Jensen's Functionals on Time Scales

Abstract

We consider Jensen’s functionals on time scales and discuss its properties and applications. Further, we define weighted generalized and power means on time scales. By applying the properties of Jensen’s functionals on these means, we obtain several refinements and converses of Hölder’s inequality on time scales.

1. 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 [𝑎,𝑏)={𝑡∈𝕋∶𝑎≤𝑡<𝑏}with𝑎,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)

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.

Theorem 1.1 (see [4, 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^𝑏_𝑎𝑓(𝑡)Δ𝑡=𝐿_[𝑎,𝑏)𝑓(𝑡)Δ𝑡, (1.3) where 𝑅 and 𝐿 indicate the Riemann and Lebesgue integrals, respectively.

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.

Theorem 1.2 (see [5, Theorem 3.2]). If 𝑓 and 𝑔 are Δ-integrable functions on [𝑎,𝑏), then _[𝑎,𝑏)𝛼𝑓+𝛽𝑔𝑑𝜇_Δ=𝛼_[𝑎,𝑏)𝑓𝑑𝜇_Δ+𝛽_[𝑎,𝑏)𝑔𝑑𝜇_Δ𝑓(𝑡)[)∀𝛼,𝛽∈ℝ,≥0∀𝑡∈𝑎,𝑏𝑖𝑚𝑝𝑙𝑖𝑒𝑠_[𝑎,𝑏)𝑓𝑑𝜇_Δ≥0. (1.4)

Theorem 1.3 (see [5, Theorem 4.2]). Assume Φ∈𝐶(𝐼,ℝ) is convex, 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 Φ∫_[𝑎,𝑏)𝑝𝑓𝑑𝜇_Δ∫_[𝑎,𝑏)𝑝𝑑𝜇_Δ≤∫_[𝑎,𝑏)𝑝Φ∘𝑓𝑑𝜇_Δ∫_[𝑎,𝑏)𝑝𝑑𝜇_Δ. (1.5)

Theorem 1.4 (see [5, Theorem 6.2]). For 𝑝≠1, define 𝑞=𝑝/(𝑝−1). Let 𝑤, 𝑓, 𝑔 be nonnegative functions such that 𝑤𝑓^𝑝, 𝑤𝑔^𝑞, 𝑤𝑓𝑔 are Δ-integrable on [𝑎,𝑏). If 𝑝>1, then _[𝑎,𝑏)𝑤𝑓𝑔𝑑𝜇_Δ≤_[𝑎,𝑏)𝑤𝑓^𝑝𝑑𝜇_Δ^1/𝑝_[𝑎,𝑏)𝑤𝑔^𝑞𝑑𝜇_Δ^1/𝑞. (1.6)
If 0<𝑝<1 and ∫_[𝑎,𝑏)𝑤𝑔^𝑞𝑑𝜇_Δ>0, or if 𝑝<0 and ∫_[𝑎,𝑏)𝑤𝑓^𝑝𝑑𝜇_Δ>0, then (1.6) is reversed.

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 Hölder’s inequality on time scales (Theorem 1.4) and obtain several refinements and converses of this inequality.

2. Properties of Jensen’s Functionals

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 𝒥Φ,𝑓,𝑝;𝜇_Δ∶=_[)𝑎,𝑏𝑝Φ∘𝑓𝑑𝜇_Δ−_[)𝑎,𝑏𝑝𝑑𝜇_ΔΦ∫_[𝑎,𝑏)𝑝𝑓𝑑𝜇_Δ∫_[𝑎,𝑏)𝑝𝑑𝜇_Δ. (2.1)

Remark 2.2. By Theorem 1.3, the following statements are obvious. If Φ is convex, then 𝒥Φ,𝑓,𝑝;𝜇_Δ≥0, (2.2) while if Φ is concave, then 𝒥Φ,𝑓,𝑝;𝜇_Δ≤0. (2.3)

