Integral Inequalities on Time Scales via the Theory of Isotonic Linear Functionals

and Applied Analysis 3 Theorem 2.3 Jensen’s inequality 5, Theorem 2.2 . Let a, b ∈ T with a < b, and suppose I ⊂ R is an interval. Assume h ∈ Crd a, b ,R satisfies ∫b a |h t |Δt > 0. If Φ ∈ C I,R is convex and f ∈ Crd a, b , I , then Φ ⎛ ⎝ ∫b a |h t |f t Δt ∫b a |h t |Δt ⎞ ⎠ ≤ ∫b a |h t |Φ ( f t ) Δt ∫b a |h t |Δt . 2.3 In 6 , Özkan et al. proved that Theorem 2.3 is also true if we use the nabla integral see 1, Section 8.4 instead of the delta integral. In 7 , Sheng et al. introduced the so-called α-diamond integral, where 0 ≤ α ≤ 1. It is a linear combination of the delta integral and the nabla integral. When α 1, we get the usual delta integral, and when α 0, we get the usual nabla integral. The following result concerning the α-diamond integral is given by Ammi et al. in 8 see also 6 . Theorem 2.4 Jensen’s inequality 8, Theorem 3.3 . Let α ∈ 0, 1 . Let a, b ∈ T with a < b and suppose I ⊂ R is an interval. Assume h ∈ C a, b ,R satisfies ∫b a |h t |♦αt > 0. If Φ ∈ C I,R is convex and f ∈ C a, b , I , then Φ ⎛ ⎝ ∫b a |h t |f t ♦αt ∫b a |h t |♦αt ⎞ ⎠ ≤ ∫b a |h t |Φ ( f t ) ♦αt ∫b a |h t |♦αt . 2.4 3. Isotonic Linear Functionals and Time-Scale Integrals We recall the following definition from 2, page 47 . Definition 3.1 Isotonic linear functional . Let E be a nonempty set and L be a linear class of real-valued functions f : E → R having the following properties: L1 If f, g ∈ L and a, b ∈ R, then af bg ∈ L. L2 If f t 1 for all t ∈ E, then f ∈ L. An isotonic linear functional is a functional A : L → R having the following properties: A1 If f, g ∈ L and a, b ∈ R, then A af bg aA f bA g . A2 If f ∈ L and f t ≥ 0 for all t ∈ E, then A f ≥ 0. When we use the approach of isotonic linear functionals as given in Definition 3.1, it is not necessary to know many details from the calculus of dynamic equations on time-scale. We only need to know that the time-scale integral is such an isotonic linear functional. Theorem 3.2. Let T be a time scale. For a, b ∈ T with a < b, let E a, b ∩ T, L Crd a, b ,R . 3.1 4 Abstract and Applied Analysis Then (L1) and (L2) are satisfied. Moreover, let A ( f ) ∫b a f t Δt, 3.2 where the integral is the Cauchy delta time-scale integral. Then (A1) and (A2) are satisfied. Proof. This follows from 1, Definition 1.58 and Theorem 1.77 . Instead of recalling the formal definition of the time-scale integral and the definition of the set of rd-continuous functions Crd used in Theorem 3.2, which can be found in 1, Section 1.4 , we choose to only give a few examples. Example 3.3. If T R in Theorem 3.2, then L C a, b ,R and A ( f ) ∫b a f t dt. 3.3 If T Z in Theorem 3.2, then L consists of all real-valued functions defined on a, b − 1 ∩ Z and


Introduction
A time scale is an arbitrary nonempty closed subset of the real numbers.For an introduction to the theory of dynamic equations on time-scale, we refer to 1 .For functions defined on a time scale, we can consider the derivative and also the integral.For example, when the time scale is the set of all real numbers, the time-scale integral is an ordinary integral; when the time scale is the set of all integers, the time-scale integral is a sum; when the time scale is the set of all integer powers of a fixed number, the time-scale integral is a Jackson integral.
In this paper, we present a series of inequalities for the time-scale integral.Among the inequalities presented, we offer time-scale versions of Jensen's-inequality, Jensen-type inequalities, converses of Jensen's inequality, inequalities for means, H ölder's inequality, Minkowski's inequality, Dresher's inequality, Aczél's inequality, Popoviciu's inequality, and Diaz-Metcalf's inequality.

