Abstract
We consider some new Steffensen-type dynamic inequalities on an arbitrary time scale by utilizing the diamond-α dynamic integrals, which are characterized as a combination of the delta and nabla integrals. These inequalities expand some known dynamic inequalities on time scales, bind together and broaden some integral inequalities and their discrete analogs.
Highlights
The renowned integral Steffensen inequality [28] is written as b b a+λ φ(t) dt ≤ φ(t)ψ(t) dt ≤ φ(t) dt, b–λ a a (1.1)where u is nonincreasing, λ = b a ψ (t) dt and ≤ ψ (t) on [a, b].It is simple to notice that inequalities (1.1) are reversed if u is nondecreasing
1 Introduction The renowned integral Steffensen inequality [28] is written as b b a+λ φ(t) dt ≤ φ(t)ψ(t) dt ≤
The following inequality is a special case of the above inequality: if we put z(t) = M and w(t) = 1, so b b φ(t)♦αt + M φ(t) – φ(b – λ) ♦αt b–λ a b
Summary
The renowned integral Steffensen inequality [28] is written as b b a+λ φ(t) dt ≤ φ(t)ψ(t) dt ≤. A few researchers created different outcomes concerning fractional calculus on time scales to deliver related dynamic inequalities (see [5,6,7, 24]). In [8], extended Steffensen’s inequality to times scale with nabla integrals as follows:. The following inequality is a special case of the above inequality: if we put z(t) = M and w(t) = 1, so b b φ(t)♦αt + M φ(t) – φ(b – λ) ♦αt b–λ a φ(t)♦αt – M φ(t) – φ(a + λ) ♦αt, a a a, b ∈ Tκκ with a < b, λ =. This article is about to extend some Steffensen-type inequalities given in [23] to a general time scale, and build up some new generalizations of the diamond-α dynamic Steffensen inequality on time scales. We get the unique Steffensen inequalities by utilizing the diamond-α integrals on time scales. An excellent review about the diamond-α calculus can be viewed in the paper [27]
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call