Volume 2009, Article ID 357349,5pages doi:10.1155/2009/357349
Research Article
A Note on H ¨older Type Inequality for the Fermionic
p
-Adic Invariant
q
-Integral
Lee-Chae Jang
Department of Mathematics and Computer Science, KonKuk University, Chungju 380-701, South Korea
Correspondence should be addressed to Lee-Chae Jang,[email protected]
Received 11 February 2009; Accepted 22 April 2009
Recommended by Kunquan Lan
The purpose of this paper is to find H ¨older type inequality for the fermionicp-adic invariantq -integral which was defined by Kim2008.
Copyrightq2009 Lee-Chae Jang. 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.
1. Introduction
Letpbe a fixed odd prime. Throughout this paperZp, Qp,Q,C,and Cpwill, respectively, denote the ring ofp-adic rational integers, the field ofp-adic rational numbers, the rational number field, the complex number field, and the completion of algebraic closure ofQp. For a fixed positive integerdwithp, d 1, let
XXdlim← N
Z/dpNZ, X1Zp,
X∗
0<a<dp
a,p1
adpZp,
adpNZpx∈X |x≡a moddpN,
1.1
wherea∈Zlies in 0≤a < dpNcf.1–24.
LetNbe the set of natural numbers. In this paper we assume thatq∈Cp,with|1−q|p< p−1/p−1, which implies thatqxexpxlogqfor|p|
p≤1. We also use the notations
xq 1−q
x
1−q, x−q
1−−qx
for allx∈Zp. For any positive integerN,the distribution is defined by
µq
adpNZp qa
dpN q
. 1.3
We say thatfis a uniformly differentiable function at a pointa∈Zpand denote this property byf ∈ UDZp, if the difference quotients Ffx, y fx−fy/x−yhave a limit
lfaasx, y → a, a cf.1–24.
For f ∈ UDZp, the above distribution µq yields the bosonic p-adic invariant
q-integral as follows:
Iq
f
Zp
fxdµqx lim N→ ∞
1 pN
q pN−1
x0
fxqx, 1.4
representing thep-adicq-analogue of the Riemann integral forf. In the sense of fermionic, let us define the fermionicp-adic invariantq-integral onZpas
I−q
f
Zp
fxdµ−qx lim N→ ∞
1 pN
−q pN−1
x0
fx−qx, 1.5
forf ∈UDZp see16. Now, we consider the fermionicp-adic invariantq-integral onZp as
I−1
flim
q→1I−q
f
Zp
fxdµ−1x. 1.6
From1.5we note that
I−1
fI−1
f2f0, 1.7
wheref1x fx1 see16.
We also introduce the classical H ¨older inequality for the Lebesgue integral in25.
Theorem 1.1. Letm, m∈Qwith1/m1/m1. Iff∈Lmandg∈Lm, thenf·g ∈L1and
fgdx≤fmgm 1.8
wheref ∈Lm⇔|f|mdx <∞andg∈Lm ⇔|g|mdx <∞andf m{
|f|mdx}1/m.
2. H ¨older Type Inequality for Fermionic
p
-Adic Invariant
q
-Integrals
In order to investigate the H ¨older type inequality forI−1, we introduce the new concept of
the inequality as follows.
Definition 2.1. Forf, g∈UDZp, we define the inequality onUDZp resp.,Cpas follows. For f, g ∈ UDZp resp.,x, y ∈ Cp,f≤pgresp.,x ≤p yif and only if |f|p ≤ |g|p resp.,
|x|p≤ |y|p.
Letm, m∈Qwith 1/m1/m1. By substitutingfx qxandgx extinto1.3, we obtain the following equation:
Zp
fxgxµ−1x
Zp
qetxdµ−1x 2
qet1, 2.1
Zp
fxmµ−1x
Zp
qmxdµ −1x
2
qm1, 2.2
Zp
gxmµ−1x
Zp
emxtdµ−1x
2
emt1. 2.3
From2.1,2.2, and2.3, we derive
Zpfxgxdµ−1x
Zpfx mdµ
−1
1/m Zpgx
m dµ−1
1/m
emt11/mqm11/m
qet1
∞ n0
n
l0
⎛ ⎜ ⎝ 1 m
l ⎞ ⎟ ⎠elmt
⎛ ⎜ ⎝
1 m
n−l ⎞ ⎟
⎠qn−lm 1
qet1
∞ n0
n
l0
⎛ ⎜ ⎝ 1 m
l ⎞ ⎟ ⎠
⎛ ⎜ ⎝
1 m
n−l ⎞ ⎟
⎠qn−lm elmt
qet1.
2.4
We remark that thenth Frobenius-Euler numbersHnqand thenth Frobenius-Euler
polynomialsHnq, xattached to algebraic numberq/1may be defined by the exponential
generating functionssee16:
1−q et−q
∞
n0
Hn
qt
n
n!, 2.5
1−q et−qe
xt∞
n0
Hn
q, xt
n
Then, it is easy to see that
2qemlt
qex1 ∞
k0
Hn
−q−1, mlt
k
k!. 2.7
From2.4and2.7, we have the following theorem.
Theorem 2.2. Letm, m∈Qwith1/m1/m1. If one takesfx qxandgx ext, then one
has
Zpfxgxdµ−1x
Zpfx mdµ
−1
1/m Zpgx
m dµ−1
1/m
1
2q
∞
n0
n
l0
⎛ ⎜ ⎝ 1 m l ⎞ ⎟ ⎠ ⎛ ⎜ ⎝ 1 m
n−l ⎞ ⎟
⎠qn−lm∞ k0
Hk
−q−1, mltk
k!.
2.8
We note that form, m, k, l∈Qwith 1/m1/m1,
max ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1
2q p , ⎛ ⎜ ⎝ 1 m l ⎞ ⎟ ⎠ p , ⎛ ⎜ ⎝ 1 m
n−l ⎞ ⎟ ⎠ p
,qml−1
p, k!1 p ⎫ ⎪ ⎪ ⎬ ⎪ ⎪
⎭≤1, 2.9
ByTheorem 2.2and2.7and the definition ofp-adic norm, it is easy to see that
Zpfxgxdµ−1x
Zpfx mdµ
−1
1/m Zpgx
mdµ −1 1/m p
≤maxHk−q−1, ml p
, 2.10
for allm, m, k, l ∈ Q with 1/m1/m 1. We note that M max{|Hk−q−1, ml|p} lies
in0,∞. Thus byDefinition 2.1and2.10, we obtain the following H ¨older type inequality theorem for fermionicp-adic invariantq-integrals.
Theorem 2.3. Letm, m∈Qwith1/m1/m1andMmax{|Hk−q−1, ml|p}. If one takes
fx qxandgx ext, then one has
Zp
fxgxdµ−1x≤pM
Zp
fxmdµ−1
1/m
Zp
gxmdµ−1 1/m
. 2.11
Acknowledgment
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