# Jensen Functionals on Time Scales for Several Variables

Authors: Matloob Anwar; Rabia Bibi; Martin Bohner; Josip Pečarić
Journal: International Journal of Analysis
 (2014)

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## Abstract

We define Jensen functionals and concerned generalized means for several variables on time scales. We derive properties of Jensen functionals and apply them to generalized means. In this setting, we obtain generalizations, refinements, and conversions of many remarkable inequalities.


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## Body

## 1. Introduction

Jensen's inequality is well known in analysis and many other areas of mathematics. Most of the classical inequalities can be obtained by using the Jensen inequality. For time scale theory, Jensen's inequality for one variable is obtained by Agarwal et al. [1], and now there are various extensions and generalizations of it given by many researchers (see [2–8]). In [3], it is shown that the Jensen inequality for one variable holds for time scale integrals including the Cauchy delta, Cauchy nabla, diamond-α, Riemann, Lebesgue, multiple Riemann, and multiple Lebesgue integrals. Further, in [4], we give properties and applications of Jensen functionals on time scales for one variable.In this paper, we obtain the Jensen inequality for several variables and deduce Jensen functionals. We discuss several properties and applications of Jensen functionals. In the sequel, we give all the results for Lebesgue delta integrals. For other time scale integrals, as mentioned above, all those results can be obtained in a similar way. These results generalize the results given in [4] for one variable. Now, we give a brief introduction of time scale integrals; for a detailed introduction we refer to [1, 9–12]. A time scale 𝕋 is an arbitrary closed subset of ℝ, and time scale calculus provides unification and extension of classical results. For example, when 𝕋=ℝ, the time scale integral is an ordinary integral, and when 𝕋=ℤ, the time scale integral becomes a sum. In [10, Chapter 5], the Lebesgue integral is introduced: let [a,b)⊆𝕋 be a time scale interval defined by
(1)[a,b)={t∈𝕋:a≤t<b},
where a,b∈𝕋 with a≤b. Let μΔ be the Lebesgue Δ-measure on [a,b). Suppose f:[a,b)→ℝ is a μΔ-measurable function. Then the Lebesgue Δ-integral of f on [a,b) is denoted by
(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 Lebesgue Δ-integrals on 𝕋. Now, we give some properties of Lebesgue Δ-integrals and state Jensen's inequality and Hölder's inequality for Lebesgue Δ-integrals. Throughout this paper, [a,b) denotes a time scale interval otherwise is specified.Theorem 1 (see [3, Theorem 3.2]).
Iff and g are Δ-integrable functions on [a,b), then
(3)∫[a,b)‍(αf+βg)dμΔ=α∫[a,b)‍fdμΔ+β∫[a,b)‍gdμΔlllllllllllllllllllllllllllllll∀α,β∈ℝ,f(t)≥0∀t∈[a,b)implies∫[a,b)‍fdμΔ≥0.Theorem 2 (see [3, 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
(4)Φ(∫[a,b)‍pfdμΔ∫[a,b)‍pdμΔ)≤∫[a,b)‍p(Φ∘f)dμΔ∫[a,b)‍pdμΔ.Theorem 3 (see [3, 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
(5)∫[a,b)‍wfgdμΔ≤(∫[a,b)‍wfpdμΔ)1/p(∫[a,b)‍wgqdμΔ)1/q.
If 0<p<1 and ∫[a,b)‍wgqdμΔ>0, or if p<0 and ∫[a,b)‍wfpdμΔ>0, then (5) is reversed.Remark 4.
Theorem1 recalls that the Lebesgue Δ-integral is an isotonic linear functional (see [13]). So we can also use the approach of isotonic linear functionals whenever results are known for isotonic linear functionals.In the next section, we give Jensen inequality on time scales for several variables and define Jensen functionals. In Section3, we investigate properties of Jensen functionals and some of its consequences regarding superadditivity and monotonicity. In Section 4, we apply these results to weighted general means, defined on time scales, and give many applications. Finally in Section 5, we give applications to Hölder's inequality on time scales.


## 2. Jensen Inequality and Jensen Functionals

Letf(t)=(f1(t),…,fn(t)) be an n-tuple of functions such that f1,…,fn are Δ-integrable on [a,b). Then ∫[a,b)‍fdμΔ denotes the n-tuple:
(6)(∫[a,b)‍f1dμΔ,…,∫[a,b)‍fndμΔ).
That is, Δ-integral acts on each component of f.Theorem 5 (Jensen inequality).
AssumeΦ∈C(K,ℝ) is convex, where K⊆ℝn is closed and convex. Suppose fi, i=1,2,…,n, are Δ-integrable on [a,b) such that f(t)=(f1(t),…,fn(t))∈K for all t∈[a,b). Moreover, let p:[a,b)→ℝ be nonnegative and Δ-integrable such that ∫[a,b)‍pdμΔ>0. Then
(7)Φ(∫[a,b)‍pfdμΔ∫[a,b)‍pdμΔ)≤∫[a,b)‍pΦ(f)dμΔ∫[a,b)‍pdμΔ.Proof.
SupposeΦ is convex on K⊂ℝn. Therefore, for every point x0∈K, there exists a point λ∈ℝn (see [13, Theorem 1.31]) such that
(8)Φ(x)-Φ(x0)≥〈λ,x-x0〉.
Let λ=(λ1,…,λn). By (8), we get
(9)∫[a,b)‍pΦ(f)dμΔ∫[a,b)‍pdμΔ-Φ(∫[a,b)‍pfdμΔ∫[a,b)‍pdμΔ)=∫[a,b)‍p{Φ(f)-Φ(∫[a,b)‍pfdμΔ/∫[a,b)‍pdμΔ)}dμΔ∫[a,b)‍pdμΔ≥∫[a,b)‍p〈λ,f-(∫[a,b)‍pfdμΔ/∫[a,b)‍pdμΔ)〉dμΔ∫[a,b)‍pdμΔ=∫[a,b)‍p∑i=1n‍λi(fi-(∫[a,b)‍pfidμΔ/∫[a,b)‍pdμΔ))dμΔ∫[a,b)‍pdμΔ=0,
and hence the proof is completed.Remark 6.
By using the fact that the time scale integral is an isotonic linear functional, Theorem5 can also be obtained by using Theorem 1 and [13, Theorem 2.6].Definition 7.
AssumeΦ∈C(K,ℝ), where K⊆ℝn is closed and convex. Suppose fi, i=1,2,…,n, are Δ-integrable on [a,b) such that f(t)=(f1(t),…,fn(t))∈K for all t∈[a,b). Moreover, let p:[a,b)→ℝ be nonnegative and Δ-integrable such that ∫[a,b)‍pdμΔ>0. Then one defines the Jensen functional on time scales for several variables by
(10)JΔ(Φ,f,p)=∫[a,b)‍pΦ(f)dμΔ-∫[a,b)‍pdμΔΦ(∫[a,b)‍pfdμΔ∫[a,b)‍pdμΔ)Remark 8.
By Theorem5, the following statements are obvious. If Φ is convex, then
(11)JΔ(Φ,f,p)≥0
while if Φ is concave, then
(12)JΔ(Φ,f,p)≤0.Example 9.
Let[a,b)={1,2,…,n}, f1(i)=x1i,…,fn(i)=xni, and p(i)=pi, i=1,2,…,n in (10). Then the Jensen functional (10) becomes
(13)Jn(Φ,X,p)=∑i=1n‍piΦ(xi)-PnΦ(∑i=1n‍pixiPn),
where X=(x1,x2,…,xn) with xi=(xi1,xi2,…,xin), p=(p1,…,pn), and Pn=∑i=1n‍pi>0. Some properties of the Jensen functional Jn are investigated in [14, 15].Example 10.
If[a,b) is a real interval, then Jensen's functional (10) becomes
(14)∫[a,b)‍p(t)Φ(f1(t),f2(t),…,fn(t))dμ(t)-∫[a,b)‍p(t)dμ(t)×Φ(∫[a,b)‍p(t)f1(t)dμ(t)∫[a,b)‍p(t)dμ(t),∫[a,b)‍p(t)f2(t)dμ(t)∫[a,b)‍p(t)dμ(t),…,∫[a,b)‍p(t)fn(t)dμ(t)∫[a,b)‍p(t)dμ(t)).


