THROREMS OF DISTORTION FOR SCHLICHT FUNCTIONS

Let \[f(z) = z + \sum\limits_{n = 1}^\infty {{a_n}{z^n} \in S} {\kern 1pt} {\kern 1pt} {\kern 1pt} and{\kern 1pt} {\kern 1pt} {\kern 1pt} \log \frac{{f(z) - f(\xi )}}{{z - \xi }} - \frac{{z\xi }}{{f(z)f(\xi )}} = \sum\limits_{m,n = 1}^\infty {{d_{m,n}}{z^m}{\xi ^n},} \], we denote \[{f_v} = f({z_v})\] , \[\begin{array}{l} {\varphi _\varepsilon }({z_u}{z_v}) = {\left| {\frac{{{f_u} - {f_v}}}{{{z_u} - {z_v}}}} \right|^\varepsilon }\frac{1}{{(1 - {z_u}{{\bar z}_v})}},\g_m^\varepsilon (z) = - {F_m}(\frac{1}{{f(z)}}) + \frac{1}{{{z^m}}} + \varepsilon {{\bar z}^m}, \end{array}\], where \({F_m}(t)\) is a Faber polynomial of degree m. Theorem 1. If \[f(z) \in S{\kern 1pt} {\kern 1pt} {\kern 1pt} and{\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{u,v = 1}^N {{A_{u,v}}{x_u}{{\bar x}_v} \ge 0} \] and then \[\begin{array}{l} \sum\limits_{u,v = 1}^N {{A_{u,v}}{\lambda _u}{{\bar \lambda }_v}} {\left| {\frac{{{f_u} - {f_v}}}{{{z_u} - {z_v}}}} \right|^\varepsilon }\exp \{ \alpha {F_l}({z_u},{z_v})\} \ \le \sum\limits_{u,v = 1}^N {{A_{u,v}}{\lambda _u}{{\bar \lambda }_v}} \varphi _\varepsilon ^\alpha ({z_u}{z_v})l = 1,2,3, \end{array}\], where \[\begin{array}{l} {F_1}({z_u},{z_v}) = \frac{1}{2}\sum\limits_{n = 1}^\infty {\frac{1}{n}} g_n^\varepsilon ({z_u})\bar g_n^\varepsilon ({z_v}),\{F_2}({z_u},{z_v}) = \frac{1}{{1 + {\varepsilon _n}R{d_{n,n}}}}Rg_n^\varepsilon ({z_u})Rg_n^\varepsilon ({z_v}),\{F_3}({z_u},{z_v}) = \frac{1}{{1 - {\varepsilon _n}R{d_{n,n}}}}Rg_n^\varepsilon ({z_u})Rg_n^\varepsilon ({z_v}). \end{array}\] The \[F({z_u},{z_v}) = \frac{1}{2}{g_1}({z_u}){{\bar g}_2}({z_v})\] is due to Kungsun. Theorem 2. If \(f(z) \in S\) ,then \[P(z) + \left| {\sum\limits_{u,v = 1}^N {{A_{u,v}}{\lambda _u}{{\bar \lambda }_v}} {{\left| {\frac{{{f_u} - {f_v}}}{{{z_u} - {z_v}}}\frac{{{z_u}{z_v}}}{{{f_u}{f_v}}}} \right|}^\varepsilon }} \right| \le \sum\limits_{u,v = 1}^N {{\lambda _u}{{\bar \lambda }_v}} \frac{1}{{1 - {z_u}{{\bar z}_v}}}\], where \[\begin{array}{l} P(z) = \frac{1}{2}\sum\limits_{n = 1}^\infty {\frac{1}{n}} {G_n}(z),\{G_n}(z) = {\left| {\left| {\sum\limits_{n = 1}^N {{\beta _u}({F_n}(\frac{1}{{f({z_u})}}) - \frac{1}{{z_u^n}})} } \right| - \left| {\sum\limits_{n = 1}^N {{\beta _u}z_u^n} } \right|} \right|^2}, \end{array}\], \(P(z) \equiv 0\) is due to Xia Daoxing.     

