Inequalities of Hermite–Hadamard Type for GA-Convex Functions

Abstract Some inequalities of Hermite-Hadamard type for GA-convex functions defined on positive intervals are given.


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 ([37]).
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 Some further properties of this class of functions can be found in [28], [29], [31], [43], [46] and [47]. Among others, its has been noted that non-negative monotone and non-negative convex functions belong to this class of functions.
De…nition 2 ( [31]). 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 [31] and [44] while for quasi convex functions, the reader can consult [30].
In order to unify the above concepts for functions of real variable, S. Varošanec introduced the concept of h-convex functions as follows.
Assume that I and J are intervals in R; (0; 1) J and functions h and f are real non-negative functions de…ned in J and I; respectively. De…nition 4 ( [52]). Let h : J ! [0; 1) with h not identical to 0. We say that f : I ! [0; 1) is an h-convex function if for all x; y 2 I we have for all t 2 (0; 1) : For some results concerning this class of functions see [52], [6], [41], [50], [48] and [51].
We can introduce now another class of functions.
De…nition 5. We say that the function f : 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 (I) the class of s-Godunova-Levin functions de…ned on I, then we obviously have for 0 s 1 s 2 1: The following inequality holds for any convex function f de…ned on R It was …rstly discovered by Ch. Hermite in 1881 in the journal Mathesis (see [42]). But this result was nowhere mentioned in the mathematical literature and was not widely known as Hermite's result. E. F. Beckenbach, a leading expert on the history and the theory of convex functions, wrote that this inequality was proven by J. Hadamard in 1893 [5]. In 1974, D. S. Mitrinović found Hermite's note in Mathesis [42]. Since (1.6) was known as Hadamard's inequality, the inequality is now commonly referred as the Hermite-Hadamard inequality.
If f is Breckner s-convex on I; for s 2 (0; 1) ; then by taking h (t) = t s in (1.7) we get that was obtained for functions of a real variable in [26].
We notice that for s = 1 the …rst inequality in (1.11) still holds, i.e. (1.12) The case for functions of real variables was obtained for the …rst time in [31].

'-Convex Functions
We introduce the following class of h-convex functions.
De…nition 6. Let ' : (0; 1) ! (0; 1) a measurable function. We say that the function f : I ! [0; 1) is a '-convex function on the interval I if for all x; y 2 I we have for all t 2 (0; 1) : If we denote`(t) = t; the identity function, then it is obvious that f is h-convex with h =`': Also, all the examples from the introduction can be seen as '-convex functions with appropriate choices of ': If we take ' (t) = 1 t s+1 with s 2 [0; 1] then we get the class of s-Godunova-Levin functions. Also, if we put ' (t) = t s 1 with s 2 (0; 1) ; then we get the concept of Breckner s-convexity. We notice that for all these examples we have The case of convex functions, i.e. when ' (t) = 1 is the only example from above for which ' + (0) is …nite, namely ' + (0) = 1: Consider the family of functions, for p > 1 and k > 0 We observe that + (p; k) (0) = (p; k) (0) = k + 1; (p; k) is strictly decreasing on [0; 1] and (p; k) (t) (p; k) (1) = 1: We say that the function f : for all t 2 (0; 1) : It is obvious that any nonnegative convex function is a (p;k) -convex function for any p > 1 and k > 0: For m > 0 we consider the family of functions for all t 2 (0; 1) : It is obvious that any nonnegative convex function is a (m)-convex function for any m > 0: There are many other examples one can consider. In fact any continuos function ' : [0; 1] ! [1; 1) can generate a class of '-convex function that contains the class of nonnegative convex functions.
Utilising Theorem 1 we can state the following result.
The proof follows from (1.7) by taking h (t) = t' (t) ; t 2 (0; 1) : Remark 1. We notice that, since R 1 0 t' (t) dt can be seen as the expectation of a random variable X with the density function '; the inequality (2.5) provides a connection to Probability Theory and motivates the introduction of '-convex function as a natural concept, having available many examples of density functions ' that arise in applications.
We have the following particular cases: Assume that the function f : I ! [0; 1) is a a (p; k)-convex function on the interval I with p > 1 and k > 0: Let y; x 2 I with y 6 = x and assume that the mapping and utilizing (2.5) we get (2.6).

Some Results for Differentiable Functions
If we assume that the function f : I ! [0; 1) is di¤erentiable on the interior of I denoted by I then we have the following "gradient inequality" that will play an essential role in the following.

Remark 2. If we assume that
for any x; y 2 I: Proof. Assume that y > x with x; y 2 I: From (3.1) we get for any u 2 [x; y] with u 6 = x+y 2 : Integrating this inequality over u on [x; y] we get The case y < x goes likewise and the proof of the second inequality in (3.6) is completed.
Assume that y > x with x; y 2 I: From (3.1) we get for any t 2 (0; 1) and for any t 2 (0; 1) : Now, if we multiply (3.7) by 1 t, (3.8) by t and add the obtained inequalities, then we get (3.9) ' for any t 2 (0; 1) ; that is of interest in itself as well. Now, if we integrate this inequality on [0; 1] we get and then by (3.11) we get the desired inequality (3.7).

Conclusion 1.
The inequalities (2.5) and (3.6) are not comparable, meaning that some time one is better then the other, depending on the '-convex function involved.

Some Related Results
If we apply Theorem 2 on the subintervals x; x+y 2 and x+y 2 ; y (provided x < y) and add the corresponding inequalities we get: 3x + y 4 + f x + 3y 4 (4.1) Also, by Theorem 4 we have Proposition 2. Let ' : (0; 1) ! (0; 1) a measurable function and such that the right limit ' + (0) exists and is …nite, the left limit ' (1) = 1 and the left derivative in 1 denoted ' 0 (1) exists and is …nite. Assume also that ' 0 (1) > 1: If the function f : I ! [0; 1) is di¤ erentiable on I and '-convex, then for any x; y 2 I: Now we can prove the following result as well: Theorem 5. Let ' : (0; 1) ! (0; 1) a measurable function and such that the right limit ' + (0) exists and is …nite, the left limit ' (1) = 1 and the left derivative in 1 denoted ' 0 (1) exists and is …nite. Assume also that ' 0 (1) > 2: If the function f : I ! [0; 1) is di¤ erentiable on I and '-convex, then for any x; y 2 I: Proof. Assume that x < y: From the inequality (3.1) we have for any u 2 [x; y] with u 6 = x+y 2 : Integrating over u 2 [x; y] and dividing by y x we have x + y 2 u du: Integrating by parts, we have Z y and by (4.5) we get which is equivalent to Since ' 0 (1) + 2 > 0; then on dividing by ' 0 (1) + 2 we get the desired result (4.3).