# Extensions of Hölder’s Inequality via Pseudo-Integral

Authors: Jing-Feng Tian; Ming-Hu Ha
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

Hölder’s inequality and its various extensions are playing very important roles in many branches of modern mathematics and physics. In this paper, we present some extensions of Hölder’s inequality via pseudo-integral. Moreover, some related Hölder type inequalities are also given.


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## Body

## 1. Introduction

The classical Hölder’s inequality states that ifX is a measure space, with measure μ, if p>0, q>0 with 1/p+1/q=1, and if f and g are measurable nonnegative functions on X, with range in (0,∞), then(1)∫Xfgdμ≤∫Xfpdμ1/p∫Xgqdμ1/q.As is well known, the above Hölder’s inequality plays an important role in many branches of modern mathematics and physics. For example, Agahi et al.[1–3] presented some interesting extensions of Hölder’s inequality (1) for decomposition integral, Sugeno integral, and pseudo-integral. Benmerrouche et al. [4] obtained a new technique based on Hölder’s integral inequality (1) which provides fundamental constraints on the QCD sum-rules. Steele et al. [5] employed the Hölder inequality to place lower bounds on the sum of u and d 1.0 GeV  MS¯ running masses. Tian et al. [6–13] gave some new properties, generalizations, refinements, and applications of Hölder’s inequalities. Wu [14, 15] presented some new sharpened and generalized versions of Hölder’s inequalities in classical real analysis. For more detail expositions, the interested reader may consult [16–22] and the references therein.Among various extensions of (1), Agahi et al. in [3] established the following interesting Hölder’s inequality and reverse Hölder’s inequality for pseudo-integral (for details on pseudo-integral, see([23–25]).Theorem A.
Let1<p<∞ and 1/p+1/q=1, and let (X,A) be a given measurable space. Suppose that u,v:X→[a,b] are two measurable functions and that a generator g:[a,b]→[0,∞] of the pseudo-addition ⊕ and the pseudo-multiplication ⊙ is an increasing function. Then for any σ-⊕-measure m, we have(2)∫X⊕u⊙v⊙dm≤∫X⊕u⊙p⊙dm⊙1/p⊙∫X⊕v⊙q⊙dm⊙1/q.Theorem B.
Let1<p<∞ and 1/p+1/q=1, and let (X,A) be a given measurable space. Suppose that u,v:X→[a,b] are two measurable functions and that a generator g:[a,b]→[0,∞] of the pseudo-addition ⊕ and the pseudo-multiplication ⊙ is a decreasing function. Then for any σ-⊕-measure m, we have(3)∫X⊕u⊙v⊙dm≥∫X⊕u⊙p⊙dm⊙1/p⊙∫X⊕v⊙q⊙dm⊙1/q.The aim of this paper is to present some extensions of the above Hölder’s inequalities (2) and (3). Some related Hölder type inequalities are also considered.


