# Jensen's Functionals on Time Scales

Authors: Matloob Anwar; Rabia Bibi; Martin Bohner; Josip Pečarić
Journal: Journal of Function Spaces and Applications
 (2012)

---

## 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.


---

## Body

## 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
(1.1)[a,b)={t∈𝕋:a≤t<b}witha,b∈𝕋,a≤b.SupposeμΔ is the Lebesgue Δ-measure on [a,b) and f:[a,b)→ℝ is a μΔ-measurable function. Then the Lebesgue Δ-integral of f on [a,b) is denoted by
(1.2)∫[a,b)‍fdμΔ,∫[a,b)‍f(t)dμΔ(t),or∫[a,b)‍f(t)Δt.All theorems of the general Lebesgue integration theory, including the Lebesgue-dominated convergence theorem, hold also for LebesgueΔ-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
(1.3)R∫ab‍f(t)Δt=L∫[a,b)‍f(t)Δt,
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]).
Iff and g are Δ-integrable functions on [a,b), then
(1.4)∫[a,b)‍(αf+βg)dμΔ=α∫[a,b)‍fdμΔ+β∫[a,b)‍gdμΔ∀α,β∈ℝ,f(t)≥0∀t∈[a,b)implies∫[a,b)‍fdμΔ≥0.Theorem 1.3 (see [5, Theorem 4.2]).
AssumeΦ∈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)‍pdμΔ>0. Then
(1.5)Φ(∫[a,b)‍pfdμΔ∫[a,b)‍pdμΔ)≤∫[a,b)‍p(Φ∘f)dμΔ∫[a,b)‍pdμΔ.Theorem 1.4 (see [5, Theorem 6.2]).
Forp≠1, define q=p/(p-1). Let w, f, g be nonnegative functions such that wfp, wgq, wfg are Δ-integrable on [a,b). If p>1, then
(1.6)∫[a,b)‍wfgdμΔ≤(∫[a,b)‍wfpdμΔ)1/p(∫[a,b)‍wgqdμΔ)1/q.
If0<p<1 and ∫[a,b)‍wgqdμΔ>0, or if p<0 and ∫[a,b)‍wfpdμΔ>0, then (1.6) is reversed.In Section2, 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).
AssumeΦ∈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)‍pdμΔ>0. Then we define Jensen’s functional on time scales by
(2.1)𝒥(Φ,f,p;μΔ):=∫[a,b)p(Φ∘f)dμΔ-∫[a,b)pdμΔΦ(∫[a,b)‍pfdμΔ∫[a,b)‍pdμΔ).Remark 2.2.
By Theorem1.3, the following statements are obvious. If Φ is convex, then
(2.2)𝒥(Φ,f,p;μΔ)≥0,
while if Φ is concave, then
(2.3)𝒥(Φ,f,p;μΔ)≤0.Theorem 2.3.
AssumeΦ∈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)‍pdμΔ>0 and ∫[a,b)‍qdμΔ>0. If Φ is convex, then 𝒥(Φ,f,·;μΔ) is superadditive, that is,
(2.4)𝒥(Φ,f,p+q;μΔ)≥𝒥(Φ,f,p;μΔ)+𝒥(Φ,f,q;μΔ),
and 𝒥(Φ,f,·;μΔ) is increasing, that is, p≥q with ∫[a,b)‍pdμΔ>∫[a,b)‍qdμΔ implies
(2.5)𝒥(Φ,f,p;μΔ)≥𝒥(Φ,f,q;μΔ).
Moreover, ifΦ is concave, then 𝒥(Φ,f,·;μΔ) is subadditive and decreasing, that is, (2.4) and (2.5) hold in reverse order.Proof.
