Generalized Steffensen’s inequality by Lidstone interpolation and Montogomery’s identity

In this paper, first we present some interesting identities associated with Green’s functions and Fink’s identity, and further we present some interesting inequalities for r-convex functions. We also present refinements of some Hardy–Littlewood–Pólya type inequalities and give an application to the Shannon entropy. Furthermore, we use the Čebyšev functional and Grüss type inequalities and present the bounds for the remainder in the obtained identities. Finally, we use the obtained identities together with Hölder’s inequality for integrals and present Ostrowski type inequalities.

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“…Now if we use the values of G δ (x,ũ) over the intervals [s, x] and [x, t], then the following result is valid (see [4] and [13]).…”

Section: Theorem 15 Let Q Imentioning

confidence: 95%

Refinements of some Hardy–Littlewood–Pólya type inequalities via Green’s functions and Fink’s identity and related results

Khalid

Pečarić

2020

J Inequal Appl

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“…(A 2 ) For n ∈ N, n ≥ 3, let ψ : [0, d] → R be an n times differentiable function with ψ(0) = 0 and ψ (n-1) absolutely continuous on [0, d]. The first part of this section is the generalization of (5). For this, we start with the following theorem.…”

Section: Generalization Of Steffensen's Inequality By Generalized Monmentioning

confidence: 99%

Generalized Steffensen’s inequality by Montgomery identity

et al. 2019

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By using generalized Montgomery identity and Green functions we proved several identities which assist in developing connections with Steffensen's inequality. Under the assumptions of n-convexity and n-concavity many inequalities, which generalize Steffensen's inequality, inequalities from (Fahad et al. in their reverse, have been proved. Generalization of some inequalities (and their reverse) which are related to Hardy-type inequality (Fahad et al. in J. Math. Inequal. 9:481-487, 2015) have also been proved. New bounds of Ostrowski and Grüss type inequalities have been developed. Moreover, we formulate generalized Steffensen-type linear functionals and prove their monotonicity for the generalized class of (n + 1)-convex functions at a point. At the end, we present some applications of our study to the theory of exponentially convex functions. MSC: Primary 26D10; secondary 26D20Keywords: Steffensen's inequality; Green's function; Montgomery's identity; (n + 1)-convex function at a point c+θ c ψ(z) dz, where θ = d c f (z) dz.( 1 )A massive literature body dealing with several variants and improvements of (1) can be seen in [14,16] and the references therein. Pečarić [13] gave a nice generalization of (1) as follows.

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“…To proceed further, we recall a nice generalization of Steffensen's inequality proved by Pečarić, see [11]. Following two lemmas will be useful in our construction as well, see [14,15].…”

Section: Introductionmentioning

confidence: 99%

Generalized Steffensen’s Inequality by Fink’s Identity

2019

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By using Fink’s Identity, Green functions, and Montgomery identities we prove some identities related to Steffensen’s inequality. Under the assumptions of n-convexity and n-concavity, we give new generalizations of Steffensen’s inequality and its reverse. Generalizations of some inequalities (and their reverse), which are related to Hardy-type inequality. New bounds of Gr u ¨ ss and Ostrowski-type inequalities have been proved. Moreover, we formulate generalized Steffensen’s-type linear functionals and prove their monotonicity for the generalized class of ( n + 1 ) -convex functions at a point. At the end, we present some applications of our study to the theory of exponentially convex functions. .

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Generalized Steffensen’s inequality by Lidstone interpolation and Montogomery’s identity

Cited by 4 publications

References 9 publications

Refinements of some Hardy–Littlewood–Pólya type inequalities via Green’s functions and Fink’s identity and related results

Refinements of some Hardy–Littlewood–Pólya type inequalities via Green’s functions and Fink’s identity and related results

Generalized Steffensen’s inequality by Montgomery identity

Generalized Steffensen’s Inequality by Fink’s Identity

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