## 2. Extensions of Hölder’s Inequality via Pseudo-Integral

In this section, we shall give the following lemma firstly before we give our results.Lemma 1 (see [26]).
LetX be a measure space, with measure μ; let 1<λ1,λ2,⋯,λn<∞ with ∑i=1n(1/λi)=1, and let fi(i=1,2,⋯,n) be measurable nonnegative functions on X, with range in (0,∞). Then(4)∫X∏i=1nfidμ≤∏i=1n∫Xfλidμ1/λi.Next, we give the following extension of inequality (2).Theorem 2.
Let1<λ1,λ2,⋯,λn<∞ and ∑i=1n(1/λi)=1, and let (X,A) be a given measurable space. Suppose that ui:X→[a,b](i=1,2,⋯,n) are n measurable functions and that a generator g:[a,b]→[0,∞] of the pseudo-addition ⊕ and the pseudo-multiplication ⊙ is an increasing function. Then for any σ-⊕-measure m, we have(5)∫X⊕u1⊙u2⊙⋯⊙un⊙dm≤∫X⊕u1⊙λ1⊙dm⊙1/λ1⊙∫X⊕u2⊙λ2⊙dm⊙1/λ2⊙⋯⊙∫X⊕un⊙λn⊙dm⊙1/λn.Proof.
From Lemma1 we obtain (6)∫X∏i=1ng∘uidg∘m≤∏i=1n∫Xg∘uiλidg∘m1/λi.Since function g is increasing function, then g-1 is also increasing function and we find (7)g-1∫X∏i=1ng∘uidg∘m≤g-1∏i=1n∫Xg∘uiλidg∘m1/λi,which implies(8)g-1∫Xgg-1∏i=1ng∘uidg∘m≤g-1gg-1∏i=1n∫Xg∘uiλidg∘m1/λi.
If pseudo-operations are generated by a monotone and continuous functiong:[a,b]→[0,∞], then x⊙y=g-1(g(x)·g(y)).
Therefore(9)x1⊙x2⊙⋯⊙xn=g-1gx1·gx2⋯gxn.
From inequality (8) and the pseudo-integral for a measurable function f:X→[a,b] given by(10)∫X⊕f⊙dm=g-1∫Xg∘fdg∘m,we have(11)∫X⊕u1⊙u2⊙⋯⊙un⊙dm=g-1∫Xgg-1∏i=1ng∘uidg∘m≤g-1gg-1∏i=1n∫Xg∘uiλidg∘m1/λi=g-1∫Xg∘u1λ1dg∘m1/λ1⊙g-1∫Xg∘u2λ2dg∘m1/λ2⊙⋯⊙g-1∫Xg∘unλndg∘m1/λn.On the other hand(12)g-1∫Xg∘u1λ1dg∘m1/λ1⊙g-1∫Xg∘u2λ2dg∘m1/λ2⊙⋯⊙g-1∫Xg∘unλndg∘m1/λn=g-1∫Xgg-1g∘u1λ1dg∘m1/λ1⊙g-1∫Xgg-1g∘u2λ2dg∘m1/λ2⊙⋯⊙g-1∫Xgg-1g∘unλndg∘m1/λn=g-1∫Xgu1⊙λ1dg∘m1/λ1⊙g-1∫Xgu2⊙λ2dg∘m1/λ2⊙⋯⊙g-1∫Xgun⊙λndg∘m1/λn=g-1gg-1∫Xgu1⊙λ1dg∘m1/λ1⊙g-1gg-1∫Xgu2⊙λ2dg∘m1/λ2⊙⋯⊙g-1gg-1∫Xgun⊙λndg∘m1/λn=g-1g∫Xgu1⊙λ1∘dm1/λ1⊙g-1g∫Xgu2⊙λ2∘dm1/λ2⊙⋯⊙g-1g∫Xgun⊙λn∘dm1/λn=∫X⊕u1⊙λ1⊙dm⊙1/λ1⊙∫X⊕u2⊙λ2⊙dm⊙1/λ2⊙⋯⊙∫X⊕un⊙λn⊙dm⊙1/λn. Therefore by inequalities (11) and (12), we obtain immediately the desired inequality (5).Example 3.
If we set[a,b]=[0,∞] and g(x)=xα for some α∈[1,∞[, then from the definition of pseudo-operations [23–25], we find that x⊕y=xα+yαα and x⊙y=xy. Therefore we obtain the following Hölder type inequality: (13)∫a,b∏i=1nfiαxdxα≤∏i=1n∫c,dfiλiαxdx1/λiα.Example 4.
If we set[a,b]=[-∞,+∞] and g(x)=ex for some α∈[1,∞[ then from the definition of pseudo-operations [23–25], we find that x⊕y=ln⁡ex+ey and x⊙y=x+y. Therefore we obtain the following Hölder type inequality: (14)ln⁡∫c,def1x+f2x+⋯+fnxdx≤∑i=1n1λiln⁡∫c,deλifixdx,that is (15)∫c,def1x+f2x+⋯+fnxdx≤∑i=1n∫c,deλifixdx1/λi.
By the same methods as in the proof of Theorem2, we can obtain the following generalization of inequality (3).Theorem 5.
