Trigonometric convexity of the multidimensional indicator

Abstract

The notion of indicator of an analytic function, that describes the function’s growth along rays, was introduced by Phragmen and Lindelöf. Trigonometric convexity is a defining property of the indicator. For multivariate cases, an analogous property of trigonometric convexity was not known so far. We prove the property of trigonometric convexity for the indicator of multivariate analytic functions, introduced by Ivanov. The results that we obtain are sharp. Derivation of a multidimensional analogue of the inverse Fourier transform in a sector and obtaining estimates on its decay is an important step of our proof.

1 Introduction

1.1 Main results

We review the notion of indicator for analytic functions in one complex variable (see Section 1.2) and its multidimensional analogues for functions in several complex variables (see Section 1.3); in particular, we treat the notion of multidimensional indicator after Ivanov. We also consider a multidimensional analogue of the inverse Fourier transform in a sector and obtain estimates on its decay (see Section 2).

We remind that the indicator is a characteristic of a function, that describes a function’s growth on directions (axes starting from the origin). In particular, the indicator and its property of trigonometric convexity plays an important role in certain methods of analytic continuation, and allows to avoid some extra hypotheses and cumbersome formulations in theorems. This claim is highlighted by Polya’s theorem [22], stated later in this article, and the theorems of Lindelöf, Arakeyan [1], Carlson [5, 6], and others. All of these results are concerned with analytic continuation of power series in one complex variable by means of interpolating its coefficients by entire functions.

Meanwhile, as far as power series in several complex variables are concerned, many of the corresponding questions are still open. We remark that the latter questions are motivated by applications in mathematical physics, in particular by applications in thermodynamics [19, 27]. They are also related to multidimensional hypergeometric series [17, 24].

As explained in Remark 1.8, our main Theorem 1.1 establishes an analogue of trigonometric convexity for the multidimensional indicator after Ivanov.

We remark that derivation of a multidimensional analogue of the inverse Fourier transform in a sector and obtaining an estimate on its decay is an important step of our proof. For a general survey on decay of the Fourier transform, we refer the reader to the book of Iosevich and Liflyand [9]. We preface the formulation of Theorem 1.1 by the following definitions:

Definition 1.1 Denote by $\Delta _{\alpha _j}\subset \mathbb {C}$ the open sector determined by the angle $ 0<\alpha _j<\pi /2$ as follows:

$$ \begin{align*} \Delta_{\alpha_j}=\lbrace z_j\in\mathbb{C}\setminus\lbrace 0\rbrace\colon \left|\arg\left(z_j\right)\right|<\alpha_j\rbrace. \end{align*} $$

Definition 1.2 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

(1.1) $$ \begin{align} &\left|f\left(z_1,\dots,z_n\right)\right|\leq k_{\varepsilon}e^{(h_1+\varepsilon)\left|z_1\right|+\dots+\left(h_n+\varepsilon\right)\left|z_n\right|},\\&\nonumber\text{ for all } z_j\in \Delta_{\alpha_j},\; 1\leq j\leq n. \end{align} $$

Note that, in this article, we tacitly assume that $h_1,\dots ,h_n\geq 0$ .

Definition 1.3 Denote by $Exp\left (\alpha _1,\dots ,\alpha _n\right )$ the class of functions f that are analytic and are of finite exponential type in $\Delta _{\alpha _1}\times \dots \times \Delta _{\alpha _n}.$

Definition 1.4 Following Ivanov [10], for $f \in Exp\left (\alpha _1,\dots ,\alpha _n\right )$ , denote by $T_{f}\left (\vec \theta \right )$ the following:

(1.2) $$ \begin{align} T_{f}\left(\vec \theta\right)= \{ & \vec \nu\in \mathbb R^n: \ln{\left|f\left(\vec re^{i\vec \theta}\right)\right|}\leq \nu_1 r_1+\cdots+\nu_nr_n+C_{\vec \nu,\vec \theta}, \\ \nonumber & \text{ for some } C_{\vec \nu,\vec\theta}, \text{ for all }\vec r\in \mathbb R^n_+ \}\,; \end{align} $$

here, $\vec re^{i\vec \theta }$ denotes the vector $\left (r_1e^{i\theta _1},\ldots ,r_ne^{i\theta _n}\right ).$

Theorem 1.1 Let a function $f \in Exp\left (\alpha _1,\alpha _2\right )$ and the numbers $A^+_1,\ A^+_2,\ A^-_1,\ A^-_2$ satisfy

$$ \begin{align*}& \left(A^+_1,A^+_2\right) \in \overline {T_f\left(\alpha_1,\alpha_2\right)}, \\& \left(A^-_1,A^-_2\right) \in \overline {T_f\left(-\alpha_1,-\alpha_2\right)} , \\& \left(A^+_1,A^-_2\right) \in \overline {T_f\left(\alpha_1,-\alpha_2\right)}, \\& \left(A^-_1,A^+_2\right) \in \overline {T_f\left(-\alpha_1,\alpha_2\right)}. \end{align*} $$

