On the K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {K}$$\end{document}-Riemann integral and Hermite–Hadamard inequalities for K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {K}$$\end{document}-convex functions

In the present paper we introduce a notion of the K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {K}$$\end{document}-Riemann integral as a natural generalization of a usual Riemann integral and study its properties. The aim of this paper is to extend the classical Hermite–Hadamard inequalities to the case when the usual Riemann integral is replaced by the K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {K}$$\end{document}-Riemann integral and the convexity notion is replaced by K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {K}$$\end{document}-convexity.


Introduction
Throughout this paper I ⊆ R stands for an interval and K denotes a subfield of the field of real numbers R. Clearly, Q ⊆ K, where Q denotes the field of rational numbers. We denote the set of the positive elements of K by K + . In the sequel the symbol [a, b] A will denote an A-convex hull of the set {a, b}, where A ⊆ R i.e.

Construction of the K-Riemann integral
Now, we introduce a notion of the K-Riemann integral as a natural generalization of the classical Riemann integral. For the theory of the classical Riemann integral see for instance [10,14,15].
Let P [a,b] denote the set of partitions of the interval [a, b] i.e.
Following Zs. Páles [12] we define the set of K-partitions of the interval [a, b] in the following way For a given K-partition π = (t 0 , t 1 , . .
These suprema and infima are well-defined, finite real numbers since f is bounded on [a, b] K . Moreover, We define the upper K-Riemann sum of f with respect to the partition π by and the lower K-Riemann sum of f with respect to the partition π by Note that if its upper and lower integrals are equal. In that case, the K- In the case when K = R we will use the standard symbol The following theorem gives a criterion for K-Riemann integrability.
Proof. Let ε > 0 and choose a partition π ∈ P K [a,b] that satisfies the above condition.
Since this inequality holds for every Conversely, suppose that f is K-Riemann integrable. Given any ε > 0, there are partitions π 1 , π 2 ∈ P K [a,b] such that Vol. 91 (2017) On K-Riemann integral and the  Now, let π := π 1 ∪ π 2 be the common refinement. Keeping in mind that the Using the above theorem we can easily obtain the following By the K 2 -Riemann integrability, for any choice s As an immediate consequence of the above proposition we obtain the following.

Corollary 11. If a function f : [a, b] → R is Riemann integrable in the usual sense, then for an arbitrary field
It is easy to observe that f is K 1 -Riemann integrable, and b a f (t)d K1 t = 0. On the other hand for every partition π ∈ P K2 [a,b] \ P K1 [a,b] one can check that S K2 (π, f ) = 1, and L K2 (π, f ) = 0.
Observe that if we replace in the formula on f the set K 1 by the set D of diadic numbers from the interval [0, 1] i.e. D := x ∈ [0, 1] | x = k 2 n , k ∈ Z, n ∈ N , then we obtain an example of a function which is non-K-Riemann integrable for any subfield K ⊆ R.

Properties of the K-Riemann integral
We start our investigation with the following.
Fix an arbitrary sequence of partitions Hence, summing a telescoping series, we get It follows that U (f, π n ) − L(f, π n ) → 0 as n → ∞ and Corollary 9 implies that f is K-Riemann integrable. The proof for a monotonic decreasing function f is similar.
In our next result we use a well-known fact from mathematical analysis that every uniformly continuous function on a set A ⊂ R n can be uniquely extended onto clA to a continuous function (see for instance [4] On account of Corollary 11 f is K-Riemann integrable, moreover, In the sequel we will use the following well-known theorems (see [8] p.147) (actually these theorems were proved for Jensen-convex functions, but the proof in our case runs without any essential changes).  for arbitrary a, b ∈ I, a < b the function f |[a, The function g ab satisfies the inequality for every x, y ∈ [a, b], in particular g ab is a convex function.
Now, we calculate an integral of a K-linear function. Note that such a function can be discontinuous at every point and non-measurable in the Lebesgue sense (see [8]), so the usual Riemann integral may not exist.
Proof. Suppose that f is a K-linear function. On account of Proposition 13 and Theorem 14 it is K-Riemann integrable on every interval [a, b]. Consider the following sequence of partitions: Now, we record some basic properties of K-Riemann integration. We omit the proofs of these properties because they run in a similar way as for the usual Riemann integral.

Vol. 91 (2017)
On K-Riemann integral and the  Theorem 17. Let f, g be K-Riemann integrable on [a, b] and let c, d ∈ R. Then Let π be the refinement of π obtained by adding c to the endpoints of π. Then π = π 1 ∪ π 2 , where Obviously, π 1 ∈ P K [a,c] and π 2 ∈ P K [c,b] , moreover, It follows that which proves that f is K-Riemann integrable on [a, c]. Exchanging π 1 and π 2 , we get the proof for [c, b].
Conversely, if f is K-Riemann integrable on [a, c] and [c, b] then there are partitions π 1 ∈ P K [a,c] and π 2 ∈ P K [c,b] such that We will say that f is radially K-differentiable at a point x whenever D K f (x, u) does exist for every u ∈ R. A function f : It is known that each K-convex function f : I → R is radially K-differentiable. In particular, such is every K-linear function a : R → R with On the other hand, if a function f : I → R is differentiable in the usual sense at a point x ∈ I then it is radially K-differentiable at x with We have the following relationship between the radial K-derivative and the K-Riemann integral. Then, if f is radially K-continuous at a point x ∈ (a, b] then F is radially K-differentiable at x in the direction x − a, moreover, Proof. Fix x ∈ [a, b] and ε > 0 arbitrarily. For α ∈ K + , since x ∈ [a, x + α(x − a)] K , on account of Theorem 18 and condition (iii) from Theorem 17 we obtain Let α ∈ K + be so small that