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1.
Let β
0=0.308443… denote the Littlewood-Salem-Izumi number, i.e., the unique solution of the equation
In this paper it is proved that if a
0≥a
1≥⋅⋅⋅≥a
n
>0 and
, k≥1 then for all θ∈(0,π)
and furthermore, the number β
0 is best possible in the sense that for any β∈(0,β
0)
where the coefficients c
k
(β) are defined as
Results for the sine sums are obtained as well.
These results generalize and sharpen the well known trigonometric inequalities of Vietoris.
This research was supported by a grant from the Australian Research Council. The second author was also supported in part
by the NSERC Canada under grant G121211001. The third author was also supported in part by the NSFC of China under grant 10471010. 相似文献
2.
Jeffrey L. Meyer 《The Ramanujan Journal》2007,14(1):79-88
This paper considers a generalization of an integral introduced by S. Ramanujan in his third notebook. Ramanujan’s integral
is itself a version of the dilogarithm,
We prove various functional equations and properties of the generalized integral.
2000 Mathematics Subject Classification Primary–33B30 相似文献
3.
Cubic elliptic functions 总被引：1，自引：1，他引：0
Shaun Cooper 《The Ramanujan Journal》2006,11(3):355-397
The function
occurs in one of Ramanujan’s inversion formulas for elliptic integrals. In this article, a common generalization of the cubic
elliptic functions
is given. The function g1 is the derivative of Ramanujan’s function Φ (after rescaling), and χ3(n) = 0, 1 or −1 according as n≡ 0, 1 or 2 (mod 3), respectively, and |q| < 1. Many properties of the common generalization, as well as the functions g1 and g2, are proved.
2000 Mathematics Subject Classification Primary—33E05; Secondary—11F11, 11F27 相似文献
4.
We consider the following Liouville equation in
For each fixed and a
j
> 0 for 1 ≤ j ≤ k, we construct a solution to the above equation with the following asymptotic behavior:
相似文献
5.
Xiaomei Wu 《分析论及其应用》2008,24(2):139-148
Let→b=(b1,b2,…,bm),bi∈∧βi(Rn),1≤I≤m,βi＞0,m∑I=1βi=β,0＜β＜1,μΩ→b(f)(x)=(∫∞0|F→b,t(f)(x)|2dt/t3)1/2,F→b,t(f)(x)=∫|x-y|≤t Ω(x,x-y)/|x-y|n-1 mΠi=1[bi(x)-bi(y)dy.We consider the boundedness of μΩ,→b on Hardy type space Hp→b(Rn). 相似文献
6.
Cristinel Mortici 《Archiv der Mathematik》2009,93(1):37-45
We prove in this paper that for every x ≥ 0,
where and α = 1.072042464..., then
where and β = 0.988503589... Besides the simplicity, our new formulas are very accurate, if we take into account that they are much stronger
than Burnside’s formula, which is considered one of the best approximation formulas ever known having a simple form.
相似文献
7.
8.
Horst Alzer 《Mediterranean Journal of Mathematics》2008,5(4):395-413
We present several sharp inequalities for the volume of the unit ball in ,
. One of our theorems states that the double-inequality
holds for all n ≥ 2 with the best possible constants
This refines and complements a result of Klain and Rota.
相似文献
9.
Irrationality measures are given for the values of the series
, where
and Wn is a rational valued Fibonacci or Lucas form, satisfying a second order linear recurrence. In particular, we prove irrationality
of all the numbers
where fn and ln are the Fibonacci and Lucas numbers, respectively.
2000 Mathematics Subject Classification Primary—11J82, 11B39 相似文献
10.
Aleksandar Ivić 《Central European Journal of Mathematics》2005,3(2):203-214
Let Δ(x) denote the error term in the Dirichlet divisor problem, and E(T) the error term in the asymptotic formula for the mean square of
. If E
*(t)=E(t)-2πΔ*(t/2π) with
, then we obtain
and
It is also shown how bounds for moments of | E
*(t)| lead to bounds for moments of
. 相似文献
11.
In what follows, $C$ is the space of
-periodic continuous functions; P is a seminorm defined on C, shift-invariant, and majorized by the uniform norm;
is the mth modulus of continuity of a function f with step h and calculated with respect to P;
,
(
),
,
,
Theorem 1.
Let
. Then
For some values of
and seminorms related to best approximations by trigonometric polynomials and splines in the uniform and integral metrics, the inequalities are sharp. Bibliography: 6 titles. 相似文献
12.
We determine exact values for the k-error linear complexity L
k
over the finite field
of the Legendre sequence
of period p and the Sidelnikov sequence
of period p
m
− 1. The results are
for 1 ≤ k ≤ (p
m
− 3)/2 and
for k≥ (p
m
− 1)/2. In particular, we prove
相似文献
13.
In this paper we establish dispersive estimates for solutions to the linear Schrödinger equation in three dimensions
where V(t,x) is a time-dependent potential that satisfies the conditions
Here c0 is some small constant and
denotes the Fourier transform with respect to the first variable. We show that under these conditions (0.1) admits solutions (·)Lt(L2x(3))L2t(L6x(3)) for any fL2(3) satisfying the dispersive inequality
For the case of time independent potentials V(x), (0.2) remains true if
We also establish the dispersive estimate with an -loss for large energies provided
.Finally, we prove Strichartz estimates for the Schrödinger equations with potentials that decay like |x|-2- in dimensions n3, thus solving an open problem posed by Journé, Soffer, and Sogge. 相似文献
| (0.1) |
| (0.2) |
14.
Mariano Giaquinta Min-Chun Hong 《NoDEA : Nonlinear Differential Equations and Applications》2004,11(4):469-490
We discuss the partial regularity of minimizers of energy functionals such as
where u is a map from a domain
into the m-dimensional unit sphere of
and A is a differential one-form in . 相似文献
15.
Martin Flucher 《Commentarii Mathematici Helvetici》1992,67(1):471-497
We prove that theTrudinger-Moser constant
|
相似文献
16.
Gensun Fang 《中国科学A辑(英文版)》1998,41(12):1272-1277
The Nikolskii type inequality for cardinal splines
17.
G. Kuba 《Archiv der Mathematik》2002,79(6):534-542
Let
(z ∈ ℝ). Further let λ denote a large real parameter. We show that for arbitrary real numbersk and α withk>=2.7013 and 0<α≦1,
18.
Let n,p and k be three non negative integers. We prove that the apparently rational fractions of q:
19.
Kolitsch and Sellers showed recently that a8(n), the number of 8-core partitions of n, is even when n belongs to certain arithmetic progressions. We prove a similar result for 16-cores. In doing so, we prove the surprising result that the a16(n), given by
20.
We investigate limiting behavior as γ tends to ∞ of the best polynomial approximations in the Sobolev-Laguerre space WN,2([0, ∞); e−x) and the Sobolev-Legendre space WN,2([−1, 1]) with respect to the Sobolev-Laguerre inner product
|