Jensen ’ s Functionals on Time Scales

Time scales theory was initiated by Hilger 1 , and now there is a lot of work in this field. For an introduction to the theory of dynamic equations on time scales, we refer to 2–4 . Time scales calculus provides unification, extension, and generalization of classical continuous and discrete results. In this paper, we give results only for LebesgueΔ-integrals, but all the results obtained are also true if we take instead certain other time scales integrals such as the Cauchy delta, Cauchy nabla, α-diamond, multiple Riemann, or multiple Lebesgue integral. Now, using the same notations as in 4, Chapter 5 , we briefly give an introduction of Lebesgue Δ-integrals. Let a, b ⊆ T be a time scales interval defined by


Introduction
Time scales theory was initiated by Hilger 1 , and now there is a lot of work in this field.For an introduction to the theory of dynamic equations on time scales, we refer to 2-4 .Time scales calculus provides unification, extension, and generalization of classical continuous and discrete results.In this paper, we give results only for Lebesgue Δ-integrals, but all the results obtained are also true if we take instead certain other time scales integrals such as the Cauchy delta, Cauchy nabla, α-diamond, multiple Riemann, or multiple Lebesgue integral.Now, using the same notations as in 4, Chapter 5 , we briefly give an introduction of Lebesgue Δ-integrals.Let a, b ⊆ T be a time scales interval defined by All theorems of the general Lebesgue integration theory, including the Lebesgue-dominated convergence theorem, hold also for Lebesgue Δ-integrals on T. The following theorem compares the Lebesgue Δ-integral with the Riemann Δ-integral.The results in this paper are based on the authors' results given in 5 .For related results we refer the reader to 6, 7 .The remaining theorems in this section are taken from 5 .Theorem 1.2 shows that the Lebesgue Δ-integral is a so-called isotonic linear functional.Theorem 1.3 recalls Jensen's inequality for Lebesgue Δ-integrals, while Theorem 1.4 states Hölder's inequality for Lebesgue Δ-integrals.These three results are used in the remainder of this paper.
In Section 2, we define Jensen's functionals and, by using Jensen's inequality on time scales Theorem 1.3 , give some of their properties concerning superadditivity and monotonicity.In Section 3, we apply the properties of Jensen's functionals to generalized means, defined on time scales, and obtain improvements of several classical inequalities on time scales.Finally, in Section 4, we give applications of H ölder's inequality on time scales Theorem 1.4 and obtain several refinements and converses of this inequality.

Properties of Jensen's Functionals
Moreover, if Φ is concave, then J Φ, f, •; μ Δ is subadditive and decreasing, that is, 2.4 and 2.5 hold in reverse order.
Proof.Let Φ be convex.Because the time scales integral is linear see Theorem 1.2 , it follows from Definition 2.1 that

2.6
If p ≥ q, we have p − q ≥ 0. Now, because Jensen's functional is superadditive see above and nonnegative see Theorem 1.2 , we have

2.7
On the other hand, if Φ is concave, then the reversed inequalities of 2.4 and 2.5 can be obtained in a similar way.
Corollary 2.4.Let Φ, f, p, q satisfy the hypotheses of Theorem 2.3.Further, suppose there exist nonnegative constants m and M such that while if Φ is concave, then the inequalities in 2.9 hold in reverse order.
Proof.By using Definition 2.1, we have Now the result follows from the second property of Theorem 2.3.
Corollary 2.5.Let Φ, f, p satisfy the hypotheses of Theorem 2.3.Further, assume that p attains its minimum value and its maximum value on its domain. where Moreover, if Φ is concave, then the inequalities in 2.11 hold in reverse order.
Proof.Let p attain its minimum value p and its maximum value p on its domain a, b .Then

2.13
By Definition 2.1, we have

2.14
Now the result follows from the second property of Theorem 2. where
Then we define the weighted generalized mean on time scales by
Proof.The functional defined in 3.2 is obtained by replacing Φ with χ • ψ −1 and f with ψ • f in Jensen's functional 2.1 , that is,

