On Some New Weighted Steffensen-Type Inequalities on Time Scales

: The motivation of this paper is to explore some new inequalities of Steffensen-type which were demonstrated by Peˇcari´c and Kalamir in 2014. The main idea is to investigate a class of certain inequalities by employing diamond- α dynamic integral on time scales. In addition, to obtain some new inequalities as special cases, we also extend our results to continuous and discrete calculations.


Introduction
In 2014, Pečarić and Kalamir in [1] have established the following interesting theorem: Theorem 1. Let h : [a, b] → R be a positive integrable function, f : [a, b] → R be an integrable function, and let c ∈ (a, b). Further, assume that g : [a, b] → R is integrable function such that 0 ≤ g(t) ≤ 1 for all t ∈ [a, b]. Moreover, let λ 1 be the solution of the equation  In 1988, Stefan Hilger [2] initiated the theory of time scales in their PhD thesis [3] in order to unify discrete and continuous analysis. The book by Bohner and Peterson [4] on the subject of time scales briefs and organizes much of time scales calculus.
We define the open intervals and half-closed intervals similarly. We will use the following crucial relations between calculus on time scales T and either differential calculus on R or difference calculus on Z. Note that: (1) where ∆ and ∇ are the forward and backward difference operators, respectively. the diamond-α dynamic derivative of f at t is defined by For more details on the diamond-α calculus on time scales, we refer the interested reader to [5].
Anderson [16] was the first to extend the Steffensen inequality to a general time scale. In particular, he gave the following result. Theorem 2. Suppose that a, b ∈ T κ with a < b, and f , g : [a, b] T → R are ∇-integrable functions such that f is of one sign and nonincreasing and 0 ≤ g(t) ≤ 1 on [a, b] T . Further, assume that In [17], Özkan and Yildirim established the following results regarding diamond-α dynamic Steffensen-type inequalities.
where λ is the solution of the equation If f /h is nondecreasing, then the reverse inequality in (4) holds.
where λ is the solution of the equation If f /h is nondecreasing, then the reverse inequality in (5) holds.

Theorem 5.
Let h be a positive integrable function on [a, b] T and f , g, ψ be integrable functions on [a, b] T such that f is nonincreasing and where λ is the solution of the equation Theorem 6. Let f , g and h be ♦ α -integrable functions defined on [a, b] T with f nonincreasing.
where λ is given by In this paper, we extend some generalizations of integral Steffensen's inequality given in [1] to a general time scale, and establish several new sharpened versions of diamond-α dynamic Steffensen's inequality on time scales. As special cases of our results, we recover the integral inequalities given in these papers. Our results also give some new discrete Steffensen's inequalities. We obtain the new dynamic Steffensen inequalities using the diamond-α integrals on time scales. For α = 1, the diamond-α integral becomes delta integral and for α = 0 it becomes nabla integral. Now, we are ready to state and prove the main results of this paper.

Main Results
Let us begin by introducing a class of functions that extends the class of convex functions. We shall need the following lemmas in the proof of our results.

Lemma 1. Let h be a positive integrable function on [a, b] T and f , g be integrable functions on
where λ is the solution of the equation If f /h is nondecreasing, then the reverse inequality in (6) holds.

Lemma 2.
Let h be a positive integrable function on [a, b] T and f , g be integrable functions on where λ is the solution of the equation If f /h is nondecreasing, then the reverse inequality in (7) holds.
where λ is the solution of the equation where λ is given by and let λ 2 be the solution of the equation (8) holds, the inequality in (9) is reversed. Since Now, from (10) and (11) we obtain Hence, if (8) is satisfied, then (9) holds. It is similar for

Corollary 1.
Setting α = 1 in Theorem 7, we obtain the delta version of inequality (9) as follows: Corollary 2. Setting α = 0 in Theorem 7, we obtain the nabla version of inequality (9) as follows: and let λ 2 be the solution of the equation If f σ /h ∈ AH c 2 [a, b] and (15) holds, the inequality in (16) is reversed.
F/h : [c, b] T → R is nondecreasing, so from Lemma 1 we obtain Hence, from (17) and (18) and let λ 2 be the solution of the equation If f σ /h ∈ AH c 2 [a, b] and (22) holds, the inequality in (23) is reversed.
Proof. Take g → g/h and f → f h in Theorem 7.

Corollary 9.
Setting α = 1 in Theorem 9, we obtain the delta version of inequality (23) as follows: Corollary 10. Setting α = 0 in Theorem 9, we obtain the nabla version of inequality (23) as follows: Theorem 10. Let h : [a, b] T → R be a positive ♦ α -integrable function, let f : [a, b] T → R be a ♦ α -integrable function, and let c ∈ (a, b). Let g : [a, b] T → R be a ♦ α -integrable function such that 0 ≤ g(t) ≤ h(t) for all t ∈ [a, b] T . Let λ 1 be the solution of the equation and let λ 2 be the solution of the equation (27) holds, the inequality in (28) is reversed.
Proof. Take g → g/h and f → f h in Theorem 8.

Corollary 13.
Setting α = 1 in Theorem 10, we obtain the delta version of inequality (28) as follows: Theorem 11. Let h : [a, b] T → R be a positive ♦ α -integrable function, let f : [a, b] T → R be a ♦ α -integrable function, and let c ∈ (a, b). Let g, k : [a, b] T → R be a ♦ α -integrable function such that 0 ≤ g(t) ≤ k(t) for all t ∈ [a, b] T . Let λ 1 be the solution of the equation If f σ /h ∈ AH c 2 [a, b] and (32) holds, the inequality in (33) is reversed.
Proof. Take h → kh, g → g/k and f → f k in Theorem 7.
Corollary 17. Setting α = 1 in Theorem 11, we obtain the delta version of inequality (33) as follows: Corollary 18. Setting α = 0 in Theorem 11, we obtain the nabla version of inequality (33) as follows: and let λ 2 be the solution of the equation If f σ /h ∈ AH c 2 [a, b] and (37) holds, the inequality in (38) is reversed.
Proof. Take h → kh, g → g/k and f → f k in Theorem 8.

Corollary 21.
Setting α = 1 in Theorem 12, we obtain the delta version of inequality (38) as follows: Corollary 22. Setting α = 0 in Theorem 12, we obtain the nabla version of inequality (38) as follows: and let λ 2 be the solution of the equation Corollary 26. Setting α = 0 in Theorem 13, we obtain the nabla version of inequality (43) as follows: