M ϕ A - h -Convexity and Hermite-Hadamard Type Inequalities

. We investigate a family of M ϕ A - h -convex functions, give some properties of it and several inequalities which are counterparts to the classical inequalities such as the Jensen inequality and the Schur inequality. We give the weighted Hermite-Hadamard inequalities for an M ϕ A - h -convex function and several estimations for the product of two functions.


Preliminaries
It is known that the classical convexity can be generalized to an MN-convexity, where M and N are means which is described in [8]. The other direction of generalization leads to the concept of h-convexity, [13]. It is interesting to see properties of a function which definition combines some elements of MN-convexity and of h-convexity.
Let M and N be two means in two variables. We say that a function f : for every x, y ∈ I.
In this paper we will focus on a somewhat special type of means.
It is obvious that the power mean M p corresponds to ϕ(x) = x p if p = 0 and to ϕ(x) = log x if p = 0.
Let ϕ and ψ be two continuous, strictly monotone functions defined on intervals I and K respectively.
Let h : J → R be a non-negative function, (0, 1) ⊆ J and let f : I → K such that h(t)ψ(f (x)) + h(1 − t)ψ(f (y )) ∈ ψ(K) for all x, y ∈ I, t ∈ (0, 1). We say that a function f is for all x, y ∈ I and all t ∈ (0, 1). Especially, a function f : for all x, y ∈ I and t ∈ (0, 1). The M ϕ M ψ -h-concavity is defined on a natural way.
Some particular cases of M ϕ M ψ -h-convex functions are recently investigated. If h(t) = t, then the M ϕ M ψ -h-convexity collapses to the M ϕ M ψ -convexity which is described in [8]. If M ϕ , M ψ are an arithmetic mean (A), a geometric mean (G) or a harmonic mean (H), then we can find several results.
For example, AA-h-convexity or simply h-convexity firstly appeared in [13]. An HA-h-convexity or harmonic-h-convexity is described in [2] and [10]. HG-h-convexity investigated in [10] and AG-hconvexity or log-h-convexity in [9]. AM p -h-convexity or (h, p)-convexity is described in [6] while some properties of M p A-h-convex functions are given in [4]. Also, we have to mention article [1]  h be a non-negative function defined on the interval J, Proof. Let us suppose thatf is M ϕ A-h-convex on I and let u, v ∈ ϕ(I), t ∈ (0, 1). Since ϕ is continuous and strictly monotone on I, there exist x, y ∈ I such that u = ϕ(x), v = ϕ(y ). Then which means that f • ϕ −1 is h-convex. The second case is proved similarly. (i) Let h 1 and h 2 have a property (ii) If f , g are M ϕ A-h-convex functions, λ > 0, then f + g and λf are M ϕ A-h-convex.
Proof. The proof is based on the known results for h-convex functions and characterization given in Other parts are proved similarly by applying Propositions 9 and 10 from [13].
The following theorem gives a counterpart of the Schur inequality.
If ϕ is increasing, then for any x 1 , x 2 , x 3 ∈ I such that x 1 < x 2 < x 3 and ϕ( If ϕ is decreasing, then for any Since a function g := f • ϕ −1 is h-convex, using Proposition 16 from [13], we get and after appropriate substitutions we obtain inequality (2.1). Inequality (2.2) is proved in a similar way.
The following theorem is a counterpart of the discrete Jensen inequality and its converse for an Theorem 2.2. Let h : J → R be a non-negative supermultiplicative function, (0, 1) ⊆ J. Let ϕ be a continuous, strictly monotone function defined on the interval I. Let f : . . , w n be non-negative real numbers such that W n = n i=1 w i = 0 and w i Wn ∈ J, i = 1, . . . , n.
(i) Then for all x 1 , . . . , x n ∈ I the following holds (ii) Then for all x 1 , . . . , x n ∈ (a, b) ⊆ I the following holds The following result is a property of subadditivity for an index set function. Let K be a finite non-empty set of positive integers. Let us define the index set function F by Let ϕ be a continuous, strictly monotone function defined on the interval I.
Furthermore, if M k := {1, . . . , k}, k = 2, . . . , n and W M k ∈ J, then Proof. Let us consider the following difference , and x, y and supermultiplicativity of h, we get Remark 2.1. If M ϕ = A, then the above results related to the Jensen inequality, its converse and to the index set function for an h-function were proved in [13].
If M ϕ = H, then the Jensen type inequality for HA-h-convex function is given in [2]. If M ϕ = M p and h(t) = t, then the Jensen inequality for M p A-convex was proved in [4]. If M ϕ ∈ {A, G, H}, then results from this section are given in [1].

