Generalization of Montgomery identity via Taylor formula on time scales

Sumaiya Malik,Khalid Mahmood Awan,Khuram Ali Khan,Ammara Nosheen

doi:10.1186/s13660-022-02759-3

Sumaiya Malik, Khalid Mahmood Awan + Show 2 more

Open Access

https://doi.org/10.1186/s13660-022-02759-3

Copy DOI

Abstract

In the current paper, a generalized Montgomery identity is obtained with the help of Taylor’s formula on time scales. The obtained identity is used to establish Ostrowski inequality, mid-point inequality, and trapezoid inequality. Moreover, the weighted versions of generalized Montgomery identity and respective Ostrowski inequality are also discussed. Special cases are obtained for different time scales to obtain new and existing results.

Highlights

IntroductionAn identity due to Montgomery is used to acquire various novel inequalities, for example, Ostrowski type inequality, trapezoid inequality, Mohajani inequality, Čebysěv and Grüss inequalities
An identity due to Montgomery is used to acquire various novel inequalities, for example, Ostrowski type inequality, trapezoid inequality, Mohajani inequality, Čebysěv and Grüss inequalities.The Montgomery identity given by Pečaríc in [18] is expressed as follows: Let g : [c1, d1] → R and g : [c1, d1] → R be integrable, d1 d1 g(x) =g(p) dp + R(x, p)g (p) dp, (1) d1 – c1 c1 c1 where R(x, p) = p–c1 d1 –c1, p–d1 d1 –c1c1 ≤ p ≤ x, . x < p ≤ d1
The obtained Montgomery identity [3, 14,15,16,17] is further used for time-scaled trapezoid and Ostrowski type inequalities

Summary

Introduction

An identity due to Montgomery is used to acquire various novel inequalities, for example, Ostrowski type inequality, trapezoid inequality, Mohajani inequality, Čebysěv and Grüss inequalities. The Montgomery identity given by Pečaríc in [18] is expressed as follows: Let g : [c1, d1] → R and g : [c1, d1] → R be integrable, d1. Bohner and Matthews [8] used time scale theory as a reference to obtain the time-scaled Montgomery identity and particular Ostrowski inequality. The weighted Montgomery identity given in [21] on time scales is stated as follows. An extension of Montgomery identity (3) is obtained by using the time scale versions of Taylor series which can be found in [1, 2, 11]. The obtained Montgomery identity [3, 14,15,16,17] is further used for time-scaled trapezoid and Ostrowski type inequalities. The extension of (4) and the respective Ostrowski inequality is discussed

Preliminary results

Ostrowski type inequality

Conclusion