Properties and inequalities for the (h1, h2)- and (h1, h2, m)-GA-convex functions

ABOUT THE AUTHORS The Inner Mongolia University for Nationalities locates at the Horqin (Khorchin, Horchin) grassland in China and was set up in 1958. The College of Mathematics is one of the eldest specialties and subjects in the university. Currently, there are 10 members in the research group of the Theory of Convexity and Applications. The members have introduced several new notions of convex functions and contributed much to the theory of convexity and applications. Feng Qi received his PhD degree of Science in mathematics from University of Science and Technology of China. He is being a full professor at Henan Polytechnic University and Tianjin Polytechnic University in China. He was the founder of the School of Mathematics and Informatics at Henan Polytechnic University in China. He was visiting professors at Victoria University in Australia and at 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 visited Copenhagen University in Denmark, Kyungpook National University and other six universities in South Korea, and Antalya in Turkey to attend an academic conference held by the Ağrı İbrahim ÇeÇen University in April 2016. He is or was editors of over 20 international respected journals. From 1993 to 2016, he published over 460 academic articles in reputed international journals. PUBLIC INTEREST STATEMENT


PUBLIC INTEREST STATEMENT
The theory of convex functions has important applications in many mathematical sciences. The notion of h-convex functions can be used to derive plenty of convex functions familiar to common mathematicians. In current paper, the authors extend the notion of h-convex functions, introduce a more general notion of h-convex functions, improve existed notions for convex functions, establish several inequalities for the general h-convex functions, and find their applications.
The rest may be proved similarly. Theorem 3.1 is thus proved.

Jensen type inequalities
Now we are in a position to establish inequalities of Jensen type for (h 1 , h 2 , m)-GA-convex functions.

then the inequality (Equation 4.1) is reversed.
Proof When n = 2, taking t = w 1 and 1 − t = w 2 in Definition (2.2) means that the inequality (Equation 4.1) holds.
Suppose that the inequality (Equation 4.1) holds for n = k, that is,

by Definition (2.2) and the hypothesis (Equation 4.2), we have
Since h 2 is a super-multiplicative function, we obtain h 2 (Δ k )h 2    for all x i ∈ (0, b] and w i > 0 such that ∑ n i=1 w i = 1.

Hermite-Hadamard type inequalities
Now we are in a position to establish some new Hermite-Hadamard type inequalities for (h 1 , h 2 , m)-GA-convex functions. ) and h 1 , If replacing a 1−t b t and a t b 1−t for 0 ≤ t ≤ 1 by x, then

and
The proof of Theorem 5.1 is complete.

)-GA-convexity of f and (Equation 5.1), we obtain
The proof of Theorem 5.2 is complete.
Corollary 5.2.1 Let h 1 (t) = t s 1 and h 2 (t) = t s 2 for all t ∈ (0, 1), let s 1 , s 2 ∈ (−1, 1] and m ∈ (0, 1], and let f : Proof From the (h 1 , h 2 , m)-GA convexity of f on 0, b m 2 , we obtain Substituting a 1−t b t and a t b 1−t for 0 ≤ t ≤ 1 by x and integrating on both sides of the above inequality with respect to t ∈ [0, 1] lead to Theorem 5.3 is proved.   Theorem 5.5 Proof Let x = a t b 1−t for 0 ∈ [0, 1]. By the (h 1 , h 2 , m)-GA-convexity of f and g, we have The proof of Theorem 5.5 is complete. In particular, if h(t) = t s for t ∈ (0, 1), s ∈ − 1 2 , 1 , and m 1 = m 2 = m, then where B denotes the well-known Beta function.

Applications
In what follows we will apply theorems and corollaries in the above section to establish inequalities for (h 1 , h 2 , m)-GA-convex functions.

Proof Using the inequality (Equation 4.1), it follows that
When h 2 1 n ≠ 1, we have