Theorem 2.3. Assume Φ∈𝐶(𝐼,ℝ), where 𝐼⊆ℝ is an interval. Suppose 𝑓∶[𝑎,𝑏)→𝐼 is Δ-integrable. Also, let 𝑝,𝑞∶[𝑎,𝑏)→ℝ be nonnegative and Δ-integrable such that ∫_[𝑎,𝑏)𝑝𝑑𝜇_Δ>0 and ∫_[𝑎,𝑏)𝑞𝑑𝜇_Δ>0. If Φ is convex, then 𝒥(Φ,𝑓,⋅;𝜇_Δ) is superadditive, that is, 𝒥Φ,𝑓,𝑝+𝑞;𝜇_Δ≥𝒥Φ,𝑓,𝑝;𝜇_Δ+𝒥Φ,𝑓,𝑞;𝜇_Δ, (2.4) and 𝒥(Φ,𝑓,⋅;𝜇_Δ) is increasing, that is, 𝑝≥𝑞 with ∫_[𝑎,𝑏)𝑝𝑑𝜇_Δ>∫_[𝑎,𝑏)𝑞𝑑𝜇_Δ implies 𝒥Φ,𝑓,𝑝;𝜇_Δ≥𝒥Φ,𝑓,𝑞;𝜇_Δ. (2.5)
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 𝒥Φ,𝑓,𝑝+𝑞;𝜇_Δ=_[𝑎,𝑏)𝑝+𝑞Φ∘𝑓𝑑𝜇_Δ−_[𝑎,𝑏)𝑝+𝑞𝑑𝜇_ΔΦ∫_[𝑎,𝑏)𝑝+𝑞𝑓𝑑𝜇_Δ∫_[𝑎,𝑏)𝑝+𝑞𝑑𝜇_Δ=_[𝑎,𝑏)𝑝+𝑞Φ∘𝑓𝑑𝜇_Δ−_[𝑎,𝑏)𝑝+𝑞𝑑𝜇_Δ×∫×Φ^[𝑎,𝑏)𝑝𝑑𝜇_Δ∫_[𝑎,𝑏)𝑝+𝑞𝑑𝜇_Δ∫_[𝑎,𝑏)𝑝𝑓𝑑𝜇_Δ∫_[𝑎,𝑏)𝑝𝑑𝜇_Δ+∫_[𝑎,𝑏)𝑞𝑑𝜇_Δ∫_[𝑎,𝑏)𝑝+𝑞𝑑𝜇_Δ∫_[𝑎,𝑏)𝑞𝑓𝑑𝜇_Δ∫_[𝑎,𝑏)𝑞𝑑𝜇_Δ≥_[𝑎,𝑏)𝑝Φ∘𝑓𝑑𝜇_Δ+_[𝑎,𝑏)𝑞Φ∘𝑓𝑑𝜇_Δ−_[𝑎,𝑏)𝑝𝑑𝜇_ΔΦ∫_[𝑎,𝑏)𝑝𝑓𝑑𝜇_Δ∫_[𝑎,𝑏)𝑝𝑑𝜇_Δ−_[𝑎,𝑏)𝑞𝑑𝜇_ΔΦ∫_[𝑎,𝑏)𝑞𝑓𝑑𝜇_Δ∫_[𝑎,𝑏)𝑞𝑑𝜇_Δ=𝒥Φ,𝑓,𝑝;𝜇_Δ+𝒥Φ,𝑓,𝑞;𝜇_Δ. (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 𝒥Φ,𝑓,𝑝;𝜇_Δ=𝒥Φ,𝑓,𝑞+𝑝−𝑞;𝜇_Δ≥𝒥Φ,𝑓,𝑞;𝜇_Δ+𝒥Φ,𝑓,𝑝−𝑞;𝜇_Δ≥𝒥Φ,𝑓,𝑞;𝜇_Δ. (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 (𝑡)(𝑡)(𝑡)[),𝑀𝑀𝑞≥𝑝≥𝑚𝑞∀𝑡∈𝑎,𝑏_[𝑎,b)𝑞𝑑𝜇_Δ>_[)𝑎,𝑏𝑝𝑑𝜇_Δ>𝑚_[)𝑎,𝑏𝑞𝑑𝜇_Δ. (2.8)
If Φ is convex, then 𝑀𝒥Φ,𝑓,𝑞;𝜇_Δ≥𝒥Φ,𝑓,𝑝;𝜇_Δ≥𝑚𝒥Φ,𝑓,𝑞;𝜇_Δ, (2.9) while if Φ is concave, then the inequalities in (2.9) hold in reverse order.

Proof. By using Definition 2.1, we have 𝒥Φ,𝑓,𝑚𝑞;𝜇_Δ=𝑚𝒥Φ,𝑓,𝑞;𝜇_Δ,𝒥Φ,𝑓,𝑀𝑞;𝜇_Δ=𝑀𝒥Φ,𝑓,𝑞;𝜇_Δ. (2.10)
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. If Φ is convex, then max_{[)𝑡∈𝑎,𝑏}𝑝(𝑡)𝔍Φ,𝑓;𝜇_Δ≥𝒥Φ,𝑓,𝑝;𝜇_Δ≥min_{[)𝑡∈𝑎,𝑏}𝑝(𝑡)𝔍Φ,𝑓;𝜇_Δ, (2.11) where 𝔍Φ,𝑓;𝜇_Δ∶=_[𝑎,𝑏)Φ∘𝑓𝑑𝜇_Δ−()Φ∫𝑏−𝑎^[𝑎,𝑏)𝑓𝑑𝜇_Δ.𝑏−𝑎 (2.12)
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 𝑝=max_{[)𝑡∈𝑎,𝑏}𝑝(𝑡)≥𝑝(𝑥)≥min_{[)𝑡∈𝑎,𝑏}𝑝(𝑡)=𝑝. (2.13)
By Definition 2.1, we have 𝒥Φ,𝑓,𝑝;𝜇_Δ=𝑝𝔍Φ,𝑓;𝜇_Δ,𝒥Φ,𝑓,𝑝;𝜇_Δ=𝑝𝔍Φ,𝑓;𝜇_Δ. (2.14)
Now the result follows from the second property of Theorem 2.3.

Remark 2.6. The first inequality in (2.11) gives a converse of Jensen’s inequality on time scales, and the second inequality in (2.11) gives a refinement of the observed inequality.

Example 2.7 (see [8, Remark 4]). Let us take the discrete form of Jensen’s functional (2.1). For this, let 𝕋=ℤ, 𝑛∈ℕ, 𝑎=1, 𝑏=𝑛+1, and 𝑓(𝑖)=𝑥_𝑖, 𝑝(𝑖)=𝑝_𝑖 for 𝑖∈[𝑎,𝑏)={1,2,…,𝑛}. Then (2.1) becomes 𝒥Φ,𝑥,𝑝=^𝑛_𝑖=1𝑝_𝑖Φ𝑥_𝑖−𝑃_𝑛Φ∑^𝑛_𝑖=1𝑝_𝑖𝑥_𝑖𝑃_𝑛, (2.15) where 𝑥∶=𝑥₁,𝑥₂,…,𝑥_𝑛∈𝐼^𝑛,𝑝∶=𝑝₁,𝑝₂,…,𝑝_𝑛∈ℝ^𝑛₊,𝑃_𝑛∶=^𝑛_𝑖=1𝑝_𝑖. (2.16)
Under these notations, (2.11) takes the form max_{1≤𝑖≤𝑛}𝑝_𝑖𝔍(Φ,𝑥)≥𝒥Φ,𝑥,𝑝≥min_{1≤𝑖≤𝑛}𝑝_𝑖𝔍(Φ,𝑥), (2.17) where 𝔍(Φ,𝑥)=^𝑛_𝑖=1Φ𝑥_𝑖∑−𝑛Φ^𝑛𝑖=1𝑥_𝑖𝑛. (2.18)

