- Research Article
- Open Access
- Published:

# 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

## References

Cenkci M, Simsek Y, Kurt V:

**Further remarks on multiple -adic --function of two variables.***Advanced Studies in Contemporary Mathematics*2007,**14**(1):49–68.Jang L-C:

**A new -analogue of Bernoulli polynomials associated with -adic -integrals.***Abstract and Applied Analysis*2008,**2008:**-6.Jang L-C, Kim S-D, Park D-W, Ro Y-S:

**A note on Euler number and polynomials.***Journal of Inequalities and Applications*2006,**2006:**-5.Kim T:

**-Volkenborn integration.***Russian Journal of Mathematical Physics*2002,**9**(3):288–299.Kim T:

**0 -integrals associated with multiple Changhee -Bernoulli polynomials.***Russian Journal of Mathematical Physics*2003,**10**(1):91–98.Kim T:

**On Euler-Barnes multiple zeta functions.***Russian Journal of Mathematical Physics*2003,**10**(3):261–267.Kim T:

**Analytic continuation of multiple -zeta functions and their values at negative integers.***Russian Journal of Mathematical Physics*2004,**11**(1):71–76.Kim T:

**Power series and asymptotic series associated with the -analog of the two-variable -adic -function.***Russian Journal of Mathematical Physics*2005,**12**(2):186–196.Kim T:

**Multiple -adic -function.***Russian Journal of Mathematical Physics*2006,**13**(2):151–157. 10.1134/S1061920806020038Kim T:

**-generalized Euler numbers and polynomials.***Russian Journal of Mathematical Physics*2006,**13**(3):293–298. 10.1134/S1061920806030058Kim T:

**Lebesgue-Radon-Nikod\'ym theorem with respect to -Volkenborn distribution on .***Applied Mathematics and Computation*2007,**187**(1):266–271. 10.1016/j.amc.2006.08.123Kim T:

**-extension of the Euler formula and trigonometric functions.***Russian Journal of Mathematical Physics*2007,**14**(3):275–278. 10.1134/S1061920807030041Kim T:

**-Bernoulli numbers and polynomials associated with Gaussian binomial coefficients.***Russian Journal of Mathematical Physics*2008,**15**(1):51–57.Kim T:

**An invariant -adic -integral on .***Applied Mathematics Letters*2008,**21**(2):105–108. 10.1016/j.aml.2006.11.011Kim T:

**An identity of the symmetry for the Frobenius-Euler polynomials associated with the Fermionic -adic invariant -integrals on .**to appear in*Rocky Mountain Journal of Mathematics*, http://arxiv.org/abs/0804.4605 to appear in Rocky Mountain Journal of Mathematics,Kim T:

**Symmetry -adic invariant integral on for Bernoulli and Euler polynomials.***Journal of Difference Equations and Applications*2008,**14**(12):1267–1277. 10.1080/10236190801943220Kim T, Choi JY, Sug JY:

**Extended -Euler numbers and polynomials associated with fermionic -adic -integral on .***Russian Journal of Mathematical Physics*2007,**14**(2):160–163. 10.1134/S1061920807020045Kim T, Kim M-S, Jang L-C, Rim S-H:

**New -Euler numbers and polynomials associated with -adic -integrals.***Advanced Studies in Contemporary Mathematics*2007,**15**(2):243–252.Kim T, Simsek Y:

**Analytic continuation of the multiple Daehee --functions associated with Daehee numbers.***Russian Journal of Mathematical Physics*2008,**15**(1):58–65.Ozden H, Simsek Y, Rim S-H, Cangul IN:

**A note on -adic -Euler measure.***Advanced Studies in Contemporary Mathematics*2007,**14**(2):233–239.Rim S-H, Kim T:

**A note on -adic Euler measure on .***Russian Journal of Mathematical Physics*2006,**13**(3):358–361. 10.1134/S1061920806030113Simsek Y:

**On -adic twisted --functions related to generalized twisted Bernoulli numbers.***Russian Journal of Mathematical Physics*2006,**13**(3):340–348. 10.1134/S1061920806030095Simsek Y:

**Generating functions of the twisted Bernoulli numbers and polynomials associated with their interpolation functions.***Advanced Studies in Contemporary Mathematics*2008,**16**(2):251–278.Srivastava HM, Kim T, Simsek Y:

**-Bernoulli numbers and polynomials associated with multiple -zeta functions and basic -series.***Russian Journal of Mathematical Physics*2005,**12**(2):241–268.Royden HL:

*Real Analysis*. Prentice-Hall, Englewood Cliffs, NJ, USA; 1998.

## Acknowledgment

This paper was supported by the KOSEF 2009-0073396.

## Author information

### Authors and Affiliations

### Corresponding author

## Rights and permissions

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## About this article

### Cite this article

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

Received:

Accepted:

Published:

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

### Keywords

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