FORMATION OF VERSIONS OF SOME DYNAMIC INEQUALITIES UNIFIED ON TIME SCALE CALCULUS

Abstract: The aim of this paper is to present some comprehensive and extended versions of classical inequalities such as Radon’s Inequality, Bergström’s Inequality, the weighted power mean inequality, Schlömilch’s Inequality and Nesbitt’s Inequality on time scale calculus. In time scale calculus, results are unified and extended. The theory of time scale calculus is applied to unify discrete and continuous analysis and to combine them in one comprehensive form. This hybrid theory is also widely applied on dynamic inequalities. The study of dynamic inequalities has received a lot of attention in the literature and has become a major field in pure and applied mathematics.


Introduction
The time scale calculus has a scope for many applications in the field of dynamic inequalities.The time scale calculus was initiated by Stefan Hilger as given in [11].A time scale is an arbitrary nonempty closed subset of the real numbers.The time scale calculus is studied as delta calculus, nabla calculus and diamond-α calculus.Basic work on dynamic inequalities is done by Ravi Agarwal, George Anastassiou, Martin Bohner, Allan Peterson, Donal O'Regan, Samir Saker and many other authors.
We will prove the following results given in (1.1), (1.2) and (1.3) on time scales.The inequality from (1.1) is called Bergström's Inequality as given in [4][5][6]13].with equality if and only if The upcoming result is called Radon's Inequality as given in [15].(1.2) 2) is widely studied by many authors because it has many applications.
The following inequality is generalized Radon's Inequality as given in [10].
In this paper, it is assumed that all considerable integrals exist and are finite and T is a time scale, a, b ∈ T with a < b and an interval [a, b] T means the intersection of a real interval with the given time scale.

Preliminaries
We need here basic concepts of delta calculus.The results of delta calculus are adapted from monographs [7,8].
For t ∈ T, the forward jump operator σ : T → T is defined by σ(t) := inf{s ∈ T : s > t}.

The mapping ν
For a function f : T → R, the delta derivative f ∆ is defined as follows: Let t ∈ T k , if there exists f ∆ (t) ∈ R such that for all ǫ > 0, there exists a neighborhood U of t, for all s ∈ U , then f is said to be delta differentiable at t, and f ∆ (t) is called the delta derivative of f at t.A function f : T → R is said to be right-dense continuous (rd-continuous), if it is continuous at each right-dense point and there exists a finite left limit at every left-dense point.The set of all rd-continuous functions is denoted by C rd (T, R).
The next definition is given in [7,8].
The following results of nabla calculus are taken from [3,7,8].If T has a right-scattered minimum m, then T k = T − {m}, otherwise T k = T.A function f : T k → R is called nabla differentiable at t ∈ T k , with nabla derivative f ∇ (t), if there exists f ∇ (t) ∈ R such that for any ǫ > 0, there exists a neighborhood V of t, such that A function f : T → R is said to be left-dense continuous (ld-continuous), provided it is continuous at all left-dense points in T and its right-sided limits exist (finite) at all right-dense points in T. The set of all ld-continuous functions is denoted by C ld (T, R).
Now we present short introduction of diamond-α derivative as given in [1,19].Let T be a time scale and f (t) be differentiable on T in the ∆ and ∇ senses.For t ∈ T k k , where Thus f is diamond-α differentiable if and only if f is ∆ and ∇ differentiable.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).
provided that there exist delta and nabla integrals of h on T.
We need the following results.
The function f is strictly convex on I T if the inequality (2.1) is strict for distinct t, s ∈ I T and λ ∈ (0, 1).
The function f is concave (respectively, strictly concave) on I T , if −f is convex (respectively, strictly convex).
If Φ is strictly convex, then the inequality ≤ can be replaced by <.
[1] One of the three most popular examples of calculus on time scales is quantum calculus, i. e., q N 0 = {q n : n ∈ N 0 }, where N 0 is the set of nonnegative integers.
If we set T = q N 0 for q > 1 and m < n, then for m, n ∈ N 0 .

Main Results
In order to present our main results, first we present an extension of Radon's Inequality via time scales.
Equality holds in (3.1), when f (x) ≡ g(x) ≡ c, where c is a nonzero real constant.
P r o o f.The right hand side of (3.1) can be written as where  Clearly equality holds in (3.1), when f (x) ≡ g(x) ≡ c, where c is a nonzero real constant, which completes the proof.If we set [a, b] T = [q m , q n ] q N 0 for q > 1 and m < n, where m, n ∈ N 0 and N 0 is the set of nonnegative integers, then thus inequality (3.7) holds.Clearly the equality holds in (3.7), when f (x) ≡ g(x) ≡ c, where c is a nonzero real constant.
Equality holds in (3.8), when f (x) = cg(x), where c is a real constant.
P r o o f.If we put β = γ in (3.1), then we get (3.8), which is Radon's Inequality on dynamic time scales.Clearly the equality holds in (3.8), if f (x) = cg(x), where c is a real constant.
Equality holds in (3.9), when f (x) = cg(x), where c is a nonzero real constant.
thus inequality (3.9) holds.Clearly the equality holds in (3.9), if f (x) = cg(x), where c is a nonzero real constant.
Remark 6.If we set α = 1, then we get delta versions of dynamic inequalities and if we set α = 0, then we get nabla versions of dynamic inequalities presented in this article.
Further if we put T = Z, then we get discrete versions of dynamic inequalities and if we put T = R, then we get continuous versions of dynamic inequalities presented in this article.

Conclusion and Future Work
In this research article, we have presented dynamic inequalities on diamond-α calculus such as Radon's Inequality, Bergström's Inequality, the weighted power mean inequality, Schlömilch's Inequality and Nesbitt's Inequality.
We can generalize dynamic inequalities using functional generalization as given in [20].We can present dynamic inequalities on fractional calculus as given in [16] and on quantum calculus.We can develop dynamic inequalities using fractional Riemann-Liouville integral on time scale calculus in a similar fashion as given in [2] and [17].Similarly we can generalize dynamic inequalities of this article using time scales fractional derivative as given in [2].It will be interesting to present dynamic inequalities in two or more dimensions.
Recently it has found that many dynamic inequalities such as Radon's Inequality, the weighted power mean inequality, Schlömilch's Inequality, Rogers-Hölder's Inequality and Bernoulli's Inequality are equivalent on time scales as given in [18], so we can find more equivalent dynamic inequalities on time scales.