## 3. Properties of Jensen Functionals

In the following theorem, we give our main result concerning the properties of the Jensen functional (10).Theorem 11.
AssumeΦ∈C(K,ℝ), where K⊆ℝn is closed and convex. Suppose fi, i=1,2,…,n, are Δ-integrable on [a,b) such that f(t)=(f1(t),…,fn(t))∈K for all t∈[a,b). Moreover, 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 JΔ(Φ,f,·) is superadditive; that is,
(15)JΔ(Φ,f,p+q)≥JΔ(Φ,f,p)+JΔ(Φ,f,q),
and JΔ(Φ,f,·) is increasing; that is, p≥q with ∫[a,b)‍pdμΔ>∫[a,b)‍qdμΔ implies
(16)JΔ(Φ,f,p)≥JΔ(Φ,f,q).
Moreover, if Φ is concave, then JΔ(Φ,f,·) is subadditive and decreasing; that is, (15) and (16) hold in reverse order.Proof.
LetΦ be convex. Because the time scales integral is linear (see Theorem 1), it follows from Definition 7 that
(17)JΔ(Φ,f,p+q)=∫[a,b)‍(p+q)Φ(f)dμΔ-∫[a,b)‍(p+q)dμΔΦ(∫[a,b)‍(p+q)fdμΔ∫[a,b)‍(p+q)dμΔ)=∫[a,b)‍pΦ(f)dμΔ+∫[a,b)‍qΦ(f)dμΔ-(∫[a,b)‍pdμΔ+∫[a,b)‍qdμΔ)×Φ(∫[a,b)‍pfdμΔ+∫[a,b)‍qfdμΔ∫[a,b)‍pdμΔ+∫[a,b)‍qdμΔ)≥∫[a,b)‍pΦ(f)dμΔ+∫[a,b)‍qΦ(f)dμΔ-∫[a,b)‍pdμΔΦ(∫[a,b)‍pfdμΔ∫[a,b)‍pdμΔ)-∫[a,b)‍qdμΔΦ(∫[a,b)‍qfdμΔ∫[a,b)‍qdμΔ)=JΔ(Φ,f,p)+JΔ(Φ,f,q).
If p≥q, we have p-q≥0. Now, because Jensen's functional is superadditive and nonnegative, we have
(18)JΔ(Φ,f,p)=JΔ(Φ,f,p-q+q)≥JΔ(Φ,f,p-q)+JΔ(Φ,f,q)≥JΔ(Φ,f,q).
On the other hand, if Φ is concave, then the reversed inequalities of (15) and (16) can be obtained in a similar way.Corollary 12.
LetΦ, f, p, q satisfy the hypotheses of Theorem 11. Further, suppose there exist nonnegative constants m and M such that
(19)Mq(t)≥p(t)≥mq(t)∀t∈[a,b),M∫[a,b)‍qdμΔ>∫[a,b)‍pdμΔ>m∫[a,b)‍qdμΔ.
If Φ is convex, then
(20)MJΔ(Φ,f,q)≥JΔ(Φ,f,p)≥mJΔ(Φ,f,q),
while if Φ is concave, then the inequalities in (20) hold in reverse order.Proof.
By using (10), we have
(21)JΔ(Φ,f,mq)=mJΔ(Φ,f,q),JΔ(Φ,f,Mq)=MJΔ(Φ,f,q).
Now the result follows from the second property of Theorem 11.Corollary 13.
LetΦ, f, p satisfy the hypotheses of Theorem 11. Further, assume that p attains its minimum value and its maximum value on its domain. If Φ is convex, then
(22)[max⁡t∈[a,b)p(t)]𝔍Δ(Φ,f)≥JΔ(Φ,f,p)≥[min⁡t∈[a,b)p(t)]𝔍Δ(Φ,f),
Where
(23)𝔍Δ(Φ,f)=∫[a,b)‍Φ(f)dμΔ-(b-a)Φ(∫[a,b)‍fdμΔb-a).
Moreover, if Φ is concave, then the inequalities in (22) hold in reverse order.Proof.
Letp attain its minimum and maximum values on its domain [a,b). Then
(24)max⁡t∈[a,b)p(t)≥p(t)≥min⁡t∈[a,b)p(t).
Let
(25)p¯(t)=max⁡t∈[a,b)p(t),p_(t)=min⁡t∈[a,b)p(t).
By using (10), we have
(26)JΔ(Φ,f,p¯)=[max⁡t∈[a,b)p(t)]𝔍Δ(Φ,f),JΔ(Φ,f,p_)=[min⁡t∈[a,b)p(t)]𝔍Δ(Φ,f).
Now the result follows from the second property of Theorem 11.Example 14.
Let the functionalJn(Φ,X,p) be defined as in Example 9. Let q=(q1,…,qn) with qi≥0 and ∑i=1nqi=Qn>0. If Φ is convex, then Theorem 11 implies Jn(Φ,X,·) is superadditive; that is,
(27)Jn(Φ,X,p+q)≥Jn(Φ,X,p)+Jn(Φ,X,q),
and Jn(Φ,X,·) is increasing; that is, if p≥q such that Pn>Qn, then
(28)Jn(Φ,X,p)≥Jn(Φ,X,q).
Moreover, if Φ is concave, then the inequalities in (27) and (28) hold in reverse order. If p attains its minimum and maximum values on its domain, then Corollary 13 yields
(29)max⁡1≤i≤n{pi}𝔍n(Φ,X)≥Jn(Φ,X,p)≥min⁡1≤i≤n{pi}𝔍n(Φ,X),
where
(30)𝔍n(Φ,X)=∑i=1n‍Φ(xi)-nΦ(∑i=1n‍xin),
if Φ is convex. Further, the inequalities in (29) hold in reverse order if Φ is concave.