LetΦ be convex. Because the time scales integral is linear (see Theorem 1.2), it follows from Definition 2.1 that
(2.6)𝒥(Φ,f,p+q;μΔ)=∫[a,b)‍(p+q)(Φ∘f)dμΔ-∫[a,b)‍(p+q)dμΔΦ(∫[a,b)‍(p+q)fdμΔ∫[a,b)‍(p+q)dμΔ)=∫[a,b)‍(p+q)(Φ∘f)dμΔ-∫[a,b)‍(p+q)dμΔ××Φ(∫[a,b)‍pdμΔ∫[a,b)‍(p+q)dμΔ∫[a,b)‍pfdμΔ∫[a,b)‍pdμΔ+∫[a,b)‍qdμΔ∫[a,b)‍(p+q)dμΔ∫[a,b)‍qfdμΔ∫[a,b)‍qdμΔ)≥∫[a,b)‍p(Φ∘f)dμΔ+∫[a,b)‍q(Φ∘f)dμΔ-∫[a,b)‍pdμΔΦ(∫[a,b)‍pfdμΔ∫[a,b)‍pdμΔ)-∫[a,b)‍qdμΔΦ(∫[a,b)‍qfdμΔ∫[a,b)‍qdμΔ)=𝒥(Φ,f,p;μΔ)+𝒥(Φ,f,q;μΔ).
Ifp≥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)𝒥(Φ,f,p;μΔ)=𝒥(Φ,f,q+p-q;μΔ)≥𝒥(Φ,f,q;μΔ)+𝒥(Φ,f,p-q;μΔ)≥𝒥(Φ,f,q;μΔ).
On the other hand, ifΦ is concave, then the reversed inequalities of (2.4) and (2.5) can be obtained in a similar way.Corollary 2.4.
LetΦ, f, p, q satisfy the hypotheses of Theorem 2.3. Further, suppose there exist nonnegative constants m and M such that
(2.8)Mq(t)≥p(t)≥mq(t)∀t∈[a,b),M∫[a,b)qdμΔ>∫[a,b)pdμΔ>m∫[a,b)qdμΔ.
IfΦ is convex, then
(2.9)M𝒥(Φ,f,q;μΔ)≥𝒥(Φ,f,p;μΔ)≥m𝒥(Φ,f,q;μΔ),
while if Φ is concave, then the inequalities in (2.9) hold in reverse order.Proof.
By using Definition2.1, we have
(2.10)𝒥(Φ,f,mq;μΔ)=m𝒥(Φ,f,q;μΔ),𝒥(Φ,f,Mq;μΔ)=M𝒥(Φ,f,q;μΔ).
Now the result follows from the second property of Theorem2.3.Corollary 2.5.
LetΦ, 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
(2.11)[maxt∈[a,b)p(t)]𝔍(Φ,f;μΔ)≥𝒥(Φ,f,p;μΔ)≥[mint∈[a,b)p(t)]𝔍(Φ,f;μΔ),
where
(2.12)𝔍(Φ,f;μΔ):=∫[a,b)‍(Φ∘f)dμΔ-(b-a)Φ(∫[a,b)‍fdμΔb-a).
Moreover, ifΦ is concave, then the inequalities in (2.11) hold in reverse order.Proof.
Letp attain its minimum value p_ and its maximum value p¯ on its domain [a,b). Then
(2.13)p¯=maxt∈[a,b)p(t)≥p(x)≥mint∈[a,b)p(t)=p_.
By Definition2.1, we have
(2.14)𝒥(Φ,f,p¯;μΔ)=p¯𝔍(Φ,f;μΔ),𝒥(Φ,f,p_;μΔ)=p_𝔍(Φ,f;μΔ).
Now the result follows from the second property of Theorem2.3.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)=xi, p(i)=pi for i∈[a,b)={1,2,…,n}. Then (2.1) becomes
(2.15)𝒥(Φ,x,p)=∑i=1npiΦ(xi)-PnΦ(∑i=1npixiPn),
where
(2.16)x:=(x1,x2,…,xn)∈In,p:=(p1,p2,…,pn)∈ℝ+n,Pn:=∑i=1n‍pi.
Under these notations, (2.11) takes the form
(2.17)[max1≤i≤npi]𝔍(Φ,x)≥𝒥(Φ,x,p)≥[min1≤i≤npi]𝔍(Φ,x),
where
(2.18)𝔍(Φ,x)=∑i=1nΦ(xi)-nΦ(∑i=1nxin).Example 2.8 (see [8, Remark 5]).
