Diamond-$\alpha$ Jensen's Inequality on Time Scales

The theory and applications of dynamic derivatives on time scales has recently received considerable attention. The primary purpose of this paper is to give basic properties of diamond-$\alpha$ derivatives which are a linear combination of delta and nabla dynamic derivatives on time scales. We prove a generalized version of Jensen's inequality on time scales via the diamond-$\alpha$ integral and present some corollaries, including H\"{o}lder's and Minkowski's diamond-$\alpha$ integral inequalities.


Introduction
Jensen's inequality is of great interest in the theory of differential and difference equations, as well as other areas of mathematics. The original Jensen's inequality can be stated as follows: Theorem 1.1. [13] If g ∈ C([a, b], (c, d)) and f ∈ C((c, d), R) is convex, then Jensen's inequality on time scales via ∆-integral has been recently obtained by Agarwal, Bohner and Peterson.
Theorem 1.2. [3] If g ∈ C rd ([a, b], (c, d)) and f ∈ C((c, d), R) is convex, then Under similar hypotheses, we may replace the ∆-integral by the ∇-integral and get a completely analogous result [14]. The aim of this paper is to extend Jensen's inequality to an arbitrary time scale via the diamond-α integral [18].
There has been recent developments of the theory and applications of dynamic derivatives on time scales. From the theoretical point of view, the study provide a unification and extension of traditional differential and difference equations. Moreover, it is a crucial tool in many computational and numerical applications. Based on the well-known ∆ (delta) and ∇ (nabla) dynamic derivatives, a combined dynamic derivative, so called ♦ α (diamond-α) dynamic derivative, was introduced as a linear combination of ∆ and ∇ dynamic derivatives on time scales [18]. The diamond-α derivative reduces to the ∆ derivative for α = 1 and to the ∇ derivative for α = 0. On the other hand, it represents a "weighted dynamic derivative" on any uniformly discrete time scale when α = 1 2 . We refer the reader to [15,17,18] for an account of the calculus associated with the diamond-α dynamic derivatives.
The paper is organized as follows. In Section 2 we briefly give the basic definitions and theorems of time scales as introduced in Hilger's thesis [10] (see also [11,12]). In Section 3 we present our main results, which are generalizations of Jensen's inequality on time scales. Some examples and applications are given in Section 4.

Preliminaries
A time scale T is an arbitrary nonempty closed subset of real numbers. The calculus of time scales was initiated by S. Hilger in his Ph.D. thesis [10] in order to unify discrete and continuous analysis. Let T be a time scale. T has the topology that inherits from the real numbers with the standard topology. For t ∈ T, we define the forward jump operator σ : T → T by σ(t) = inf {s ∈ T : s > t}, and the backward jump operator ρ : T → T by ρ(t) = sup {s ∈ T : s < t}.
If σ(t) > t, we say that t is right-scattered, while if ρ(t) < t, we say that t is left-scattered. Points that are simultaneously right-scattered and left-scattered are called isolated. If σ(t) = t, then t is called right-dense, and if ρ(t) = t, then t is called left-dense. Points that are simultaneously right-dense and left-dense are called dense. Let t ∈ T, then two mappings µ, ν : T → [0, +∞) are defined as follows: We introduce the sets T k , T k , and T k k , which are derived from the time scale T, as follows. If T has a left-scattered maximum t 1 , then T k = T − {t 1 }, Throughout the text we will denote a time scales interval by Let f : T → R be a real function on a time scale T. Then, for t ∈ T k we define f ∆ (t) to be the number, if one exists, such that for all ǫ > 0 there is a neighborhood U of t such that for all s ∈ U , We say that f is delta differentiable on T k , provided f ∆ (t) exists for all t ∈ T k . Similarly, for t ∈ T k , we define f ∇ (t) to be the number value, if one exists, such that for all ǫ > 0, there is a neighborhood V of t such that for all s ∈ V , We say that f is nabla differentiable on T k , provided that f ∇ (t) exists for all t ∈ T k .
Given a function f : The following properties hold for all t ∈ T k : (i) If f is delta differentiable at t, then f is continuous at t.
(ii) If f is continuous at t and t is right-scattered, then f is delta differentiable .
(iii) If f is right-dense, then f is delta differentiable at t if and only if the limit lim s→t exists as a finite number. In this case, f ∆ (t) = lim s→t Similarly, given a function f : T → R, the following is true for all t ∈ T k : (a) If f is nabla differentiable at t, then f is continuous at t. .
(c) If f is left-dense, then f is nabla differentiable at t if and only if the limit lim s→t exists as a finite number. In this case, f ∇ (t) = lim s→t .
A function f : T → R is called rd-continuous, provided it is continuous at all right-dense points in T and its left-sided limits exist at all left-dense points in T.
A function f : T → R is called 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.
Let T be a time scale, and t, s ∈ T. Following [16], we define µ ts = σ(t) − s, is differentiable on t ∈ T k k both in the delta and nabla senses, then f is diamond-α differentiable at t and the dynamic derivative f ♦α (t) is given by (see [16,Theorem 3.2]). Equality (1) is the definition of f ♦α (t) found in [18]. The diamond-α derivative reduces to the standard ∆ derivative for α = 1, or the standard ∇ derivative for α = 0. On the other hand, it represents a "weighted dynamic derivative" for α ∈ (0, 1). Furthermore, the combined dynamic derivative offers a centralized derivative formula on any uniformly discrete time scale Let a, t ∈ T, and h : T → R. Then, the diamond-α integral of h from a to t is defined by provided that there exist delta and nabla integrals of h on T. It is clear that the diamond-α integral of h exists when h is a continuous function. We may notice that the ♦ α combined derivative is not a dynamic derivative for the absence of its anti-derivative [16,Sec. 4]. Moreover, in general we do not have (2) Next lemma provides some straightforward but useful results for what follows.

