INEQUALITIES OF HERMITE-HADAMARD TYPE FOR COMPOSITE h-CONVEX FUNCTIONS

In this paper we obtain some inequalities of Hermite-Hadamard type for composite convex functions. Applications for AG, AH-h-convex functions, GA; GG; GH-h-convex functions and HA; HG; HH-h-convex function are given. 1. Introduction We recall here some concepts of convexity that are well known in the literature. Let I be an interval in R. Denition 1 ([52]). We say that f : I ! R is a Godunova-Levin function or that f belongs to the class Q (I) if f is non-negative and for all x; y 2 I and t 2 (0; 1) we have (1.1) f (tx+ (1 t) y) 1 t f (x) + 1 1 t (y) : Some further properties of this class of functions can be found in [42], [43], [45], [58], [64] and [65]. Among others, its has been noted that non-negative monotone and non-negative convex functions belong to this class of functions. Denition 2 ([45]). We say that a function f : I ! R belongs to the class P (I) if it is nonnegative and for all x; y 2 I and t 2 [0; 1] we have (1.2) f (tx+ (1 t) y) f (x) + f (y) : Obviously Q (I) contains P (I) and for applications it is important to note that also P (I) contain all nonnegative monotone, convex and quasi convex functions, i. e. nonnegative functions satisfying (1.3) f (tx+ (1 t) y) max ff (x) ; f (y)g for all x; y 2 I and t 2 [0; 1] : For some results on P -functions see [45] and [62] while for quasi convex functions, the reader can consult [44]. Denition 3 ([10]). Let s be a real number, s 2 (0; 1]: A function f : [0;1) ! [0;1) is said to be s-convex (in the second sense) or Breckner s-convex if f (tx+ (1 t) y) tf (x) + (1 t) f (y) for all x; y 2 [0;1) and t 2 [0; 1] : 1991 Mathematics Subject Classication. 26D15; 26D10.


Introduction
We recall here some concepts of convexity that are well known in the literature.Let I be an interval in R.
In order to unify the above concepts for functions of real variable, S. Varošanec introduced the concept of h-convex functions as follows.
We can introduce now another class of functions.
De…nition 5. We say that the function f : I ![0; 1) is of s-Godunova-Levin type, with s 2 [0; 1] ; if for all t 2 (0; 1) and x; y 2 I: We observe that for s = 0 we obtain the class of P -functions while for s = 1 we obtain the class of Godunova-Levin.If we denote by Q s (C) the class of s-Godunova-Levin functions de…ned on C, then we obviously have For an extension of this result to functions de…ned on convex subsets of linear spaces and re…nements, see [31].
In order to extend this result for other classes of functions, we need the following preparations.
If we take g (t) =  1) which is the concept of LogExp h-convex function on [a; b] : For h (t) = t; the concept was considered in [28].Further, assume that f : [a; b] !J ; J an interval of real numbers and k : J ! R a continuous function on J that is strictly increasing (decreasing) on J : De…nition 7. We say that the function f : ] a continuous strictly increasing function that is di¤ erentiable on (a; b) ; f : [a; b] !J ; J an interval of real numbers and k : J ! R a continuous function on J that is strictly increasing (decreasing) on J ; we can also consider the following concept: De…nition 8. We say that the function f : This de…nition is equivalent to the condition (1.13) k f g 1 (( 1) for any t; s 2 [a; b] and 2 [0; 1] : If k : J ! R is strictly increasing (decreasing) on J ; then the condition (1.13) is equivalent to: (1.14) f g 1 (( 1) If k (t) = ln t; t > 0 and f : [a; b] !(0; 1), then the fact that f is k-composite h-convex on [a; b] is equivalent to the fact that f is log-convex or multiplicatively convex or AG-h-convex, namely, for all x; y 2 I and t 2 [0; 1] one has the inequality: A function f : I !Rn f0g is called AH-h-convex (concave) on the interval I if the following inequality holds [1] (1. 16) f ((1 for any x; y 2 I and 2 [0; 1] : An important case that provides many examples is that one in which the function is assumed to be positive for any x 2 I: In that situation the inequality (1.16) is equivalent to for any x; y 2 I and 2 [0; 1] : Taking into account this fact, we can conclude that the function f : and g (t) = ln t; t 2 I: Following [1], we say that the function We observe that the inequality (1.18) is equivalent to The function f : I (0; 1) !(0; 1) is called GG-h-convex on the interval I of real umbers R if [5] (1.20) for any x; y 2 I and 2 [0; 1] : If the inequality is reversed in (1.20) then the function is called GG-h-concave.
For h (t) = t; this concept was introduced in 1928 by P. Montel [59], however, the roots of the research in this area can be traced long before him [60].It is easy to see that [60], the function f : Following [1] we say that the function for all x; y 2 I and t 2 where k (t) = t r ; t > 0: In this paper we obtain some inequalities of Hermite-Hadamard type for composite convex functions.Applications for AG, AH-h-convex functions, GA; GG; GH-h-convex functions and HA; HG; HH-h-convex function are given.

