Abstract. The aim of this work is to establish several inequalities for functions whose first derivative in absolute value are (s,m) -convex in the second sense. Some estimates to the left hand side of the Hermite-Hadamard type inequality for (s,m) -convex functions in the second sense are given.Mathematics subject classification (2010): 26D10, 39B62.

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In this work we investigate a natural preorder on c0, the Banach space of all real sequences tend to zero with the supremum norm, which is said to be ?convex majorization?. Some interesting properties of all bounded linear operators T : c0 ? c0, preserving the convex majorization, are given and we characterize such operators.

In this paper, we consider an equivalence relation ∼ c on p (I), which is said to be "convex equivalent" for p ∈ [1, +∞) and a nonempty set I. We characterize the structure of all bounded linear operators T : p (I) → p (I) that strongly preserve the convex equivalence relation. We prove that the rows of the operator which preserve convex equivalent, belong to 1 (I). Also, we show that any bounded linear operators T : p (I) −→ p (I) which preserve convex equivalent, also preserve convex majorization.