  In this paper, we primarily deal with approximately monotone and convex
sequences. We start by showing that any sequence can be expressed as the
difference between two nondecreasing sequences. One of these two monotone
sequences act as the majorant of the original sequence, while the other
possesses non-negativity. Another result establishes that an approximately
monotone(increasing) sequence can be closely approximated by a non-decreasing
sequence. A similar assertion can be made for approximately convex sequence. A
sequence $\big<u_n\big>_{n=0}^{\infty}$ is said to be approximately convex (or
$\varepsilon$-convex) if the following inequality holds under the mentioned
assumptions \begin{equation*} u_{i}- u_{i-1}\leq
u_{j}-u_{j-1}+\varepsilon\quad\quad\mbox{ where}\quad\quad i,j\in\mathbb{N}
\quad \mbox{with} \quad i<j. \end{equation*} We proved that an approximately
convex sequence can be written as the algebraic difference of two specific
types of sequences. The initial sequence contains sequential convexity
property, while the other sequence possesses the Lipschitz property. Moreover,
we introduce an operator $\mathbb{T}$, that termed as a twisting operator. In a
compact interval $I(\subseteq\mathbb{R})$; we characterized the convex function
with this newly introduced operator. Besides various sequence decomposition
results and study related to $\mathbb{T}$-operator on convex function; a
characterization regarding non-negative sequential convexity, a fractional
inequality, implication of $\mathbb{T}$ operator on various types of functions,
relationship between a convex function and a convex sequence are also included.