Refinements of HH-Inequality
The following representation result holds.
Proof.For = 0 and = 1 the equality (2.1) is obvious.Let 2 (0; 1) : Observe that ] dt: If we make the change of variable u := (1 t) + t then we have ) and du = (1 ) du: ] du: If we make the change of variable u := t then we have du = dt and  1) g (x) + g (y)) + ( 1) function function, then by Hermite-Hadamard type inequality (1.6) we have 1) g (x) + g (y)) + tg (y)] dt f g 1 ((1 ) g (x) + g (y)) + f (y) Now, if we multiply the inequality (2.3) by 1 0 and (2.4) by 0 and add the obtained inequalities, then we get and by (2.1) we obtain where the last inequality follows by the de…nition of composite-g 1 h-convexity and performing the required calculation.By using the change of variable u = (1 t) g (x)+tg (y) ; we have du = (g (y) g (x)) dt and then f g 1 (u) du: If we change the variable t = g 1 (u) ; then u = g (t) ; which gives that du = g 0 (t) dt and then and the inequality (2.2) is obtained.
Remark 1.With the assumptions from Theorem 1, we observe that if we take either = 0 or = 1 in the …rst two inequalities in (2.2), then we get (1.6).If we take = 1 2 and use the h-convexity of f g 1 ; then we get from (2.2) that 1 4h 2 1 2 f g 1 g (x) + g (y) 2 (2.7) where y; x 2 I with y 6 = x: Remark 2. In general, if h ( ) > 0 for 2 (0; 1) ; then for y; x 2 I with y 6 = x and from (2.2) we get the sequence of inequalities 1) g (x) + g (y)) + ( 1) for y; x 2 I with y 6 = x: In particular, we have In a similar way, if In particular, 1) be a composite-g 1 convex function on the interval I in R: Then for any y; x 2 I with y 6 = x and for any 2 [0; 1] we have the inequalities We have: 1) be a composite-g 1 Breckner s-convex function on the interval I with s 2 (0; 1].Then for any y; x 2 I with y 6 = x and for any 2 [0; 1] we have the inequalities (2.14) We also have: 1) be a composite-g 1 of s-Godunova-Levin type on the interval I with s 2 (0; 1).Then for any y; x 2 I with y 6 = x and for any 2 (0; 1) we have the inequalities More generally, we have: ] is a continuous strictly increasing function that is di¤ erentiable on (a; b) ; f : [a; b] !J ; J an interval of real numbers and k : J ! R is a continuous function on J that is strictly increasing.
The proof follows by the inequalities (2.8) and (2.10) and we omit the details.In 1906, Fejér [51], while studying trigonometric polynomials, obtained the following inequalities which generalize that of Hermite & Hadamard: i.e., y = w (x) is a symmetric curve with respect to the straight line which contains the point 1  2 (a + b) ; 0 and is normal to the x-axis.Under those conditions the following inequalities are valid: If h is concave on (a; b), then the inequalities reverse in (2.18).; h ( ) for any 2 [0; 1] and x; y 2 [a; b] with y 6 = x:

Applications for GA; GG and GH-h-Convex Functions
If we take g for any 2 [0; 1] and for y; x 2 [a; b] with y 6 = x: The function f : for any 2 [0; 1] and for y; x 2 [a; b] with y 6 = x: We also have that f t and g (t) = ln t; t 2 I: By making use of Corollary 4 we get function on an interval I of real numbers with h 2 L [0; 1] and f 2 L [a; b] with a; b 2 I; a < b; then we have the Hermite-Hadamard type inequality obtained by Sarikaya et al. in[66] 1=r for any x; y 2 [a; b] and 2 [0; 1].For h (t) = t; the concept was considered in [61], If r > 0; then the condition (1.22) is equivalent to

Theorem 2 (
Fejér's Inequality).Consider the integral R b a h (x) w (x) dx, where h is a convex function in the interval (a; b) and w is a positive function in the same interval such that w (x) = w (a + b x) ; for any x 2 [a; b] If w : [a; b] !R is continuous and positive on the interval [a; b] ; then the function W : [a; b] ![0; 1); W (x) := R x a w (s) ds is strictly increasing and di¤erentiable on (a; b) and the inverse W 1 : h a; R b a w (s) ds i ![a; b] exists.function on J that is strictly increasing.If the function f : [a; b] !J is kcomposite-W 1 h-convex on [a; b] ; then we have the weighted inequality

Z 1 0h 3 .
(t) dt; for any 2 [0; 1] and for y; x 2 [a; b] with y 6 = x: Applications for AG and AH-h-Convex Functions The function f : [a; b] !(0; 1) is AG-h-convex means that f is k-composite h-convex on [a; b] with k (t) = ln t; t > 0: By making use of Corollary 4 for g (t) = t; we get (3.1)

Z 1 0h 5 .
(t) dt; for any 2 [0; 1] and for y; x 2 [a; b] with y 6 = x: Applications for HA; HG and HH-h-Convex Functions Let f : [a; b] (0; 1) !R be an HA-h-convex function on the interval [a; b] : This is equivalent to the fact that f is composite-g 1 h-convex on [a; b] with the increasing function g (t) =