## 4. Applications to Weighted Generalized Means

In the sequel,I⊂ℝ is an interval and K⊂ℝn is closed and convex.Definition 15.
Assumeχ∈C(I,ℝ) is strictly monotone and φ:K→I is a function of n variables. Suppose fi, i=1,2,…,n, are Δ-integrable on [a,b) such that f(t)=(f1(t),…,fn(t))∈K for all t∈[a,b). Moreover, let p:[a,b)→ℝ be nonnegative and Δ-integrable such that pχ(φ(f)) is Δ-integrable and ∫[a,b)‍pdμΔ>0. Then one defines the weighted generalized mean on time scales by
(31)MΔ(χ,φ(f),p)=χ-1(∫[a,b)‍pχ(φ(f))dμΔ∫[a,b)‍pdμΔ).Theorem 16.
Assumeχ,ψi∈C(I,ℝ), i=1,2,…,n, are strictly monotone and φ:K→I⊆ℝ is a function of n variables. Suppose fi:[a,b)→I, i=1,2,…,n, are Δ-integrable such that f(t)=(f1(t),…,fn(t))∈K for all t∈[a,b). Moreover, let p,q:[a,b)→ℝ be nonnegative and Δ-integrable such that pχ(φ(f)), qχ(φ(f)), pψi(fi)), qψi(fi)), i=1,2,…,n, are Δ-integrable and ∫[a,b)‍pdμΔ>0, ∫[a,b)‍qdμΔ>0. If H defined by
(32)H(s1,…,sn)=χ∘φ(ψ1-1(s1),…,ψn-1(sn))
is convex, then the functional
(33)∫[a,b)‍pdμΔ[χ(MΔ(χ,φ(f),p))-χ∘φ(MΔ(ψ1,f1,p),…,MΔ(ψn,fn,p))]
is superadditive, that is,
(34)∫[a,b)‍(p+q)dμΔ[χ(MΔ(χ,φ(f),p+q))-χ∘φ(MΔ(ψ1,f1,p+q),…,MΔ(ψn,fn,p+q))]≥∫[a,b)‍pdμΔ[χ(MΔ(χ,φ(f),p))-χ∘φ(MΔ(ψ1,f1,p),…,MΔ(ψn,fn,p))]+∫[a,b)‍qdμΔ[χ(MΔ(χ,φ(f),q))-χ∘φ(MΔ(ψ1,f1,q),…,MΔ(ψn,fn,q))],
and increasing; that is, p≥q with ∫[a,b)‍pdμΔ>∫[a,b)‍qdμΔ implies
(35)∫[a,b)‍pdμΔ[χ(MΔ(χ,φ(f),p))-χ∘φ(MΔ(ψ1,f1,p),…,MΔ(ψn,fn,p))]≥∫[a,b)‍qdμΔ[χ(MΔ(χ,φ(f),q))-χ∘φ(MΔ(ψ1,f1,q),…,MΔ(ψn,fn,q))].
Moreover, if H is continuous and concave, then (33) is subadditive and decreasing; that is, (34) and (35) hold in reverse order.Proof.
The functional defined in (33) is obtained by replacing Φ with H and fi with ψi(fi), i=1,2,…,n, in the Jensen functional (10) and letting Ψ(f)=(ψ1(f1),…,ψn(fn)); that is,
(36)JΔ(H,Ψ(f),p)=∫[a,b)‍pχ∘φ(f1,…,fn)dμΔ-∫[a,b)‍pdμΔ×H(∫[a,b)‍pψ1(f1)dμΔ∫[a,b)‍pdμΔ,…,∫[a,b)‍pψn(fn)dμΔ∫[a,b)‍pdμΔ)=∫[a,b)‍pdμΔχ(MΔ(χ,φ(f),p))-∫[a,b)‍pdμΔχ∘φ(MΔ(ψ1,f1,p),…,MΔ(ψn,fn,p))=∫[a,b)‍pdμΔ[χ(MΔ(χ,φ(f),p))-χ∘φ(MΔ(ψ1,f1,p),…,MΔ(ψn,fn,p))].
Now, all claims follow immediately from Theorem 11.Corollary 17.
LetH, φ,f, p, χ, fi, and ψi, i=1,…,n, satisfy the hypothesis of Theorem 16. Further, assume that p attains its minimum value and its maximum value on its domain. If H is convex, then
(37)[max⁡t∈[a,b)p(t)](b-a)×[χ(𝔐Δ(χ,φ(f)))-χ∘φ(𝔐Δ(ψ1,f1),…,𝔐Δ(ψn,fn))(𝔐Δ(χ,φ(f)))]≥∫[a,b)‍pdμΔ[χ(MΔ(χ,φ(f),p))-χ∘φ(MΔ(ψ1,f1,p),…,MΔ(ψn,fn,p))]≥[min⁡t∈[a,b)p(t)](b-a)×[χ(𝔐Δ(χ,φ(f)))-χ∘φ(𝔐Δ(ψ1,f1),…,𝔐Δ(ψn,fn))(𝔐Δ(χ,φ(f)))],
where
(38)𝔐Δ(χ,φ(f))=χ-1(∫[a,b)‍χ(φ(f))dμΔb-a).
Moreover, if H is concave, then the inequalities in (37) hold in reverse order.Proof.
The proof is omitted as it is similar to the proof of Corollary13.Remark 18.
If we take the discrete form of the weighted generalized mean (31) with ∫[a,b)‍pdμΔ=1, then we obtain the quasiarithmetic mean. Namely, let ψ:I⊆ℝ¯→ℝ¯ be continuous and strictly monotone, a=(a1,…,an) with ak∈I, k=1,…,n, and w=(w1,…,wn) with wk≥0 and ∑k=1n‍wk=1. Then the quasiarithmetic mean of a with weight w is defined by
(39)Mn=ψ-1(∑k=1n‍wkψ(ak)).Now the following examples connect the quasiarithmetic mean (39) and the properties of Jensen functionals.Example 19 (see [16, Corollary 3]).
Letw and ψ be defined as in Remark 18 and let ψ be strictly increasing and strictly convex with continuous derivatives of second order such that ψ′/ψ′′ is concave. Further, let X, p, xi, i=1,…,n, be defined as in Example 9, and q=(q1,…,qn) with qi≥0, i=1,…,n, and ∑i=1n‍qi=Qn>0. Then, ΦMn(xi)=ψ-1(∑k=1n‍wkψ(xik)) is a convex function (see [17, Theorem 1, page 197]). Hence by Theorem 11, the functional