3.5
Now, all claims follow immediately from Theorem 2.3.
Corollary 3.3.Let f, p, χ, ψ satisfy the hypotheses of Theorem 3.2.Further, assume that p attains its minimum value and its maximum value on its domain. where Moreover, if χ • ψ −1 is concave, then the inequalities in 3.6 hold in reverse order.
Proof.The proof is omitted as it is similar to the proof of Corollary 2.5.
provided 3.8 is well defined.
Remark 3.5.The weighted generalized power mean defined in 3.8 follows from the weighted generalized mean defined in 3.1 by taking χ x x r x > 0 in the weighted generalized mean.Theorem 3.6.Assume r, s ∈ R with r / 0. Suppose f : a, b → I is positive and Δ-integrable, where I ⊆ R is an interval.Moreover, let p, q : a, b → R be nonnegative and Δ-integrable such that the functional

3.11
Moreover, if r > s > 0 or 0 > s > r, then 3.9 is subadditive and decreasing, that is, 3.10 and 3.11 hold in reverse order.
Proof.If r / 0, then let χ x x s and ψ x x r x > 0 in Theorem 3.2.Then χ • ψ −1 x x s/r and therefore

3.14
Moreover, if r > s > 0 or 0 > s > r, then the inequalities in 3.13 hold in reverse order.
Proof.The proof is omitted as it is similar to the proof of Corollary 2.5 followed by Theorem 3.6.
Example 3.8 See 8, Remark 7 .From the discrete form of Corollary 3.7, that is, by using T Z, we get a refinement and a converse of the arithmetic-geometric mean inequality.Using the notation as introduced in Example 2.7, let x i > 0 for all i ∈ a, b and s 1, r 0. Then 3.13 becomes

3.16
The first inequality in 3.15 gives a converse and the second one gives a refinement of the arithmetic geometric-mean inequality of M 1 x, p and M 0 x, p .Theorem 3.9.Let r, f, p, q satisfy the hypotheses of Theorem 3.6.Suppose that the functional is well defined.If r < 0, then 3.17 is superadditive, that is,

3.19
Moreover, if r > 0, then 3.17 is subadditive and decreasing, that is, 3.18 and 3.19 hold in reverse order.
Proof.Let χ x ln x and ψ x x r in Theorem 3.2.Then χ • ψ −1 x 1/r ln x .Thus χ • ψ −1 is convex if r < 0 and concave if r > 0. Now the rest of the proof follows immediately from Theorem 3.2.
Corollary 3.10.Let r, f, p satisfy the hypotheses of Theorem 3.6.Further, assume that p attains its minimum value and its maximum value on its domain.If r < 0, then

3.20
where M r f; μ Δ is defined in 3.14 .Moreover, if r > 0, then the inequalities in 3.20 hold in reverse order.
Proof.The proof is omitted as it is similar to the proof of Corollary 2.5 followed by Theorem 3.9.

3.22
The inequalities in 3.22 provide a refinement and a converse of the arithmetic-geometric mean inequality in quotient form.
Example 3.12 See 8, Remark 9 .The relations 3.15 and 3.22 also yield refinements and converses of Young's inequality.To see this, consider again T Z.Using the notation as introduced in Example 3.8, define where x and p are positive n-tuples such that n i 1 1/p i 1.Then, 3.15 and 3.22 become

3.25
The inequalities in 3.24 and 3.25 provide the refinements and converses of Young's inequality in difference and quotient form.

4.3
Now, by applying the Δ-integral to the last two equations, we get

4.4
By applying the Δ-integral to the series of inequalities in 3.24 , we obtain the required inequalities. Remark provided that all expressions are well defined.
Proof.We consider relation 3.25 in the same settings as in Theorem 4.1.By inverting, 3.25 can be rewritten in the form Now, if we consider the n-tuple x x 1 , x 2 , . . ., x n , where then the expressions that represent the means in 4.6 become

4.8
Now, by taking the Δ-integral on 4.6 in described setting, we obtain the required inequalities.
Remark 4.4.The first inequality in Theorem 4.3 gives a refinement and the second one gives a converse of H ölder's inequality on time scales.
Corollary 4.5.Let r, s ∈ R such that 1/r 1/s 1.Further, assume that f, g are positive and Δ-integrable such that f attains its minimum value and its maximum value on its domain.

