共查询到20条相似文献,搜索用时 406 毫秒
1.
Alon Nishry 《Israel Journal of Mathematics》2011,186(1):197-220
Consider the random entire function , where the ? n are independent standard complex Gaussian coefficients, and the a n are positive constants, which satisfy .
$f(z) = \sum\limits_{n = 0}^\infty {{\phi _n}{a_n}{z^n}} $
$\mathop {\lim }\limits_{x \to \infty } {{\log {a_n}} \over n} = - \infty $
We study the probability P H (r) that f has no zeroes in the disk{|z| < r} (hole probability). Assuming that the sequence a n is logarithmically concave, we prove that , where , and r tends to ∞ outside a (deterministic) exceptional set of finite logarithmic measure.
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$\log {P_H}(r) = - S(r) + o(S(r))$
$S(r) = 2 \cdot \sum\limits_{n:{a_n}{r^n} \ge 1} {\log ({a_n}{r^n})} $
2.
For ?1≤B<A≤1, let \(\mathcal {S}^{*}(A,B)\) denote the class of normalized analytic functions \(f(z)= z+{\sum }_{n=2}^{\infty }a_{n} z^{n}\) in |z|<1 which satisfy the subordination relation z f ′(z)/f(z)?(1 + A z)/(1 + B z) and Σ?(A,B) be the corresponding class of meromorphic functions in |z|>1. For \(f\in \mathcal {S}^{*}(A,B)\) and λ>0, we shall estimate the absolute value of the Taylor coefficients a n (?λ,f) of the analytic function (f(z)/z)?λ . Using this we shall determine the coefficient estimate for inverses of functions in the classes \(\mathcal {S}^{*}(A,B)\) and Σ?(A,B). 相似文献
3.
The old result due to[Ozaki,S.:On the theory of multivalent functions Ⅱ.Sci.Rep.Tokyo Bunrika Daigaku Sect.A,45-87(1941)],says that if f(z) = z~p + ∑_(n=p+1~(a_nz~n))~∞ is analytic in a convex domain D and for some real α we have Re{exp(iα)f~((p))(z)} 0 in D,then f(z) is at most p-valent in ED.In this paper,we consider similar problems in the unit disc B = {z ∈ C:|z| 1}. 相似文献
4.
Fixed points of meromorphic functions and of their differences,divided differences and shifts 下载免费PDF全文
Let f(z) be a finite order meromorphic function and let c∈C\{0} be a constant.If f(z)has a Borel exceptional value a∈C,it is proved that max{τ(f(z)),τ(△_cf(z))}=max{τ(f(z)),τ(f(z+c))}=max{τ(△_cf(z)),τ(f(z+c))}=σ(f(z)).If f(z) has a Borel exceptional value b∈(C\{0})∪{∞},it is proved that max{τ(f(z)),τ(△cf(z)/f(z))}=max{τ(△cf(z)/f(z)),τ(f(z+c))}=σ(f(z)) unless f(z) takes a special form.Here τ(g(z)) denotes the exponent of convergence of fixed points of the meromorphic function g(z),and σ(g(z)) denotes the order of growth of g(z). 相似文献
5.
K. Boussaf A. Boutabaa A. Escassut 《P-Adic Numbers, Ultrametric Analysis, and Applications》2016,8(4):280-297
Let IK be a complete ultrametric algebraically closed field and let A(IK) be the IK-algebra of entire functions on IK. For an f ∈ A(IK), similarly to complex analysis, one can define the order of growth as \(\rho \left( f \right) = \mathop {\lim }\limits_{r \to + \infty } \sup \frac{{\log \left( {\log |f|\left( r \right)} \right)}}{{\log r}}\). When ρ(f) ≠ 0,+∞, one can define the type of growth as \(\sigma \left( f \right) = \mathop {\lim }\limits_{r \to + \infty } \sup \frac{{\log \left( {|f|\left( r \right)} \right)}}{{{r^\rho }\left( f \right)}}\). But here, we can also define the cotype of growth as \(\psi \left( f \right) = \mathop {\lim }\limits_{r \to + \infty } \sup \frac{{q\left( {f,r} \right)}}{{{r^\rho }\left( f \right)}}\) where q(f, r) is the number of zeros of f in the disk of center 0 and radius r. Many properties described here were first given in the Houston Journal, but new inequalities linking the order, type and cotype are given in this paper: we show that ρ(f)σ(f) ≤ ψ(f) ≤ eρ(f)σ(f). Moreover, if ψ or σ are veritable limits, then ρ(f)σ(f) = ψ(f) and this relation is conjectured in the general case. Several other properties are examined concerning ρ, σ, ψ for f and f’. Particularly,we show that if an entire function f has finite order, then \(\frac{{f'}}{{{f^2}}}\) takes every value infinitely many times. 相似文献
6.
