Eat number of dynamic inequalities on time WZ8040 References scales has been establishedConsume variety of

Eat number of dynamic inequalities on time WZ8040 References scales has been established
Consume variety of dynamic inequalities on time scales has been established by many researchers who have been motivated by some applications (see [4,61]). Some researchers created various outcomes regarding fractional calculus on time scales to make connected dynamic inequalities (see [125]).Mathematics 2021, 9,3 ofAnderson [16] was the initial to extend the Steffensen inequality to a common time scale. In specific, he gave the following outcome. Theorem two. Suppose that a, b T using a b, and f , g : [ a, b]T R are -integrable functions such that f is of a single sign and nonincreasing and 0 g(t) 1 on [ a, b]T . Additional, assume that b = a g(t) t such that b – , a T. Thenb b-f (t) tb af (t) g(t) ta af (t) t.In [17], kan and Yildirim established the following results relating to diamond- dynamic Steffensen-type inequalities. Theorem three. Let h be a constructive integrable YC-001 MedChemExpress function on [ a, b]T and f , g be integrable functions on [ a, b]T such that f is nonincreasing and 0 g(t) h(t) for all t [ a, b]T . Thenb af (t) g(t) ta af (t)h(t) t,(4)where would be the option in the equationb a ag(t) t =ah(t) t.If f /h is nondecreasing, then the reverse inequality in (4) holds. Theorem 4. Let h be a good integrable function on [ a, b]T and f , g be integrable functions on [ a, b]T such that f is nonincreasing and 0 g(t) h(t) for all t [ a, b]T . Thenb b-f (t)h(t) tb af (t) g(t) t,(five)where may be the solution in the equationb b- bh(t) t =ag(t) t.If f /h is nondecreasing, then the reverse inequality in (five) holds. Theorem five. Let h be a good integrable function on [ a, b]T and f , g, be integrable functions on [ a, b]T such that f is nonincreasing and 0 (t) g(t) h(t) – (t) for all t [ a, b]T . Thenb b- bf (t)h(t) t baf (t) – f (b – ) (t) taf (t) g(t) taaf (t)h(t) t -b af (t) – f ( a ) (t) t,where is definitely the answer in the equationa a b bh(t) t =ag(t) t =b-h(t) t.Mathematics 2021, 9,four ofTheorem 6. Let f , g and h be -integrable functions defined on [ a, b]T with f nonincreasing. Furthermore, let 0 g(t) h(t) for all t [ a, b]T . Thenb b-f (t)h(t) tb b- b af (t)h(t) – f (t) – f (b – )h(t) – g(t) tf (t) g(t) taaf (t)h(t) – f (t) – f ( a ) f (t)h(t) t,h(t) – g(t) taawhere is given bya a b bh(t) t =ag(t) t =b-h(t) t.In this paper, we extend some generalizations of integral Steffensen’s inequality given in [1] to a general time scale, and establish various new sharpened versions of diamond- dynamic Steffensen’s inequality on time scales. As specific situations of our final results, we recover the integral inequalities provided in these papers. Our benefits also give some new discrete Steffensen’s inequalities. We obtain the new dynamic Steffensen inequalities utilizing the diamond- integrals on time scales. For = 1, the diamond- integral becomes delta integral and for = 0 it becomes nabla integral. Now, we are ready to state and prove the key benefits of this paper. two. Principal Results Let us begin by introducing a class of functions that extends the class of convex functions. Definition 1. Let , h : [ a, b]T R be good functions, f : [ a, b]T R be a function, and c c c ( a, b). We say that f /h belongs to the class AH1 [ a, b] (respectively, AH2 [ a, b]) if there ( t ) /h ( t ) – A ( t ) is nonincreasing (respectively, exists a constant A such that the function f nondecreasing) on [ a, c]T and nondecreasing (respectively, nonincreasing) on [c, b]T . We shall want the following lemmas in the proof of our benefits. Lemma 1. Let h be a positive integrable function on [ a, b]T and f , g be integrable functi.