Abstract
The main objective of the present article is to prove some new ∇ dynamic inequalities of Hardy–Hilbert type on time scales. We present and prove very important generalized results with the help of Fenchel–Legendre transform, submultiplicative functions. We prove the (γ,a)-nabla conformable Hölder’s and Jensen’s inequality on time scales. We prove several inequalities due to Hardy–Hilbert inequalities on time scales. Furthermore, we introduce the continuous inequalities and discrete inequalities as special case.
Highlights
We give several foundational definitions and notations of basic calculus of time scales
We begin with the definition of time scale
By implying (24), we study some new dynamic inequalities of Hardy–Hilbert type using nabla-integral on time scales
Summary
We give several foundational definitions and notations of basic calculus of time scales. Let f : T → R be a function defined on a time scale T. We introduce the nabla derivative of a function f : T → R at a point t ∈ Tκ as follows: Definition 6. For a function f : T → R, the nabla conformable fractional derivative, T∇,α f (t) ∈ R of order α ∈ Rahmat et al [11] presented a new type of conformable nabla derivative and integral bn (t, s) for s, t ∈ T. which involve the time-scale power function G bn : Definition 9. These inequalities maybe be used to obtain more generalized results of several obtained inequalities before by replace ψ, ψ∗ by specific substitution
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