Main Results
We now prove Jensen's diamond-α integral inequalities.
Remark 3.1. In the particular case α = 1, inequality (3) reduces to that of Theorem 1.2. If T = R, then Theorem 3.1 gives the classical Jensen inequality, i.e. Theorem 1.1. However, if T = Z and f (x) = − ln(x), then one gets the well-known arithmetic-mean geometric-mean inequality (7).
Proof. Since f is convex we have Using now Jensen's inequality on time scales (see Theorem 1.2), we get Remark 3.2. Theorem 3.2 is the same as [14,Theorem 3.17]. However, we prove Theorem 3.2 using a different approach than that proposed in [14]: in [14] it is stated that such result follows from the analog nabla-inequality. As we have seen, diamond-alpha integrals have different properties than those of delta or nabla integrals (cf. Example 2.1). On the other hand, there is an inconsistency in [14]: a very simple example showing this fact is given below in Remark 3.6.
Remark 3.3. In the particular case h = 1, Theorem 3.2 reduces to Theorem 3.1.
Remark 3.4. If f is strictly convex, the inequality sign "≤" in (4) can be replaced by "<". Similar result to Theorem 3.2 holds if one changes the condition "f is convex" to "f is concave", by replacing the inequality sign "≤" in (4) by "≥".
Proof. Since f is convex, it follows, for example from [9, Exercise 3.42C], that for t ∈ (c, d) there exists a t ∈ R such that This leads to the desired inequality.  When α = 1, we obtain the well-known arithmetic-mean geometric-mean inequality: When α = 0, we also have (vi) Let T = 2 N0 and N ∈ N. We can apply Theorem 3.2 with a = 1, b = 2 N and g : {2 k : 0 ≤ k ≤ N } → (0, ∞). Then, we get: We conclude that On the other hand, It follows that In the particular case when α = 1 we have and when α = 0 we get the inequality

Related Diamond-α Integral Inequalities
The usual proof of Hölder's inequality use the basic Young inequality x Proof. Choosing f (x) = x p in Theorem 3.2, which for p > 1 is obviously a convex function on [0, Inequality (8) is trivially true in the case when g is identically zero. We consider two cases: (i) g(x) > 0 for all x ∈ [a, b] T ; (ii) there exists at least one x ∈ [a, b] T such that g(x) = 0. We begin with situation (i). Replacing g by f g −q p and |h(x)| by hg q in inequality (9), we get: Using the fact that 1 We now consider situation (ii).
g(x) > 0, and it follows from (10) where p > 1 and q = p p−1 . Remark 4.2. In the special case p = q = 2, (8) reduces to the following diamondα Cauchy-Schwarz integral inequality on time scales: We are now in position to prove a Minkowski inequality using our Hölder's inequality (8).
As another application of Theorem 3.2, we have:   Proof. We prove only (i). The proof of (ii) is similar. Inequality (i) is trivially true when f is zero: both the left and right hand sides reduce to b a hg♦ α x.
Otherwise, applying Theorem 3.2 with f (x) = (1 + x p ) 1 p , which is clearly convex on (0, ∞), we obtain Changing h and f by hf R b a hf ♦αx and g f in the last inequality, respectively, we obtain directly the inequality (i) of Theorem 4.3.