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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)
The purpose of this paper is to find Hölder type inequality for the fermionic -adic invariant -integral which was defined by Kim (2008).
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
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 ). Now, we consider the fermionic -adic invariant -integral on as
From (1.5) we note that
where (see ).
We also introduce the classical Hölder inequality for the Lebesgue integral in .
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.
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 ):
Then, it is easy to see that
From (2.4) and (2.7), we have the following theorem.
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.
Let with and . If one takes and , then one has
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/S1061920806020038
Kim T: -generalized Euler numbers and polynomials. Russian Journal of Mathematical Physics 2006,13(3):293–298. 10.1134/S1061920806030058
Kim 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.2006.08.123
Kim T: -extension of the Euler formula and trigonometric functions. Russian Journal of Mathematical Physics 2007,14(3):275–278. 10.1134/S1061920807030041
Kim 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.2006.11.011
Kim 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. 10.1080/10236190801943220
Kim 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/S1061920807020045
Kim 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/S1061920806030113
Simsek Y: On -adic twisted --functions related to generalized twisted Bernoulli numbers. Russian Journal of Mathematical Physics 2006,13(3):340–348. 10.1134/S1061920806030095
Simsek 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.
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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