THE ASYMPTOTIC BEHAVIOUR OF ANALYTIC FUNCTIONS

AIn this paper, the author obtains the following results:(1) If Taylor coeffiients of a function satisfy the conditions:(i),(ii),(iii)A_k=O(1/k) the for any h>0 the function φ(z)=exp{w(z)} satisfies the asymptotic equality the case h>1/2 was proved by Milin.(2) If f(z)=z α_2z~2 …∈S~* and,then for λ>1/2     

On analytic continuation of a hypergeometric series using the Euler-Knopp transformation

The Euler-Knopp transformation is considered in terms of the problems of regularity and acceleration of the rate of convergence. The object of study is the hypergeometric series

Powers of the Catalan Generating Function and Lagrange''s 1770 Trinomial Equation Series

The Catalan numbers $1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862,\ldots$ are given by $C(n)=\frac{1}{n+1}\binom{2n}{n}$ for $n\geq 0$. They are named for Eugene Catalan who studied them as early as 1838. They were also found by Leonhard Euler (1758), Nicholas von Fuss (1795), and Andreas von Segner (1758). The Catalan numbers have the binomial generating function $$\mathbf{C}(z) = \sum_{n=0}^{\infty}C(n)z^n = \frac{1 - \sqrt{1-4z}}{2z}$$ It is known that powers of the generating function $\mathbf{C}(z)$ are given by $$\mathbf{C}^a(z) = \sum_{n=0}^{\infty}\frac{a}{a+2n}\binom{a+2n}{n}z^n.$$ The above formula is not as widely known as it should be. We observe that it is an immediate, simple consequence of expansions first studied by J. L. Lagrange. Such series were used later by Heinrich August Rothe in 1793 to find remarkable generalizations of the Vandermonde convolution. For the equation $x^3 - 3x + 1 =0$, the numbers $\frac{1}{2k+1}\binom{3k}{k}$ analogous to Catalan numbers occur of course. Here we discuss the history of these expansions. and formulas due to L. C. Hsu and the author.     

Periodic points of algebraic functions and Deuring’s class number formula

The Ramanujan Journal - In this paper, transformation formulas for the function $$\begin{aligned} A_{1}\left( z,s:\chi \right) =\sum \limits _{n=1}^{\infty }\sum \limits _{m=1} ^{\infty }\chi...     

Projektive Skalen,Tschebyscheff-Polynome und Fibonacci-Zahlen

Summary. Let $\widehat{\widehat T}_n$ and $\overline U_n$ denote the modified Chebyshev polynomials defined by $\widehat{\widehat T}_n (x) = {T_{2n + 1} \left(\sqrt{x + 3 \over 4} \right) \over \sqrt{x + 3 \over 4}}, \quad \overline U_{n}(x) = U_{n} \left({x + 1 \over 2}\right) \qquad (n \in \mathbb{N}_{0},\ x \in \mathbb{R}).$ For all $n \in \mathbb{N}_{0}$ define $\widehat{\widehat T}_{-(n + 1)} = \widehat{\widehat T}_n$ and $\overline U_{-(n + 2)} = - \overline U_n$, furthermore $\overline U_{-1} = 0$. In this paper, summation formulae for sums of type $\sum\limits^{+\infty}_{k = -\infty} \mathbf a_{\mathbf k}(\nu; x)$ are given, where $\bigl(\mathbf a_{\mathbf k}(\nu; x)\bigr)^{-1} = (-1)^k \cdot \Bigl( x \cdot \widehat{\widehat T}_{\left[k + 1 \over 2\right] - 1} (\nu) +\widehat{\widehat T}_{\left[k + 1 \over 2\right]}(\nu)\Bigr) \cdot \Bigl(x \cdot \overline U_{\left[k \over 2\right] - 1} (\nu) + \overline U_{\left[k \over 2\right]} (\nu)\Bigr)$ with real constants $ x, \nu $. The above sums will turn out to be telescope sums. They appear in connection with projective geometry. The directed euclidean measures of the line segments of a projective scale form a sequence of type $(\mathbf a_{\mathbf k} (\nu;x))_{k \in \mathbb{Z}}$ where $ \nu $ is the cross-ratio of the scale, and x is the ratio of two consecutive line segments once chosen. In case of hyperbolic $(\nu \in \mathbb{R} \setminus] - 3,1[)$ and parabolic $\nu = -3$ scales, the formula $\sum\limits^{+\infty}_{k = -\infty} \mathbf a_{\mathbf k} (\nu; x) = {\frac{1}{x - q_{{+}\atop(-)}}} - {\frac{1}{x - q_{{-}\atop(+)}}} \eqno (1)$ holds for $\nu > 1$ (resp. $\nu \leq - 3$), unless the scale is geometric, that is unless $x = q_+$ or $x = q_-$. By $q_{\pm} = {-(\nu + 1) \pm \sqrt{(\nu - 1)(\nu + 3)} \over 2}$ we denote the quotient of the associated geometric sequence.

