Video library: A. J. Mkrtchyan, Trigonometric Convexity for the Multidimensional Indicator after Ivanov

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October 27, 2022 16:50

Trigonometric Convexity for the Multidimensional Indicator after Ivanov A. J. Mkrtchyan^ab ^a Institute of Mathematics and Computer Science, Siberian Federal University, Krasnoyarsk ^b Institute of Mathematics, National Academy of Sciences of Armenia, Yerevan
Abstract: 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).