In addition to the notation introduced in Example2.7, let q(i)=qi>0 for i∈[a,b)={1,2,…,n} and put q:=(q1,q2,…,qn). Using
(2.19)m=min1≤i≤npiqi,M=max1≤i≤npiqi
in Corollary 2.4, (2.9) becomes
(2.20)[max1≤i≤npiqi]𝒥(Φ,x,q)≥𝒥(Φ,x,p)≥[min1≤i≤npiqi]𝒥(Φ,x,q)≥0.Example 2.9.
Suppose𝕋=ℝ and a,b∈ℝ. Then Jensen’s functional (2.1) becomes
(2.21)∫[a,b)p(t)Φ(f(t))dμ(t)-∫[a,b)p(t)dμ(t)Φ(∫[a,b)‍p(t)f(t)dμ(t)∫[a,b)‍p(t)dμ(t)).


## 3. Applications to Weighted Generalized Means

Definition 3.1 (weighted generalized mean).
Assumeχ∈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)‍pdμΔ>0. Then we define the weighted generalized mean on time scales by
(3.1)ℳχ(f,p;μΔ):=χ-1(∫[a,b)‍p(χ∘f)dμΔ∫[a,b)‍pdμΔ),
provided (3.1) is well defined.Theorem 3.2.
Assumeχ,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
(3.2)∫[a,b)pdμΔ[χ(ℳχ(f,p;μΔ))-χ(ℳψ(f,p;μΔ))]
is well defined. If χ∘ψ-1 is convex, then (3.2) is superadditive, that is,
(3.3)∫[a,b)(p+q)dμΔ[χ(ℳχ(f,p+q;μΔ))-χ(ℳψ(f,p+q;μΔ))]≥∫[a,b)pdμΔ[χ(ℳχ(f,p;μΔ))-χ(ℳψ(f,p;μΔ))]+∫[a,b)qdμΔ[χ(ℳχ(f,q;μΔ))-χ(ℳψ(f,q;μΔ))],
and (3.2) is increasing, that is, p≥q with ∫[a,b)‍pdμΔ>∫[a,b)‍qdμΔ implies
(3.4)∫[a,b)pdμΔ[χ(ℳχ(f,p;μΔ))-χ(ℳψ(f,p;μΔ))]≥∫[a,b)qdμΔ[χ(ℳχ(f,q;μΔ))-χ(ℳψ(f,q;μΔ))].
Moreover, ifχ∘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,
(3.5)𝒥(χ∘ψ-1,ψ∘f,p;μΔ)=∫[a,b)p(χ∘ψ-1∘ψ∘f)dμΔ-∫[a,b)pdμΔ(χ∘ψ-1(∫[a,b)‍p(ψ∘f)dμΔ∫[a,b)‍pdμΔ))=∫[a,b)p(χ∘f)dμΔ-∫[a,b)pdμΔχ(ℳψ(f,p;μΔ))=∫[a,b)pdμΔχ(ℳχ(f,p;μΔ))-∫[a,b)pdμΔχ(ℳψ(f,p;μΔ))=∫[a,b)pdμΔ[χ(ℳχ(f,p;μΔ))-χ(ℳψ(f,p;μΔ))].
Now, all claims follow immediately from Theorem2.3.Corollary 3.3.
Letf, 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
(3.6)[maxt∈[a,b)p(t)](b-a)[χ(𝔐χ(f;μΔ))-χ(𝔐ψ(f;μΔ))]≥∫[a,b)pdμΔ[χ(ℳχ(f,p;μΔ))-χ(ℳψ(f,p;μΔ))]≥[mint∈[a,b)p(t)](b-a)[χ(𝔐χ(f;μΔ))-χ(𝔐ψ(f;μΔ))],
where
(3.7)𝔐η(f;μΔ):=η-1(∫[a,b)‍(η∘f)dμΔb-a),η∈{χ,ψ}.
Moreover, ifχ∘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 Corollary2.5.Definition 3.4 (weighted generalized power mean).