     

The zero set $$\Lambda $$ of exponential polynomials and the Riesz property of its exponential system $$E_{\Lambda }$$ in $$L^2(0,T)$$

Archiv der Mathematik - Consider the class of exponential polynomials of the form $$\begin{aligned} f(z)=\sum _{n=0}^{N}\left( \sum _{k=0}^{m_n}c_{n,k}z^k\right) e^{h_n z}, \qquad 0=h_0<...     

Precise Asymptotics in the Baum-Katz and Davis Laws of Large Numbers of ρ-mixing Sequences

Let {X,Xn;n ≥ 1} be a strictly stationary sequence of ρ-mixing random variables with mean zeros and finite variances. Set Sn =∑k=1^n Xk, Mn=maxk≤n｜Sk｜,n≥1.Suppose limn→∞ESn^2/n=：σ^2〉0 and ∑n^∞=1 ρ^2/d（2^n）〈∞,where d=2 if 1≤r〈2 and d〉r if r≥2.We prove that if E｜X｜^r 〈∞,for 1≤p〈2 and r〉p,then limε→0ε^2（r-p）/2-p ∑∞n=1 n^r/p-2 P{Mn≥εn^1/p}=2p/r-p ∑∞k=1（-1）^k/（2k＋1）^2（r-p）/（2-p）E｜Z｜^2（r-p）/2-p,where Z has a normal distribution with mean 0 and variance σ^2.     

ON THE EXPECTED SAMPLE SIZES OF SOME POWER ONE TESTS FOR NORMAL MEAN WITH UNKNOWN VARIANCE

Suppose that $\[{x_1},{x_2}, \cdots \]$ are i i d. random variables on a probability space $\[(\Omega ,F,P)\]$ and $\[{x_1}\]$ is normally distributed with mean $\[\theta \]$ and variance $\[{\sigma ^2}\]$, both of which are unknown. Given $\[{\theta _0}\]$ and $\[0 < \alpha < 1\]$, we propose a concrete stopping rule T w. r. e.the $\[\{ {x_n},n \ge 1\} \]$ such that $$\[{P_{\theta \sigma }}(T < \infty ) \le \alpha \begin{array}{*{20}{c}} {for}&{\begin{array}{*{20}{c}} {all}&{\theta \le {\theta _0},\sigma > 0,} \end{array}} \end{array}\]$$ $$\[{P_{\theta \sigma }}(T < \infty ) = 1\begin{array}{*{20}{c}} {for}&{\begin{array}{*{20}{c}} {all}&{\theta > {\theta _0},\sigma > 0,} \end{array}} \end{array}\]$$ $$\[\mathop {\lim }\limits_{\theta \downarrow {\theta _0}} {(\theta - {\theta _0})^2}{({\ln _2}\frac{1}{{\theta - {\theta _0}}})^{ - 1}}{E_{\theta \sigma }}T = 2{\sigma ^2}{P_{{\theta _0}\sigma }}(T = \infty )\]$$ where $\[{\ln _2}x = \ln (\ln x)\]$.     

Hardy-Sobolev类上的Bohr型不等式

Let β 〉 0 and Sβ ：= {z ∈ C ： ｜Imz｜ 〈β} be a strip in the complex plane. For an integer r ≥ 0, let H∞^Г,β denote those real-valued functions f on R, which are analytic in Sβ and satisfy the restriction ｜f^（r）（z）｜ ≤ 1, z ∈ Sβ. For σ 〉 0, denote by Bσ the class of functions f which have spectra in （-2πσ, 2πσ）. And let Bσ^⊥ be the class of functions f which have no spectrum in （-2πσ, 2πσ）. We prove an inequality of Bohr type
‖f‖∞≤π/√λ∧σ^r∑k=0^∞（-1）^k（r＋1）/（2k＋1）^rsinh（（2k＋1）2σβ）,f∈H∞^r,β∩B1/σ,
where λ∈（0,1）,∧and ∧′are the complete elliptic integrals of the first kind for the moduli λ and λ′=√1- λ^2,respectively,and λ satisfies
4∧β/π∧′=1/σ.
The constant in the above inequality is exact.     

