This paper provides a method to generalize and refine the Hadamard's inequality on a hypercube in
For a convex function
In [
In [
In this paper, we first establish a mapping in Section
In Section
In Section
All the above Hadamard’s inequalities (
Thus, by establishing a family of the mappings, we have provided a method to get Hadamard’s inequality of the convex function on a hypercube in
The main properties of the mapping
If the mapping one has the bounds:
the mapping there is a point one obtains, the classical inequality (
(i) Let
(ii) By the convexity of
(iii) Let
(iv) We define the function
(v) Considering the properties (ii), (iii), and (iv) above, we have proved the Hadamard’s inequality (
We consider a convex function with two variables defined on a square in two-dimensional Euclidean space.
If the mapping one has the bounds
the mapping there is a point one has the Hadamard’s type inequality (
For a fixed
(i) Let
(ii) By the convexity of
(iii) The monotonicity of
(iv) We define the function
(v) Considering the properties (ii), (iii), and (iv) above, we have proved the Hadamard’s type inequality (
In recent years, several authors have discussed the problem of extending the classical inequality (
It is a much more delicate problem to extend the right side of (
When the convex function
For a closed hyperball
Now, let
Some mappings associated to Hadamard’s inequality have been established in [