Some new inequalities of Hermite-Hadamard type for GA-convex functions

Some new inequalities of Hermite–Hadamard type for GA-convex functions defined on positive intervals are given. Refinements and weighted version of known inequalities are provided. Some applications for special means are also obtained.

It has been shown in [28] that the function f : (0, ∞) → R defined by is GA-concave on (0, ∞) while the function g : (0, ∞) → R defined by is GA-convex on (0, ∞). We recall that the classical Hermite-Hadamard inequality states that for any convex function f : [a, b] → R.
The differentiability of the function is not necessary in Theorem 1 for the first inequality (1.4) to hold, as shown in [10].
If we take λ = 1 2 in the definition (1.1) of GA-convex (concave) function on [a, b], then we have The following refinement of (1.5), which is an inequality of Hermite-Hadamard type, holds (see [25] for an extension for GA h-convex functions): Motivated by the above results we provide in the following a refinement of (1.6) for a division of the interval [a, b]. We also establish a weighted version that generalizes the inequalities (1.4) and (1.6) and provides upper and lower bounds for the moments 2. Some refinements. In 1994, [5] (see also [17, p. 22]) we proved the following refinement of Hermite-Hadamard inequality. For the sake of completeness we give here a direct proof that is different from the one in [5].
Then for any division c = x 0 < x 1 < · · · < x n−1 < x n = d with n ≥ 1 we have the inequalities Proof. Using the Hermite-Hadamard inequality on the interval Summing in (2.2) over i ∈ {0, . . . , n − 1} and dividing by d − c, we get the second and the third inequalities in (2.1).
Since for which proves the first inequality in (2.1). For a convex function g : [c, d] → R we have This implies that and for any i ∈ {0, . . . , n − 1}. Therefore, and the last part of (2.1) is proved.
Then for any division a = t 0 < t 1 < · · · < t n−1 < t n = b with n ≥ 1 we have the inequalities Proof.
If we write the inequality (2.1) for g = f • exp on the interval [c, d] and for the division ln a = ln t 0 < ln t 1 < · · · < ln t n−1 < ln t n = ln b, we have that is equivalent to By using the change of variable exp (x) = t, we have x = ln t, dx = dt t and and by (2.5) we get the desired result (2.3).
In the following section we establish some weighted Hermite-Hadamard type inequalities for GA-convex functions.

Weighted inequalities.
We have the following weighted inequality: By the convexity of f • exp we have and the second inequality in (3.1) is proved.
a w (t) dt and the first part of (3.1) is proved.
Proof. If we take w (t) = 1 t in (3.1), then we have Define the p-logarithmic mean as , then for any p ∈ R with p = 0, −1 we have If p = 0, then we have Proof. If we take w (t) = t p , with p = 0, −1 in (3.1), then we have From (3.7) we then have i.e.
By multiplying this inequality with L p p (a, b), we get If we perform the calculations in the above inequalities for p = 0, we get the desired inequality (3.6). We omit the details.

Remark 2.
If we take p = 1 in (3.5), then we get