- Research Article
- Open Access

# A Note on Hölder Type Inequality for the Fermionic -Adic Invariant -Integral

- Lee-Chae Jang
^{1}Email author

**2009**:357349

https://doi.org/10.1155/2009/357349

© Lee-Chae Jang. 2009

**Received:**11 February 2009**Accepted:**22 April 2009**Published:**2 June 2009

## Abstract

The purpose of this paper is to find Hölder type inequality for the fermionic -adic invariant -integral which was defined by Kim (2008).

## Keywords

- Generate Function
- Differentiable Function
- Algebraic Number
- Exponential Generate
- Difference Quotient

## 1. Introduction

Let be a fixed odd prime. Throughout this paper , , and will, respectively, denote the ring of -adic rational integers, the field of -adic rational numbers, the rational number field, the complex number field, and the completion of algebraic closure of . For a fixed positive integer with , let

Let be the set of natural numbers. In this paper we assume that with , which implies that for . We also use the notations

for all . For any positive integer the distribution is defined by

We say that is a uniformly differentiable function at a point and denote this property by , if the difference quotients have a limit as (cf. [1–24]).

For , the above distribution yields the bosonic -adic invariant -integral as follows:

representing the -adic -analogue of the Riemann integral for . In the sense of fermionic, let us define the fermionic -adic invariant -integral on as

for (see [16]). Now, we consider the fermionic -adic invariant -integral on as

From (1.5) we note that

where (see [16]).

We also introduce the classical Hölder inequality for the Lebesgue integral in [25].

Theorem 1.1.

where and and .

The purpose of this paper is to find Hölder type inequality for the fermionic -adic invariant -integral .

## 2. Hölder Type Inequality for Fermionic -Adic Invariant -Integrals

In order to investigate the Hölder type inequality for , we introduce the new concept of the inequality as follows.

Definition 2.1.

For , we define the inequality on (resp., ) as follows. For (resp., ), (resp., ) if and only if (resp., ).

From (2.1), (2.2), and (2.3), we derive

We remark that the th Frobenius-Euler numbers and the th Frobenius-Euler polynomials attached to algebraic number may be defined by the exponential generating functions (see [16]):

Then, it is easy to see that

From (2.4) and (2.7), we have the following theorem.

Theorem 2.2.

We note that for with ,

By Theorem 2.2 and (2.7) and the definition of -adic norm, it is easy to see that

for all with . We note that lies in . Thus by Definition 2.1 and (2.10), we obtain the following Hölder type inequality theorem for fermionic -adic invariant -integrals.

Theorem 2.3.

## Declarations

### Acknowledgment

This paper was supported by the KOSEF 2009-0073396.

## Authors’ Affiliations

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## Copyright

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