(40)Jn(ΦMn,X,p)=∑i=1n‍piΦMn(xi)-PnΦMn(∑i=1n‍pixiPn)
is superadditive, that is,
(41)Jn(ΦMn,X,p+q)≥Jn(ΦMn,X,p)+Jn(ΦMn,X,q),
and increasing; that is, if p≥q such that Pn>Qn, then
(42)Jn(ΦMn,X,p)≥Jn(ΦMn,X,q).
Also, by Corollary 12, we have
(43)max⁡1≤i≤n{pi}𝔍n(ΦMn,X)≥Jn(ΦMn,X,p)≥min⁡1≤i≤n{pi}𝔍n(ΦMn,X),
where
(44)𝔍n(ΦMn,X)=∑i=1n‍ΦMn(xi)-nΦMn(∑i=1n‍xin).Example 20 (see [16, Corollary 4]).
Consider (39), but with different conditions on ψ and w. Namely, if (i)
wi≥1 for i=1,…,n;(ii)
ψ:ℝ+→ℝ+;(iii)
lim⁡x→0ψ(x)=∞ or lim⁡x→∞ψ(x)=∞,then we define
(45)M~n=ψ-1(∑k=1n‍wkψ(ak)).
Let X, p, xi, i=1,…,n, be defined as in Example 9 and q=(q1,…,qn) with qi≥0 and ∑i=1n‍qi=Qn>0. Let ψ be strictly increasing and strictly convex with continuous derivatives of second order such that ψ/ψ′ is convex. Then ΦM~n(xi)=ψ-1(∑k=1n‍wkψ(xik)) is a convex function (see [17, Theorem 2, page 197]). Hence, by Theorem 11, the functional
(46)Jn(ΦM~n,X,p)=∑i=1n‍piΦM~n(xik)-PnΦM~n(∑i=1n‍pixikPn)
is superadditive, that is,
(47)Jn(ΦM~n,X,p+q)≥Jn(ΦM~n,X,p)+Jn(ΦM~n,X,q),
and increasing; that is, if p≥q, then
(48)Jn(ΦM~n,X,p)≥Jn(ΦM~n,X,q).
Also, by Corollary 12, we have
(49)max⁡1≤i≤n{pi}𝔍n(ΦM~n,X)≥Jn(ΦMn,X,p)≥min⁡1≤i≤n{pi}𝔍n(ΦM~n,X),
where
(50)𝔍n(ΦM~n,X)=∑i=1n‍ΦM~n(xik)-nΦM~n(∑i=1n‍xikn).Example 21 (see [16, Corollary 5]).
For a real-valued functionf defined on interval [a,b], an nth order divided difference of f at distinct points x0,…,xn∈[a,b] is defined recursively by
(51)[xi]f=f(xi),i=0,…,n,[x0,…,xn]f=[x1,…,xn]f-[x0,…,xn-1]fxn-x0.
Further, f is n-convex on [a,b], n≥0, if and only if, for all choices of n+1 distinct points in [a,b],
(52)[x0,…,xn]f≥0.
Let X, p, xi, i=1,…,n, be defined as in Example 9 and q=(q1,…,qn), with qi≥0 and ∑i=1n‍qi=Qn>0. Let f:I→ℝ be (n+1)-convex, where I⊂ℝ is a closed and bounded interval. Then by Theorem 11, for ΦG(xi)=[xi1,…,xin]f, the functional
(53)Jn(ΦG,X,p)=∑i=1n‍piΦG(xi)-PnΦG(∑i=1n‍pixiPn)
is superadditive, that is,
(54)Jn(ΦG,X,p+q)≥Jn(ΦG,X,p)+Jn(ΦG,X,q),
and increasing; that is, if p≥q such that Pn>Qn, then
(55)Jn(ΦG,X,p)≥Jn(ΦG,X,q).
Also, by Corollary 12, we have
(56)max⁡1≤i≤n{pi}𝔍n(ΦG,X)≥Jn(ΦG,X,p)≥min⁡1≤i≤n{pi}𝔍n(ΦG,X)
where
(57)𝔍n(ΦG,X)=∑i=1n‍ΦG(xi)-nΦG(∑i=1n‍xin).Corollary 22.
Assumeχ, ψ1, and ψ2∈C2(I,ℝ) are strictly monotone. Suppose f1,f2:[a,b)→I are Δ-integrable such that f1(t)+f2(t)∈I for all t∈[a,b) and p,q:[a,b)→ℝ are nonnegative and Δ-integrable such that pχ(f1+f2), qχ(f1+f2), pψi(fi), and qψi(fi), i=1,2, are Δ-integrable and ∫[a,b)‍pdμΔ>0, ∫[a,b)‍qdμΔ>0. Further, let
(58)E=ψ1′ψ1′′,F=ψ2′ψ2′′,G=χ′χ′′.
If ψ1′, ψ2′, and χ′ are positive and ψ1′′, ψ2′′, and χ′′ are negative, then the functional
(59)∫[a,b)‍pdμΔ[χ(MΔ(χ,f1+f2,p))-χ(MΔ(ψ1,f1,p)+MΔ(ψ2,f2,p))]
is superadditive, that is,
(60)∫[a,b)‍(p+q)dμΔ[χ(MΔ(χ,f1+f2,p+q))-χ(MΔ(ψ1,f1,p+q)+MΔ(ψ2,f2,p+q))]≥∫[a,b)‍pdμΔ[χ(MΔ(χ,f1+f2,p))-χ(MΔ(ψ1,f1,p)+MΔ(ψ2,f2,p))]+∫[a,b)‍qdμΔ[χ(MΔ(χ,f1+f2,q))-χ(MΔ(ψ1,f1,q)+MΔ(ψ2,f2,q))],
and increasing; that is, if p≥q such that ∫[a,b)‍pdμΔ>∫[a,b)‍qdμΔ, then
(61)∫[a,b)‍pdμΔ[χ(MΔ(χ,f1+f2,p))-χ(MΔ(ψ1,f1,p)+MΔ(ψ2,f2,p))]≥∫[a,b)‍qdμΔ[χ(MΔ(χ,f1+f2,q))-χ(MΔ(ψ1,f1,q)+MΔ(ψ2,f2,q))],
if and only if G(x+y)≤E(x)+F(y). If p attains its minimum and maximum values on its domain [a,b), then (61) yields
