New time scale generalizations of the Ostrowski-Grüss type inequality for k points

Two Ostrowski-Grüss type inequalities for k points with a parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\lambda\in[0, 1]$\end{document}λ∈[0,1] are hereby presented. The first generalizes a recent result due to Nwaeze and Tameru, and the second extends the result of Liu et al. to k points. Many new interesting inequalities can be derived as special cases of our results by considering different values of λ and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$k\in\mathbb{N}$\end{document}k∈N. In addition, we apply our results to the continuous, discrete, and quantum time scales to obtain several novel inequalities in this direction.


Introduction
In , Dragomir and Wang [] (see also [, ] for related results) obtained the following inequality which is today known as the Ostrowski-Grüss inequality.
Theorem  If f : [a, b] → R is differentiable on [a, b] and γ ≤ f (x) ≤ for all x ∈ [a, b] for some constants γ , ∈ R, then for all x ∈ [a, b].
With the introduction of the theory of time scales (see Section ), Tuna and Daghan [] obtained the following time scale version of the Ostrowski-Grüss type inequality. Specifically, they proved the following.
Theorem  Let a, b, x, t ∈ T, a < b, and f : [a, b] → R be differentiable. If f is rdcontinuous and γ ≤ f (t) ≤ for all t ∈ [a, b] and for some γ , ∈ R, then for all x ∈ [a, b], we have Recently, Nwaeze and Tameru [] proved the following generalization of Theorem  to k points.

Theorem  Suppose that
. a, b ∈ T, I k : a = x  < x  < · · · < x k- < x k = b is a partition of the interval [a, b] for x  , x  , . . . , x k ∈ T; . α j ∈ T (j = , , . . . , k + ) is k +  points so that α  = a, α j ∈ [x j- , x j ] (j = , . . . , k) and ]. Then we have the following inequality: Inequality () is sharp in the sense that the constant / on the right-hand side cannot be replaced by a smaller one.
Many other variants of the Ostrowski-Grüss type inequality (on time scales) and related results (see, for example, [-, , ] and the references therein) are bound in the literature. For the sake of this work, we present next a recent result due to Liu et al. [].
The aim of this work is the following: generalize Theorem  via a parameter λ ∈ [, ] such that for λ = , we recover Theorem , and for λ ∈ (, ] we get completely new results. Next, we extend Theorem  to k points. This paper is organized as follows. In Section , we recall some definitions and results of the time scale theory. Thereafter, our results are stated and proved in Section . Finally, we apply our results to different time scales in Section .

Time scale essentials
In order to unify the theory of integral and differential calculus with the calculus of finite difference, the German mathematician Stefan Hilger [] in  introduced the concept of time scales. We now present a brief overview of the theory of time scales. For an in-depth study, we invite the interested reader to see references [, ].
A time scale T is an arbitrary nonempty closed subset of R. We assume throughout that a time scale T has the topology that it inherits from the real numbers with the standard topology. Since a time scale may not be connected, we need the following concept of jump operators.
The forward jump operator σ : T → T is defined by while the backward jump operator ρ : T → T is defined by In this definition, we put inf where ∅ denotes the empty set. The jump operators σ and ρ allow the classification of points in T in this manner Points that are right-scattered and left-scattered at the same time are called isolated. Also, if t < sup T and σ (t) = t, then t is called right-dense, and if t > inf T and ρ(t) = t, then t is called left-dense. Points that are right-dense and left-dense at the same time are called dense. We also introduce the sets T k , T k , and T k k , which are derived from the time scale T as follows: if T has a left-scattered maximum t  , then Open intervals and half-open intervals are defined in the same manner.
Definition  (Delta derivative) Assume that f : T → R is a function. Then the delta derivative f (t) ∈ R at t ∈ T k is defined to be number (provided it exists) with the property that, for any > , there exists a neighborhood U of t such that Theorem  Let f , g : T → R be two differentiable functions at t ∈ T k . Then the product fg : T → R is also differentiable at t with Definition  The function f : T → R is said to be rd-continuous if it is continuous at all dense points t ∈ T and its left-sided limits exist at all left-dense points t ∈ T.
Definition  Let f be an rd-continuous function. Then g : T → R is called the antiderivative of f on T if it is differentiable on T and satisfies g (t) = f (t) for any t ∈ T k . In this case, . -When T = R, then for all s, t ∈ T, h k (t, s) = (ts) k k! .
-When T = Z, then for all s, t ∈ T,

Main results
In this section, we will state and prove two Ostrowski-Grüss type inequalities with a parameter λ. For this, we will need the following lemma which is given in [, Lemma ] but with some typos. We present here the correct version.
Lemma  (Generalized Montgomery identity with a parameter) Suppose that ()

Generalized Ostrowski-Grüss type inequality with a parameter I
We now state and prove our first result.
]. Then we have the following inequality: Inequality () is sharp in the sense that the constant / on the right-hand side cannot be replaced by a smaller one.
Proof To proceed, we will need to make the following computations. For this, we apply the items of Theorem , where applicable, to get b a K(t, Following a similar approach, one gets b a K(t, The desired inequality follows by using equations (), (), and () in inequality ().
Remark  By setting λ =  in Theorem , we regain Theorem .

Generalized Ostrowski-Grüss type inequality with a parameter II
Next, we present a generalization of Theorem  to k points.
Then we have the following inequality: where K(t, I k ) is defined by ().
Proof We start by making the following computations: () Following the same process, one gets the following identities: Using the Cauchy-Schwarz inequality on time scales, we get Inequality () is achieved by applying ()-() and the definition of definite integral (given in Section  above) to ().

Application to different time scales
In this section, we apply our theorems to different time scales to obtain completely new inequalities. We start with Theorem .
Corollary  Let T = R. Then we have Proof The proof follows by applying Theorem  and using the facts that f σ (t) = f (t) and h  (t, s) = (t-s)   (from the first item of Definition  for the case k = ).