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(