UNUNIQUENESS OF SOME HOLOMORPHIC FUNCTIONS

In the present paper, we show that there exist a bounded, holomorphic function $\[f(z) \ne 0\]$ in the domain $\[\{ z = x + iy:\left| y \right| < \alpha \} \]$ such that $\[f(z)\]$ has a Dirichlet expansion $\[\sum\limits_{n = 0}^{ + \infty } {{d_n}{e^{ - {u_n}}}} \]$ in the halfplane $\[x > {x_f}\]$ if and only if $\[\frac{a}{\pi }\log r - \sum\limits_{{u_n} < r} {\frac{2}{{{u_n}}}} \]$ has a finite upperbound on $\[[1, + \infty )\]$, where $\[\alpha \]$ is a positive constant,$\[{x_f}( < + \infty )\]$ is the abscissa of convergence of $\[\sum\limits_{n = 0}^{ + \infty } {{d_n}{e^{ - {u_n}}}} \]$ and the infinite sequence $\[\{ {u_n}\} \]$ satisfies $\[\mathop {\lim }\limits_{n \to + \infty } ({u_{n + 1}} - {u_n}) > 0\]$. We also point out some necessary conditions and sufficient ones Such that a bounded holomorphic function in an angular(or half-band) domain is identically zero if an infinite sequence of its derivatives and itself vanish at some point of the domain. Here some result are generalizations of those in [4].     

Oscillation theorems for second-order nonlinear delay difference equations

By means of Riccati transformation technique, we establish some new oscillation criteria for second-order nonlinear delay difference equation $$\Delta (p_n (\Delta x_n )^\gamma ) + q_n f(x_{n - \sigma } ) = 0,\;\;\;\;n = 0,1,2,...,$$ when $\sum\limits_{n = 0}^\infty {\left( {\frac{1}{{Pn}}} \right)^{\frac{1}{\gamma }} = \infty }$ . When $\sum\limits_{n = 0}^\infty {\left( {\frac{1}{{Pn}}} \right)^{\frac{1}{\gamma }} < \infty }$ we present some sufficient conditions which guarantee that, every solution oscillates or converges to zero. When $\sum\limits_{n = 0}^\infty {\left( {\frac{1}{{Pn}}} \right)^{\frac{1}{\gamma }} = \infty }$ holds, our results do not require the nonlinearity to be nondecreasing and are thus applicable to new classes of equations to which most previously known results are not.     

The Krzyz Problem and Polynomials with Zeros on the Unit Circle

Let $ \Pi_{n,M} $ be the class of all polynomials $ p(z) = \sum _{0}^{n} a_{k}z^{k} $ of degree n which have all their zeros on the unit circle $ |z| = 1$ , and satisfy $ M = \max _{|z| = 1}|\,p(z)| $ . Let $ \mu _{k,n} = \sup _{p\in \Pi _{n,M}} |a_{k}| $ . Saff and Sheil-Small asked for the value of $\overline {\lim }_{n\rightarrow \infty }\mu _{k,n} $ . We find an equivalence between this problem and the Krzyz problem on the coefficients of bounded non-vanishing functions. As a result we compute $$ \overline {\lim }_{n\rightarrow \infty }\mu _{k,n} = {{M} \over {e}}\quad {\rm for}\ k = 1,2,3,4,5.$$ We also obtain some bounds for polynomials with zeros on the unit circle. These are related to a problem of Hayman.     

Numerical Characteristics of a Random Variable Related to the Engel Expansions of Real Numbers

Ukrainian Mathematical Journal - It is known that any number x ∈ (0; 1] ≡ Ω has a unique Engel expansion $$ x=\sum \limits_{n=1}^{\infty}\frac{1}{\left({p}_1(x)+1\right)\dots...     

Limit theorems for counting variables based on records and extremes

Hsu and Robbins (Proc. Nat. Acad. Sci. USA 33, 25–31, 1947) introduced the concept of complete convergence as a complement to the Kolmogorov strong law, in that they proved that \( {\sum }_{n=1}^{\infty } P(|S_{n}|>n\varepsilon )<\infty \) provided the mean of the summands is zero and that the variance is finite. Later, Erd?s proved the necessity. Heyde (J. Appl. Probab. 12, 173–175, 1975) proved that, under the same conditions, \(\lim _{\varepsilon \searrow 0} \varepsilon ^{2}{\sum }_{n=1}^{\infty } P(| S_{n}| \geq n\varepsilon )=EX^{2}\), thereby opening an area of research which has been called precise asymptotics. Both results above have been extended and generalized in various directions. Some time ago, Kao proved a pointwise version of Heyde’s result, viz., for the counting process \(N(\varepsilon ) ={\sum }_{n=1}^{\infty }1\hspace *{-1.0mm}\text {{I}} \{|S_{n}|>n\varepsilon \}\), he showed that \(\lim _{\varepsilon \searrow 0} \varepsilon ^{2} N(\varepsilon )\overset {d}{\to } E\,X^{2}{\int }_{0}^{\infty } 1\hspace *{-1.0mm}\text {I}\{|W(u)|>u\}\,du\), where W(?) is the standard Wiener process. In this paper we prove analogs for extremes and records for i.i.d. random variables with a continuous distribution function.     