(62)[max⁡t∈[a,b)p(t)](b-a)[χ(𝔐Δ(χ,f1+f2))-χ(𝔐Δ(ψ1,f1)+𝔐Δ(ψ2,f2))]≥∫[a,b)‍pdμΔ[χ(MΔ(χ,f1+f2,p))-χ(MΔ(ψ1,f1,p)+MΔ(ψ2,f2,p))]≥[min⁡t∈[a,b)p(t)](b-a)[χ(𝔐Δ(χ,f1+f2))-χ(𝔐Δ(ψ1,f1)+𝔐Δ(ψ2,f2))].
Moreover, if ψ1′, ψ2′, χ′, ψ1′′, ψ2′′, and χ′′ are all positive, then the inequalities in (60), (61), and (62) are reversed if and only if G(x+y)≥E(x)+F(y).Proof.
Letn=2 in Theorem 16. By setting φ(x,y)=x+y, we have
(63)H(s1,s2)=χ(ψ1-1(s1)+ψ2-1(s2)).
If ψ1′, ψ2′, and χ′ are positive and ψ1′′, ψ2′′, and χ′′ are negative, then H is convex if and only if G(x+y)≤E(x)+F(y) (see [18]). If ψ1′, ψ2′, χ′, ψ1′′, ψ2′′, and χ′′ are all positive, then H is concave if and only if G(x+y)≥E(x)+F(y) (see [18]). Now, all claims follow immediately from Theorem 16.Corollary 23.
Assumeχ, ψ1, and ψ2∈C2(I,ℝ) are strictly monotone. Suppose f1,f2:[a,b)→I are Δ-integrable such that f1(t)f2(t)∈I for all t∈[a,b) and p,q:[a,b)→ℝ are nonnegative and Δ-integrable such that pχ(f1·f2), qχ(f1·f2), pψi(fi), and qψi(fi), i=1,2, are Δ-integrable and ∫[a,b)‍pdμΔ>0, ∫[a,b)‍qdμΔ>0. Further, let
(64)A(t)=ψ1′(t)ψ1′(t)+tψ1′′(t),B(t)=ψ2′(t)ψ2′(t)+tψ2′′(t),C(t)=χ′(t)χ′(t)+tχ′′(t).
If ψ1′, ψ2′, and χ′ are positive and A, B, and C are negative, then the functional
(65)∫[a,b)‍pdμΔ[χ(MΔ(χ,f1·f2,p))-χ(MΔ(ψ1,f1,p)·MΔ(ψ2,f2,p))]
is superadditive, that is,
(66)∫[a,b)‍(p+q)dμΔ[χ(MΔ(χ,f1·f2,p+q))-χ(MΔ(ψ1,f1,p+q)·MΔ(ψ2,f2,p+q))]≥∫[a,b)‍pdμΔ[χ(MΔ(χ,f1·f2,p))-χ(MΔ(ψ1,f1,p)·MΔ(ψ2,f2,p))]+∫[a,b)‍qdμΔ[χ(MΔ(χ,f1·f2,q))-χ(MΔ(ψ1,f1,q)·MΔ(ψ2,f2,q))],
and increasing; that is, if p≥q such that ∫[a,b)‍pdμΔ>∫[a,b)‍qdμΔ, then
(67)∫[a,b)‍pdμΔ[χ(MΔ(χ,f1·f2,p))-χ(MΔ(ψ1,f1,p)·MΔ(ψ2,f2,p))]≥∫[a,b)‍qdμΔ[χ(MΔ(χ,f1·f2,q))-χ(MΔ(ψ1,f1,q)·MΔ(ψ2,f2,q))],
if and only if C(x·y)≤A(x)+B(y). If p attains its minimum and maximum values on its domain [a,b), then (67) yields
(68)[max⁡t∈[a,b)p(t)](b-a)[χ(𝔐Δ(χ,f1·f2))-χ(𝔐Δ(ψ1,f1)·𝔐Δ(ψ2,f2))]≥∫[a,b)‍pdμΔ[χ(MΔ(χ,f1·f2,p))-χ(MΔ(ψ1,f1,p)·MΔ(ψ2,f2,p))]≥[min⁡t∈[a,b)p(t)](b-a)[χ(𝔐Δ(χ,f1·f2))-χ(𝔐Δ(ψ1,f1)·𝔐Δ(ψ2,f2))].
If ψ1′,ψ2′,χ′,A,B, and C are all positive, then the inequalities in (66), (67), and (68) are reversed if and only if C(x·y)≥A(x)+B(y).Proof.
Letn=2 in Theorem 16. By setting φ(x,y)=x·y, we have
(69)H(s1,s2)=χ(ψ1-1(s1)·ψ2-1(s2)).
If ψ1′, ψ2′, and χ′ are positive and A, B, and C are negative, then H is convex if and only if C(x·y)≤A(x)+B(y). If ψ1′, ψ2′, χ′, A, B, and C are all positive, then H is concave if and only if C(x·y)≥A(x)+B(y) (see [18]). Now, all claims follow immediately from Theorem 16.Corollary 24.
Letλ,ω,ν∈ℝ be such that (a)
λ<0<ω,ν, or ω,ν<0<λ;(b)
λ<ω,ν<0, or ν<0<ω<λ, or ω<0<ν<λ, for 1/λ≤1/ω+1/ν;(c)
λ<ω<0<ν, or λ<ν<0<ω, for 1/λ≥1/ω+1/ν.Suppose f1,f2:[a,b)→ℝ are Δ-integrable and p,q:[a,b)→ℝ are nonnegative and Δ-integrable such that p·f1λ·f2λ, q·f1λ·f2λ, pf1ω, qf1ω, pf2ν, and qf2ν are Δ-integrable and ∫[a,b)‍pdμΔ>0, ∫[a,b)‍qdμΔ>0. Then the functional
(70)∫[a,b)‍p·f1λ·f2λdμΔ-∫[a,b)‍pdμΔ[(∫[a,b)‍pf1ωdμΔ∫[a,b)‍pdμΔ)1/ω(∫[a,b)‍pf2νdμΔ∫[a,b)‍pdμΔ)1/ν]λ
is superadditive, that is,
(71)∫[a,b)‍(p+q)·f1λ·f2λdμΔ-∫[a,b)‍(p+q)dμΔ[(∫[a,b)‍(p+q)f1ωdμΔ∫[a,b)‍(p+q)dμΔ)1/ω×(∫[a,b)‍(p+q)f2νdμΔ∫[a,b)‍(p+q)dμΔ)1/ν]λ≥∫[a,b)‍p·f1λ·f2λdμΔ-∫[a,b)‍pdμΔ[(∫[a,b)‍pf1ωdμΔ∫[a,b)‍pdμΔ)1/ω×(∫[a,b)‍pf2νdμΔ∫[a,b)‍pdμΔ)1/ν]λ+∫[a,b)‍q·f1λ·f2λdμΔ-∫[a,b)‍qdμΔ[(∫[a,b)‍qf1ωdμΔ∫[a,b)‍qdμΔ)1/ω×(∫[a,b)‍qf2νdμΔ∫[a,b)‍qdμΔ)1/ν]λ,
and increasing; that is, if p≥q such that ∫[a,b)‍pdμΔ>∫[a,b)‍qdμΔ, then
