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Дни анализа в Сириусе
27 октября 2022 г. 16:50–17:30, Сочи
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Trigonometric Convexity for the Multidimensional Indicator after Ivanov
A. J. Mkrtchyanab a Institute of Mathematics and Computer Science, Siberian Federal University, Krasnoyarsk
b Institute of Mathematics, National Academy of Sciences of Armenia, Yerevan
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Количество просмотров: |
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Аннотация:
The concept of indicator is well-known for analytic functions in one complex variable.
Multidimensional indicator after Ivanov is a generalization of that concept for analytic functions in several complex variables.
We state the trigonometric convexity for n-dimensional indicator after Ivanov [1].
Definition 1.
Denote by $\Delta_{\alpha_j}\subset \mathbb{C}$ the open sector determined by the angle $ 0<\alpha_j<\pi/2$ as follows:
\begin{equation*}
\Delta_{\alpha_j}=\left\lbrace z_j\in\mathbb{C}\setminus\lbrace 0\rbrace\colon \left|\arg\left(z_j\right)\right|<\alpha_j\right\rbrace.
\end{equation*}
Definition 2.
Recall that a function $f$ is of finite exponential type $\left(h_1,\dots, h_n\right)$ in
$\Delta_{\alpha_1}\times\dots\times \Delta_{\alpha_n}$ if
for any $\varepsilon>0$ there exists a constant $k_{\varepsilon}\geq 0$ such that
\begin{equation*}
\left|f\left(z_1,\dots,z_n\right)\right|\leq k_{\varepsilon}e^{(h_1+\varepsilon)\left|z_1\right|+\dots+\left(h_n+\varepsilon\right)\left|z_n\right|},\quad \text{ for all } z_j\in \Delta_j,\; 1\leq j\leq n.
\end{equation*}
In definition 2 we tacitly assume that $h_1,\dots,h_n\geq 0$.
Here we assume $h_1,\dots,h_n\geq 0$.
Definition 3.
Denote by $Exp\left(\alpha_1,\dots,\alpha_n\right)$ the class of functions $f$ that are analytic and of finite exponential type in $\Delta_1\times \dots\times\Delta_n.$
Definition 4.
Namely, Ivanov introduced the following set:
\begin{align*}
T_{f}\left(\vec \theta\right)=\{&
\vec \nu\in \mathbb R^n:
\ln{\left|f\left(\vec re^{i\vec \theta}\right)\right|}\leq \nu_1 r_1+...+\nu_nr_n+C_{\vec \nu,\vec \theta},
\text{ for all }\vec r\in \mathbb R^n_+
\},
\end{align*}
here $\vec re^{i\vec \theta}$ is the vector $(r_1e^{i\theta_1},...,r_ne^{i\theta_n}).$
The set $T_{f}\left(\vec \theta\right)$ implicitly reflects the notion of an indicator of an entire function.
Theorem.
Let a function $f \in Exp\left(\alpha_1,\dots,\alpha_n\right)$ and
the numbers $A^+_1,A^-_1 \dots,A^+_n,A^-_n$ satisfy
\begin{align*}
\left(A^{l_1}_1, \dots,A^{l_n}_n\right) \in \overline T_f\left({l_1}\alpha_1, \dots,{l_n}\alpha_n\right),
\end{align*}
where $l_j=\pm, \;$ $\; j=1,\dots,n.$
Then
\begin{equation*}
\left(C_1,\dots,C_n\right)\in \overline T_f\left(\theta_1,\dots,\theta_n\right),
\end{equation*}
where the constants $C_1,\dots,C_n$ determine from the following formulas:
\begin{align*}
C_j\sin\left(2\alpha_j\right)=A^+_j\sin\left(\theta_j+\alpha_j\right)+A^-_j\sin\left(\alpha_j-\theta_j\right), \;\; j=1,...,n.
\end{align*}
Remark.
Theorem is sharp: that is, there exists a function $f$ for whom: the assumptions of theorem are satisfied, and the inequality is an equality.
This is a joint work with Armen Vagharshakyan.
References
[1] A. Mkrtchyan, A. Vagharshakyan,
Trigonometric convexity for the multidimensional indicator after Ivanov.
arXiv:2205.02585 (2022).
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