Generalized results of majorization inequality via Lidstone's polynomial and newly Green functions

Abstract

Generalized results of majorization inequality are obtained by using newly Green functions defined in [N. Mahmood, R. P. Agarwal, S. I. Butt, J. Pečarić, J. Inequal. Appl., 2017 (2017), 17 pages] and Lidstone's polynomial. We find new upper bounds of Grüss and Ostrowski type. We give further results of majorization inequality by making linear functionals constructed on convex functionsx . Some applications are given.

Majorization inequalities via Green functions and Fink’s identity with applications to Shannon entropy

“…. Therefore using (33) in 28, we have (34). Moreover, if inequality (33) is reversed, then using (28), we obtain (34) in the reverse direction.…”

“…Remarks 1 (i) We can give upper bounds like Grüss-and Ostrowski-type inequalities for our generalized identities. We can also obtain Lagrange-and Cauchy-type mean value theorems by using linear functionals deduced from our generalized results (see, e.g., [27,32,34]). (ii) We can use an elegant method introduced by Jakšetić and Pečarć [17,18] (see also [22,30]) to obtain n-exponential convexity, exponential convexity, and log-convexity with the help of linear functionals deduced from our generalized results on a given family with the same property for both discrete and integral cases.…”

“…where ξ 1 , ξ 4 = ϑ 2 and ξ 2 , ξ 3 = ϑ 1 . Moreover, (34) holds in the reverse direction if inequality (33) is reversed.…”

“…In 2018, Latif et al [27] studied generalized results related to majorization inequality by using Taylor's polynomial in combination with newly introduced Green functions. In the same year, Siddique et al [34] gave generalized majorization results via Lidstone's polynomial and newly defined Green functions. The theory of majorization often appears in many fields of applications.…”

Difference equations related to majorization theorems via Montgomery identity and Green’s functions with application to the Shannon entropy

“…In 2018, Latif et al [29] studied generalized results related to the majorization inequality by using Taylor's polynomial in combination with newly introduced Green's functions. In the same year, Siddique et al [44] gave generalized majorization results via Lidstone's polynomial and newly defined Green's functions. The theory of majorization is widely used in many fields of application.…”