Assumer∈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)‍pdμΔ>0. Then we define the weighted generalized power mean on time scales by
(3.8)ℳ[r](f,p;μΔ):={(∫[a,b)‍pfrdμΔ∫[a,b)‍pdμΔ)1/rifr≠0,exp(∫[a,b)‍pln(f)dμΔ∫[a,b)‍pdμΔ)ifr=0,
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)=xr (x>0) in the weighted generalized mean.Theorem 3.6.
Assumer,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
(3.9)∫[a,b)pdμΔ{[ℳ[s](f,p;μΔ)]s-[ℳ[r](f,p;μΔ)]s}
is well defined. If min{0,r}>s>max{0,r}, then (3.9) is superadditive (also if r=0), that is,
(3.10)∫[a,b)(p+q)dμΔ{[ℳ[s](f,p+q;μΔ)]s-[ℳ[r](f,p+q;μΔ)]s}≥∫[a,b)pdμΔ{[ℳ[s](f,p;μΔ)]s-[ℳ[r](f,p;μΔ)]s}+∫[a,b)qdμΔ{[ℳ[s](f,q;μΔ)]s-[ℳ[r](f,q;μΔ)]s},
and (3.9) is increasing, that is, p≥q with ∫[a,b)‍pdμΔ>∫[a,b)‍qdμΔ implies
(3.11)∫[a,b)pdμΔ{[ℳ[s](f,p;μΔ)]s-[ℳ[r](f,p;μΔ)]s}≥∫[a,b)qdμΔ{[ℳ[s](f,q;μΔ)]s-[ℳ[r](f,q;μΔ)]s}.
Moreover, ifr>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.
Ifr≠0, then let χ(x)=xs and ψ(x)=xr (x>0) in Theorem 3.2. Then (χ∘ψ-1)(x)=xs/r and therefore
(3.12)(χ∘ψ-1)′′(x)=s(s-r)r2xs/r-2.
Thusχ∘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)=xs and ψ(x)=ln(x) (x>0) in Theorem 3.2. Then (χ∘ψ-1)(x)=esx. Thus χ∘ψ-1 is convex for s≠0. In either case the result follows now immediately from Theorem 3.2.Corollary 3.7.
Letr, 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
(3.13)[maxt∈[a,b)p(t)](b-a){[𝔐[s](f;μΔ)]s-[𝔐[r](f;μΔ)]s}≥∫[a,b)pdμΔ{[ℳ[s](f,p;μΔ)]s-[ℳ[r](f,p;μΔ)]s}≥[mint∈[a,b)p(t)](b-a){[𝔐[s](f;μΔ)]s-[𝔐[r](f;μΔ)]s},
where
(3.14)𝔐[u](f;μΔ)≔{(∫[a,b)‍fudμΔb-a)1/uifu∈ℝ∖{0},exp(∫[a,b)‍ln(f)dμΔb-a)ifu=0.
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 Corollary2.5 followed by Theorem 3.6.Example 3.8 (see [8, Remark 7]).
From the discrete form of Corollary3.7, 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 xi>0 for all i∈[a,b) and s=1, r=0. Then (3.13) becomes
(3.15)n[max1≤i≤npi][An(x)-Gn(x)]≥Pn[ℳ[1](x,p)-ℳ[0](x,p)]≥n[min1≤i≤npi][An(x)-Gn(x)]≥0,
where
(3.16)ℳ[r](x,p)={(∑i=1npixirPn)1/rifr∈ℝ∖{0},(∏i=1n‍xipi)1/Pnifr=0,An(x)=∑i=1nxin,Gn(x)=(∏i=1n‍xi)1/n.
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.