Example 2.8 (see [8, Remark 5]). In addition to the notation introduced in Example 2.7, let 𝑞(𝑖)=𝑞_𝑖>0 for 𝑖∈[𝑎,𝑏)={1,2,…,𝑛} and put 𝑞∶=(𝑞₁,𝑞₂,…,𝑞_𝑛). Using 𝑚=min_{1≤𝑖≤𝑛}𝑝_𝑖𝑞_𝑖,𝑀=max_{1≤𝑖≤𝑛}𝑝_𝑖𝑞_𝑖 (2.19) in Corollary 2.4, (2.9) becomes max_{1≤𝑖≤𝑛}𝑝_𝑖𝑞_𝑖𝒥Φ,𝑥,𝑞≥𝒥Φ,𝑥,𝑝≥min_{1≤𝑖≤𝑛}𝑝_𝑖𝑞_𝑖𝒥Φ,𝑥,𝑞≥0. (2.20)

Example 2.9. Suppose 𝕋=ℝ and 𝑎,𝑏∈ℝ. Then Jensen’s functional (2.1) becomes _[)𝑎,𝑏𝑝(𝑡)Φ𝑓(𝑡)𝑑𝜇(𝑡)−_[)𝑎,𝑏𝑝(𝑡)(𝑡)Φ∫𝑑𝜇^[𝑎,𝑏)𝑝(𝑡)𝑓(𝑡)𝑑𝜇(𝑡)∫_[𝑎,𝑏)𝑝(𝑡)𝑑𝜇(𝑡). (2.21)

3. Applications to Weighted Generalized Means

Definition 3.1 (weighted generalized mean). Assume 𝜒∈𝐶(𝐼,ℝ) is strictly monotone, where 𝐼⊆ℝ is an interval. Suppose 𝑓∶[𝑎,𝑏)→𝐼 is Δ-integrable. Also, let 𝑝∶[𝑎,𝑏)→ℝ be nonnegative and Δ-integrable such that ∫_[𝑎,𝑏)𝑝𝑑𝜇_Δ>0. Then we define the weighted generalized mean on time scales by ℳ_𝜒𝑓,𝑝;𝜇_Δ∶=𝜒⁻¹∫_[𝑎,𝑏)𝑝𝜒∘𝑓𝑑𝜇_Δ∫_[𝑎,𝑏)𝑝𝑑𝜇_Δ, (3.1) provided (3.1) is well defined.

Theorem 3.2. Assume 𝜒,𝜓∈𝐶(𝐼,ℝ) are strictly monotone, where 𝐼⊆ℝ is an interval. Suppose 𝑓∶[𝑎,𝑏)→𝐼 is Δ-integrable. Moreover, let 𝑝,𝑞∶[𝑎,𝑏)→ℝ be nonnegative and Δ-integrable such that the functional _[)𝑎,𝑏𝑝𝑑𝜇_Δ𝜒ℳ_𝜒𝑓,𝑝;𝜇_Δ−𝜒ℳ_𝜓𝑓,𝑝;𝜇_Δ (3.2) is well defined. If 𝜒∘𝜓⁻¹ is convex, then (3.2) is superadditive, that is, _[)𝑎,𝑏𝑝+𝑞𝑑𝜇_Δ𝜒ℳ_𝜒𝑓,𝑝+𝑞;𝜇_Δ−𝜒ℳ_𝜓𝑓,𝑝+𝑞;𝜇_Δ≥_[)𝑎,𝑏𝑝𝑑𝜇_Δ𝜒ℳ_𝜒𝑓,𝑝;𝜇_Δ−𝜒ℳ_𝜓𝑓,𝑝;𝜇_Δ+_[)𝑎,𝑏𝑞𝑑𝜇_Δ𝜒ℳ_𝜒𝑓,𝑞;𝜇_Δ−𝜒ℳ_𝜓𝑓,𝑞;𝜇_Δ, (3.3) and (3.2) is increasing, that is, 𝑝≥𝑞 with ∫_[𝑎,𝑏)𝑝𝑑𝜇_Δ>∫_[𝑎,𝑏)𝑞𝑑𝜇_Δ implies _[)𝑎,𝑏𝑝𝑑𝜇_Δ𝜒ℳ_𝜒𝑓,𝑝;𝜇_Δ−𝜒ℳ_𝜓𝑓,𝑝;𝜇_Δ≥_[)𝑎,𝑏𝑞𝑑𝜇_Δ𝜒ℳ_𝜒𝑓,𝑞;𝜇_Δ−𝜒ℳ_𝜓𝑓,𝑞;𝜇_Δ. (3.4)
Moreover, if 𝜒∘𝜓⁻¹ is concave, then (3.2) is subadditive and decreasing, that is, (3.3) and (3.4) hold in reverse order.

