This paper investigates Hölder’s inequality under the condition of

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

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

The other refinements are Hölder’s inequality and inverse Hölder’s inequality for the pseudo-integral established by Agahi et al. [

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 [

The fundamental refinement is the improvement of Hölder’s inequality itself. As far as in 1961, Beckenbach and Bellman [

The corresponding integral form is

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

Many existing inequalities related to the Hölder inequality are special cases of inequalities (

Based on these contributions of Masjed-Jamei, Qiang and Hu [

These aforementioned literature studies have generalized Hölder’s inequality upon the condition

Putting

In this paper, our contributions are concluded as follows. We have derived generalized Hölder’s inequality under the condition of

First, we would provide some preliminary definitions for building the formulation of Hölder’s inequality with

If

It is clear that equation (

If

If

If

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

The integral of a real-valued, measurable function

If

For any nonnegative measurable function

Note that it is customary to relate all objects of the study to a basic probability space

Given a measure space

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

(generalized Young inequality). Assume that

Putting

Substituting, respectively,

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

(Hölder’s inequality with a pair of conjugate exponents). Let

One could refer to [

For every measurable function

It is trivial to prove this conclusion by putting

The simplest measure on a measurable space

Every measurable mapping

If

(Hölder’s inequality with a pair of

By Theorem

This implies

Replacing

Then, the equality holds in equation (

For any measurable function

First, note that both

Remark: Corollary

In literature [

A nondecreasing and countably subadditive set function

Assume

Since set

We first prove the left inequality of equation (

Hence, the left inequality of equation (

Furthermore, if

Now, we consider the case of

Hence, the right inequality of equation (

Therefore, equation (

Assume

Let

It follows that

Replacing

This completes the proof.

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

(Hölder’s inequality with

We prove this conclusion by induction.

If

Assume that the conclusion holds when

By the aforementioned assumption, it follows that

Thus, combining the two inequalities above, we derive

This proves the case

(generalized Hölder’s inequality with

Assume the positive function sequence

By simple computation and rearrangement, it follows that

Then, by using Hölder’s inequality with

This proves the final result.

Under the assumption of Theorem

Remarks:

Taking

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

Instituting each

Similar to equation (

Let

The classical Hölder’s inequality of the form

On the one hand, in modern probability theory, it is customary to relate all objects of the study to a fundamental probability space

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.

No data were used to support this study.

The authors declare that they have no conflicts of interest.

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).