B. Simon 《Functional Analysis and Its Applications》2007,41(2):143-153
We show that the parameters a n , b n of a Jacobi matrix have a complete asymptotic expansion , where 1 < |µj| < R for j ? K(R) and all R, if and only if the Jost function, u, written in terms of z (where E = z + z ?1) is an entire meromorphic function. We relate the poles of u to the µj’s.
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$a_n^2 - 1 = \sum\limits_{k = 1}^{K(R)} {p_k (n)\mu _k^{ - 2n} + O(R^{ - 2n} ),} b_n = \sum\limits_{k = 1}^{K(R)} {p_k (n)\mu _k^{ - 2n + 1} + O(R^{ - 2n} )} $
7.
Li Songying 《数学学报(英文版)》1988,4(2):97-110
In this paper, we shall prove the existence of the singular directions related to Hayman's problems[1]. The results are as follows.
- Suppose that f(z) is a transcendental integral function in the finite plane, then there exists a direction H: argz= θ0 (0≤θ0>2π) such that for every positive ε, every integer p(≠0, ?1) and every finite complex number b(≠0), we have $$\mathop {\lim }\limits_{r \to \infty } \left\{ {n(r,\theta _0 ,\varepsilon ,f' \cdot \{ f\} ^p = b)} \right\} = + \infty $$
- Suppose that f(z) is a transcendental integral function in the finite plane, then there exists a direction H:z= θ0 (0≤θ0>2π) such that for every positive ε, every integrer p(≥3) and any finite complex numbers a(≠0) and b, we have $$\mathop {\lim }\limits_{r \to \infty } \left\{ {n(r,\theta _0 ,\varepsilon ,f' - a\{ f\} ^p = b)} \right\} = + \infty $$
- Suppose that f(z) is a meromorphic function in the finite plane and satisfies the following condition $$\mathop {\lim }\limits_{r \to \infty } \frac{{T(r,f)}}{{(\log r)^3 }} = + \infty $$ then there exists a direction H:z= θ0 (0≤θ0>2π) such that for every positive ε, every integer p(≥5) and every two finite complex numbers a(≠0) and b, we have $$\mathop {\lim }\limits_{r \to \infty } \left\{ {n(r,\theta _0 ,\varepsilon ,f' - a\{ f\} ^p = b)} \right\} = + \infty $$
8.
Let f and g be multiplicative functions of modulus 1. Assume that \( {\lim_{x \to \infty }}\frac{1}{x}\left| {\sum\nolimits_{n \leqslant x} {f(n)} } \right| = A > 0 \) and \( {\lim_{x \to \infty }}\frac{1}{x}\left| {\sum\nolimits_{n \leqslant x} {g(n)} } \right| = 0 \). We prove that, under these conditions,Concerning the Liouville function λ, we find an upper estimate for \( \frac{1}{x}\left| {\sum\limits_{n \leqslant x} {\lambda (n)\lambda (n + 1)} } \right| \) under the unproved hypothesis that L(s, χ) have Siegel zeros for an infinite sequence of L-functions.
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$ \mathop {\lim }\limits_{x \to \infty } \frac{1}{x}\sum\limits_{n \leqslant x} {f(n)g(n + 1) = 0.}$
9.
