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1.
We describe sequences of zeros of functionsf≢0 that are analytic in the half-plane ℂ+={z:Rez> and satisfy the condition
where 0≤σ<+∞ and η is a positive function continuously differentiable on [0; +∞) and such thatxη′(x)/η(x)→0 asx→+∞.
Drohobych Pedagogic University, Drohobych. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 7, pp. 904–909,
July, 1999. 相似文献
2.
For given analytic functions ϕ(z) = z + Σ
n=2∞ λ
n
z
n
, Ψ(z) = z + Σ
n=2∞ μ with λ
n
≥ 0, μ
n
≥ 0, and λ
n
≥ μ
n
and for α, β (0≤α<1, 0<β≤1), let E(φ,ψ; α, β) be of analytic functions ƒ(z) = z + Σ
n=2∞
a
n
z
n
in U such that f(z)*ψ(z)≠0 and
for z∈U; here, * denotes the Hadamard product. Let T be the class of functions ƒ(z) = z - Σ
n=2∞|a
n
| that are analytic and univalent in U, and let E
T
(φ,ψ;α,β)=E(φ,ψ;α,β)∩T. Coefficient estimates, extreme points, distortion properties, etc. are determined for the class E
T
(φ,ψ;α,β) in the case where the second coefficient is fixed. The results thus obtained, for particular choices of φ(z) and ψ(z), not only generalize various known results but also give rise to several new results.
University of Bahrain, Isa Town, Bahrain. Published in Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 9, pp. 1162–1170,
September, 1997. 相似文献
3.
Walter Bergweiler 《Journal d'Analyse Mathématique》1994,63(1):121-129
Let (zj) be a sequence of complex numbers satisfying |zj|→ ∞ asj → ∞ and denote by n(r) the number of zj satisfying |zj|≤ r. Suppose that lim infr → ⇈ log n(r)/ logr > 0. Let ϕ be a positive, non-decreasing function satisfying ∫∞ (ϕ(t)t logt)−1
dt < ∞. It is proved that there exists an entire functionf whose zeros are the zj such that log log M(r,f) = o((log n(r))2ϕ(log n(r))) asr → ∞ outside some exceptional set of finite logarithmic measure, and that the integral condition on ϕ is best possible here.
These results answer a question by A. A. Gol’dberg. 相似文献
4.
E. G. Goluzina 《Journal of Mathematical Sciences》2006,137(3):4774-4779
The paper studies the region of values Dm,1(T) of the system {ƒ(z1), ƒ(z2), …, ƒ(zm), ƒ(r)}, m e 1, where zj (j = 1, 2, …,m) are arbitrary fixed points of the disk U = {z: |z| < 1} with Im zj ≠ 0 (j = 1, 2, …,m), and r, 0 < r < 1, is fixed, in the class T of functions ƒ(z) = z+a2z2+ ⋯ regular in the disk U and satisfying in the latter the condition Im ƒ(z) Imz > 0 for Im z ≠ 0. An algebraic characterization of the set Dm,1(T) in terms of nonnegative-definite Hermitian forms is given, and all the boundary functions are described. As an implication,
the region of values of ƒ(zm) in the subclass of functions from the class T with prescribed values ƒ(zk) (k = 1, 2, …,m − 1) and ƒ(r) is determined. Bibliography: 5 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 323, 2005, pp. 24–33. Original article submitted June 13, 2005. 相似文献
5.
M. M. Sheremeta 《Ukrainian Mathematical Journal》1996,48(3):460-466
We prove that, for every sequence (a
k) of complex numbers satisfying the conditions Σ(1/|a
k
|) < ∞ and |a
k+1| − |a
k
| ↗ ∞ (k → ∞), there exists a continuous functionl decreasing to 0 on [0, + ∞] and such thatf(z) = Π(1 −z/|a
k
|) is an entire function of finitel-index. 相似文献
6.
Letu be a function harmonic in the unit disc or in the plane, and letu(z)≤M(|z|) for a majorantM. We formulate conditions onM that guarantee thatu(z)≥−(1+o(1))M(|z|) for |z|→1 in the disc and for |z|→∞ in the plane. 相似文献
7.
We establish conditions under which, for a Dirichlet series F(z) = Σ
n = 1
∞
d
n
exp(λ
n
z), the inequality ⋎F(x)⋎≤y(x),x≥x
o, implies the relation Σ
n = 1
∞ |d
n
exp(λ
n
z)| ⪯ γ((1 + o(1))x) as x→+∞, where γ is a nondecreasing function on (−∞,+∞).