(72)∫[a,b)‍p·f1λ·f2λdμΔ-∫[a,b)‍pdμΔ[(∫[a,b)‍pf1ωdμΔ∫[a,b)‍pdμΔ)1/ω(∫[a,b)‍pf2νdμΔ∫[a,b)‍pdμΔ)1/ν]λ≥∫[a,b)‍q·f1λ·f2λdμΔ-∫[a,b)‍qdμΔ[(∫[a,b)‍qf1ωdμΔ∫[a,b)‍qdμΔ)1/ω(∫[a,b)‍qf2νdμΔ∫[a,b)‍qdμΔ)1/ν]λ.
If p attains its minimum and maximum values on its domain, then
(73)max⁡t∈[a,b)p(t)[[(∫[a,b)‍f2νdμΔb-a)1/ν]λ∫[a,b)‍f1λ·f2λdμΔ-(b-a)[(∫[a,b)‍f1ωdμΔb-a)1/ω×(∫[a,b)‍f2νdμΔb-a)1/ν]λ]≥∫[a,b)‍p·f1λ·f2λdμΔ-∫[a,b)‍pdμΔ[(∫[a,b)‍pf1ωdμΔ∫[a,b)‍pdμΔ)1/ω×(∫[a,b)‍pf2νdμΔ∫[a,b)‍pdμΔ)1/ν]λ≥min⁡t∈[a,b)p(t)[[(∫[a,b)‍f2νdμΔb-a)1/ν]λ∫[a,b)‍f1λ·f2λdμΔ-(b-a)[(∫[a,b)‍f1ωdμΔb-a)1/ω×(∫[a,b)‍f2νdμΔb-a)1/ν]λ].
Moreover, the inequalities in (71), (72), and (73) are reversed provided that (a′)
ω,ν>λ>0, for 1/λ≥1/ω+1/ν;(b′)
ω,ν<λ<0, for 1/λ≤1/ω+1/ν.Proof.
Letn=2 in Theorem 16. By setting φ(x,y)=x·y, χ(t)=tλ, ψ1(t)=tω, and ψ2(t)=tν, we have
(74)H(s1,s2)=χ(ψ1-1(s1)·ψ2-1(s2))=(s11/ωs21/ν)λ.
Now, H is convex if and only if d2H≥0, which implies
(75)λω(λω-1)≥0,λν(λν-1)≥0,λ3ων(1λ-1ω-1ν)≥0,
and these are satisfied if λ, ω, and ν satisfy conditions (a), (b), and (c). H is concave if and only if d2H≤0, and this implies
(76)λω(λω-1)≤0,λν(λν-1)≤0,λ3ων(1λ-1ω-1ν)≥0.
These are satisfied if λ, ω, and ν satisfy conditions (a′) and (b′). Now, all claims follow immediately from Theorem 16.Corollary 25.
Letλ,ω,ν∈ℝ be such that λ,ω,ν>0, λ,ω,ν≠1 and (a)
λ<1<ω,ν, or ω,ν<1<λ;(b)
λ<ω,ν<1, or ν<1<ω<λ, or ω<1<ν<λ, for 1/log⁡λ≤1/log⁡ω+1/log⁡ν;(c)
λ<ω<1<ν, or λ<ν<1<ω, for 1/log⁡λ≥1/log⁡ω+1/log⁡ν.Suppose f1,f2:[a,b)→ℝ are Δ-integrable and p,q:[a,b)→ℝ are nonnegative and Δ-integrable such that pλf1+f2, qλf1+f2, pωf1, qωf1, pνf2, and qνf2 are Δ-integrable and ∫[a,b)‍pdμΔ>0, ∫[a,b)‍qdμΔ>0. Then the functional
(77)∫[a,b)‍pλf1+f2dμΔ-∫[a,b)‍pdμΔ×λlog⁡ω(∫[a,b)‍p·ωf1dμΔ/∫[a,b)‍pdμΔ)+log⁡ν(∫[a,b)‍p·νf2dμΔ/∫[a,b)‍pdμΔ)
is superadditive, that is,
(78)∫[a,b)‍(p+q)λf1+f2dμΔ-∫[a,b)‍(p+q)dμΔ(λlog⁡ω(∫[a,b)‍(p+q)·ωf1dμΔ/∫[a,b)‍(p+q)dμΔ)×λlog⁡ν(∫[a,b)‍(p+q)·νf2dμΔ/∫[a,b)‍(p+q)dμΔ))≥∫[a,b)‍pλf1+f2dμΔ-∫[a,b)‍pdμΔ×λlog⁡ω(∫[a,b)‍p·ωf1dμΔ/∫[a,b)‍pdμΔ)+log⁡ν(∫[a,b)‍p·νf2dμΔ/∫[a,b)‍pdμΔ)+∫[a,b)‍qλf1+f2dμΔ-∫[a,b)‍qdμΔ×λlog⁡ω(∫[a,b)‍q·ωf1dμΔ/∫[a,b)‍qdμΔ)+log⁡ν(∫[a,b)‍q·νf2dμΔ/∫[a,b)‍qdμΔ),
and increasing; that is, if p≥q such that ∫[a,b)‍pdμΔ>∫[a,b)‍qdμΔ, then
(79)∫[a,b)‍pλf1+f2dμΔ-∫[a,b)‍pdμΔ×λlog⁡ω(∫[a,b)‍p·ωf1dμΔ/∫[a,b)‍pdμΔ)+log⁡ν(∫[a,b)‍p·νf2dμΔ/∫[a,b)‍pdμΔ)≥∫[a,b)‍qλf1+f2dμΔ-∫[a,b)‍qdμΔ×λlog⁡ω(∫[a,b)‍q·ωf1dμΔ/∫[a,b)‍qdμΔ)+log⁡ν(∫[a,b)‍q·νf2dμΔ/∫[a,b)‍qdμΔ).
If p attains its minimum and maximum values on its domain, then
(80)max⁡t∈[a,b)p(t)[∫[a,b)‍λf1+f2dμΔ-(b-a)×λlog⁡ω(∫[a,b)‍ωf1dμΔ/(b-a))+log⁡ν(∫[a,b)‍νf2dμΔ/(b-a))∫[a,b)‍]≥∫[a,b)‍pλf1+f2dμΔ-∫[a,b)‍pdμΔ×λlog⁡ω(∫[a,b)‍p·ωf1dμΔ/∫[a,b)‍pdμΔ)+log⁡ν(∫[a,b)‍p·νf2dμΔ/∫[a,b)‍pdμΔ)≥min⁡t∈[a,b)p(t)[∫[a,b)‍λf1+f2dμΔ-(b-a)×λlog⁡ω(∫[a,b)‍ωf1dμΔ/(b-a))+log⁡ν(∫[a,b)‍νf2dμΔ/(b-a))∫[a,b)‍].
Moreover, the inequalities in (78), (79), and (80) are reversed provided that (a′)
ω,ν>λ>1, for 1/log⁡λ≥1/log⁡ω+1/log⁡ν;(b′)
ω,ν<λ<0, for 1/log⁡λ≤1/log⁡ω+1/log⁡ν.Proof.
Letn=2 in Theorem 16. By setting φ(x,y)=x+y, χ(t)=λt, ψ1(t)=ωt, and ψ2(t)=νt, we have
(81)H(s1,s2)=(s11/log⁡ω·s21/log⁡ν)log⁡λ.
Now, the proof is similar to the proof of Corollary 24.Corollary 26.
Letλ,ω,ν∈ℝ be such that (a)
0<ω,ν≤λ<1, for all f1,f2>0;(b)
0<ν≤λ≤ω<1, for f2≥(((ω-λ)(1-ν))/((λ-ν)(1-ω)))f1≥0;(c)
0<ω≤λ≤ν<1, for (((λ-ω)(1-ν))/((ν-λ)(1-ω)))f1≥f2≥0.Suppose f1,f2:[a,b)→ℝ are Δ-integrable and p,q:[a,b)→ℝ are nonnegative and Δ-integrable such that p(f1+f2)λ, q(f1+f2)λ, pf1ω, qf1ω, pf2ν, and qf2ν are Δ-integrable and ∫[a,b)‍pdμΔ>0,∫[a,b)‍qdμΔ>0. Then the functional