Rafael de la Llave 《Regular and Chaotic Dynamics》2017,22(6):650-676
We present simple proofs of a result of L.D. Pustylnikov extending to nonautonomous dynamics the Siegel theorem of linearization of analytic mappings. We show that if a sequence f n of analytic mappings of C d has a common fixed point f n (0) = 0, and the maps f n converge to a linear mapping A∞ so fast that then f n is nonautonomously conjugate to the linearization. That is, there exists a sequence h n of analytic mappings fixing the origin satisfying The key point of the result is that the functions hn are defined in a large domain and they are bounded. We show that We also provide results when f n converges to a nonlinearizable mapping f∞ or to a nonelliptic linear mapping. In the case that the mappings f n preserve a geometric structure (e. g., symplectic, volume, contact, Poisson, etc.), we show that the hn can be chosen so that they preserve the same geometric structure as the f n . We present five elementary proofs based on different methods and compare them. Notably, we consider the results in the light of scattering theory. We hope that including different methods can serve as an introduction to methods to study conjugacy equations.
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$$\sum\limits_n {{{\left\| {{f_m} - {A_\infty }} \right\|}_{L\infty \left( B \right)}} < \infty } $$
$${A_\infty } = diag\left( {{e^{2\pi i{\omega _1}}},...,{e^{2\pi i{\omega _d}}}} \right)\omega = \left( {{\omega _1},...,{\omega _q}} \right) \in {\mathbb{R}^d},$$
$${h_{n + 1}} \circ {f_n} = {A_\infty }{h_n}.$$
$${\sum\nolimits_n {\left\| {{h_n} - Id} \right\|} _{L\infty (B)}} < \infty .$$
10.
This paper is concerned with the existence of positive solutions of the third-order boundary value problem with full nonlinearity where \(f:[0,1]\times \mathbb {R}^+\times \mathbb {R}^+\times \mathbb {R}^-\rightarrow \mathbb {R}^+\) is continuous. Under some inequality conditions on f as |(x, y, z)| small or large enough, the existence results of positive solution are obtained. These inequality conditions allow that f(t, x, y, z) may be superlinear, sublinear or asymptotically linear on x, y and z as \(|(x,y,z)|\rightarrow 0\) and \(|(x,y,z)|\rightarrow \infty \). For the superlinear case as \(|(x,y,z)|\rightarrow \infty \), a Nagumo-type growth condition is presented to restrict the growth of f on y and z. Our discussion is based on the fixed point index theory in cones.
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$$\begin{aligned} \left\{ \begin{array}{lll} u'''(t)&{}=f(t,u(t),u'(t),u''(t)),\quad t\in [0,1],\\ u(0)&{}=u'(1)=u''(1)=0, \end{array}\right. \end{aligned}$$
11.
Miroslav Pavlović 《Archiv der Mathematik》2011,97(5):467-473
We improve a recent result of Yang and Xu (Arch. Math. 96 (2011), 151–160) by proving that if ψ is a normal function on [1, ∞) and \({f(z)=\sum_{n=0}^\infty a_n z^{k_n}}\) (|z| < 1) is an analytic function with Hadamard gaps, thenwhere C is a constant independent of ζ and {a n }.
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$\frac 1C \sup_{n\ge 0} \frac{|a_n|}{\psi(k_n)} \le \sup_{0 < r < 1} \frac{|f(r\zeta)|}{\psi(1/(1-r))} \le C\sup_{n\ge 0}\frac{|a_n|}{\psi(k_n)}, \quad |\zeta|=1,$
12.