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, 𝒥𝜒∘𝜓⁻¹,𝜓∘𝑓,𝑝;𝜇_Δ=_[)𝑎,𝑏𝑝𝜒∘𝜓⁻¹∘𝜓∘𝑓𝑑𝜇_Δ−_[)𝑎,𝑏𝑝𝑑𝜇_Δ𝜒∘𝜓⁻¹∫_[𝑎,𝑏)𝑝𝜓∘𝑓𝑑𝜇_Δ∫_[𝑎,𝑏)𝑝𝑑𝜇_Δ=_[)𝑎,𝑏𝑝𝜒∘𝑓𝑑𝜇_Δ−_[)𝑎,𝑏𝑝𝑑𝜇_Δ𝜒ℳ_𝜓𝑓,𝑝;𝜇_Δ=_[)𝑎,𝑏𝑝𝑑𝜇_Δ𝜒ℳ_𝜒𝑓,𝑝;𝜇_Δ−_[)𝑎,𝑏𝑝𝑑𝜇_Δ𝜒ℳ_𝜓𝑓,𝑝;𝜇_Δ=_[)𝑎,𝑏𝑝𝑑𝜇_Δ𝜒ℳ_𝜒𝑓,𝑝;𝜇_Δ−𝜒ℳ_𝜓𝑓,𝑝;𝜇_Δ. (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. If 𝜒∘𝜓⁻¹ is convex, then max_{[)𝑡∈𝑎,𝑏}𝑝(𝑡)(𝑏−𝑎)𝜒𝔐_𝜒𝑓;𝜇_Δ−𝜒𝔐_𝜓𝑓;𝜇_Δ≥_[)𝑎,𝑏𝑝𝑑𝜇_Δ𝜒ℳ_𝜒𝑓,𝑝;𝜇_Δ−𝜒ℳ_𝜓𝑓,𝑝;𝜇_Δ≥min_{[)𝑡∈𝑎,𝑏}𝑝(𝑡)(𝑏−𝑎)𝜒𝔐_𝜒𝑓;𝜇_Δ−𝜒𝔐_𝜓𝑓;𝜇_Δ, (3.6) where 𝔐_𝜂𝑓;𝜇_Δ∶=𝜂⁻¹∫_[𝑎,𝑏)𝜂∘𝑓𝑑𝜇_Δ.𝑏−𝑎,𝜂∈𝜒,𝜓 (3.7)
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.

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∫∶=^[𝑎,𝑏)𝑝𝑓^𝑟𝑑𝜇_Δ∫_[𝑎,𝑏)𝑝𝑑𝜇_Δ^1/𝑟∫if𝑟≠0,exp^[𝑎,𝑏)𝑝ln𝑓𝑑𝜇_Δ∫_[𝑎,𝑏)𝑝𝑑𝜇_Δif𝑟=0, (3.8) 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 _[)𝑎,𝑏𝑝𝑑𝜇_Δℳ^[𝑠]𝑓,𝑝;𝜇_Δ^𝑠−ℳ^[𝑟]𝑓,𝑝;𝜇_Δ^𝑠 (3.9) is well defined. If min{0,r}>𝑠>max{0,𝑟}, then (3.9) is superadditive (also if 𝑟=0), that is, _[)𝑎,𝑏𝑝+𝑞𝑑𝜇_Δℳ^[𝑠]𝑓,𝑝+𝑞;𝜇_Δ^𝑠−ℳ^[𝑟]𝑓,𝑝+𝑞;𝜇_Δ^𝑠≥_[)𝑎,𝑏𝑝𝑑𝜇_Δℳ^[𝑠]𝑓,𝑝;𝜇_Δ^𝑠−ℳ^[𝑟]𝑓,𝑝;𝜇_Δ^𝑠+_[)𝑎,𝑏𝑞𝑑𝜇_Δℳ^[𝑠]𝑓,𝑞;𝜇_Δ^𝑠−ℳ^[𝑟]𝑓,𝑞;𝜇_Δ^𝑠, (3.10) and (3.9) is increasing, that is, 𝑝≥𝑞 with ∫_[𝑎,𝑏)𝑝𝑑𝜇_Δ>∫_[𝑎,𝑏)𝑞𝑑𝜇_Δ implies _[)𝑎,𝑏𝑝𝑑𝜇_Δℳ^[𝑠]𝑓,𝑝;𝜇_Δ^𝑠−ℳ^[𝑟]𝑓,𝑝;𝜇_Δ^𝑠≥_[)𝑎,𝑏𝑞𝑑𝜇_Δℳ^[𝑠](𝑓,𝑞;𝜇_Δ)^𝑠−ℳ^[𝑟](𝑓,𝑞;𝜇_Δ)^𝑠. (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 𝜒∘𝜓^−1′′(𝑥)=𝑠(𝑠−𝑟)𝑟²𝑥^{𝑠/𝑟−2}. (3.12)
Thus 𝜒∘𝜓⁻¹ is convex if min{0,𝑟}>𝑠>max{0,𝑟} and concave if 𝑟>𝑠>0 or 0>𝑠>𝑟. If, however, 𝑟=0, then let 𝜒(𝑥)=𝑥^𝑠 and 𝜓(𝑥)=ln(𝑥) (𝑥>0) in Theorem 3.2. Then (𝜒∘𝜓⁻¹)(𝑥)=𝑒^𝑠𝑥. Thus 𝜒∘𝜓⁻¹ is convex for 𝑠≠0. In either case the result follows now immediately from Theorem 3.2.