(82)∫[a,b)‍p·(f1+f2)λdμΔ-∫[a,b)‍pdμΔ×[(∫[a,b)‍pf1ωdμΔ∫[a,b)‍pdμΔ)1/ω+(∫[a,b)‍pf2νdμΔ∫[a,b)‍pdμΔ)1/ν]λ
is superadditive, that is,
(83)∫[a,b)‍(p+q)·(f1+f2)λdμΔ-∫[a,b)‍(p+q)dμΔ[(∫[a,b)‍(p+q)f1ωdμΔ∫[a,b)‍(p+q)dμΔ)1/ω+(∫[a,b)‍(p+q)f2νdμΔ∫[a,b)‍(p+q)dμΔ)1/ν]λ≥∫[a,b)‍p·(f1+f2)λdμΔ-∫[a,b)‍pdμΔ×[(∫[a,b)‍pf1ωdμΔ∫[a,b)‍pdμΔ)1/ω+(∫[a,b)‍pf2νdμΔ∫[a,b)‍pdμΔ)1/ν]λ+∫[a,b)‍q·(f1+f2)λdμΔ-∫[a,b)‍qdμΔ×[(∫[a,b)‍qf1ωdμΔ∫[a,b)‍qdμΔ)1/ω+(∫[a,b)‍qf2νdμΔ∫[a,b)‍qdμΔ)1/ν]λ,
and increasing; that is, if p≥q such that ∫[a,b)‍pdμΔ>∫[a,b)‍qdμΔ, then
(84)∫[a,b)‍p·(f1+f2)λdμΔ-∫[a,b)‍pdμΔ×[(∫[a,b)‍pf1ωdμΔ∫[a,b)‍pdμΔ)1/ω+(∫[a,b)‍pf2νdμΔ∫[a,b)‍pdμΔ)1/ν]λ≥∫[a,b)‍q·(f1+f2)λdμΔ-∫[a,b)‍qdμΔ×[(∫[a,b)‍qf1ωdμΔ∫[a,b)‍qdμΔ)1/ω+(∫[a,b)‍qf2νdμΔ∫[a,b)‍qdμΔ)1/ν]λ.
If p attains its minimum and maximum values on its domain, then
(85)max⁡t∈[a,b)p(t)[[(∫[a,b)‍pf2νdμΔb-a)1/ν]λ∫[a,b)‍(f1+f2)λdμΔ-(b-a)[(∫[a,b)‍pf1ωdμΔb-a)1/ω+(∫[a,b)‍pf2νdμΔb-a)1/ν]λ]≥∫[a,b)‍p·(f1+f2)λdμΔ-∫[a,b)‍pdμΔ×[(∫[a,b)‍pf1ωdμΔ∫[a,b)‍pdμΔ)1/ω+(∫[a,b)‍pf2νdμΔ∫[a,b)‍pdμΔ)1/ν]λ≥min⁡t∈[a,b)p(t)[[(∫[a,b)‍pf2νdμΔb-a)1/ν]λ∫[a,b)‍(f1+f2)λdμΔ-(b-a)[(∫[a,b)‍pf1ωdμΔb-a)1/ω+(∫[a,b)‍pf2νdμΔb-a)1/ν]λ].
Moreover, the inequalities in (83), (84), and (85) are reversed provided that (a′)
1<λ≤ω,ν, for all f1,f2>0;(b′)
1<ν≤λ≤ω, for 0≤f2≤(((ω-λ)(ν-1))/((λ-ν)(ω-1)))f1;(c′)
1<ω≤λ≤ν, for f2≥(((λ-ω)(ν-1))/((ν-λ)(ω-1)))f1≥0.Proof.
Letn=2 in Theorem 16. By setting φ(x,y)=x+y, χ(t)=tλ, ψ1(t)=tω, and ψ2(t)=tν, we have
(86)H(s1,s2)=(s11/ω+s21/ν)λ.
Now, the proof is similar to the proof of Corollary 22, with some extra considerations of the definitions of E, F, and G.Corollary 27.
Supposef1,f2:[a,b)→[0,π/4] are Δ-integrable. Moreover, let p,q:[a,b)→ℝ be nonnegative and Δ-integrable such that pcos⁡(f1+f2), qcos⁡(f1+f2), pcos⁡(fi), and qcos⁡(fi), i=1,2, are Δ-integrable and ∫[a,b)‍pdμΔ>0, ∫[a,b)‍qdμΔ>0. Then the functional
(87)∫[a,b)‍pdμΔ·cos⁡[arccos(∫[a,b)‍p·cos⁡(f1)dμΔ∫[a,b)‍pdμΔ)+arccos(∫[a,b)‍p·cos⁡(f2)dμΔ∫[a,b)‍pdμΔ)]-∫[a,b)‍pcos⁡(f1+f2)dμΔ
is subadditive, that is,
(88)∫[a,b)‍(p+q)dμΔ·cos⁡[arccos(∫[a,b)‍(p+q)·cos⁡(f1)dμΔ∫[a,b)‍(p+q)dμΔ)+arccos(∫[a,b)‍(p+q)·cos⁡(f2)dμΔ∫[a,b)‍(p+q)dμΔ)]-∫[a,b)‍(p+q)·cos⁡(f1+f2)dμΔ≤∫[a,b)‍pdμΔ·cos⁡[arccos(∫[a,b)‍p·cos⁡(f1)dμΔ∫[a,b)‍pdμΔ)+arccos(∫[a,b)‍p·cos⁡(f2)dμΔ∫[a,b)‍pdμΔ)]-∫[a,b)‍p·cos⁡(f1+f2)dμΔ+∫[a,b)‍qdμΔ·cos⁡[arccos(∫[a,b)‍q·cos⁡(f1)dμΔ∫[a,b)‍qdμΔ)+arccos(∫[a,b)‍q·cos⁡(f2)dμΔ∫[a,b)‍qdμΔ)]-∫[a,b)‍q·cos⁡(f1+f2)dμΔ,
and decreasing; that is, if p≥q such that ∫[a,b)‍pdμΔ>∫[a,b)‍qdμΔ, then
(89)∫[a,b)‍pdμΔ·cos⁡[arccos(∫[a,b)‍p·cos⁡(f1)dμΔ∫[a,b)‍pdμΔ)+arccos(∫[a,b)‍p·cos⁡(f2)dμΔ∫[a,b)‍pdμΔ)]-∫[a,b)‍p·cos⁡(f1+f2)dμΔ≤∫[a,b)‍qdμΔ·cos⁡[arccos(∫[a,b)‍q·cos⁡(f1)dμΔ∫[a,b)‍qdμΔ)+arccos(∫[a,b)‍q·cos⁡(f2)dμΔ∫[a,b)‍qdμΔ)]-∫[a,b)‍q·cos⁡(f1+f2)dμΔ.
If p attains its minimum and maximum values on its domain, then
(90)[max⁡t∈[a,b)p(t)](b-a)cos⁡[arccos(∫[a,b)‍cos⁡(f1)dμΔb-a)+arccos(∫[a,b)‍cos⁡(f2)dμΔb-a)]-∫[a,b)‍cos⁡(f1+f2)dμΔ≤∫[a,b)‍pdμΔ·cos⁡[arccos(∫[a,b)‍p·cos⁡(f1)dμΔ∫[a,b)‍pdμΔ)+arccos(∫[a,b)‍p·cos⁡(f2)dμΔ∫[a,b)‍pdμΔ)]-∫[a,b)‍p·cos⁡(f1+f2)dμΔ≤[min⁡t∈[a,b)p(t)](b-a)cos⁡[arccos(∫[a,b)‍cos⁡(f1)dμΔb-a)+arccos(∫[a,b)‍cos⁡(f2)dμΔb-a)]-∫[a,b)‍cos⁡(f1+f2)dμΔ.Proof.
Letn=2 in Theorem 16. By setting φ(x,y)=x+y and χ(t)=ψ1(t)=ψ2(t)=-cos⁡(t), we have
(91)H(s1,s2)=-cos⁡(arccos(-s1)+arccos(-s2)).
Now, the proof is similar to the proof of Corollary 22.


