The subject of functions with non-decreasing increments has gained prominence in many branches of mathematics in recent decades, and due to its relevance, it has attracted the attention of many mathematicians. We’d like to figure out what the relationship(s) are between functions with non-decreasing increments and the arithmetic integral mean, Wright convex functions, convex functions, and arithmetic integral mean. ∇−convex functions, Jensen m−convex functions, m−convex functions, m−∇−convex functions, k−monotonic functions, absolutely monotonic functions, completely monotonic functions, Laplace Transform and exponentially convex functions, by using the ﬁnite diﬀerence operator as diﬀerent cases of . We also consider function with nondecreasing increments of third order and obtain the generalizations of the Levinson’s-type inequality and Jensen-Mercer’s-type inequality by using Jensen-Boas inequality. We will deduce some general identities of Popoviciu type for discrete case for sums for function and sequence in two dimensions using higher order ∇ divided difference, positivity of these expressions are characterized for higher order ∇−convex functions. We will also obtain some general identities of Popoviciu type for integral of higher order diﬀerentiable function and positivity of these expressions are characterized for higher order ∇−convex and completely monotonic functions. We would discuss some applications in terms of generalized Cauchy-type means and exponential convexity as well. We would get the generalization of discrete identity and inequality of Čebyšev-type and discuss generalization of integral identities and inequalities of Čebyšev-type and Ky Fan-type for higher order ∇−convex functions with two variables. The subject of functions with non-decreasing increments has gained prominence in many branches of mathematics in recent decades, and due to its relevance, it has attracted the attention of many mathematicians. We’d like to figure out what the relationship(s) are between functions with non-decreasing increments and the arithmetic integral mean, Wright convex functions, convex functions, and arithmetic integral mean.

Author(s) Name

**Faraz Mehmood
**Assistant Professor of Mathematics, Dawood University of Engineering and Technology, Karachi, Pakistan.

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