Known Results Concerning Jensen's Inequality
Jensen's inequality is of great interest in the theories of differential and difference equations as well as other areas of mathematics.The original Jensen inequality can be stated as follows.The Jensen inequality on time-scale has been obtained by Agarwal et al. 4 .
Theorem 2.2 Jensen's inequality 1, Theorem 6.17 .Let a, b ∈ T with a < b, and suppose When T R in Theorem 2.2, we obtain Theorem 2.1.When T Z in Theorem 2.2, we get the usual geometric-arithmetic mean inequality.
The following result is given by Wong et al. in 5 .When h t ≡ 1 in Theorem 2.3, we obtain Theorem 2.2.Theorem 2. 3 Jensen's inequality 5, Theorem 2.2 .Let a, b ∈ T with a < b, and suppose In 6 , Özkan et al. proved that Theorem 2.3 is also true if we use the nabla integral see 1, Section 8.4 instead of the delta integral.In 7 , Sheng et al. introduced the so-called α-diamond integral, where 0 ≤ α ≤ 1.It is a linear combination of the delta integral and the nabla integral.When α 1, we get the usual delta integral, and when α 0, we get the usual nabla integral.The following result concerning the α-diamond integral is given by Ammi et al. in 8 see also 6 . 2.4

Isotonic Linear Functionals and Time-Scale Integrals
We recall the following definition from 2, page 47 .
Definition 3.1 Isotonic linear functional .Let E be a nonempty set and L be a linear class of real-valued functions f : E → R having the following properties: An isotonic linear functional is a functional A : L → R having the following properties: When we use the approach of isotonic linear functionals as given in Definition 3.1, it is not necessary to know many details from the calculus of dynamic equations on time-scale.We only need to know that the time-scale integral is such an isotonic linear functional.
Let q > 1.If T q N 0 in Theorem 3.2, then L consists of all real-valued functions defined on a, b/q ∩ q N 0 and Note that Theorem 3.2 also has corresponding versions for the nabla and α-diamond integral, which are given next for completeness.Proof.This follows from 7, Definition 3.2 and Theorem 3.7 .
Multiple Riemann integration on time-scale was introduced in 9 .The Riemann integral introduced there is also an isotonic linear functional.Theorem 3.6.Let T 1 , . . ., T n be time-scale.For be Jordan Δ-measurable and let L be the set of all bounded Δ-integrable functions from E to R. Then (L 1 ) and (L 2 ) are satisfied.Moreover, let where the integral is the multiple Riemann delta time-scale integral.Then (A 1 ) and (A 2 ) are satisfied.
Proof.This follows from 9, Definition 4.13 and Theorem 3.4 .
From 9, Remark 2.18 , it is also clear that a theorem similar to Theorem 3.6 is also true for the nabla case or any mixture of delta and nabla integrals in the multiple variable case.
The multiple Lebesgue integration on time-scale was introduced in 10 .The Lebesgue integral introduced there is also an isotonic linear functional.
Abstract and Applied Analysis be Lebesgue Δ-measurable and let L be the set of all Δ-measurable functions from E to R. Then (L 1 ) and (L 2 ) are satisfied.Moreover, let where the integral is the multiple Lebesgue delta time-scale integral.Then (A 1 ) and (A 2 ) are satisfied.
Theorem 3.8.Under the assumptions of Theorem 3.7, let A f be replaced by 15 Then A is an isotonic linear functional satisfying A 1 1.

Jensen's Inequality
Jessen in 11 gave the following generalization of Jensen's inequality for isotonic linear functionals.
Theorem 4.1 Jessen's inequality 2, Theorem 2.4 .Let L satisfy properties (L 1 ) and (L 2 ).Assume Φ ∈ C I, R is convex, where Now our first result is the following generalization of Jensen's inequality.
Remark 4.3 Jensen's inequality .Note that the known results from Section 2 follow from Theorem 4.1 in the same way as Theorem 4.2 does: Theorem 2.3 follows as in Theorems 3.2 and 2.4 follows as in Theorem 3.5.Note also that a similar theorem for the multiple Riemann integral can be stated and proved using Theorem 3.6.This will be the case for all inequalities stated in this paper; however, we only explicitly state each time the case for the multiple Lebesgue integral.