Letr, f, p, q satisfy the hypotheses of Theorem 3.6. Suppose that the functional
(3.17)∫[a,b)pdμΔ{∫[a,b)pln(f)dμΔ∫[a,b)pdμΔ-ln(ℳ[r](f,p;μΔ))}
is well defined. If r<0, then (3.17) is superadditive, that is,
(3.18)∫[a,b)(p+q)dμΔ{∫[a,b)(p+q)ln(f)dμΔ∫[a,b)(p+q)dμΔ-ln(ℳ[r](f,p+q;μΔ))}≥∫[a,b)pdμΔ{∫[a,b)pln(f)dμΔ∫[a,b)pdμΔ-ln(ℳ[r](f,p;μΔ))}+∫[a,b)qdμΔ{∫[a,b)qln(f)dμΔ∫[a,b)qdμΔ-ln(ℳ[r](f,q;μΔ))},
and (3.17) is increasing, that is, p≥q with ∫[a,b)‍pdμΔ>∫[a,b)‍qdμΔ implies
(3.19)∫[a,b)pdμΔ{∫[a,b)pln(f)dμΔ∫[a,b)pdμΔ-ln(ℳ[r](f,p;μΔ))}≥∫[a,b)qdμΔ{∫[a,b)qln(f)dμΔ∫[a,b)qdμΔ-ln(ℳ[r](f,q;μΔ))}.
Moreover, ifr>0, then (3.17) is subadditive and decreasing, that is, (3.18) and (3.19) hold in reverse order.Proof.
Letχ(x)=ln(x) and ψ(x)=xr 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.
Letr, 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)[maxt∈[a,b)p(t)](b-a){∫[a,b)ln(f)dμΔb-a-ln(𝔐[r](f;μΔ))}≥∫[a,b)pdμΔ{∫[a,b)pln(f)dμΔ∫[a,b)pdμΔ-ln(ℳ[r](f,p;μΔ))}≥[mint∈[a,b)p(t)](b-a){∫[a,b)ln(f)dμΔb-a-ln(𝔐[r](f;μΔ))},
where 𝔐[r](f;μΔ) 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 Corollary2.5 followed by Theorem 3.9.Example 3.11 (see [8, Remark 8]).
Again we consider𝕋=ℤ. Using the notation as introduced in Example 3.8, the term ∫[a,b)‍pln(f)dμΔ/∫[a,b)‍pdμΔ takes the form
(3.21)∑i=1npiln(xi)∑i=1npi=ln(∏i=1n‍xipi)1/Pn=ln(ℳ[0](x,p)),
and (3.20) becomes
(3.22)[Gn(x)An(x)]nmax1≤i≤npi≥[ℳ[0](x,p)ℳ[1](x,p)]Pn≥[Gn(x)An(x)]nmin1≤i≤npi.
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
(3.23)xp:=(x1p1,x2p2,…,xnpn),p-1:=(1p1,1p2,…,1pn),
where x and p are positive n-tuples such that ∑i=1n(1/pi)=1. Then, (3.15) and (3.22) become
(3.24)n[max1≤i≤n1pi][An(xp)-Gn(xp)]≥ℳ[1](xp,p-1)-ℳ[0](xp,p-1)≥n[min1≤i≤n1pi][An(xp)-Gn(xp)],(3.25)[Gn(xp)An(xp)]nmax1≤i≤n(1/pi)≥ℳ[0](xp,p-1)ℳ[1](xp,p-1)≥[Gn(xp)An(xp)]nmin1≤i≤n(1pi).
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

Letn∈ℕ and let fi:[a,b)→ℝ be Δ-integrable for all i∈{1,2,…,n}. Assume pi>1 for all i∈{1,2,…,n} are conjugate exponents, that is, ∑i=1n(1/pi)=1, and ∏i=1nfi1/pi is Δ-integrable on [a,b). Hölder’s inequality on time scales (Theorem 1.4) asserts that
(4.1)∫[a,b)∏i=1n‍fi1/pidμΔ≤∏i=1n‍(∫[a,b)fidμΔ)1/pi.Theorem 4.1.