## 5. Applications to Hölder's Inequality

Supposefi, i=1,2,…,n, are nonnegative Δ-integrable functions on [a,b) such that ∏i=1n‍fiαi is Δ-integrable, where αi≥0, i=1,…,n, are such that ∑i=1n‍αi=1. Then, by using Theorem 3 (Hölder's inequality on time scales), we have
(92)∫[a,b)‍∏i=1n‍fiαidμΔ≤∏i=1n‍(∫[a,b)‍fidμΔ)αi.
If ∑i=1n‍αi=An>0, then (92) implies
(93)∫[a,b)‍∏i=1n‍fiαi/AndμΔ≤∏i=1n‍(∫[a,b)‍fidμΔ)αi/An
or
(94)(∫[a,b)‍∏i=1n‍fiαi/AndμΔ)An≤∏i=1n‍(∫[a,b)‍fidμΔ)αi.
In this section, we discuss properties of the functional, deduced from the Hölder inequality (93), defined in the following way.Definition 28.
Supposef=(f1,…,fn) is such that fi, i=1,…,n, are nonnegative Δ-integrable functions on [a,b). Let α=(α1,…,αn) be such that αi≥0 and ∑i=1n‍αi=An>0. Then one defines the functional HΔ by
(95)HΔ(f,α)=∏i=1n‍(∫[a,b)‍fidμΔ)αi(∫[a,b)‍∏i=1n‍fiαi/AndμΔ)An.Theorem 29.
Letα=(α1,…,αn) and β=(β1,…,βn) be real n-tuples with αi≥0, βi≥0 and ∑i=1n‍αi=An>0, ∑i=1n‍βi=Bn>0. Suppose fi, i=1,…,n, are nonnegative Δ-integrable on [a,b) such that ∏i=1n‍fiαi/An and ∏i=1n‍fiβi/Bn are Δ-integrable. Then
(96)HΔ(f,α+β)≥HΔ(f,α)·HΔ(f,β),
and  HΔ(f,·,μΔ) is increasing; that is, if α≥β such that An>Bn, then
(97)HΔ(f,α)≥HΔ(f,β).Proof.
By Definition28, we have
(98)HΔ(f,α+β)=∏i=1n‍(∫[a,b)‍fidμΔ)αi+βi(∫[a,b)‍∏i=1n‍fi(αi+βi)/(An+Bn)dμΔ)An+Bn,
where
(99)(∫[a,b)‍∏i=1n‍fi(αi+βi)/(An+Bn)dμΔ)An+Bn=[∫[a,b)‍(∏i=1n‍fiαi/An)An/(An+Bn)×(∏i=1n‍fiβi/Bn)Bn/(An+Bn)dμΔ]An+Bn≤(∫[a,b)‍∏i=1n‍fiαi/AndμΔ)An(∫[a,b)‍∏i=1n‍fiβi/BndμΔ)Bn.
Now, by combining (98) and (99), we have
(100)HΔ(f,α+β)≥∏i=1n‍(∫[a,b)‍fidμΔ)αi∏i=1n‍(∫[a,b)‍fidμΔ)βi(∫[a,b)‍∏i=1n‍fiαi/AndμΔ)An(∫[a,b)‍∏i=1n‍fiβi/BndμΔ)Bn=HΔ(f,α)·HΔ(f,β).
If α≥β, then α-β≥0, and therefore
(101)HΔ(f,α)=HΔ(f,(α-β)+β)≥HΔ(f,α-β)·HΔ(f,β)≥HΔ(f,β).
This completes the proof.Corollary 30.
Letf and α satisfy the hypothesis of Theorem 29. Then
(102)[∏i=1n‍∫[a,b)‍fidμΔ(∫[a,b)‍∏i=1n‍fi1/ndμΔ)n]max⁡1≤i≤n{αi}≥HΔ(f,α)≥[∏i=1n‍∫[a,b)‍fidμΔ(∫[a,b)‍∏i=1n‍fi1/ndμΔ)n]min⁡1≤i≤n{αi}.Proof.
Let(103)αmax⁡=(max⁡1≤i≤n{αi},…,max⁡1≤i≤n{αi}),αmin⁡=(min⁡1≤i≤n{αi},…,min⁡1≤i≤n{αi}).
By Definition 28, we have
(104)HΔ(f,αmax⁡)=[∏i=1n‍∫[a,b)‍fidμΔ(∫[a,b)‍∏i=1n‍fi1/ndμΔ)n]max⁡1≤i≤n{αi},HΔ(f,αmin⁡)=[∏i=1n‍∫[a,b)‍fidμΔ(∫[a,b)‍∏i=1n‍fi1/ndμΔ)n]min⁡1≤i≤n{αi}.
Since αmax⁡≥α≥αmin⁡, the result follows from the second property of Theorem 29.Corollary 31.
Letf, α, and β satisfy the hypothesis of Theorem 29 with An=Bn=1. If there exist constants M>1>m such that Mβ≥α≥mβ, then
(105)HΔ(f,Mβ)≥HΔ(f,α)≥HΔ(f,mβ).Proof.
By Definition28, we have
(106)HΔ(f,Mβ)=MHΔ(f,β),HΔ(f,mβ)=mHΔ(f,β).
Now the result follows from the second property of Theorem 29.Remark 32.
Some results for isotonic linear functionals related to the results given in this paper can be found in [16].



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