Hermite-Hadamard's Inequality
Beesack and Pečarić in 12 gave the following generalization of Hermite-Hadamard's inequality for isotonic linear functionals.
Let p, q ≥ 0 be such that p q > 0 and Proof.Just apply Theorems 5.4 and 3.8.

H ölder's Inequality
We first recall H ölder's inequality for isotonic linear functionals as given in 2 .
This inequality is reversed if 0 < p < 1 and E |w t ||g t | q Δt > 0, and it is also reversed if p < 0 and Proof.Just apply Theorems 6.1 and 3.7.

Minkowski's Inequality
Another classical inequality is Minkowski's inequality.We first recall Minkowski's inequality for isotonic linear functionals as given in 2 . Theorem

Dresher's Inequality
If n 2 in the result of this section, then one has the so-called Dresher inequality see 19, Section 7 .We first present the generalization of this inequality for isotonic linear functionals as given in 2 .
Remark 8.3 Dresher's inequality .Dresher's inequality on time-scale is new even for the cases of a single-variable Cauchy delta and nabla integral and also for the α-diamond integral.

Popoviciu's Inequality
We first recall Popoviciu's inequality for isotonic linear functionals as given in 2 .
Abstract and Applied Analysis 11 Proof.Just let p 2 in Theorem 9.2.
Remark 9.4 Aczél's and Popoviciu's inequalities .Aczél's and Popoviciu's inequalities on time-scale are new even for the cases of a single-variable Cauchy delta and nabla integral and also for the α-diamond integral.The original Aczél inequality can be found in 20 .For a version of Aczél's inequality for isotonic linear functionals, we refer to 2, Theorem 4.26 .

Bellman's Inequality
We first recall Bellman's inequality for isotonic linear functionals as given in 2 .

Diaz-Metcalf's Inequality
If p q 2 and w 1 in the first result of this section, then one has the so-called Diaz-Metcalf inequality.We first present the generalization of this inequality for isotonic linear functionals as given in 2 .

11.3
If p > 1, or if p < 0 and at least one of the two integrals on the left-hand side of the following inequality is positive, then

11.4
This inequality is reversed if 0 < p < 1 and at least one of the two integrals on the left-hand side is positive.

Abstract and Applied Analysis 13
The last two results in this section follow from 2, Theorem 4.16 and Theorem 4.18 in the same way as Theorem 11.2 follows from Theorem 11.1.
Theorem 11.3.Let E, p, q, w, f, g, m, M be as in Theorem 11.2 This inequality is reversed if p < 0 or 0 < p < 1, provided at least one of the two integrals on the right-hand side is positive.
Theorem 11.4.Let E, p, q, w, f, g, m, M be as in Theorem 11.2 and assume where F f f g −q/p and G g f g −q/p .Let K p, q, m, M denote the constant on the righthand side of the inequality in Theorem 11.

Further Converses of Jensen's Inequality
Several results from the previous sections are sometimes also called converses of Jensen's inequality.This section is concerned with some further converses of Jensen's inequality.The five results presented follow from the specified results in 2 in the same way as Theorem 5.2 follows from Theorem 5.1.Let E ⊂ R n be as in Theorem 3.

Theorem 2 . 1
Jensen's inequality 3, Formula 5 .Let a, b ∈ R with a < b, and suppose I ⊂ R is an interval.If Φ ∈ C I, R is convex and f ∈ C a, b , I , then Φ

Remark 5 . 3 1 .Theorem 5 . 5
Hermite-Hadamard's inequality .Note that the known result 13, Theorem 3.14 see also 14, 15 follows from Theorem 5.1 in the same way as Theorem 5.2 does, this time applying Theorem 3.5.A combination of Theorem 4.1 and Theorem 5.1 in a slightly different form is given by Pečarić and Beesack in 16 as follows.Theorem 5.4 Pečarić-Beesack's inequality 2, Theorem 5.13 .Let L satisfy properties (L 1 ) and (L 2 ).Assume Φ ∈ C I, R is convex, where m, M ⊂ I with m < M and I ⊂ R is an interval.Suppose A satisfies (A 1 ) and (A 2 ) such that A 1 Let f ∈ L such that f E ⊂ m, M and Φ f ∈ L, and define p, q ≥ 0 such that p q > 0 and Hermite-Hadamard's inequality .Assume Φ ∈ C I, R is convex, where m, M ⊂ I with m < M and I ⊂ R is an interval.Let E ⊂ R n be as in Theorem 3.7 and suppose f is