Letpi>1, i∈{1,2,…,n}, be conjugate exponents, and let fi, i∈{1,2,…,n}, be nonnegative Δ-integrable functions such that ∏i=1nfi1/pi and ∏i=1nfi1/n are nonnegative and Δ-integrable. Then the following inequalities hold:
(4.2)n[max1≤i≤n1pi][∏i=1n‍(∫[a,b)fidμΔ)1/pi-∏i=1n‍(∫[a,b)fidμΔ)1/pi-1/n(∫[a,b)∏i=1n‍fi1/ndμΔ)]≥∏i=1n‍(∫[a,b)fidμΔ)1/pi-(∫[a,b)∏i=1n‍fi1/pidμΔ)≥n[min1≤i≤n1pi][∏i=1n‍(∫[a,b)fidμΔ)1/pi-∏i=1n‍(∫[a,b)fidμΔ)1/pi-1/n(∫[a,b)∏i=1n‍fi1ndμΔ)].Proof.
Letxi=[fi/∫[a,b)‍fidμΔ]1/pi, i∈{1,2,…,n}, in Example 3.12. Then the expressions in (3.24) become
(4.3)ℳ1(xp,p-1)-ℳ0(xp,p-1)=∑i=1nfipi∫[a,b)‍fidμΔ-∏i=1n‍fi1/pi(∫[a,b)‍fidμΔ)1/pi,An(xp)-Gn(xp)=1n∑i=1nfi∫[a,b)‍fidμΔ-∏i=1n‍fi1/n(∫[a,b)‍fidμΔ)1/n.
Now, by applying theΔ-integral to the last two equations, we get
(4.4)∫[a,b)[ℳ1(xp,p-1)-ℳ0(xp,p-1)]dμΔ=∑i=1n‍∫[a,b)fidμΔpi∫[a,b)fidμΔ-∫[a,b)‍(∏i=1nfi1/pi)dμΔ∏i=1n(∫[a,b)fidμΔ)1/pi=1-∫[a,b)(∏i=1nfi1/pi)dμΔ∏i=1n(∫[a,b)fidμΔ)1/pi,∫[a,b)‍[An(xp)-Gn(xp)]dμΔ=1n∑i=1n∫[a,b)fidμΔ∫[a,b)fidμΔ-∫[a,b)(∏i=1nfi1/n)dμΔ∏i=1n‍(∫[a,b)fidμΔ)1/n=1-∫[a,b)(∏i=1nfi1/n)dμΔ∏i=1n(∫[a,b)fidμΔ)1/n.
By applying theΔ-integral to the series of inequalities in (3.24), we obtain the required inequalities.Remark 4.2.
The first inequality in Theorem4.1 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 Theorem4.1, the following inequalities hold:
(4.5)[nn∏i=1n∫[a,b)fidμΔ]min1≤i≤n(1/pi)××∫[a,b)[∑i=1nfipi∫[a,b)fidμΔ][∏i=1nfi1/n∑i=1n(fi/∫[a,b)fidμΔ)]nmin1≤i≤n(1/pi)dμΔ≥∫[a,b)∏i=1nfi1/pidμΔ∏i=1n(∫[a,b)‍fidμΔ)1/pi≥[nn∏i=1n∫[a,b)‍fidμΔ]min1≤i≤n(1/pi)××∫[a,b)‍[∑i=1nfipi∫[a,b)‍fidμΔ][∏i=1nfi1/n∑i=1n(fi/∫[a,b)fidμΔ)]nmax1≤i≤n(1/pi)dμΔ
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
(4.6)ℳ1(xp,p-1)[Gn(xp)An(xp)]nmin1≤i≤n(1/pi)≥ℳ0(xp,p-1)≥ℳ1(xp,p-1)[Gn(xp)An(xp)]nmax1≤i≤n(1/pi).
Now, if we consider then-tuple x=(x1,x2,…,xn), where
(4.7)xi=[fi∫[a,b)fidμΔ]1/pi∀i∈{1,2,…,n},
then the expressions that represent the means in (4.6) become
(4.8)ℳ1(xp,p-1)=∑i=1nfipi∫[a,b)‍fidμΔ,ℳ0(xp,p-1)=∏i=1nfi1/pi(∫[a,b)‍fidμΔ)1/pi,An(xp)=1n∑i=1nfi∫[a,b)‍fidμΔ,Gn(xp)=∏i=1nfi1/n(∫[a,b)‍fidμΔ)1/n.
