BOUNDARY PROPERTIES OF ANALYTIC FUNCTIONS WITH GAP POWER SERIES

1970 ◽  
Vol 21 (2) ◽  
pp. 247-256 ◽  
Author(s):  
J. M. ANDERSON
Keyword(s):  
Power Series ◽  
Analytic Functions ◽  
Boundary Properties

Formal power series and nearly analytic functions

1991 ◽  
Vol 57 (1) ◽  
pp. 61-70 ◽  
Author(s):  
H�kan Hedenmalm
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Analytic Functions ◽  
Formal Power

Some boundary properties of analytic functions

1954 ◽  
Vol 61 (1) ◽  
pp. 186-199 ◽  
Author(s):  
F. Bagemihl ◽  
W. Seidel
Keyword(s):  
Analytic Functions ◽  
Boundary Properties

scholarly journals Free Function Theory Through Matrix Invariants

2017 ◽  
Vol 69 (02) ◽  
pp. 408-433 ◽  
Author(s):  
Igor Klep ◽  
Špela Špenko
Keyword(s):  
Power Series ◽  
Analytic Functions ◽  
Function Theory ◽  
Rational Functions ◽  
Generalized Polynomials ◽  
Direct Sums ◽  
Real Analytic Functions ◽  
Power Series Expansions ◽  
Noncommutative Polynomials ◽  
Free Function

Abstract This paper concerns free function theory. Freemaps are free analogs of analytic functions in several complex variables and are defined in terms of freely noncommuting variables. A function of g noncommuting variables is a function on g-tuples of square matrices of all sizes that respects direct sums and simultaneous conjugation. Examples of such maps include noncommutative polynomials, noncommutative rational functions, and convergent noncommutative power series. In sharp contrast to the existing literature in free analysis, this article investigates free maps with involution, free analogs of real analytic functions. To get a grip on these, techniques and tools from invariant theory are developed and applied to free analysis. Here is a sample of the results obtained. A characterization of polynomial free maps via properties of their finite-dimensional slices is presented and then used to establish power series expansions for analytic free maps about scalar and non-scalar points; the latter are series of generalized polynomials for which an invarianttheoretic characterization is given. Furthermore, an inverse and implicit function theorem for free maps with involution is obtained. Finally, with a selection of carefully chosen examples it is shown that free maps with involution do not exhibit strong rigidity properties enjoyed by their involutionfree counterparts.

s-power series: an alternative to Poisson expansions for representing analytic functions

2005 ◽  
Vol 22 (2) ◽  
pp. 103-119 ◽  
Author(s):  
J. Sánchez-Reyes ◽  
J.M. Chacón
Keyword(s):  
Power Series ◽  
Analytic Functions

scholarly journals WIMAN’S TYPE INEQUALITY FOR SOME DOUBLE POWER SERIES

2021 ◽  
Vol 9 (1) ◽  
pp. 56-63
Author(s):  
O. Skaskiv ◽  
A. Kuryliak
Keyword(s):  
Power Series ◽  
Analytic Functions ◽  
Type Inequality ◽  
Double Power Series

By $\mathcal{A}^2$ denote the class of analytic functions of the formBy $\mathcal{A}^2$ denote the class of analytic functions of the form$f(z)=\sum_{n+m=0}^{+\infty}a_{nm}z_1^nz_2^m,$with {the} domain of convergence $\mathbb{T}=\{z=(z_1,z_2)\in\mathbb C^2\colon|z_1|<1,\ |z_2|<+\infty\}=\mathbb{D}\times\mathbb{C}$ and$\frac{\partial}{\partial z_2}f(z_1,z_2)\not\equiv0$ in $\mathbb{T}.$ In this paper we prove some analogue of Wiman's inequalityfor analytic functions $f\in\mathcal{A}^2$. Let a function $h\colon \mathbb R^2_+\to \mathbb R_+$ be such that$h$ is nondecreasing with respect to each variables and $h(r)\geq 10$ for all $r\in T:=(0,1)\times (0,+\infty)$and $\iint_{\Delta_\varepsilon}\frac{h(r)dr_1dr_2}{(1-r_1)r_2}=+\infty$ for some $\varepsilon\in(0,1)$, where $\Delta_{\varepsilon}=\{(t_1, t_2)\in T\colon t_1>\varepsilon,\ t_2> \varepsilon\}$.We say that $E\subset T$ is a set of asymptotically  finite $h$-measure on\ ${T}$if $\nu_{h}(E){:=}\iint\limits_{E\cap\Delta_{\varepsilon}}\frac{h(r)dr_1dr_2}{(1-r_1)r_2}<+\infty$ for some $\varepsilon>0$. For $r=(r_1,r_2)\in T$ and a function $f\in\mathcal{A}^2$ denote\begin{gather*}M_f(r)=\max \{|f(z)|\colon  |z_1|\leq r_1,|z_2|\leq r_2\},\\mu_f(r)=\max\{|a_{nm}|r_1^{n} r_2^{m}\colon(n,m)\in{\mathbb{Z}}_+^2\}.\end{gather*}We prove the following theorem:{\sl Let $f\in\mathcal{A}^2$. For every $\delta>0$ there exists a set $E=E(\delta,f)$ of asymptotically  finite $h$-measure on\ ${T}$ such that for all $r\in (T\cap\Delta_{\varepsilon})\backslash E$ we have \begin{equation*} M_f(r)\leq\frac{h^{3/2}(r)\mu_f(r)}{(1-r_1)^{1+\delta}}\ln^{1+\delta} \Bigl(\frac{h(r)\mu_f(r)}{1-r_1}\Bigl)\cdot\ln^{1/2+\delta}\frac{er_2}{\varepsilon}. \end{equation*}}

