Diversity of several estimates transformed on time scales

In this research article, we prove several generalizations of reverse Callebaut, Rogers–Hölder, and Cauchy–Schwarz inequalities via reverses of Young inequalities on time scales. Discrete, continuous, and quantum versions of the results are unified and extended on time scales.


Introduction
The calculus of time scales was accomplished by Stefan Hilger [7].A time scale is an arbitrary nonempty closed subset of the real numbers.Let T be a time scale, ξ , ω ∈ T with ξ < ω, and an interval [ξ , ω] T means the intersection of the real interval with the given time scale.The major aim of the calculus of time scales is to establish results in general, comprehensive, unified, and extended forms.This hybrid theory is also widely applied in dynamic inequalities, see [2,[8][9][10][11][12].The basic ideas about time scale calculus are given in the monographs [3,4].
We state here the different versions of reverses of Callebaut, Rogers-Hölder, and Cauchy-Schwarz inequalities, see [5].

Preliminaries
First, we present a short introduction to the diamond-α derivative as given in [1,13].
Let T be a time scale and f (τ ) be differentiable on T in the and ∇ sense.For τ ∈ T, the diamond-α dynamic derivative f α (τ ) is defined by The diamond-α derivative reduces to the standard -derivative for α = 1, or the standard ∇-derivative for α = 0.It represents a weighted dynamic derivative for α ∈ (0, 1).
The following definition is given in [13].
Let ξ , τ ∈ T and h : T → R. Then the diamond-α integral from ξ to τ of h is defined by provided that there exist delta and nabla integrals of h on T.
The following well-known Young inequality holds: For , > 0 and v ∈ [0, 1], we have The following inequalities are given in [5].
In this paper, it is assumed that all considered integrals exist and are finite.

Main results
In the following, we give an extension of reverse Callebaut inequality on time scales.Throughout this section, we assume that neither s ≡ 0 nor t ≡ 0.
Then the following inequalities hold true: Then using the inequalities (2.1) and (2.2), we have (3.4) Multiplying by |z(λ)| and integrating (3.4) with respect to λ from ξ to ω, we obtain The following reverse of Callebaut inequality holds: Then the following inequalities hold true: Proof Take v = 1 2 in Theorem 3.1, and the result follows.
The following another reverse of Callebaut inequality holds: Then the following inequalities hold true: The following another reverse of Callebaut inequality holds: Then the following inequalities hold true: (3.8) Proof Take v = 1 2 ν in Theorem 3.1, and the result follows.
In the following, we give an extension of reverse Rogers-Hölder inequality on time scales.
Then the following inequality holds true: Proof Using the given conditions, for λ ∈ [ξ , ω] T , we have (3.12) Integrating (3.13) with respect to λ from ξ to ω, we obtain This completes the proof of Theorem 3.2.
Next, we give an extension of reverse Cauchy-Schwarz inequality on time scales.
Then the following inequality holds true:

.15)
Proof Take p = q = 2 in Theorem 3.2, and the result follows.
Remark 3.1 We have the following:   Finally, we give another extension of reverse Rogers-Hölder dynamic inequality.

3). (iv) Let
Then the following inequalities hold true: , and v = 1 q .Then using the inequalities (2.1) and (2.4), respectively, we have Multiplying by |z(λ)||u 1 (λ)| and integrating (3.18) with respect to λ from ξ ω, we obtain Then the following inequalities hold true: Now, we give another extension of reverse Rogers-Hölder inequality on time scales.
Then the following inequalities hold true: Next, we give another extension of reverse Rogers-Hölder inequality on time scales.
Then the following inequalities hold true:

Proof Replace v by 1 2 ( 1 -
v) in Theorem 3.1, and the result follows.