Corollary 3.7. Let 𝑟, 𝑠, 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 min{0,𝑟}>𝑠>max{0,𝑟}, then max_{[)𝑡∈𝑎,𝑏}𝑝(𝑡)(𝑏−𝑎)𝔐^[𝑠]𝑓;𝜇_Δ^𝑠−𝔐^[𝑟]𝑓;𝜇_Δ^𝑠≥_[𝑎,𝑏)𝑝𝑑𝜇_Δℳ^[𝑠]𝑓,𝑝;𝜇_Δ^𝑠−ℳ^[𝑟]𝑓,𝑝;𝜇_Δ^𝑠≥min_{[)𝑡∈𝑎,𝑏}𝑝(𝑡)(𝑏−𝑎)𝔐^[𝑠]𝑓;𝜇_Δ^𝑠−𝔐^[𝑟]𝑓;𝜇_Δ^𝑠, (3.13) where 𝔐^[𝑢]𝑓;𝜇_Δ∫∶=^[𝑎,𝑏)𝑓^𝑢𝑑𝜇_Δ𝑏−𝑎^1/𝑢{0},∫𝑖𝑓𝑢∈ℝ⧵exp^[𝑎,𝑏)ln𝑓𝑑𝜇_Δ𝑏−𝑎𝑖𝑓𝑢=0. (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 𝑛max_{1≤𝑖≤𝑛}𝑝_𝑖𝐴_𝑛(𝑥)−𝐺_𝑛(𝑥)≥𝑃_𝑛ℳ^[1]𝑥,𝑝−ℳ^[0]𝑥,𝑝≥𝑛min_{1≤𝑖≤𝑛}𝑝_𝑖𝐴_𝑛(𝑥)−𝐺_𝑛(𝑥)≥0, (3.15) where ℳ^[𝑟]=∑𝑥,𝑝^𝑛𝑖=1𝑝_𝑖𝑥^𝑟_𝑖𝑃_𝑛^1/𝑟if𝑟∈ℝ⧵_𝑛{0},∏_𝑖=1𝑥^𝑝𝑖_𝑖^1/𝑃𝑛𝐴if𝑟=0,_𝑛(𝑥)=∑^𝑛_𝑖=1𝑥_𝑖𝑛,𝐺_𝑛(𝑥)=^𝑛_𝑖=1𝑥_𝑖^1/𝑛. (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 _[)𝑎,𝑏𝑝𝑑𝜇_Δ∫_[)𝑎,𝑏𝑝ln𝑓𝑑𝜇_Δ∫_[)𝑎,𝑏𝑝𝑑𝜇_Δ−lnℳ^[𝑟]𝑓,𝑝;𝜇_Δ (3.17) is well defined. If 𝑟<0, then (3.17) is superadditive, that is, _[)𝑎,𝑏𝑝+𝑞𝑑𝜇_Δ∫_[)𝑎,𝑏𝑝+𝑞ln𝑓𝑑𝜇_Δ∫_[)𝑎,𝑏𝑝+𝑞𝑑𝜇_Δ−lnℳ^[𝑟]𝑓,𝑝+𝑞;𝜇_Δ≥_[)𝑎,𝑏𝑝𝑑𝜇_Δ∫_[)𝑎,𝑏𝑝ln𝑓𝑑𝜇_Δ∫_[)𝑎,𝑏𝑝𝑑𝜇_Δ−lnℳ^[𝑟]𝑓,𝑝;𝜇_Δ+_[)𝑎,𝑏𝑞𝑑𝜇_Δ∫_[)𝑎,𝑏𝑞ln𝑓𝑑𝜇_Δ∫_[)𝑎,𝑏𝑞𝑑𝜇_Δ−lnℳ^[𝑟]𝑓,𝑞;𝜇_Δ, (3.18) and (3.17) is increasing, that is, 𝑝≥𝑞 with ∫_[𝑎,𝑏)𝑝𝑑𝜇_Δ>∫_[𝑎,𝑏)𝑞𝑑𝜇_Δ implies _[)𝑎,𝑏𝑝𝑑𝜇_Δ∫_[)𝑎,𝑏𝑝ln𝑓𝑑𝜇_Δ∫_[)𝑎,𝑏𝑝𝑑𝜇_Δ−lnℳ^[𝑟]𝑓,𝑝;𝜇_Δ≥_[)𝑎,𝑏𝑞𝑑𝜇_Δ∫_[)𝑎,𝑏𝑞ln𝑓𝑑𝜇_Δ∫_[)𝑎,𝑏𝑞𝑑𝜇_Δ−lnℳ^[𝑟]𝑓,𝑞;𝜇_Δ. (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 max_{[)𝑡∈𝑎,𝑏}𝑝(𝑡)∫)(𝑏−𝑎^[𝑎,𝑏)ln𝑓𝑑𝜇_Δ𝔐𝑏−𝑎−ln^[𝑟]𝑓;𝜇_Δ≥_[)𝑎,𝑏𝑝𝑑𝜇_Δ∫_[)𝑎,𝑏𝑝ln𝑓𝑑𝜇_Δ∫_[)𝑎,𝑏𝑝𝑑𝜇_Δ−lnℳ^[𝑟]𝑓,𝑝;𝜇_Δ≥min_{[)𝑡∈𝑎,𝑏}𝑝(𝑡)∫)(𝑏−𝑎^[)𝑎,𝑏ln𝑓𝑑𝜇_Δ𝔐𝑏−𝑎−ln^[𝑟]𝑓;𝜇_Δ, (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.