Now, by taking theΔ-integral on (4.6) in described setting, we obtain the required inequalities.Remark 4.4.
The first inequality in Theorem4.3 gives a refinement and the second one gives a converse of Hölder’s inequality on time scales.Corollary 4.5.
Letr,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
(4.9)[maxt∈[a,b)f(t)][(b-a)1/r(∫[a,b)gfdμΔ)1/s-∫[a,b)(gf)1/sdμΔ]≥(∫[a,b)fdμΔ)1/r(∫[a,b)gdμΔ)1/s-∫[a,b)f1/rg1/sdμΔ≥[mint∈[a,b)f(t)][(b-a)1/r(∫[a,b)gfdμΔ)1/s-∫[a,b)(gf)1/sdμΔ].
Moreover, if0<r<1, then the inequalities in (4.9) hold in reverse order.Proof.
The result follows from Corollary2.5 by replacing f with g/f, p with f, and letting Φ(x)=-rsx1/s. Then Φ is convex on (0,∞), and we have
(4.10)𝒥(Φ,gf,f;μΔ)=∫[a,b)fΦ(gf)dμΔ-∫[a,b)fdμΔΦ(∫[a,b)gdμΔ∫[a,b)fdμΔ)=rs[(∫[a,b)fdμΔ)1-1/s(∫[a,b)gdμΔ)1/s-∫[a,b)f1-1/sg1/sdμΔ]=rs[(∫[a,b)fdμΔ)1/r(∫[a,b)gdμΔ)1/s-∫[a,b)f1/rg1/sdμΔ],𝔍(Φ,gf;μΔ)=∫[a,b)Φ(gf)dμΔ-(b-a)Φ(∫[a,b)g/fdμΔb-a)=rs[(b-a)1-1/s(∫[a,b)gfdμΔ)1/s-∫[a,b)(gf)1/sdμΔ]=rs[(b-a)1/r(∫[a,b)gfdμΔ)1/s-∫[a,b)(gf)1/sdμΔ].
If r>1, then by substituting 𝒥(Φ,g/f,f;μΔ) and 𝔍(Φ,g/f;μΔ) in (2.11), we get (4.9). If 0<r<1, then rs<0, and since the expressions 𝒥(Φ,g/f,f;μΔ) and 𝔍(Φ,g/f;μΔ) 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.
Letr,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)[maxt∈[a,b)f(t)]××[(∫[a,b)fdμΔ)s-1∫[a,b)gfdμΔ-(∫[a,b)fdμΔb-a)s-1(∫[a,b)(gf)1/sdμΔ)s]≥[(∫[a,b)fdμΔ)1/r(∫[a,b)gdμΔ)1/s]s-[∫[a,b)f1/rg1/sdμΔ]s≥[mint∈[a,b)f(t)]××[(∫[a,b)fdμΔ)s-1∫[a,b)gfdμΔ-(∫[a,b)fdμΔb-a)s-1(∫[a,b)(gf)1/sdμΔ)s].Proof.
In Corollary2.5, replace f with (g/f)1/s, p with f, and let Φ(x)=xs/(s(s-1)). Then Φ is convex on (0,∞). We get
(4.12)𝒥(Φ,(gf)1/s,f;μΔ)=∫[a,b)fΦ((gf)1/s)dμΔ-∫[a,b)fdμΔΦ(∫[a,b)f1/rg1/sdμΔ∫[a,b)fdμΔ)=1s(s-1)[∫[a,b)gdμΔ-(∫[a,b)fdμΔ)1-s(∫[a,b)f1/rg1/sdμΔ)s],𝔍(Φ,(gf)1/s;μΔ)=∫[a,b)Φ((gf)1/s)dμΔ-(b-a)Φ(∫[a,b)(g/f)1/sdμΔb-a)=1s(s-1)[∫[a,b)gfdμΔ-(b-a)1-s(∫[a,b)(gf)1/sdμΔ)s].
Now, the result follows immediately from (2.11).



---

*Source: 384045-2012-12-04.xml*