We consider the following two problems. Problem 1: what conditions on a sequence of finite subsets A k ? ? and a sequence of functions λ k : A k → ? provide the existence of a number C such that any function f ∈ L 1 satisfies the inequality ‖U A,Λ(f)‖ p ≤ C‖f‖1 and what is the exact constant in this inequality? Here, \(U_{\mathcal{A},\Lambda } \left( f \right)\left( x \right) = \sum\nolimits_{k = 1}^\infty {\left| {\sum\nolimits_{m \in A_k } {\lambda _k \left( m \right)c_m \left( f \right)e^{imx} } } \right|}\) and c m (f) are Fourier coefficients of the function f ∈ L 1. Problem 2: what conditions on a sequence of finite subsets A k ? ? guarantee that the function \(\sum\nolimits_{k = 1}^\infty {\left| {\sum\nolimits_{m \in A_k } {c_m \left( h \right)e^{imx} } } \right|}\) belongs to L p for every function h of bounded variation? 相似文献
13.
Let L be a Schrdinger operator of the form L =-? + V acting on L~2(R~n), n≥3, where the nonnegative potential V belongs to the reverse Hlder class B_q for some q≥n. Let BMO_L(R~n) denote the BMO space associated to the Schrdinger operator L on R~n. In this article, we show that for every f ∈ BMO_L(R~n) with compact support, then there exist g ∈ L~∞(R~n) and a finite Carleson measure μ such that f(x) = g(x) + S_(μ,P)(x) with ∥g∥∞ + |||μ|||c≤ C∥f∥BMO_L(R~n), where S_(μ,P)=∫(R_+~(n+1))Pt(x,y)dμ(y, t),and Pt(x, y) is the kernel of the Poisson semigroup {e-~(t(L)~(1/2))}t0 on L~2(R~n). Conversely, if μ is a Carleson measure, then S_(μ,P) belongs to the space BMO_L(R~n). This extends the result for the classical John-Nirenberg BMO space by Carleson(1976)(see also Garnett and Jones(1982), Uchiyama(1980) and Wilson(1988)) to the BMO setting associated to Schrdinger operators. 相似文献
14.
M. G. Grigoryan K. A. Navasardyan 《Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences)》2016,51(1):21-33
The paper proves that for any ε > 0 there exists ameasurable set E ? [0, 1] with measure |E| > 1 ? ε such that for each f ∈ L1[0, 1] there is a function \(\tilde f \in {L^1}\left[ {0,1} \right]\) coinciding with f on E whose Fourier-Walsh series converges to \(\tilde f\) in L1[0, 1]-norm, and the sequence \(\left\{ {\left| {{c_k}\left( {\tilde f} \right)} \right|} \right\}_{n = 0}^\infty \) is monotonically decreasing, where \(\left\{ {{c_k}\left( {\tilde f} \right)} \right\}\) is the sequence of Fourier-Walsh coefficients of \(\left\{ {\left| {{c_k}\left( {\tilde f} \right)} \right|} \right\}_{n = 0}^\infty \). 相似文献
15.
A. V. Egorov 《Moscow University Mathematics Bulletin》2007,62(4):141-147
A generalization of the notion of expansion over the signum system to the case of a sigma-finite space is considered in the paper. Let {E k } k=1 ∞ be an exhaustion of X, then under the condition \(\sum\limits_{k = 1}^\infty {\tfrac{1}{{\mu E_k }}} = \infty \) the expansion converges to the expanded function in the metric of L 2, and under the conditions \(\sum\limits_{k = 1}^\infty {\tfrac{1}{{\mu E_k }}} = \infty \) and \(\mathop {\lim }\limits_{k \to \infty } \tfrac{{\mu E_{k - 1} }}{{\mu E_k }} = 1\) the convergence holds almost everywhere for functions from L ∞(X). 相似文献
16.
We prove the existence of infinitely many solutions for where V(x) satisfies \(\lim _{|x| \rightarrow \infty } V(x) = V_\infty >0\) and some conditions. We require conditions on f(u) only around 0 and at \(\infty \).
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$$\begin{aligned} - \Delta u + V(x) u = f(u) \quad \text { in } \mathbb {R}^N, \quad u \in H^1(\mathbb {R}^N), \end{aligned}$$
17.