Example 3.11 (see [8, Remark 8]). Again we consider 𝕋=ℤ. Using the notation as introduced in Example 3.8, the term ∫_[𝑎,𝑏)𝑝ln(𝑓)𝑑𝜇_Δ/∫_[𝑎,𝑏)𝑝𝑑𝜇_Δ takes the form ∑^𝑛_𝑖=1𝑝_𝑖ln𝑥_𝑖∑^𝑛_𝑖=1𝑝_𝑖=ln^𝑛_𝑖=1𝑥^𝑝𝑖_𝑖^1/𝑃𝑛=lnℳ^[0]𝑥,𝑝, (3.21) and (3.20) becomes 𝐺_𝑛(𝑥)𝐴_𝑛(𝑥)^{𝑛max1≤𝑖≤𝑛𝑝i}≥ℳ^[0]𝑥,𝑝ℳ^[1]𝑥,𝑝^𝑃𝑛≥𝐺_𝑛(𝑥)𝐴_𝑛(𝑥)^{𝑛min1≤𝑖≤𝑛𝑝i}. (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 𝑥^𝑝∶=𝑥^𝑝1₁,𝑥^𝑝2₂,…,𝑥^𝑝𝑛_𝑛,𝑝⁻¹1∶=𝑝₁,1𝑝₂1,…,𝑝_𝑛, (3.23) where 𝑥 and 𝑝 are positive 𝑛-tuples such that ∑^𝑛_𝑖=1(1/𝑝_𝑖)=1. Then, (3.15) and (3.22) become 𝑛max_{1≤𝑖≤𝑛}1𝑝_𝑖𝐴_𝑛𝑥^𝑝−𝐺_𝑛𝑥^𝑝≥ℳ^[1]𝑥^𝑝,𝑝⁻¹−ℳ^[0]𝑥^𝑝,𝑝⁻¹≥𝑛min_{1≤𝑖≤𝑛}1𝑝_𝑖𝐴_𝑛𝑥^𝑝−𝐺_𝑛𝑥^𝑝,𝐺_𝑛𝑥^𝑝𝐴_𝑛(𝑥^𝑝)^{𝑛max1≤𝑖≤𝑛(1/𝑝𝑖)}≥ℳ^[0]𝑥^𝑝,𝑝⁻¹ℳ^[1]𝑥^𝑝,𝑝⁻¹≥𝐺_𝑛𝑥^𝑝𝐴_𝑛(𝑥^𝑝)^{𝑛min1≤𝑖≤𝑛(}1𝑝_𝑖⁾. (3.24)(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. Improvements of Hölder’s Inequality

Let 𝑛∈ℕ and let 𝑓_𝑖∶[𝑎,𝑏)→ℝ be Δ-integrable for all 𝑖∈{1,2,…,𝑛}. Assume 𝑝_𝑖>1 for all 𝑖∈{1,2,…,𝑛} are conjugate exponents, that is, ∑^𝑛_𝑖=1(1/𝑝_𝑖)=1, and ∏^𝑛_𝑖=1𝑓^1/𝑝𝑖_𝑖 is Δ-integrable on [𝑎,𝑏). Hölder’s inequality on time scales (Theorem 1.4) asserts that _[)^𝑛_{𝑎,𝑏𝑖=1}𝑓^1/𝑝𝑖_𝑖𝑑𝜇_Δ≤^𝑛_{𝑖=1[𝑎,𝑏)}𝑓_𝑖𝑑𝜇_Δ^1/𝑝𝑖. (4.1)

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=1𝑓^1/𝑝𝑖_𝑖 and ∏^𝑛_𝑖=1𝑓_𝑖^1/𝑛 are nonnegative and Δ-integrable. Then the following inequalities hold: 𝑛max_{1≤𝑖≤𝑛}1𝑝_𝑖^𝑛_{𝑖=1[)𝑎,𝑏}𝑓_𝑖𝑑𝜇_Δ^1/𝑝𝑖−^𝑛_{i=1[)𝑎,𝑏}𝑓_𝑖𝑑𝜇_Δ^{1/𝑝𝑖−1/𝑛}_[)^𝑛_{𝑎,𝑏𝑖=1}𝑓_𝑖^1/𝑛𝑑𝜇_Δ≥^𝑛_{𝑖=1[)𝑎,𝑏}𝑓_𝑖𝑑𝜇_Δ^1/𝑝𝑖−_[)^𝑛_{𝑎,𝑏𝑖=1}𝑓^1/𝑝𝑖_𝑖𝑑𝜇_Δ≥𝑛min_{1≤𝑖≤𝑛}1𝑝_𝑖^𝑛_{𝑖=1[)𝑎,𝑏}𝑓_𝑖𝑑𝜇_Δ^1/𝑝𝑖−^𝑛_{𝑖=1[)𝑎,𝑏}𝑓_𝑖𝑑𝜇_Δ^{1/𝑝𝑖−1/𝑛}_[)^𝑛_{𝑎,𝑏𝑖=1}𝑓1_𝑖𝑛𝑑𝜇_Δ. (4.2)

