#### Abstract

This paper investigates Hölder’s inequality under the condition of -conjugate exponents in the sense that . Successively, we have, under -conjugate exponents relative to the -norm, investigated generalized Hölder’s inequality, the interpolation of Hölder’s inequality, and generalized -order Hölder’s inequality which is an expansion of the known Hölder’s inequality.

#### 1. Introduction

The celebrated Hölder inequality is one of the most important inequalities in mathematics and statistics. It is applied widely in dealing with many problems from social science, management science, and natural science. Its classical version for integrals is of the following formulation: for any measurable functions and on some measurable space , it follows thatfor all with , where for all measurable functions (for details, see [1, 2]).

Since Hölder’s inequality has been extensively investigated and applied to some new fields, many literature studies are contributed to the refinement of Hölder’s inequality according to specific applied fields. These improvements mainly incorporate in the following fields.

The first refinement is relative to the higher integrability of gradients of solutions to various partial differential equations. It characterizes the following style: there exist , only depending on , , and so thatfor all time-space cylinders with , where . The recent representative articles with respect to this topic are referred to in [3–6].

The other refinements are Hölder’s inequality and inverse Hölder’s inequality for the pseudo-integral established by Agahi et al. [7]. Assume that and are a pair of conjugate exponents and that is a measure space. Suppose that are two measurable functions and that a generator of the pseudo-addition and the pseudo-multiplication is an increasing function. Then, for any --measure , it follows that

The recent research studies contribute to interesting extensions of Hölder’s inequality for the decomposition integral, Sugeno integral, and pseudo-integral (for more details, see [7–12]) since fuzzy measures and Sugeno integral have been successfully applied to various fields such as to decision-making [13] and to artificial intelligence [14]. Besides, Khan et al. investigated the converses of the Jensen inequality in their articles such as [15–18].

The fundamental refinement is the improvement of Hölder’s inequality itself. As far as in 1961, Beckenbach and Bellman [19] derived generalized Hölder’s inequality of the following form: let and be positive integers, and let , , and . Then, it follows that

The corresponding integral form iswhere , , , and . Note that, by taking , , and , inequality (5) reduces to inequality (1). Qiang and Hu [20], then, derived further contributions on this topic as follows: let , , , , , and . Then,

Moreover, for the integral form of the above inequality, if , and , then

Many existing inequalities related to the Hölder inequality are special cases of inequalities (6) and (7). For example, putting for , and , inequalities (6) and (7) are reduced to (4) and (5), respectively. The condition may be regarded as Hölder’s inequality with a system of generalized conjugate exponents which will be introduced in Section 2. Recently, Masjed-Jamei [21] established an extension of the Callebaut inequality [22], that is,where and , .

Based on these contributions of Masjed-Jamei, Qiang and Hu [20] derived the following interesting result:whereis a nonincreasing function with .

These aforementioned literature studies have generalized Hölder’s inequality upon the condition . On the one hand, however, this condition lacks consistency in the form. On the other hand, these inequalities lack relevance on the two sides of the inequalities. Recall the mean value theorem of integrals: let be measurable on and be a monotone function. Then, there exists such that

Putting , and , equation (11) will reduce to the mean value theorem of integrals. Furthermore, Tian et al. [9, 23, 24] investigated the monotone relationship between adjoining exponents of or .

In this paper, our contributions are concluded as follows. We have derived generalized Hölder’s inequality under the condition of -conjugate exponents in the sense that . Successively, we have, under -conjugate exponents relative to the -norm, checked generalized Hölder’s inequality, the interpolation of Hölder’s inequality, and generalized -order Hölder’s inequality which is an expansion of the known Hölder’s inequality.

#### 2. Preliminary Definitions and Notations

First, we would provide some preliminary definitions for building the formulation of Hölder’s inequality with -order -conjugate exponents.

*Definition 1. *If and are a pair of positive real numbers such that or equivalentlythen the positive real numbers and are referred to as a pair of conjugate exponents.

It is clear that equation (12) implies that and and that as . Hence, it is reasonable to regard 1 and as a pair of conjugate exponents [25].

*Definition 2. *If are positive real numbers such thatthen the sequence , is known as -order conjugate exponents.

*Definition 3. *If and are positive real numbers such that or equivalentlythen the numbers and are referred to as a pair of -conjugate exponents.

*Definition 4. *If are positive real numbers such thatthen the positive real numbers , are referred to as -order -conjugate exponents.

Given the definitions of conjugate exponents, our next aim is to define the integral and to further recall the -norm.

*Definition 5. *The integral of a real-valued, measurable function on some measure space is defined as

If is simple and nonnegative, hence of the form for some , , and , is defined as

For any nonnegative measurable function , we may choose some simple measurable functions with , and define [2].

Note that it is customary to relate all objects of the study to a basic probability space , which is nothing more than a normalized measure space.

*Definition 6. *Given a measure space and , we write for the class of all measurable functions on withand call the -norm of and the -space.

#### 3. Main Results

##### 3.1. Generalized Hölder’s Inequality with a Pair of -Conjugate Exponents

In this part, we will generalize the celebrated Young inequality and Hölder inequality for integrals to those with -conjugate exponents.

Theorem 1. *(generalized Young inequality). Assume that and are a pair of -conjugate exponents. Then, for positive real numbers and ,with equality if and only if .*

*Proof. *Putting , we have for any real number . Thus, is convex for . Assuming and , due to Jensen’s inequality, we obtainSubstituting, respectively, and for and , it follows thatthat is, . The equality holds if and only if or equally, .

Then, we will derive the generalized Hölder inequality with a pair of conjugate exponents.