CONSTRUCTIVE PROPERTIES OF A KIND OF UNIFORMLY ALMOST PERIODIC FUNCTIONS

In this paper,we have discussed constructive properties of a kind of uniformly almost periodic functions, of which the sequence of its Fourier exponents has unique limit point at infinity. \[\begin{gathered} f(x) \sim \sum\limits_{k = - \infty }^\infty {{A_k}} {e^{i{\Lambda _k}x}} \hfill \ {\Lambda _0} = \alpha ,0 < \alpha \leqslant {\Lambda _k} < {\Lambda _{k + 1}}(k = 0,1,2,...) \hfill \ \mathop {\lim }\limits_{k \to \infty } {\Lambda _k} = \infty ,{\Lambda _k} = - {\Lambda _k} \hfill \ |{\Lambda _k}| + |{\Lambda _{ - k}}| > 0{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} (k \ne 0) \hfill \\ \end{gathered} \] Analogons to the approximation theory of periodic functioiis, we get some theorems similar to the Jackson theorem, Bernstein theorem and Zygmund theorem of periodio functions.     

Complete asymptotic expansions associated with Epstein zeta-functions

Let Q(u,v)=|u+vz|² be a positive-definite quadratic form with a complex parameter z=x+iy in the upper-half plane. The Epstein zeta-function attached to Q is initially defined by for Re s>1, where the term with m=n=0 is to be omitted. We deduce complete asymptotic expansions of as y→+∞ (Theorem 1 in Sect. 2), and of its weighted mean value (with respect to y) in the form of a Laplace-Mellin transform of (Theorem 2 in Sect. 2). Prior to the proofs of these asymptotic expansions, the meromorphic continuation of over the whole s-plane is prepared by means of Mellin-Barnes integral transformations (Proposition 1 in Sect. 3). This procedure, differs slightly from other previously known methods of the analytic continuation, gives a new alternative proof of the Fourier expansion of (Proposition 2 in Sect. 3). The use of Mellin-Barnes type of integral formulae is crucial in all aspects of the proofs; several transformation properties of hypergeometric functions are especially applied with manipulation of these integrals. Research supported in part by Grant-in-Aid for Scientific Research (No. 13640041), the Ministry of Education, Culture, Sports, Science and Technology of Japan.     

Degree-one maps between hyperbolic 3-manifolds with the same volume limit

Suppose that are degree-one maps between closed hyperbolic 3-manifolds with

Then, our main theorem, Theorem 2, shows that, for all but finitely many , is homotopic to an isometry. A special case of our argument gives a new proof of Gromov-Thurston's rigidity theorem for hyperbolic 3-manifolds without invoking any ergodic theory. An example in §3 implies that, if the degree of these maps is greater than 1, the assertion corresponding to our theorem does not hold.

     

Limiting Behavior of Infinite Products Scaled by Pisot Numbers

Journal of Fourier Analysis and Applications - For $$\theta &gt;1$$ , the infinite product $$\Gamma _{\theta }(x)=\prod _{n=0}^{\infty }\cos (\pi \theta ^{-j}x)$$ is the Fourier transform of...     

On Lipschitz continuity of solutions of hyperbolic Poisson’s equation

In this paper, we investigate solutions of the hyperbolic Poisson equation \(\Delta _{h}u(x)=\psi (x)\), where \(\psi \in L^{\infty }(\mathbb {B}^{n}, {\mathbb R}^n)\) and

$$\begin{aligned} \Delta _{h}u(x)= (1-|x|^2)^2\Delta u(x)+2(n-2)\left( 1-|x|^2\right) \sum _{i=1}^{n} x_{i} \frac{\partial u}{\partial x_{i}}(x) \end{aligned}$$

is the hyperbolic Laplace operator in the n-dimensional space \(\mathbb {R}^n\) for \(n\ge 2\). We show that if \(n\ge 3\) and \(u\in C^{2}(\mathbb {B}^{n},{\mathbb R}^n) \cap C(\overline{\mathbb {B}^{n}},{\mathbb R}^n )\) is a solution to the hyperbolic Poisson equation, then it has the representation \(u=P_{h}[\phi ]-G_{ h}[\psi ]\) provided that \(u\mid _{\mathbb {S}^{n-1}}=\phi \) and \(\int _{\mathbb {B}^{n}}(1-|x|^{2})^{n-1} |\psi (x)|\,d\tau (x)<\infty \). Here \(P_{h}\) and \(G_{h}\) denote Poisson and Green integrals with respect to \(\Delta _{h}\), respectively. Furthermore, we prove that functions of the form \(u=P_{h}[\phi ]-G_{h}[\psi ]\) are Lipschitz continuous.