Proof. Let 𝑥_𝑖=[𝑓_𝑖/∫_[𝑎,𝑏)𝑓_𝑖𝑑𝜇_Δ]^1/𝑝𝑖, 𝑖∈{1,2,…,𝑛}, in Example 3.12. Then the expressions in (3.24) become ℳ₁𝑥^𝑝,𝑝⁻¹−ℳ₀𝑥^𝑝,𝑝⁻¹=^𝑛_𝑖=1𝑓_𝑖𝑝_𝑖∫_[𝑎,𝑏)𝑓_𝑖𝑑𝜇_Δ−^𝑛_𝑖=1𝑓^1/𝑝𝑖_𝑖∫_[𝑎,𝑏)𝑓_𝑖𝑑𝜇_Δ^1/𝑝𝑖,𝐴_𝑛𝑥^𝑝−𝐺_𝑛𝑥^𝑝=1𝑛^𝑛_𝑖=1𝑓_𝑖∫_[𝑎,𝑏)𝑓_𝑖𝑑𝜇_Δ−^𝑛_𝑖=1𝑓_𝑖^1/𝑛∫_[𝑎,𝑏)𝑓_𝑖𝑑𝜇_Δ^1/𝑛. (4.3)
Now, by applying the Δ-integral to the last two equations, we get _[)𝑎,𝑏ℳ₁𝑥^𝑝,𝑝⁻¹−ℳ₀𝑥^𝑝,𝑝⁻¹𝑑𝜇_Δ=^𝑛_𝑖=1∫_[)𝑎,𝑏𝑓_𝑖𝑑𝜇_Δ𝑝_𝑖∫_[)𝑎,𝑏𝑓_𝑖𝑑𝜇_Δ−∫_[𝑎,𝑏)∏^𝑛_𝑖=1𝑓^1/𝑝𝑖_𝑖𝑑𝜇_Δ∏^𝑛_𝑖=1∫_[)𝑎,𝑏𝑓_𝑖𝑑𝜇_Δ^1/𝑝𝑖∫=1−^[)𝑎,𝑏∏^𝑛_𝑖=1𝑓^1/𝑝𝑖_𝑖𝑑𝜇_Δ∏^𝑛_𝑖=1∫_[)𝑎,𝑏𝑓_𝑖𝑑𝜇_Δ^1/𝑝𝑖_[𝑎,𝑏),𝐴_𝑛𝑥^𝑝−𝐺_𝑛𝑥^𝑝𝑑𝜇_Δ=1𝑛^𝑛_𝑖=1∫_[)𝑎,𝑏𝑓_𝑖𝑑𝜇_Δ∫_[)𝑎,𝑏𝑓_𝑖𝑑𝜇_Δ−∫_[)𝑎,𝑏∏^𝑛_𝑖=1𝑓_𝑖^1/𝑛𝑑𝜇_Δ𝑛∏_𝑖=1∫_[)𝑎,𝑏𝑓_𝑖𝑑𝜇_Δ^1/𝑛∫=1−^[)𝑎,𝑏∏^𝑛_𝑖=1𝑓_𝑖^1/𝑛𝑑𝜇_Δ∏^𝑛_𝑖=1∫_[)𝑎,𝑏𝑓_𝑖𝑑𝜇_Δ^1/𝑛. (4.4)
By applying the Δ-integral to the series of inequalities in (3.24), we obtain the required inequalities.

Remark 4.2. The first inequality in Theorem 4.1 gives a converse and the second one gives a refinement of Hölder’s inequality on time scales.

