On Hadamard Type Inequalities Involving Several Kind of Convexity

In this paper, we not only give the extensions of the results given in [7] by Gill et al. for log-convex functions, but also obtain some new Hadamard type inequalities for log-convex, m-convex and (alpha,m)-convex functions.


Introduction
The following inequality is well known in the literature as Hadamard's inequality: where f : I → R is a convex function on the interval I of real numbers and a, b ∈ I with a < b. This inequality is one of the most useful inequalities in mathematical analysis. For new proofs, note worthy extension, generalizations and numerous applications on this inequality, see ( [1], [4], [5], [6], [9], [12]) where further references are given. Let I be on interval in R. Then f : I → R is said to be convex if for all x, y ∈ I and λ ∈ [0, 1], f (λx + (1 − λ) y) ≤ λf (x) + (1 − λ) f (y) (see [9, P.1]). Geometrically, this means that if K, L and M are three distinct points on the graph of f with L between K and M , then L is on or below chord KM.
Recall that a function f : I → (0, ∞) is said to be log-convex function, if for all x, y ∈ I and t ∈ [0, 1], one has the inequality (see [9, P.3]) It is said to be log-concave if the inequality in (1.2) is reversed.
In [7], P.M. Gill et al. established the following results: is the Logarithmic mean of the positive real numbers p, q (for p = q, we put L (p, p) = p).
For f a positive log-concave function, the inequality is reversed.
If f is a positive log-concave function, then For some recent results related to the Hadamard's inequalities involving two logconvex functions, see [11] and the references cited therein. The main purpose of this paper is to establish the general version of the inequalities (1.3) and new Hadamard type inequalities involving two log-convex functions or two m-convex functions or two (α, m)-convex functions using elementary analysis.

Main Results
We start with the following Theorem.
Theorem 2. Let f i : I ⊂ R → (0, ∞) (i = 1, 2, ..., n) be log-convex functions on I and a, b ∈ I with a < b. Then the following inequality holds: where L is a logarithmic mean of positive real numbers.
For f a positive log-concave function, the inequality is reversed.
We will now point out some new results of the Hadamard type for log-convex, m−convex and (α, m)-convex functions, respectively.
Theorem 3. Let f, g : I → (0, ∞) be log-convex functions on I and a, b ∈ I with a < b. Then the following inequalities hold: Proof. We can write Using the elementary inequality cd ≤ 1 2 c 2 + d 2 (c, d ≥ 0 reals) and equality (2.7), we have Since f, g are log-convex functions, we obtain Rewriting (2.8) and (2.9), we have Integrating both sides of (2.10) and (2.11) on [0, 1] over t, respectively, we obtain Combining (2.12) and (2.13), we get the desired inequalities (2.6). The proof is complete.
Theorem 4. Let f, g : I → (0, ∞) be log-convex functions on I and a, b ∈ I with a < b. Then the following inequalities hold: where L (. , .) is a logarithmic mean of positive real numbers.
Proof. From the inequality (2.10), we have for all a, b ∈ I and t ∈ [0, 1] .
Using the elementary inequality cd ≤ 1 2 c 2 + d 2 (c, d ≥ 0 reals) on the right side of the above inequality, we have Since f, g are log-convex functions, then we get .
If f is non-increasing m 1 −convex function and g is non-increasing m 2 −convex function on [a, b] for some fixed m 1 , m 2 ∈ (0, 1] , then the following inequality holds: Proof. Since f is m 1 -convex function and g is m 2 -convex function, we have for all t ∈ [0, 1]. It is easy to observe that Using the elementary inequality cd ≤ 1 2 c 2 + d 2 (c, d ≥ 0 reals), (2.20) and (2.21) on the right side of (2.22) and making the charge of variable and since f, g is non- Rewriting (2.23) and (2.24), we get the required inequality in (2.19). The proof is complete.