Five Mappings in Connection to Hadamard ’ s Inequality

In this paper we point out five new inequalities of the Hadamard’s type and use a simple new technique in the proof.


Introduction
Let f : I ⊂ ℜ → ℜ be a convex mapping of the interval I of real numbers and a, b ∈ I with a < b.The following double inequality is known in the literature as Hadamard's inequality.In [1], Fejer generalized the inequality (1.1) by proving that if g : The 0-convex functions are simply the log-convex functions and 1-convex functions are the ordinary convex functions.We define the following mappings: .
(vi) We have the inequality

Main Result
We prove the following Theorem 2.1.If f is r-convex (r = 0), then m(t) has the following properties: By the r-convexity of f , we have Therefore, we have By the convexity of m, we have which implies that m(t 2 ) ≥ m(t 1 ).

Proof. (i)
In this case we have to show that log n is convex.For this let t 1 , This implies Also, and this implies log n(t) ≤ log n (1).
Therefore, we have inf (iii) This follows exactly as in the case of Theorem 2.1.

Proof. (i)
It is similar to that given in Theorem 2.1.
Also, we have Therefore, we get inf Then, we have, by the convexity of M M (t 2 ) − M (t 1 ) Therefore M (t 2 ) ≥ M (t 1 ).That M (t) decreases on 0, 1  2 follows from (iii).
Theorem 2.4.If f is log-convex, then N (t) has the following properties (iv) N decreases on 0, 1 2 and increases on 1 Proof.(i) This is similar to that given in Theorem 2.2.
Also, we have Therefore, we get inf (iii) and (iv) Similar cases are proved before.