- Research Article
- Open Access

# Stability of a Jensen Type Logarithmic Functional Equation on Restricted Domains and Its Asymptotic Behaviors

- Jae-Young Chung
^{1}Email author

**2010**:432796

https://doi.org/10.1155/2010/432796

© Jae-Young Chung. 2010

**Received:**28 June 2010**Accepted:**25 December 2010**Published:**30 December 2010

## Abstract

## Keywords

- Banach Space
- Asymptotic Behavior
- Functional Equation
- Rational Number
- Unique Function

## 1. Introduction

The stability problems of functional equations have been originated by Ulam in 1940 (see [1]). One of the first assertions to be obtained is the following result, essentially due to Hyers [2], that gives an answer for the question of Ulam.

Theorem 1.1.

In 1950-1951 this result was generalized by the authors Aoki [3] and Bourgin [4, 5]. Unfortunately, no results appeared until 1978 when Th. M. Rassias generalized the Hyers' result to a new approximately linear mappings [6]. Following the Rassias' result, a great number of the papers on the subject have been published concerning numerous functional equations in various directions [6–16]. For more precise descriptions of the Hyers-Ulam stability and related results, we refer the reader to the paper of Moszner [17]. Among the results, the stability problem in a restricted domain was investigated by Skof, who proved the stability problem of the inequality (1.1) in a restricted domain [16]. Developing this result, Jung considered the stability problems in restricted domains for the Jensen functional equation [11] and Jensen type functional equations [14]. The results can be summarized as follows: let and be a real normed space and a real Banach space, respectively. For fixed , if satisfies the functional inequalities (such as that of Cauchy, Jensen and Jensen type, etc.) for all with , the inequalities hold for all . We also refer the reader to [18–26] for some interesting results on functional equations and their Hyers-Ulam stabilities in restricted conditions.

in the restricted domains of the form for fixed with or , and . Also we obtain asymptotic behaviors of the inequalities (1.4) and (1.9) as and , respectively.

## 2. Hyers-Ulam Stability in Restricted Domains

We call the functions satisfying (1.5)*logarithmic functions*. As a direct consequence of Theorem 1.1, we obtain the stability of the logarithmic functional equation, viewing
as a multiplicative group (see also the result of Forti [9]).

Theorem A.

We first consider the usual logarithmic functional inequality (2.1) in the restricted domains .

Theorem 2.1.

Proof.

This completes the proof.

Now we consider the Hyers-Ulam stability of the Jensen type logarithmic functional inequality (1.4) in the restricted domains .

Theorem 2.2.

Proof.

for all . This completes the proof.

As a matter of fact, we obtain that in Theorem 2.2 provided that and or is a rational number, or and or is a rational number.

Theorem 2.3.

Proof.

for all . Similarly, using (2.19) we can show that . Thus the inequality (2.12) follows from (2.7). This completes the proof.

Theorem 2.4.

Proof.

for all . This completes the proof.

From Theorem 2.4, using the same approach as in the proof of Theorem 2.3 we have the following.

Theorem 2.5.

for all . Using Theorem 2.2 we have the following.

Corollary 2.6 (see [22]).

Proof.

for all . Letting we get the result.

Using Theorem 2.3, we have the following.

Corollary 2.7.

Using Theorem 2.4, we have the following.

Corollary 2.8.

Using Theorem 2.5, we have the following.

Corollary 2.9.

## 3. Asymptotic Behavior of the Inequality

In this section, we consider asymptotic behaviors of the inequalities (1.4) and (2.1).

Theorem 3.1.

as . Then is a logarithmic function.

Proof.

Letting in (3.5), we have . Thus, letting in (3.3), we get the result.

Theorem 3.2.

Proof.

for all and all positive integers . Now as in the proof of Theorem 3.1, it follows from (3.12) that for all . Letting in (3.9), (3.10), and (3.11) we get the result.

Similarly using Theorems 2.3 and 2.5, we have the following.

Theorem 3.3.

as . Then is a constant function.

Using Corollaries 2.6 and 2.8 we have the following.

Corollary 3.4.

Using Corollaries 2.7 and 2.9 we have the following.

Corollary 3.5.

## Declarations

### Acknowledgments

The author expresses his sincere gratitude to a referee of the paper for many useful comments and introducing the interesting related recent results including the papers [17–26]. This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (MEST) (no. 2010-0016963).

## Authors’ Affiliations

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