he 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.
Faraz Mehmood
Assistant Professor of Mathematics, Dawood University of Engineering and Technology, Karachi, Pakistan.
View Book:- https://stm.bookpi.org/FNIPTIISI/issue/view/17
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