William A. Veech 《Journal d'Analyse Mathématique》2018,135(2):413-436
Let S be the set of square-free natural numbers. A Hilbert-Schmidt operator, A, associated to the Möbius function has the property that it maps from \({ \cup _{0 < r < \infty }}{l^r}(s)\) to \({ \cap _{0 < r < \infty }}{l^r}(s)\), injectively. If 0 < r< 2 and ξ ∈ lr (S), the series \({f_\zeta } = \sum\nolimits_{n \in s} {A\zeta (x)cos2\pi nx} \) converges uniformly to an element of fξR0, i.e., a periodic, even, continuous function with equally spaced Riemann sums, \(\sum\nolimits_{j = 0}^{N - 1} {{f_\zeta }} (j/N) = 0,N = 1,2....\) If \({A_{\zeta \lambda }} = \lambda {\zeta _\lambda },{\zeta _\lambda }(1) = 1\), then ξλ is multiplicative. If \({f_{{\zeta _\lambda }}} \in {\Lambda _a}\), the space of α-Lipschitz continous functions, for some α > 0, and if χ is any Dirichlet character, then L(s, χ) ≠ 0, Res > 1 ? α. Conjecturally, the Generalized Riemann Hypothesis (GRH) is equivalent to fξ ∈ Λα, α < 1/2, ξ ∈ lr (S), 0 < r < 2. Using a 1991 estimate by R. C. Baker and G. Harman, one finds GRH implies fξ ∈ Λα, α < 1/4, ξ ∈ lr (S), 0 < r < 2. The question of whether R0 ∩ Λα ≠ {0} for some positive α > 0 is open. 相似文献
18.
Huixue Lao 《Acta Appl Math》2010,110(3):1127-1136
Let L(sym j f,s) be the jth symmetric power L-function attached to a holomorphic Hecke eigencuspform f(z) for the full modular group, and \(\lambda_{\mathrm{sym}^{j}f}(n)\) denote its nth coefficient. In this paper we are able to prove that and
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$\int_{1}^{x}\bigg|\sum_{n\leq y}\lambda_{\mathrm{sym}^{3}f}(n)\bigg|^{2}dy=O\bigl(x^{2}\bigr),$
$\int_{1}^{x}\bigg|\sum_{n\leq y}\lambda_{\mathrm{sym}^{4}f}(n)\bigg|^{2}dy=O\bigl(x^{\frac{11}{5}}\log x\bigr).$
19.
N. A. Izobov 《Differential Equations》2008,44(5):618-631
We prove the conditional exponential stability of the zero solution of the nonlinear differential system with L p -dichotomous linear Coppel-Conti approximation .x = A(t)x whose principal solution matrix X A (t), X A (0) = E, satisfies the condition where P 1 and P 2 are complementary projections of rank k ∈ {1, …, n ? 1} and rank n ? k, respectively, and with a higher-order infinitesimal perturbation f:[0, ∞) × U → R n that is piecewise continuous in t ≥ 0 and continuous in y in some neighborhood U of the origin.
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$$\dot y = A(t)y + f(t,y),{\mathbf{ }}y \in R^n ,{\mathbf{ }}t \geqslant 0,$$
$$\mathop \smallint \limits_0^t \left\| {X_A (t)P_1 X_A^{ - 1} (\tau )} \right\|^p d\tau + \mathop \smallint \limits_t^{ + \infty } \left\| {X_A (t)P_2 X_A^{ - 1} (\tau )} \right\|^p d\tau \leqslant C_p (A) < + \infty ,{\mathbf{ }}p \geqslant 1,{\mathbf{ }}t \geqslant 0,$$
20.
We concern the sublinear Schrödinger-Poisson equations \(\left\{ \begin{gathered}- \Delta u + \lambda V\left( x \right)u + \phi u = f\left( {x,u} \right)in{\mathbb{R}^3} \hfill \\- \Delta \phi = {u^2}in{\mathbb{R}^3} \hfill \\ \end{gathered} \right.\) where λ > 0 is a parameter, V ∈ C(R3,[0,+∞)), f ∈ C(R3×R,R) and V-1(0) has nonempty interior. We establish the existence of solution and explore the concentration of solutions on the set V-1(0) as λ → ∞ as well. Our results improve and extend some related works. 相似文献