Franko Drohobych State Pedagogic Institute, Drohobych. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 12,
pp. 1610–1616. December, 1997 相似文献
8.
We study the Cauchy problem for the nonlinear dissipative equations (0.1) uo∂u-αδu + Β|u|2/n
u = 0,x ∃ Rn,t } 0,u(0,x) = u0(x),x ∃ Rn, where α,Β ∃ C, ℜα 0. We are interested in the dissipative case ℜα 0, and ℜδ(α,Β)≥ 0, θ = |∫ u0(x)dx| ⊋ 0, where δ(α, Β) = ##|α|n-1nn/2 / ((n + 1)|α|2 + α2
n/2. Furthermore, we assume that the initial data u0 ∃ Lp are such that (1 + |x|)αu0 ∃ L1, with sufficiently small norm ∃ = (1 + |x|)α u0 1 + u0 p, wherep 1, α ∃ (0,1). Then there exists a unique solution of the Cauchy problem (0.1)u(t, x) ∃ C ((0, ∞); L∞) ∩ C ([0, ∞); L1 ∩ Lp) satisfying the time decay estimates for allt0 u(t)||∞ Cɛt-n/2(1 + η log 〈t〉)-n/2, if hg = θ2/n 2π ℜδ(α, Β) 0; u(t)||∞ Cɛt-n/2(1 + Μ log 〈t〉)-n/4, if η = 0 and Μ = θ4/n 4π)2 (ℑδ(α, Β))2 ℜ((1 + 1/n) υ1-1 υ2) 0; and u(t)||∞ Cɛt-n/2(1 + κ log 〈t〉)-n/6, if η = 0, Μ = 0, κ 0, where υl,l = 1,2 are defined in (1.2), κ is a positive constant defined in (2.31). 相似文献
9.
A. N. Tikhomirov 《Siberian Advances in Mathematics》2009,19(3):211-223
Let X:= (X
jk
) denote a Hermitian random matrix with entries X
jk
which are independent for all 1 ≤ j ≤ k. We study the rate of convergence of the expected spectral distribution function of the matrix X to the semi-circular law under the conditions E
X
jk
= 0, E
X
jk
2 = 1, and E|X
jk
|2+η
≤ M
η
< ∞, 0 < η ≤ 2. The bounds of order $
O(n^{ - \frac{\eta }
{{2 + \eta }}} )
$
O(n^{ - \frac{\eta }
{{2 + \eta }}} )
for 1 ≤ η ≤ 2, and those of order $
O(n^{ - \frac{{2\eta }}
{{(2 + \eta )(3 - \eta )}}} )
$
O(n^{ - \frac{{2\eta }}
{{(2 + \eta )(3 - \eta )}}} )
for 0 < η ≤ 1, are obtained. 相似文献
10.
V. V. Napalkov Jr. 《Mathematical Notes》1996,60(1):42-48
We study the growth of an entire functionƒ, whose Borel transformγƒ is holomorphic outside a bounded convex regionD
ƒ with boundary curvature bounded away from 0 and ∞. The functionγƒ is assumed to belong to the Dirichlet space, i.e., it satisfies
, wheredv(ξ) is the area element. It is shown that forγƒ to satisfy the above conditions, it is necessary and sufficient to have
and the growth indicatrixh
ƒ ofƒ satisfies the relation 0<m≤h″(ϕ)+h(ϕ)≤M<∞.
Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 58–65, July, 1996. 相似文献
11.
Pascale Vitse 《Rendiconti del Circolo Matematico di Palermo》2004,53(2):283-312
For Banach space operatorsT satisfying the Tadmor-Ritt condition ‖(zI−T)−1‖≤C|z−1|−1, |z|>1, we show how to use the Riesz turndown collar theorem to estimate sup
n≥0‖T
n‖. A similar estimate is shown for lim sup
n
‖T
n‖ in terms of the Ritt constantM=lim sup
z→1‖(1−z)(zI−T)−1‖. We also obtain an estimate of the functional calculus for these operators proving, in particular, that ‖f(T)‖≤C
q‖f‖
Mult
, where ‖·‖
Mult
stands for the multiplier norm of the Cauchy-Stieltjes integrals over a Lusin type cone domain depending onC and a parameterq, 0<q<1.
Notation.D denotes the open unit disc of the complex plane,D={z∈ℂ:|z|<1}, andT={z∈ℂ:|z|=1} is the unit circle.H
∞ is the Banach algebra of bounded analytic functions onD equipped with the supremum norm ‖.‖∞. 相似文献
12.
