Abstract

In the paper, we introduce new -functional inequalities related to the Jensen functional equation and some properties. The Hyers-Ulam stability of functional inequalities is proved.

1. Introduction and Preliminaries

Ulam [1] raised a stability problem for a homomorphism in metric groups. In 1941, Hyers [2] proved that if is a mapping from a normed vector space into a Banach space and satisfies , then there exists an additive function such that . Hyers’ theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by Găvruta [5] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach. The stability problems for several functional equations or inequalities have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [6–21]).

According to Theorem 6 in [22], a mapping satisfying is a solution of the Jensen functional equation if and only if . Hence, the general solution of the Jensen’s equation in is where and are arbitrary constants. The first result about the stability of the Jensen’s equation was got by Kominek [23]. Recently, the varied Jensen function have been studied by a number of authors, for instance, see [24, 25].

In this paper, we investigate the stability of the Jensen functional inequality, in some classes of noncommutative groups with . The Jensen functional equation was studied in [26–29].

2. Auxiliary Results

Suppose that is an arbitrary group and that is an arbitrary real Banach space. In this section, assume that is an arbitrary multiplicative group and is the identity element of . In the section, for an arbitrary group , the decomposition in order to prove the stability of Jensen inequality.

Definition 1. We say that a function is a -Jensen functional inequality if the function satisfies for all .
We denote the set of all -Jensen functional inequality by . Denote by , the subset of consisting of functions such that . We can prove is Jensen equation by the inequality (3), is a subspace of , and .

Example 2. If the function satisfies for all , then . The result can be proved by trigonometric inequality.

Definition 3. We say that a function is an -quasi -Jensen function if there is such that for all .

It is clear that the set of -quasi -Jensen functions is a real linear space. Denote it by . From (5), we have

Therefore, where . Now, letting in (5), we get

Hence, where . Again, letting in (5), we get

By (7) and (10), we obtain where .

Let be as in (5) and define the set as follow: , where , , , and if .

Lemma 4. Let such that Then, for any and with , the following relation holds:

Proof. The proof is by induction on . For , the lemma is established in (9). Suppose that for , the lemma has been already established, let us verify it for . Letting in (5), we have Then, and hence, Now, the lemma is proved.

Theorem 5. Let . For any , and , we have

Proof. The proof will be based on mathematical induction for . If , then (17) follows from (13). Suppose that (17) for is true, let us verify it for . Replacing by in (17), we get Now, by (13), we get and hence, This implies This completes the proof of the lemma.

From (18), it follows that the set is bounded.

Replacing by in (18), we obtain

Then, the sequence is a Cauchy sequence. Since the space is complete, the Cauchy sequence has a limit and denoted by . That is,

By (18),

Lemma 6. Let such that for all . Then, for any , .

Proof. for all .

For all , we get the equation

In fact,

Lemma 7. If , then for any .

Proof. By the definition of and , we can see that and belong to . We can define the function Then, and for all and . From (27), there are such that Thus, and , and we get.

We denote by , the space of all bounded functions on group that take values in . By , the set of -pseudo--Jensen functions, i.e., -quasi--Jensen functions such that for all and .

Theorem 8. For an arbitrary group, the following decomposition holds:

Proof. It is clear that and are subspaces of , and Now, we only need to prove that . In fact, if , then, we can define the function , and it is easy to show that . Hence, . Thus, we have and .

3. Ulam Stability

In this section, we will prove the Hyers-Ulam stability of the Jensen’s functional inequality (3) when is an integer by the so-called direct method. The stability results of Jensens functional equation can be found in, for instance, Srisawat [30]. Suppose that is a group and is a real Banach space.

Definition 9. We say that the inequality (3) is stable for the pair if for any satisfying functional inequality for some , there is a solution of the functional equation (3) such that .
Next, the inequality (3) is stable on if and only if .

Theorem 10. The inequality (3) is stable if and only if .

Proof. Suppose . Let satisfy the inequality (3) for some . By Lemma 6 and Theorem 8, there is a function such that is a bounded function on .
Now, suppose . Then, we will show that the inequality (5) is not stable. Let , by Theorem 8, there are and such that . It follows that for any , we get Thus, we can see that for all . It is a contradiction to the assumption on . So .

4. Conclusion

This paper introduces a modified version of Jensen’s inequality, which examines the characteristics of the original inequality and the robustness of Jensen’s functional inequality in various groups. Particularly, the sequence can perform the appropriate control of

Data Availability

The authors confirm that the data supporting the findings of this study are available within the article.

Conflicts of Interest

The authors declare that they have no competing interests.

Authors’ Contributions

The authors equally conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

Acknowledgments

This work was supported by National Natural Science Foundation of China (No. 11761074), Project of Jilin Science and Technology Development for Leading Talent of Science and Technology Innovation in Middle and Young and Team Project (No. 20200301053RQ), and the Scientific Research Project of Guangzhou College of Technology and Business in 2020 and 2021 (No. KA202032, KAZX2021013).