Hermite-Hadamard type inequality and related results
Counterparts of the Hermite-Hadamard inequality appear in the study of every kind of convexity.
Namely, in the classical convexity, the left-hand side or the right-hand side of the Hermite-Hadamard inequality are equivalent to the definition of convexity. The Hermite-Hadamard inequality for an hconvex function was proved in [3] and [11] and has the following form.
If h is an integrable function, h( 1 2 ) = 0, then for an integrable h-convex function f : [a, b] → R, the following sequence of inequalities hold: provided that all integrals exist. Moreover,

4)
provided that all integrals exist.

Inequality (3.4) for an M ϕ A-convex function is given in
provided that all integrals exist. Furthermore, provided that all integrals exist.
The following theorem contains estimations for the integral mean of the product of two M ϕ A-hconvex functions. If f is M ϕ A-h 1 -convex and g is M ϕ A-h 2 -convex, then the following hold: and provided that all integrals exist. ((a, b; t, 1−t)) ≤ h 1 (t)f (a)+h 1 (1−t)f (b) and g(M ϕ (a, b; t, 1−t)) ≤ h 2 (t)g(a)+h 2 (1−t)g(b).
Multiplying these two inequalities and integrating it over [0, 1], we obtain and after a substitution M ϕ (a, b; t, 1 − t) = x, we get inequality (3.5).
for t ∈ (0, 1) and since f is where in the last inequality we used the M ϕ A-h-convexity again. Integrating the above inequality and using into account that we obtain inequality (3.6).
Multiplying these two inequalities, then multiplying with ϕ (x)ϕ (y ) and integrating it over [a, b] with respect to x and y and over [0, 1] with respect to t, we obtain b a b a 1 0 ϕ (x)ϕ (y )f (M ϕ (x, y ; t, 1 − t))g(M ϕ (x, y ; t, 1 − t)) dt dy dx Using the right-hand side of inequality (3.4) to estimate b a f (x)ϕ (x) dx and b a g(x)ϕ (x) dx and some simple transformations, we get (3.7).
(iv) In this case we begin with inequalities and proceed in the similar way, i.e. multiply them mutually, then multiply with ϕ (x) and integrate with respect to x and t. We get ϕ In the next step we use the right-hand side of (3.4) to estimate . After a short calculation we get the required inequality (3.8).
Remark 3.2. Particular cases of the above results are already known. If ϕ(x) = x, then (3.5) and (3.6) for AA-h-convex functions are given in [11]. The above results for M ϕ A-convex functions, i.e.
The case p = 0 is not considered in [5], so, in the following corollary we give this, complementary result.
Corollary 3.2. Let h i , i = 1, 2 be non-negative functions defined on the interval J i , (0, 1) ⊆ J i , and If f is GA-h 1 -convex and g is GA-h 2 -convex, then    Proof. Applying the function ϕ(x) = log x in Theorem 3.2, we get results of this corollary.
The following theorem also contains some estimations for the integral mean of the product of two functions, but the proofs of these inequalities are based on the following inequality: if a ≤ b and c ≤ d, then ad + cb ≤ bd + ac.   and integrating with respect to t, we get .

Conflicts of Interest:
The author declares that there are no conflicts of interest regarding the publication of this paper.