Let1<λ1,λ2,⋯,λn<∞ and ∑i=1n(1/λi)=1, and let (X,A) be a given measurable space. Suppose that ui:X→[a,b](i=1,2,⋯,n) are n measurable functions and a generator g:[a,b]→[0,∞] of the pseudo-addition ⊕ and the pseudo-multiplication ⊙ is a decreasing function. Then for any σ-⊕-measure m, we have(16)∫X⊕u1⊙u2⊙⋯⊙un⊙dm≥∫X⊕u1⊙λ1⊙dm⊙1/λ1⊙∫X⊕u2⊙λ2⊙dm⊙1/λ2⊙⋯⊙∫X⊕un⊙λn⊙dm⊙1/λn.Next we consider theCase: ⊕=sup and ⊙=g-1(g(x)g(y)).Theorem 6.
Let1<λ1, λ2, ⋯, λn<∞ with ∑i=1n(1/λi)=1. Suppose that ⊙ is represented by an increasing generator g and m is a complete sup-measure. Then for any functions ui(i=1,2,…,n):X→[a,b], we have(17)∫Xsupu1⊙u2⊙⋯⊙un⊙dm≤∫Xsupu1⊙λ1⊙dm⊙1/λ1⊙∫Xsupu2⊙λ2⊙dm⊙1/λ2⊙⋯⊙∫Xsupun⊙λn⊙dm⊙1/λn.Proof.
Since⊕=sup and ⊙=g-1(g(x)g(y)), by the pseudo-integral of the function f:R→[a,b] given by (18)∫R⊕f⊙dm=sup⁡fx⊙ψx,we have (19)∫Xsupu1⊙u2⊙⋯⊙un⊙dm=supx∈X⁡u1x⊙u2x⊙⋯⊙unx⊙ψx=g-1supx∈X⁡∏i=1nguixgψx,where ψ:X→[a,b] is a density function related to m.
Moreover(20)∫Xsupui⊙λi⊙dm⊙1/λi=g-1supyi∈X⁡guiλiyigψyi1/λi=g-1supyi∈X⁡guiyig1/λiψyii=1,2,⋯,n.Consequently, (21)∫Xsupu1⊙λ1⊙dm⊙1/λ1⊙∫Xsupu2⊙λ2⊙dm⊙1/λ2⊙⋯⊙∫Xsupun⊙λn⊙dm⊙1/λn=g-1∏i=1nsupyi∈X⁡guiyig1/λiψyi≥g-1supx∈X⁡∏i=1nguxg1/λiψx=g-1supx∈X⁡∏i=1nguixgψx=∫Xsupu1⊙u2⊙⋯⊙un⊙dm.Thus the proof of Theorem 6 is completed.Remark 7.
For anyx∈[a,b] and λi>0, if we set ⊕=sup and ⊙=inf, x⊙λi=x, then from inequality (17) we obtain the following Hölder type inequality: (22)∫Xsupu1⊙u2⊙⋯⊙un⊙dm≤∫Xsupu1⊙⊙dm⊙∫Xsupu2⊙⊙dm⊙⋯⊙∫Xsupun⊙⊙dm,which implies (23)supx∈X⁡inf⁡u1x,u1x,⋯,unx,ψx≤inf⁡supx1∈X⁡inf⁡u1x1,ψx1,supx2∈X⁡inf⁡u2x2,ψx2,⋯,supxn∈X⁡inf⁡u2xn,ψxn.Example 8.
Let[a,b]=[-∞,+∞], and let fi,ψ be given real functions on X. Suppose that g generating ⊙ is given by g(x)=ex. Then we have x⊙y=x+y. Therefore from inequality (17) we obtain the following Hölder type inequality: (24)supx∈X⁡f1x+f2x+⋯+fnx+ψx≤∑i=1n1λisupx∈X⁡λi·ux+ψx.


## 3. Conclusions

Höder’s inequality is playing a very important and basic role in different branches of modern mathematics such as classical real and complex analysis, numerical analysis, probability and statistics, fuzzy measure theory, qualitative theory of differential equations and their applications. It is also an indispensable and basic tool in engineering technology. Pseudo-integral is applicable in many of the fields of applied sciences which helps us to interpret many problems. In this paper we have presented some extensions of Höder’s inequality via pseudo-integral. Moreover, we have obtained some related Höder type inequalities. The obtained results may have some potential for applications. For example, we may improve some results on stochastic matrix in Markov chains and their applications.



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