Theorem 4.3. Under the same assumption as in Theorem 4.1, the following inequalities hold: 𝑛^𝑛∏^𝑛_𝑖=1∫_[)𝑎,𝑏𝑓_𝑖𝑑𝜇_Δ^{min1≤𝑖≤𝑛(1/𝑝𝑖)}××_[)𝑎,𝑏^𝑛_𝑖=1𝑓_𝑖𝑝_𝑖∫_[)𝑎,𝑏𝑓_𝑖𝑑𝜇_Δ∏^𝑛_𝑖=1𝑓_𝑖^1/𝑛∑^𝑛_𝑖=1𝑓_𝑖/∫_[)𝑎,𝑏𝑓_𝑖𝑑𝜇_Δ^{𝑛min1≤𝑖≤𝑛(1/𝑝𝑖)}𝑑𝜇_Δ≥∫_[)𝑎,𝑏∏^𝑛_𝑖=1𝑓^1/𝑝𝑖_𝑖𝑑𝜇_Δ∏^𝑛_𝑖=1∫_[𝑎,𝑏)𝑓_𝑖𝑑𝜇_Δ^1/𝑝𝑖≥𝑛^𝑛∏^𝑛_𝑖=1∫_[𝑎,𝑏)𝑓_𝑖𝑑𝜇_Δ^{min1≤𝑖≤𝑛(1/𝑝𝑖)}××_[𝑎,𝑏)^𝑛_𝑖=1𝑓_𝑖𝑝_𝑖∫_[𝑎,𝑏)𝑓_𝑖𝑑𝜇_Δ∏^𝑛_𝑖=1𝑓_𝑖^1/𝑛∑^𝑛_𝑖=1𝑓_𝑖/∫_[)𝑎,𝑏𝑓_𝑖𝑑𝜇_Δ^{𝑛max1≤𝑖≤𝑛(1/𝑝𝑖)}𝑑𝜇_Δ (4.5) 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 ℳ₁𝑥^𝑝,𝑝⁻¹𝐺_𝑛𝑥^𝑝𝐴_𝑛(𝑥^𝑝)^{𝑛min1≤𝑖≤𝑛(1/𝑝𝑖)}≥ℳ₀𝑥^𝑝,𝑝⁻¹≥ℳ₁𝑥^𝑝,𝑝⁻¹𝐺_𝑛𝑥^𝑝𝐴_𝑛(𝑥^𝑝)^{𝑛max1≤𝑖≤𝑛(1/𝑝𝑖)}. (4.6)
Now, if we consider the 𝑛-tuple 𝑥=(𝑥₁,𝑥₂,…,𝑥_𝑛), where 𝑥_𝑖=𝑓_𝑖∫_[)𝑎,𝑏𝑓_𝑖𝑑𝜇_Δ^1/𝑝𝑖∀𝑖∈{1,2,…,𝑛}, (4.7) then the expressions that represent the means in (4.6) become ℳ₁𝑥^𝑝,𝑝⁻¹=^𝑛_𝑖=1𝑓_𝑖𝑝_𝑖∫_[𝑎,𝑏)𝑓_𝑖𝑑𝜇_Δ,ℳ₀𝑥^𝑝,𝑝⁻¹=^𝑛_𝑖=1𝑓^1/𝑝𝑖_𝑖∫_[𝑎,𝑏)𝑓_𝑖𝑑𝜇_Δ^1/𝑝𝑖,𝐴_𝑛𝑥^𝑝=1𝑛^𝑛_𝑖=1𝑓_𝑖∫_[𝑎,𝑏)𝑓_𝑖𝑑𝜇_Δ,𝐺_𝑛𝑥^𝑝=^𝑛_𝑖=1𝑓_𝑖^1/𝑛∫_[𝑎,𝑏)𝑓_𝑖𝑑𝜇_Δ^1/𝑛. (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 Hölder’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. If 𝑟>1, then max_{[)𝑡∈𝑎,𝑏}𝑓(𝑡)(𝑏−𝑎)^1/𝑟_[)𝑎,𝑏𝑔𝑓𝑑𝜇_Δ^1/𝑠−_[)𝑎,𝑏𝑔𝑓^1/𝑠𝑑𝜇_Δ≥_[)𝑎,𝑏𝑓𝑑𝜇_Δ^1/𝑟_[)𝑎,𝑏𝑔𝑑𝜇_Δ^1/𝑠−_[)𝑎,𝑏𝑓^1/𝑟𝑔^1/𝑠𝑑𝜇_Δ≥min_{[)𝑡∈𝑎,𝑏}𝑓(𝑡)(𝑏−𝑎)^1/𝑟_[)𝑎,𝑏𝑔𝑓𝑑𝜇_Δ^1/𝑠−_[)𝑎,𝑏𝑔𝑓^1/𝑠𝑑𝜇_Δ. (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 𝒥𝑔Φ,𝑓,𝑓;𝜇_Δ=_[)𝑎,𝑏𝑔𝑓Φ𝑓𝑑𝜇_Δ−_[)𝑎,𝑏𝑓𝑑𝜇_ΔΦ∫_[)𝑎,𝑏𝑔𝑑𝜇_Δ∫_[)𝑎,𝑏𝑓𝑑𝜇_Δ=𝑟𝑠_[)𝑎,𝑏𝑓𝑑𝜇_Δ^1−1/𝑠_[)𝑎,𝑏𝑔𝑑𝜇_Δ^1/𝑠−_[)𝑎,𝑏𝑓^1−1/𝑠𝑔^1/𝑠𝑑𝜇_Δ=𝑟𝑠_[)𝑎,𝑏𝑓𝑑𝜇_Δ^1/𝑟_[)𝑎,𝑏𝑔𝑑𝜇_Δ^1/𝑠−_[)𝑎,𝑏𝑓^1/𝑟𝑔^1/𝑠𝑑𝜇_Δ,𝔍𝑔Φ,𝑓;𝜇_Δ=_[)𝑎,𝑏Φ𝑔𝑓𝑑𝜇_Δ−()Φ∫𝑏−𝑎^[)𝑎,𝑏𝑔/𝑓𝑑𝜇_Δ)𝑏−𝑎=𝑟𝑠(𝑏−𝑎^1−1/𝑠_[)𝑎,𝑏𝑔𝑓𝑑𝜇_Δ^1/𝑠−_[)𝑎,𝑏𝑔𝑓^1/𝑠𝑑𝜇_Δ=𝑟𝑠(𝑏−𝑎)^1/𝑟_[)𝑎,𝑏𝑔𝑓𝑑𝜇_Δ^1/𝑠−_[)𝑎,𝑏𝑔𝑓^1/𝑠𝑑𝜇_Δ. (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 Hölder’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 max_{[)𝑡∈𝑎,𝑏}𝑓(𝑡××)_[)𝑎,𝑏𝑓𝑑𝜇_Δ^𝑠−1_[)𝑎,𝑏𝑔𝑓𝑑𝜇_Δ−∫_[)𝑎,𝑏𝑓𝑑𝜇_Δ𝑏−𝑎^𝑠−1_[)𝑎,𝑏𝑔𝑓^1/𝑠𝑑𝜇_Δ^𝑠≥_[)𝑎,𝑏𝑓𝑑𝜇_Δ^1/𝑟_[)𝑎,𝑏𝑔𝑑𝜇_Δ^1/𝑠𝑠−_[𝑎,𝑏)𝑓^1/𝑟𝑔^1/𝑠𝑑𝜇_Δ^𝑠≥min_{[)𝑡∈𝑎,𝑏}𝑓(𝑡××_[)𝑎,𝑏)𝑓𝑑𝜇_Δ^𝑠−1_[)𝑎,𝑏𝑔𝑓𝑑𝜇_Δ−∫_[)𝑎,𝑏𝑓𝑑𝜇_Δ𝑏−𝑎^𝑠−1_[)𝑎,𝑏𝑔𝑓^1/𝑠𝑑𝜇_Δ^𝑠. (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 𝒥𝑔Φ,𝑓^1/𝑠,𝑓;𝜇_Δ=_[)𝑎,𝑏𝑔𝑓Φ𝑓^1/𝑠𝑑𝜇_Δ−_[)𝑎,𝑏𝑓𝑑𝜇_ΔΦ∫_[)𝑎,𝑏𝑓^1/𝑟𝑔^1/𝑠𝑑𝜇_Δ∫_[)𝑎,𝑏𝑓𝑑𝜇_Δ=1𝑠(𝑠−1)_[)𝑎,𝑏𝑔𝑑𝜇_Δ−_[)𝑎,𝑏𝑓𝑑𝜇_Δ^1−𝑠_[)𝑎,𝑏𝑓^1/𝑟𝑔^1/𝑠𝑑𝜇_Δ^𝑠,𝔍𝑔Φ,𝑓^1/𝑠;𝜇_Δ=_[)𝑎,𝑏Φ𝑔𝑓^1/𝑠𝑑𝜇_Δ−()Φ∫𝑏−𝑎^[)𝑎,𝑏𝑔/𝑓^1/𝑠𝑑𝜇_Δ=1𝑏−𝑎𝑠(𝑠−1)_[)𝑎,𝑏𝑔𝑓𝑑𝜇_Δ−(𝑏−𝑎)^1−𝑠_[)𝑎,𝑏𝑔𝑓^1/𝑠𝑑𝜇_Δ^𝑠. (4.12)
Now, the result follows immediately from (2.11).