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

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 725987, 11 pages http://dx.doi.org/10.1155/2014/725987 Research Article New Hermite-Hadamard Type Inequalities for 𝑛-Times Differentiable and 𝑠-Logarithmically Preinvex Functions Shuhong Wang1,2 and Ximin Liu1 1 2 School of Mathematical Sciences, Dalian University of Technology, Dalian 11602, China College of Mathematics, Inner Mongolia University for Nationalities, Tongliao 028043, China Correspondence should be addressed to Shuhong Wang; Received 6 March 2014; Revised 27 June 2014; Accepted 29 June 2014; Published 20 July 2014 Academic Editor: Cristina Pignotti Copyright © 2014 S. Wang and X. Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 R𝑛 , 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 𝑓 (𝜆𝑥 + (1 − 𝜆) 𝑦) ≤ 𝜆𝑓 (𝑥) + (1 − 𝜆) 𝑓 (𝑦) (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 ⊆ R𝑛 is said to be invex with respect to the map 𝜂 : 𝐴 × 𝐴 → R𝑛 , if for every 𝑥, 𝑦 ∈ 𝐴 and 𝑡 ∈ [0, 1] 𝑦 + 𝑡𝜂 (𝑥, 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 ⊆ R𝑛 be an invex set with respect to 𝜂 : 𝐴×𝐴 → R𝑛 . For every 𝑥, 𝑦 ∈ 𝐴, the 𝜂-path 𝑃𝑥V joining the points 𝑥 and V = 𝑥 + 𝜂(𝑦, 𝑥) is defined by 𝑃𝑥V = {𝑧 | 𝑧 = 𝑥 + 𝑡𝜂 (𝑦, 𝑥) , 𝑡 ∈ [0, 1]} . (3) Definition 4 (see [2]). Let 𝐴 ⊆ R𝑛 be an invex set with respect to 𝜂 : 𝐴 × 𝐴 → R𝑛 . A function 𝑓 : 𝐴 → R is said to be preinvex with respect to 𝜂, if for every 𝑥, 𝑦 ∈ 𝐴 and 𝑡 ∈ [0, 1] 𝑓 (𝑦 + 𝑡𝜂 (𝑥, 𝑦)) ≤ 𝑡𝑓 (𝑥) + (1 − 𝑡) 𝑓 (𝑦) . (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 ⊆ R𝑛 be an invex set with respect to 𝜂 : 𝐴 × 𝐴 → R𝑛 . 