4.9
Moreover, if 0 < r < 1, then the inequalities in 4.9 hold in reverse order.
Proof.The result follows from Corollary 2.5 by replacing f with g/f, p with f, and letting Φ x −rsx 1/s .Then Φ is convex on 0, ∞ , and we have

4.10
If r > 1, then by substituting J Φ, g/f, f; μ Δ and J Φ, g/f; μ Δ in 2.11 , we get 4.9 .If 0 < r < 1, then rs < 0, and since the expressions J Φ, g/f, f; μ Δ and J Φ, g/f; μ Δ contain the factor rs, we conclude that the inequalities in 4.9 hold in reverse order in that case.
Remark 4.6.The first inequality in 4.9 gives a converse and the second one gives a refinement of H ölder's inequality on time scales.
Corollary 4.7.Let r, s ∈ R such that r > 0 and 1/r 1/s 1.Further, assume that f, g are positive and Δ-integrable such that f attains its minimum value and its maximum value on its domain.Then

4.11
Proof.In Corollary 2.5, replace f with g/f 1/s , p with f, and let Φ x x s / s s − 1 .Then Φ is convex on 0, ∞ .We get J Φ, g f

4.12
Now, the result follows immediately from 2.11 .
a, b {t ∈ T : a ≤ t < b} with a, b ∈ T, a ≤ b. 1.1 Suppose μ Δ is the Lebesgue Δ-measure on a, b and f : a, b → R is a μ Δ -measurable function.Then the Lebesgue Δ-integral of f on a, b is denoted by a,b fdμ Δ , a,b f t dμ Δ t , or a,b f t Δt. 1.2

Theorem 1 .1 see 4 ,where
Theorem 5.81 .Let a, b be a closed bounded interval in T, and let f be a bounded real-valued function defined on a, b .If f is Riemann Δ-integrable from a to b, then f is Lebesgue Δ-integrable on a, b , and R and L indicate the Riemann and Lebesgue integrals, respectively.

Definition 2 . 1
Jensen's functional .Assume Φ ∈ C I, R , where I ⊆ R is an interval.Suppose f : a, b → I is Δ-integrable.Moreover, let p : a, b → R be nonnegative and Δ-integrable such that a,b pdμ Δ > 0. Then we define Jensen's functional on time scales by

Definition 3 . 4
weighted generalized power mean .Assume r ∈ R. Suppose f : a, b → I is positive and Δ-integrable, where I ⊆ R is an interval.Moreover, let p : a, b → R be nonnegative and Δ-integrable such that a,b pdμ Δ > 0. Then we define the weighted generalized power mean on time scales by M r f, p; μ Δ :

1
Remark 2.2.By Theorem 1.3, the following statements are obvious.If Φ is convex, then

Theorem 2.3. Assume
Φ ∈ C I, R , where I ⊆ R is an interval.Suppose f : a, b → I is Δintegrable.Also, let p, q : a, b → R be nonnegative and Δ-integrable such that a,b pdμ Δ > 0 and a,b qdμ Δ Example 3.11see 8, Remark 8 .Again we consider T Z.Using the notation as introduced in Example 3.8, the term a,b p ln f dμ Δ / a,b pdμ Δ takes the form Theorem 4.1.Let p i > 1, i ∈ {1, 2, . . ., n}, be conjugate exponents, and let f i , i ∈ {1, 2, . . ., n}, be nonnegative Δ-integrable functions such that n i 1 f .1 4.2.The first inequality in Theorem 4.1 gives a converse and the second one gives a refinement of H ölder's inequality on time scales.Under the same assumption as in Theorem 4.1, the following inequalities hold: