𝑛𝑞+𝑞−𝑟−2𝑎,𝑏−2𝑛𝑞−𝑟−1𝑥−𝑏)^1−1/𝑞×𝐻𝑟+1;𝜉^𝑞,𝑥−𝑏𝜂(𝑎,𝑏)^1/𝑞+𝑓^(𝑛)(𝑎×𝑛𝜂()())𝑛𝑞+𝑞−𝑟−2𝑎,𝑏−2𝑛𝑞−𝑟−1𝑏+𝜂𝑎,𝑏−𝑥^1−1/𝑞×𝑏+𝜂(𝑎,𝑏)−𝑥^{(𝑛𝑞−𝑟−1)/𝑞}×𝐻𝑟+1;𝜉^−𝑞,𝑏+𝜂(𝑎,𝑏)−𝑥𝜂(𝑎,𝑏)^1/𝑞. (49)

Conflict of Interests

The authors declare that they have no conflict of interests.

Authors’ Contribution

All authors contributed equally to the paper and read and approved the final paper.

Acknowledgments

This work was supported by the Fundamental Research Funds for the Central Universities (no. DUT14ZD208) and by the NNSF of China (no. 11361038).