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] 𝑡 1−𝑡 𝑓 (𝑦 + 𝑡𝜂 (𝑥, 𝑦)) ≤ [𝑓(𝑥)] [𝑓(𝑦)] . (5) For properties and applications of preinvex and logarithmically preinvex functions, please refer to [1–8] and closely related references therein. 2 Abstract and Applied Analysis 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 𝑓( 𝑏 𝑓 (𝑎) + 𝑓 (𝑏) 1 𝑎+𝑏 )≤ . ∫ 𝑓 (𝑥) 𝑑𝑥 ≤ 2 𝑏−𝑎 𝑎 2 (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 󵄨󵄨 󵄨󵄨 𝑏 1 󵄨󵄨 󵄨󵄨 𝑓 (𝑎) + 𝑓 (𝑏) 󵄨󵄨 󵄨󵄨 𝑓 𝑑𝑥 − ∫ (𝑥) 󵄨󵄨 󵄨󵄨 2 𝑏 − 𝑎 𝑎 󵄨 󵄨 󵄨󵄨 󸀠 󵄨󵄨 󵄨󵄨 󸀠 󵄨󵄨 (𝑏 − 𝑎) [󵄨󵄨󵄨𝑓 (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 󵄨󵄨 󵄨󵄨 𝑏 1 󵄨󵄨 𝑓 (𝑎) + 𝑓 (𝑏) 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑓 𝑑𝑥 − ∫ (𝑥) 󵄨󵄨 󵄨󵄨 2 𝑏 − 𝑎 𝑎 󵄨 󵄨 󵄨󵄨 󸀠 󵄨󵄨𝑞 󵄨󵄨 󸀠 󵄨󵄨𝑞 1/𝑞 𝑏 − 𝑎 [ 󵄨󵄨󵄨𝑓 (𝑎)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑓 (𝑏)󵄨󵄨󵄨 ] ≤ . 4 2 [ ] 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 𝑃𝑏𝑐 for 𝑐 = 𝑏 + 𝜂(𝑎, 𝑏), then 󵄨󵄨 󵄨󵄨 𝑓 (𝑏) + 𝑓 (𝑏 + 𝜂 (𝑎, 𝑏)) 𝑏+𝜂(𝑎,𝑏) 1 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨󵄨 𝑓 𝑑𝑥 − ∫ (𝑥) 󵄨󵄨 󵄨󵄨 2 𝜂 𝑏) (𝑎, 𝑏 󵄨 󵄨 2 [𝜂(𝑎, 𝑏)] 1 1/𝑞 󵄨󵄨 󸀠󸀠 󵄨󵄨𝑞 󵄨󵄨 󸀠󸀠 󵄨󵄨𝑞 1/𝑞 ≤ ( ) [󵄨󵄨󵄨𝑓 (𝑎)󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑓 (𝑏)󵄨󵄨󵄨 ] . 12 2 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 𝑃𝑏𝑐 for 𝑐 = 𝑏 + 𝜂(𝑎, 𝑏). If |𝑓(𝑛) |𝑞 is preinvex on 𝐴 for 𝑞 ≥ 1, then 󵄨󵄨 𝑓 (𝑏) + 𝑓 (𝑏 + 𝜂 (𝑎, 𝑏)) 𝑏+𝜂(𝑎,𝑏) 1 󵄨󵄨 󵄨󵄨 𝑓 (𝑥) 𝑑𝑥 − ∫ 󵄨󵄨 2 𝜂 (𝑎, 𝑏) 𝑏 󵄨 [𝜂 (𝑎, 𝑏)] (1 − 𝑘) 4 [(𝑘 + 1)!] 