A Variety of Dynamic Steffensen-Type Inequalities on a General Time Scale

: This work is motivated by the work of Josip Peˇcari´c in 2013 and 1982 and the work of Srivastava in 2017. By the utilization of the diamond- α dynamic inequalities, which are deﬁned as a linear mixture of the delta and nabla integrals, we present and prove very important generalized results of diamond- α Steffensen-type inequalities on a general time scale. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities.

Wu and Srivastava in [2] acquired the accompanying result.
Theorem 4. Suppose the integrability ofĝ,ĥ,f , ψ : [r 1 , r 2 ] → R such thatf is nonincreasing. Additionally, suppose that 0 ≤ψ(ı) ≤ĝ(ı) ≤ĥ(ı) −ψ(ı) for all ı ∈ [r 1 , r 2 ]. Then, where℘ is given by where℘ is given by The calculus of time scales with the intention to unify discrete and continuous analysis (see [4]) was proposed by Hilger [5]. For additional subtleties on time scales, we refer the reader to the book by Bohner and Peterson [6]. Additionally, understanding of diamond-α calculus on time scales is assumed, and we refer the interested reader to [7] for further details.
We devote the remaining part of this section to the diamond-α calculus on time scales, and we refer the interested reader to [7] for further details.
If T is a time scale, and ζ is a function that is delta and nabla differentiable on T, then, for any ı ∈ T, the diamond-α dynamic derivative of ζ at ı, denoted by ζ ♦ α (ı), is defined by We conclude from the last relation that a function ζ is diamond-α differentiable if and only if it is both delta and nabla differentiable. For α = 1, the diamond-α derivative boils down to a delta derivative, and for α = 0 it boils down to a nabla derivative.
Assume ζ, g : T → R are diamond-α differentiable functions at ı ∈ T, and let k ∈ R. Then, Let ζ : T → R be a continuous function. Then, the definite diamond-α integral of ζ is defined by Let a, b, c ∈ T, k ∈ R. Then, In this article, we explore new generalizations of the integral Steffensen inequality given in [1][2][3] via diamond − α integral on general time scale measure space. We also retrieve some of the integral inequalities known in the literature as special cases of our tests.

Main Results
Next, we use the accompanying suppositions for the verifications of our primary outcomes: ℘ is the solution of the equations listed below: Presently, we are prepared to state and explain the principle results that have had more effect effect from the literature. Theorem 6. Let S 1 , S 2 , S 3 , S 4 and S 11 be satisfied. Then, We obtain the reverse of (5) if ζ/Ξ is nondecreasing.

Proof.
[r 1 , The proof is complete.

Corollary 1. Delta version obtained from Theorem 6 by taking
Corollary 2. Nabla version obtained from Theorem 6 by taking α = 0
We obtain the reverse of (6) if ζ/Ξ is nondecreasing.
We will need the following lemma to prove the subsequent results. Then, and Proof. The suppositions of the Lemma imply that Now we have proved (9), we see that Since Combination of (11) and (12) led to the required integral identity (9) asserted by the Lemma. The integral identity (16) can be proved similarly. The proof is complete.
Theorem 11. If S 1 , S 2 , S 6 , S 7 and S 14 hold. Then, Proof. This proof is similar to the proof of the right-hand side inequality in Theorem 9.
Proof. Carry out the same proof of the left-hand side inequality in Theorem 9.

Conclusions
In this article, we explore new generalizations of the integral Steffensen inequality given in [1][2][3] by the utilization of the diamond-α dynamic inequalities which are used in various problems involving symmetry. We generalize a number of those inequalities to a general time scale measure space. In addition to this, in order to obtain some new inequalities as special cases, we also extend our inequalities to a discrete and constant calculus.

Conflicts of Interest:
The authors declare no conflict of interest.