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