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

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. [1–24]).

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. [1–24]).

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)