𝑘=1 (8) 󵄨󵄨 󵄨 × [𝑓(𝑘) (𝑏) + (−1)𝑘 𝑓(𝑘) (𝑏 + 𝜂 (𝑎, 𝑏))] 󵄨󵄨󵄨󵄨 󵄨󵄨 󵄨𝑛 󵄨󵄨 󵄨𝜂(𝑎, 𝑏)󵄨󵄨󵄨 (𝑛 − 1)1−1/𝑞 ≤ 󵄨 4 [(𝑛 + 1)!] (𝑛 + 2)1/𝑞 󵄨 󵄨𝑝/(𝑝−1) 󵄨 󵄨𝑝/(𝑝−1) 1−1/𝑝 × {[󵄨󵄨󵄨󵄨𝑓󸀠 (𝑎)󵄨󵄨󵄨󵄨 + 3󵄨󵄨󵄨󵄨𝑓󸀠 (𝑏)󵄨󵄨󵄨󵄨 ] 󵄨 󵄨𝑞 󵄨 󵄨𝑞 1/𝑞 + [(𝑛2 − 2)󵄨󵄨󵄨󵄨𝑓(𝑛) (𝑎)󵄨󵄨󵄨󵄨 + 𝑛󵄨󵄨󵄨󵄨𝑓(𝑛) (𝑏)󵄨󵄨󵄨󵄨 ] } . (9) 󵄨 󵄨𝑝/(𝑝−1) 󵄨󵄨 󸀠 󵄨󵄨𝑝/(𝑝−1) 1−1/𝑝 + [3󵄨󵄨󵄨󵄨𝑓󸀠 (𝑎)󵄨󵄨󵄨󵄨 + 󵄨󵄨󵄨𝑓 (𝑏)󵄨󵄨󵄨 ] }. 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: 𝑓( 𝑏+𝜂(𝑎,𝑏) 2𝑏 + 𝜂 (𝑎, 𝑏) 1 𝑓 (𝑥) 𝑑𝑥 )≤ ∫ 2 𝜂 (𝑎, 𝑏) 𝑏 𝑓 (𝑎) + 𝑓 (𝑏) ≤ . 2 (12) 󵄨 󵄨𝑞 󵄨 󵄨𝑞 1/𝑞 × {[𝑛󵄨󵄨󵄨󵄨𝑓(𝑛) (𝑎)󵄨󵄨󵄨󵄨 + (𝑛2 − 2)󵄨󵄨󵄨󵄨𝑓(𝑛) (𝑏)󵄨󵄨󵄨󵄨 ] 󵄨󵄨 󵄨󵄨 𝑏 𝑎+𝑏 1 󵄨󵄨 󵄨󵄨 󵄨󵄨𝑓 ( 󵄨󵄨 𝑓 𝑑𝑥 ) − ∫ (𝑥) 󵄨󵄨 󵄨󵄨 2 𝑏 − 𝑎 𝑎 󵄨 󵄨 1/𝑝 𝑏−𝑎 4 ( ) 16 𝑝 + 1 𝑘 𝑛−1 +∑ 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 ≤ (11) (10) 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 󵄨󵄨 𝑎+𝜂(𝑏,𝑎) 󵄨󵄨 𝑓 (𝑎) + 𝑓 (𝑎 + 𝜂 (𝑏, 𝑎)) 1 󵄨󵄨 𝑓 (𝑥) 𝑑𝑥 − ∫ 󵄨󵄨 2 𝜂 (𝑏, 𝑎) 𝑎 󵄨󵄨 󵄨󵄨 𝑘 𝑛−1 󵄨󵄨 (−1)𝑘 (𝑘 − 1) (𝜂 (𝑏, 𝑎)) (𝑘) −∑ 𝑓 (𝑎 + 𝜂 (𝑏, 𝑎))󵄨󵄨󵄨 󵄨󵄨 2 + 1)! (𝑘 𝑘=2 󵄨 𝑛 ≤ (𝜂(𝑏, 𝑎)) 𝑛 − 1 1−1/𝑞 1/𝑞 [𝐸1 (𝑛, 𝑞)] , ( ) 2𝑛! 𝑛+1 (13) Abstract and Applied Analysis 3 (𝑥−𝑏)/(𝜂(𝑎,𝑏)) where × [∫ 0 𝐸1 (𝑛, 𝑞) 󵄨 󵄨 󵄨 󵄨 󵄨𝑞 󵄨 (−1)𝑛 𝑛! {𝑞 [ln (󵄨󵄨󵄨󵄨𝑓(𝑛) (𝑏)󵄨󵄨󵄨󵄨) − ln (󵄨󵄨󵄨󵄨𝑓(𝑛) (𝑎)󵄨󵄨󵄨󵄨)] + 2} 󵄨󵄨󵄨󵄨𝑓(𝑛) (𝑎)󵄨󵄨󵄨󵄨 = 󵄨 󵄨 󵄨 𝑛+1 󵄨 𝑞𝑛+1 [ln (󵄨󵄨󵄨𝑓(𝑛) (𝑏)󵄨󵄨󵄨) − ln (󵄨󵄨󵄨𝑓(𝑛) (𝑎)󵄨󵄨󵄨)] 󵄨𝑞 󵄨 2󵄨󵄨󵄨󵄨𝑓(𝑛) (𝑏)󵄨󵄨󵄨󵄨 󵄨󵄨 (𝑛) 󵄨󵄨𝑞 − 󵄨 󵄨 󵄨 − 𝑛!󵄨󵄨󵄨𝑓 (𝑏)󵄨󵄨󵄨 󵄨 𝑞 [ln (󵄨󵄨󵄨𝑓(𝑛) (𝑏)󵄨󵄨󵄨) − ln (󵄨󵄨󵄨𝑓(𝑛) (𝑎)󵄨󵄨󵄨)] 󵄨 󵄨 󵄨 󵄨 𝑛 (−1)𝑘 {𝑞 [ln (󵄨󵄨𝑓(𝑛) (𝑏)󵄨󵄨) − ln (󵄨󵄨𝑓(𝑛) (𝑎)󵄨󵄨 (...truncated)