Ki-Ho Kwon 《Israel Journal of Mathematics》1994,86(1-3):409-427
Suppose thatα>1, 0<R<∞ and thatf is analytic in |z|≤αR with |f(0)|≥1. It is shown that for a constant dα depending only onα,
. Therefore iff is entire of order λ<∞, logM(r,f)/T(r,f) has order at most λ/2. These results are shown by example to be quite precise. 相似文献
13.
By using a specially constructed cone and the fixed point index theory, this paper investigates the existence of multiple
positive solutions for the third-order threepoint singular semipositone BVP:
where 1/2 < η < 1, the non-linear term ƒ(t, x): (0, 1) × (0, + ∞) → (-∞, + ∞) is continuous and may be singular att = 0,t = 1, andx = 0, also may be negative for some values oft andx, λ is a positive parameter. 相似文献
14.
We obtain asymptotic representations as t ↑ ω, ω ≤ + ∞, for all possible types of P
ω(Y
0, λ
0)-solutions (where Y
0 is zero or ±∞ and −∞ ≤ λ0 ≤ +∞) of nonlinear differential equations y
(n) = α
0
p(t)φ(y), where α
0 ∈ {−1, 1}, p: [a, ω[→]0,+∞[ is a continuous function, and φ is a continuous regularly varying function in a one-sided neighborhood of Y
0. 相似文献
15.
I. E. Chyzhykov 《Ukrainian Mathematical Journal》2007,59(7):1088-1109
For arbitrary 0 ≤ σ ≤ ρ ≤ σ + 1, we describe the class A
σ
ρ
of functions g(z) analytic in the unit disk
= {z : ∣z∣ < 1} and such that g(z) ≠ 0, ρT[g] = σ, and ρM[g] = ρ, where
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 7, pp. 979–995, July, 2007. 相似文献
16.
For a scale of spaces X of functions analytic in the unit disc, including the Korenblum space, and for some natural families ɛ of uniqueness subsets
for X, we describe minorants for (X, ɛ), that is, non-decreasing functions M: (0, 1) → (0, ∞) such that f ∈ X, E ∈ ɛ, and log |f(z)| ≤ −M(|z|) on E imply f = 0. We give an application of this result to approximation by simple fractions with restrictions on the coefficients.
The first author was partially supported by the ANR project DYNOP.
The second author was partially supported by the Research Council of Norway, grant 160192/V30. 相似文献
17.
We study univalent holomorphic functions in the unit diskU = {z: |z| < 1} of the formf(z)=z+∑
n=2
∞
a
n
z
n
that satisfy the condition Re zf’(z)/f(z) > α with α∈ [0, 1) inU. Some integral means of such funcions are estimated. 相似文献
18.
YU JIARONG 《高校应用数学学报(英文版)》1995,10(4):361-366
WEIGHTEDAPPROXIMATIONOFRANDOMFUNCTIONSYUJIARONGAbstract:Let(Ω,A,P)beaprobabilityspace,X(t,ω)arandomfunctioncontinuousinprobab... 相似文献
19.
In the paper, we characterize the coefficient multiplier spaces of mixed norm spaces (H
p,q
(φ1), H
u,v
(φ2)) for the values of p, q, u, v in three cases: (i) 0 < p ≤ u ≤ ∞, 0 < q ≤ min(1, v) ≤ ∞. (ii) v = ∞, 0 < p ≤ u ≤ ∞, 1 ≤ u, q ≤ ∞. (iii) 1 ≤ v ≤ 2 ≤ q ≤ ∞, and 0 < p ≤ u ≤ ∞ or 1 ≤ p, u ≤ ∞. The first case extends the result of Blasco, Jevtić, and Pavlović in one variable. The third case generalizes partly
the results of Jevtić, Jovanović, and Wojtaszczyk to higher dimensions.
Dedicated to Professor Sheng GONG on the occasion of his 75th birthday 相似文献
20.
Jacob Korevaar 《Journal d'Analyse Mathématique》2001,85(1):177-194
For entire functionsf whose power series have Hadamard gaps with ratio ≥1+α>1, Gaier has shown that the condition |f(x)|≤e
x forx≥0 implies |f(z)|≤C
αe|z| (*) for allz. Here the result is extended to the case of square root gaps, that is,
, with
, where α>0. Smaller gaps cannot work. In connection with his proof of the general high indices theorem for Borel summability,
Gaier had shown that square root gaps imply
. Having such an estimate, one can adapt Pitt’s Tauberian method for the restricted Borel high indices theorem to show that,
in fact,
, which implies (*). The author also states an equivalent distance formula involving monomialsx
pke−xinL
∞ (0, ∞). 相似文献