Theorem 12 . 1 .
(a) Assume Φ ∈ C I, R is convex, where I m, M with m < M such that Φ x ≥ 0 with equality for at most isolated points of I. Assume further that either i Φ x > 0 for all x ∈ I, i Φ x > 0 for all m < x < M with either Φ m 0, Φ m / 0, or Φ M 0, Φ M / 0, 14 Abstract and Applied Analysis ii Φ x < 0 for all x ∈ I, ii Φ x < 0 for all m < x < M with precisely one of Φ m 0, Φ M 0.
6.3 H ölder's inequality .Note that the known results from the time-scale literature follow from Theorem 6.1 in the same way as Theorem 6.2 does: 1, Theorem 6.13 follows as in Theorem 3.2 and 8, Theorem 4.1 see also 17, 18 follows as in Theorem 3.5.Cauchy-Schwarz's inequality .Let E ⊂ R n be as in Theorem 3.7.If |w|f 2 , |w|g 2 , |wfg| are Δ-integrable on E, then E w t f t g t Δt ≤ E |w t |f 2 t Δt E |w t |g 2 t Δt .6.3 Proof.Just let p 2 in Theorem 6.2.
This inequality is reversed if 0 < p < 1 or p < 0 provided A |w||f| p > 0 and A |w||g| p > 0 hold.Minkowski's inequality .Let E ⊂ R n be as in Theorem 3.7.For p ∈ R, assume |w||f| p , |w||g| p , |w||f g| p are Δ-integrable on E. If p > 1, then This 7.1 Minkowski's inequality 2, Theorem 4.13 .Let E, L, and A be such that (L 1 ), (L 2 ), (A 1 ), and (A 2 ) are satisfied.For p ∈ R, assume |w||f| p , |w||g| p , |w||f g| p ∈ L. If p > 1, then E |w t | f t g t p Δt inequality is reversed for 0 < p < 1 or p < 0 provided each of the two terms on the right-hand side is positive.Proof.Just apply Theorems 7.1 and 3.7.Remark 7.3 Minkowski's inequality .Note that the known results from the time-scale literature follow from Theorem 7.1 in the same way as Theorem 7.2 does: 1, Theorem 6.16 follows as in Theorem 3.2 and 8, Theorem 4.4 see also 17, 18 follows as in Theorem 3.5.
Dresher's inequality .Let E ⊂ R n be as inTheorem 3.7.If are Δ-integrable on E, where p ≥ 1 > r > 0 andE |w t ||g i t | r Δt > 0 for 1 ≤ i ≤ n, then 3. If p > 1, then This inequality is reversed if 0 < p < 1, or if p < 0 and the integral on the left-hand side is positive.Proof.Just apply 2, Theorem 4.18 and Theorem 3.7.
7, and suppose f is Δ-integrable on E such that f E I.Moreover, let h : E → R be Δ-integrable such that E |h t |Δt > 0. Then Φ, may be determined as follows: define νΦ M − Φ m / M − m .If ν 0, let x ∈ m, M be the unique solution of the equation Φ x 0; then λ Φ m /Φ x .If ν / 0, let x ∈ m, M be the unique solution of the equation νΦ x − Φ x Φ m ν x − m 0; then λ ν/Φ x .Moreover, one has x ∈ m, M in the cases (i), (ii).(b)Letall the hypotheses of (a) hold except that Φ is concave on I with Φ x ≤ 0 with equality for at most isolated points of I. Then , where x ∈ m, M is the unique solution of the equation Φ x ν.(b) Let all the hypotheses of (a) hold except that Φ is strictly decreasing on I. Then Φ E |h t |f t Δt Just apply 2, Theorem 3.41 and Theorem 3.8.In addition to the assumptions of Theorem 5.2, let J ⊂ R be an interval such that J ⊃ Φ I and assume that F : J × J → R is increasing in the first variable.ThenF E |h t |Φ f t Δt E |h t |Δt , Φ E |h t |ft Δt E |h t |Δt and the right-hand side of the inequality is an increasing function of M and a decreasing function of m.Proof.Just apply 2, Theorem 3.42 and Theorem 3.8.Under the same hypotheses as in Theorem 12.3 except that F is decreasing in its first variable, one has F Proof.Just apply 2, Lemma 4.25 and Theorem 3.7.