Theorem 2. *(Hölder’s inequality with a pair of conjugate exponents). Let and be a pair of conjugate exponents. For any measurable functions and on some measure space , we havewith equality if and only if there exist two constants and which need not vanish such that a.e.*

*Proof. *One could refer to [22] when the need arises.

Corollary 1. *For every measurable function on some measure space , we havefor .*

*Proof. *It is trivial to prove this conclusion by putting in Theorem 2.

The simplest measure on a measurable space is the unit masses or Dirac measure , defined by . We may form, for any countable set , the associated counting measure .

Every measurable mapping of into some measurable space is referred to as a random element in . A random element in is called a random variable when , a random vector when , and a random sequence when , respectively.

If is Lebesgue measure on , we write instead of . If is the counting measure on a set , it is customary to denote the corresponding -space by . In that case, the -norm of a random vector or a random sequence is or , respectively. Thus, the Hölder inequality for sums is similar as that for integrals.

Theorem 3. *(Hölder’s inequality with a pair of -conjugate exponents). Let and be a pair of -conjugate exponents. For any measurable functions and on some measure space , we havewith equality if and only if there exist two constants and which need not vanish such that a.e.*

*Proof. *By Theorem 2 with , we haveThis implies , and furthermore, .

Replacing and by and , respectively, we obtainThen, the equality holds in equation (18) if and only if . Replacing and by and , respectively, we obtain . Hence, the conclusion follows.

Corollary 2. *For any measurable function on some measure space and nonnegative real numbers and with , we have*

*Proof. *First, note that both and hold when and are a pair of -conjugate exponents. Then, putting in Theorem 5, we derive . Therefore, it follows that with .

Remark: Corollary 2 implies that -spaces are nonincreasing with respect to .

##### 3.2. Interpolation of Hölder’s Inequality with -Conjugate Exponents

In literature [26], İşcan derived the interpolation of Hölder’s inequality with a pair of conjugate exponents with respect to integrable functions on . Furthermore, Gowda [27] gave an interpolation theorem relative to spectral norms. We will extend the interpolation of Hölder’s inequality to the -conjugate exponents’ case on measure space .

*Definition 7. *A nondecreasing and countably subadditive set function with being referred to as an outer measure on a space . Furthermore, given an outer measure on , we say that a set is -measurable if

Theorem 4. *Assume and to be a pair of conjugate exponents and both and to be measurable functions on . Then, for any -measurable set , it follows that*

*Proof. *Since set is -measurable, it follows that .

We first prove the left inequality of equation (23). By Theorem 2, we may obtainHence, the left inequality of equation (23) follows.

Furthermore, if , then a.e. or a.e. Thus, we have either and or and . Therefore, the right inequality of equation (23) holds, that is,Now, we consider the case of . By Young’s inequality, we deriveHence, the right inequality of equation (23) holds, that is,Therefore, equation (23) follows.

Theorem 5. *Assume and to be a pair of -conjugate exponents and both and to be measurable functions on . Then, for any -measurable set , it follows that*

*Proof. *Let and be a pair of -conjugate exponents, i.e., . By Theorem 6, we deriveIt follows thatReplacing and by and , respectively, it follows thatthat is,This completes the proof.

##### 3.3. Generalized Hölder’s Inequality with -Order -Conjugate Exponents

In this part, we will present the most important results of this paper in the following theorems.

Theorem 6. *(Hölder’s inequality with -order -conjugate exponents). Assume that , are -order -conjugate exponents and that , are all measurable functions on . Then,*

*Proof. *We prove this conclusion by induction.

If , the result reduces to Hölder’s inequality with a pair of conjugate exponents.

Assume that the conclusion holds when . Then, putting , we havewhere .

By the aforementioned assumption, it follows thatThus, combining the two inequalities above, we deriveThis proves the case .

Theorem 7. *(generalized Hölder’s inequality with -order -conjugate exponents). Assume that , are -order -conjugate exponents, , and , are all measurable functions on . Then,*

*Proof. *Assume the positive function sequenceBy simple computation and rearrangement, it follows thatwith the assumptions and .

Then, by using Hölder’s inequality with -order -conjugate exponents, it follows thatThis proves the final result.

Corollary 3. *Under the assumption of Theorem 7, taking for , and with , we have*

Remarks:(i)Taking for , and , inequality (47) is reduced to (43)(ii)If we discuss Hölder’s inequality on the measure space with the counting measure, Hölder’s inequality for integrals is reduced to the corresponding result for sum(iii)Instituting each for , the conclusion still holds since is an induced measure space

Similar to equation (9), we derive the following theorem.

Theorem 8. *Let , and . Then,where*

#### 4. Conclusion and Future Work

The classical Hölder’s inequality of the formis applied widely in social science, management science, and natural science, where . Qiang and Hu [20] expanded this result to the case and derivedwhereis a nonincreasing function with .

On the one hand, in modern probability theory, it is customary to relate all objects of the study to a fundamental probability space ; especially, -space is a most important probability space. On the other hand, to seek the relationship between inequality (52) and the mean value theorem of integrals is interesting. Thus, we want to investigate generalized Hölder’s inequality with -conjugate exponents under the -space. Successively, based on -conjugate exponents, we derived completely generalized Young’s inequality, generalized Hölder’s inequality, the interpolation of Hölder’s inequality, and generalized -order Hölder’s inequality.

Regretfully, the relationship between the mean value theorem of integrals and the interpolation of Hölder’s inequality is left to consider intensively as the future work.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This paper was supported by Hulunbuir College Doctor Foundation Project of China (2018BS16 and 2020BS04), the Natural Science Foundation of Inner Mongolia (CN) (2018MS01023), and the National Natural Science Foundation of China (12061033).