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

Journal of Inequalities and Applications20092009:357349

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

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).

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

(1.1)

where lies in (cf. [124]).

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

(1.2)

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

(1.3)

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. [124]).

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

(1.4)

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

(1.5)

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

(1.6)

From (1.5) we note that

(1.7)

where (see [16]).

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

Theorem 1.1.

Let with . If and , then and
(1.8)

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., ).

Let with . By substituting and into , we obtain the following equation:
(2.1)
(2.2)
(2.3)

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

(2.4)

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]):

(2.5)
(2.6)

Then, it is easy to see that

(2.7)

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

Theorem 2.2.

Let with . If one takes and , then one has
(2.8)

We note that for with ,

(2.9)

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

(2.10)

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.

Let with and . If one takes and , then one has
(2.11)

Declarations

Acknowledgment

This paper was supported by the KOSEF 2009-0073396.

Authors’ Affiliations

(1)
Department of Mathematics and Computer Science, KonKuk University

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Copyright

© Lee-Chae Jang. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.