Trigonometric Convexity for the Multidimensional Indicator after Ivanov

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).