A real valued function f defined on a real open interval I is called Φ-monotone if, for all x, y ∈ I with x ≤ y it satisfies f (x) ≤ f (y) + Φ(y − x), where Φ : [0, ℓ(I)[ → R+ is a given nonnegative error function, where ℓ(I) denotes the length of the interval I. If f and −f are simultaneously Φ-monotone, then f is said to be a Φ-Hölder function. In the main results of the paper, using the notions of upper and lower interpolations, we establish a characterization for both classes of functions. This allows one to construct Φmonotone and Φ-Hölder functions from elementary ones, which could be termed the building blocks for those classes. In the second part, we deduce Ostrowski-and Hermite-Hadamard-type inequalities from the Φ-monotonicity and Φ-Hölder properties, and then we verify the sharpness of these implications. We also establish implications in the reversed direction.

A real valued function f defined on a real open interval I is called Φ-monotone if, for all x,y ∈ I with x y it satisfies f (x) f (y) + Φ(y − x), where Φ : [0, (I)[→ R + is a given nonnegative error function, where (I) denotes the length of the interval I. If f and − f are simultaneously Φ-monotone, then f is said to be a Φ-Hölder function. In the main results of the paper, using the notions of upper and lower interpolations, we establish a characterization for both classes of functions. This allows one to construct Φmonotone and Φ-Hölder functions from elementary ones, which could be termed the building blocks for those classes. In the second part, we deduce Ostrowski-and Hermite-Hadamardtype inequalities from the Φ-monotonicity and Φ-Hölder properties, and then we verify the sharpness of these implications. We also establish implications in the reversed direction.

In this paper, our primary objective is to study a possible decomposition of an approximately convex sequence.For a given ε > 0; a sequence un ∞ n=0 is said to be ε-convex, if for any i, j ∈ N with i < j there exists an n ∈]i, j] ∩ N such that the following discrete functional inequality holdsWe show that such a sequence can be represented as the algebraic summation of a convex and a controlled sequence which is bounded in between − ε 2 , ε 2 .On the other hand, if for any i, j ∈ N with i < j, if a sequence unthen we term it as ε-affine sequence. Such a sequence can be decomposed as the algebraic summation of an affine and a bounded sequence whose supremum norm doesn't exceed ε.Various definitions, backgrounds, motivations, findings, and other important things are discussed in the introduction section.

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