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A Note on Hölder Type Inequality for the Fermionic -Adic Invariant -Integral
Journal of Inequalities and Applications volume 2009, Article number: 357349 (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
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.
Let with . If and , then and
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:
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.
Let with . If one takes and , then one has
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.
Let with and . If one takes and , then one has
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Acknowledgment
This paper was supported by the KOSEF 2009-0073396.
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Jang, LC. A Note on Hölder Type Inequality for the Fermionic -Adic Invariant -Integral. J Inequal Appl 2009, 357349 (2009). https://doi.org/10.1155/2009/357349
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DOI: https://doi.org/10.1155/2009/357349