scholarly journals Boundary Properties of Analytic Functions

1967 ◽  
Vol 17 (5) ◽  
pp. 407-419
Author(s):  
J. McMillan
Keyword(s):  
Analytic Functions ◽  
Boundary Properties

scholarly journals Hankel Determinants for New Subclasses of Analytic Functions Related to a Shell Shaped Region

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 1041 ◽  
Author(s):  
Gangadharan Murugusundaramoorthy ◽  
Teodor Bulboacă
Keyword(s):  
Power Series ◽  
Upper Bound ◽  
Analytic Functions ◽  
Series Expansions ◽  
Hankel Determinant ◽  
Hankel Determinants ◽  
Power Series Expansions

Using the operator L c a defined by Carlson and Shaffer, we defined a new subclass of analytic functions ML c a ( λ ; ψ ) defined by a subordination relation to the shell shaped function ψ ( z ) = z + 1 + z 2 . We determined estimate bounds of the four coefficients of the power series expansions, we gave upper bound for the Fekete–SzegőSzegő functional and for the Hankel determinant of order two for f ∈ ML c a ( λ ; ψ ) .

scholarly journals Spectral Properties of Nonhomogenous Differential Equations with Spectral Parameter in the Boundary Condition

2014 ◽  
Vol 22 (2) ◽  
pp. 109-120
Author(s):  
Özkan Karaman
Keyword(s):  
Boundary Value Problem ◽  
Spectral Parameter ◽  
Analytic Functions ◽  
Spectral Properties ◽  
Boundary Value ◽  
Uniqueness Theorems ◽  
Spectral Singularities ◽  
Boundary Properties ◽  
Boundary Uniqueness ◽  
Complex Valued

AbstractIn this paper, using the boundary properties of the analytic functions we investigate the structure of the discrete spectrum of the boundary value problem (0.1)$$\matrix{\hfill {iy_1^\prime + q_1 \left(x \right)y_2 - \lambda y_1 = \varphi _1 \left(x \right)\;\;} & \hfill {} \cr \hfill {- iy_2^\prime + q_2 \left(x \right)y_1 - \lambda y_2 = \varphi _2 \left(x \right),} & \hfill {x \in R_ + } \cr }$$ and the condition (0.2)$$\left({a_1 \lambda + b_1 } \right)y_2 \left({0,\lambda } \right) - \left({a_2 \lambda + b_2 } \right)y_1 \left({0,\lambda } \right) = 0$$ where q1,q2, φ1, φ2 are complex valued functions, ak ≠ 0, bk ≠ 0, k = 1, 2 are complex constants and λ is a spectral parameter. In this article, we investigate the spectral singularities and eigenvalues of (0.1), (0.2) using the boundary uniqueness theorems of analytic functions. In particular, we prove that the boundary value problem (0.1), (0.2) has a finite number of spectral singularities and eigenvalues with finite multiplicities under the conditions, $$\matrix{{\mathop {\sup }\limits_{x \in R_ + } \left[ {\left| {\varphi _k \left(x \right)} \right|\exp \left({\varepsilon x^\delta } \right)} \right] < \infty ,\;\;\;k = 1.2} \hfill \cr {\mathop {\sup }\limits_{x \in R_ + } \left[ {\left| {q_k \left(x \right)} \right|\exp \left({\varepsilon x^\delta } \right)} \right] < \infty ,\;\;\;k = 1.2} \hfill \cr }$$ for some ε > 0, ${1 \over 2} < \delta < 1$

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