Integral inequalities of Hermite–Hadamard type for logarithmically h-preinvex functions

In the paper, the authors introduce the notion “logarithmically h-preinvex functions”, reveal that the class of h-preinvex functions include several new and known classes of preinvex functions, and establish several integral inequalities of Hermite–Hadamard type.


ABOUT THE AUTHOR
Feng Qi is a full professor in mathematics at Tianjin Polytechnic University in China. He received his PhD degree from University of Science and Technology of China. He was the founder and the former head of School of Mathematics and Informatics at Henan Polytechnic University. He was a visiting professors at Victoria University in Australia and University of Hong Kong in China. He was a part-time professor at Henan University, Henan Normal University, and Inner Mongolia University for Nationalities in China. He is being editor of over 15 international journals. He visited Copenhagen University in Denmark and several universities such as Dongguk University in South Korea. He published over 400 research papers in over 130 reputed journals. His research areas include the analytic combinatorics, analytic number theory, special functions, mathematical inequalities, mathematical means, differential geometry, and so on. For more detailed information, please click \url http://qifeng618. wordpress.com and related links therein.

PUBLIC INTEREST STATEMENT
The Hermite-Hadamard type inequalities for convex functions and sequences are a milestone of the theory of convex analysis. The concept of convexity for functions has been generalized and extended in many directions and in diverse forms. In the paper, the authors introduce a new notion "logarithmically h-preinvex functions", reveal that the class of h-preinvex functions include the logarithmically s-preinvex functions, logarithmically P-preinvex functions, and logarithmically Q-preinvex functions, and establish several integral inequalities of Hermite-Hadamard type for these convex functions.
Motivated by this ongoing research, we now introduce a new class of preinvex functions, which are called logarithmically h-preinvex functions, and derive several new integral inequalities of Hermite-Hadamard type for logarithmically h-preinvex functions.

Definitions and a lemma
Let K be a nonempty closed set in ℝ n , let f : K → ℝ be a continuous function, and let (⋅, ⋅) : K × K → ℝ n be a continuous bi-function.
Definition 2.1 (Weir & Mond, 1988) A set K is said to be invex with respect to (⋅, ⋅), if for a, b ∈ K and t ∈ [0, 1]. The invex set K is also called an -connected set.
Remark 2.1 (Antczak, 2005) The above Definition 2.1 has a geometric interpretation. This definition essentially says that there is a path starting from a point a which is contained in K. The point b may not be one of the end points of the path. This observation plays an important role in our analysis. If b is an end point of the path for every pair of points a, b ∈ K, then (b, a) = b − a and, consequently, invexity reduces to convexity. Thus, it is true that every convex set is also an invex set with respect to (b, a) = b − a, but not conversely (see Mohan & Neogy, 1995;Weir & Mond, 1988 and related references therein). For the sake of simplicity, we always assume that K = [a, a + (b, a)], unless otherwise specified.
Definition 2.2 (Weir & Mond, 1988) A function f is said to be preinvex with respect to an arbitrary bi-function (⋅, ⋅), if is valid for a, b ∈ K and t ∈ [0, 1].
.1, the preinvex function becomes a convex function in the classical sense.
In Noor et al. (2015), it was showed that the class of h-preinvex functions generalizes several other classes of convex functions. For example, if we take h(t) = t, h(t) = 1 t , h(t) = t s , and h(t) = 1 in (2.2), then the h-preinvex function reduces to the preinvex function in Weir and Mond (1988), the Q-preinvex function, the s-preinvex function, and the P-preinvex function, respectively. If we take (b, a) = b − a, then the definition of h-preinvex functions reduces to the definition of h-convex functions, which was introduced in Varošanec (2007). Noor (2007a) showed that a function f is preinvex if and only if The double inequality Equation 2.3 is known as the Hermite-Hadamard-Noor inequality for prepinvex functions. If (b, a) = b − a, then the double inequality Equation 2.3 reduces to the classical Hermite-Hadamard inequality for convex functions. For recent developments and applications (see Sarikaya et al., 2013).

Remark 2.2 From Definition 2.4, we may obtain
To prove some results in this paper, we need the following well-known Condition C introduced by Mohan and Neogy.
Condition C (Mohan & Neogy, 1995) Let K ⊂ ℝ be an invex set with respect to the bi-function (⋅, ⋅). Then for any a, b ∈ K and t ∈ [0, 1], we have From Condition C, it follows that for every a, b ∈ K and t 1 , t 2 ∈ [0, 1].
It is worth mentioning that Condition C plays a crucial and significant role in the development of the variational-like inequalities and optimization problems (see Farajzadeh et al., 2009;Mohan & Neogy, 1995;Noor, 1994; and related references therein).
The following lemma is also necessary for us.

Main results
We now start out to establish several new integral inequalities of Hermite-Hadamard type for logarithmically h-preinvex functions.
Theorem 3.1 Let f be a logarithmically h-preinvex function such that h 1 2 ≠ 0. Also suppose that Condition C holds for , then, for (b, a) > 0, we have

Consequently,
Proof Since f is logarithmically h-preinvex, using Condition C, we have Taking the logarithm on both sides of the above inequality yields + t (b, a))dt Theorem 3.2 Let f , g:K → ℝ be logarithmically h-preinvex functions and a, a + (b, a) ∈ K with (b, a) > 0. Then Proof Using Young's inequality ab ≤ a 1∕ + b 1∕ for , > 0 and + = 1 produces The proof of Theorem 3.2 is complete.