Generalizations of majorization inequality via Lidstone’s polynomial and their applications. (English) Zbl 1357.26030

Adil Khan, M.; Latif, N.; Pečarić, J.

Commun. Math. Anal. 19, No. 2, 101-122 (2016).

Summary: In this paper, we obtain the generalizations of majorization inequalities by using Lidstone’s interpolating polynomials and conditions on Green’s functions. We give bounds for identities related to the generalizations of majorization inequalities by using Chebyshev functionals. We also give Grüss type inequalities and Ostrowski-type inequalities for these functionals. We present mean value theorems and \(n\)-exponential convexity which leads to exponential convexity and then log-convexity for these functionals. We give some families of functions which enable us to construct a large families of functions that are exponentially convex and also give Stolarsky type means with their monotonicity.

MSC:

26D15	Inequalities for sums, series and integrals
26D20	Other analytical inequalities

Keywords:

majorization inequailty; Lidstone polynomial; Green function; (\(2n\))-convex function; Chebyshev functional; Grüss-type inequality; Ostrowski-type inequality; \(n\)-exponentially convex function; mean value theorems; Stolarsky type means