Then we have

$$ \begin{align*} \left(C_1,C_2\right)\in \overline {T_f\left(\theta_1,\theta_2\right)}, \end{align*} $$

where the constants $C_1,C_2$ are determined by the following formulas:

(1.3) $$ \begin{align} C_1\sin\left(2\alpha_1\right)=A^+_1\sin\left(\theta_1+\alpha_1\right)+A^-_1\sin\left(\alpha_1-\theta_1\right), \end{align} $$

(1.4) $$ \begin{align} C_2\sin\left(2\alpha_2\right)=A^+_2\sin\left(\theta_2+\alpha_2\right)+A^-_2\sin\left(\alpha_2-\theta_2\right). \end{align} $$

Theorem 1.1 can be paraphrased as follows:

Remark 1.2. Let a function $f \in Exp\left (\alpha _1,\alpha _2\right )$ and the numbers $A^+_1,\ A^+_2,\ A^-_1,\ A^-_2$ satisfy

(1.5) $$ \begin{align} \nonumber & \left|f\left(z_1,z_2\right)\right|\leq k_{\varepsilon}e^{(A^+_1+\varepsilon)\left|z_1\right|+\left(A^+_2+\varepsilon\right)\left|z_2\right|},\quad \text{for }\arg(z_1)=\alpha_1,\;\arg\left(z_2\right)=\alpha_2, \\\nonumber & \left|f\left(z_1,z_2\right)\right|\leq k_{\varepsilon}e^{(A^+_1+\varepsilon)\left|z_1\right|+\left(A^-_2+\varepsilon\right)\left|z_2\right|},\quad \text{for }\arg(z_1)=\alpha_1,\;\arg\left(z_2\right)=-\alpha_2, \\\nonumber & \left|f\left(z_1,z_2\right)\right|\leq k_{\varepsilon}e^{(A^-_1+\varepsilon)\left|z_1\right|+\left(A^+_2+\varepsilon\right)\left|z_2\right|},\quad \text{for }\arg(z_1)=-\alpha_1,\;\arg\left(z_2\right)=\alpha_2, \\ & \left|f\left(z_1,z_2\right)\right|\leq k_{\varepsilon}e^{(A^-_1+\varepsilon)\left|z_1\right|+\left(A^-_2+\varepsilon\right)\left|z_2\right|},\quad \text{for }\arg(z_1)=-\alpha_1,\;\arg\left(z_2\right)=-\alpha_2, \end{align} $$

where by the value of the function f on the mentioned rays we mean f’s non-tangential limit (see [25, Remark 2.1] for details). Then, for any $\varepsilon>0$ , there exists a constant $k^{\prime }_{\varepsilon }>0$ such that

(1.6) $$ \begin{align} \left|f\left(z_1,z_2\right)\right|\leq k^{\prime}_{\varepsilon}e^{(C_1+\varepsilon)\left|z_1\right|+\left(C_2+\varepsilon\right)\left|z_2\right|},\quad \text{for } \arg\left(z_1\right)=\theta_1,\;\arg\left(z_2\right)=\theta_2, \end{align} $$

where the constants $C_1,C_2$ are determined by the following formulas:

$$ \begin{align*} C_1\sin\left(2\alpha_1\right)=A^+_1\sin\left(\theta_1+\alpha_1\right)+A^-_1\sin\left(\alpha_1-\theta_1\right), \\ C_2\sin\left(2\alpha_2\right)=A^+_2\sin\left(\theta_2+\alpha_2\right)+A^-_2\sin\left(\alpha_2-\theta_2\right). \end{align*} $$

In Section 3, we derive Theorem 1.1 from Theorem 1.3 that serves as a two-dimensional analogue of the Fourier inversion formula (see [25, Theorem 1.2] and [7]) for functions of exponential type in a sector. We preface the formulation of Theorem 1.3 by the following definitions:

Definition 1.5 Let $f \in Exp\left (\alpha _1,\alpha _2\right )$ , and let the numbers $A^+_1,A^+_2,A^-_1,A^-_2$ satisfy (1.5). Define the function m as f’s two-dimensional concatenated Laplace transform. Namely, the domain of function m is the Cartesian product $\Omega _1\times \Omega _2,$ where

$$ \begin{align*}& \Omega_1=\Omega_1^+\cup\Omega_1^-, \\\nonumber & \Omega_2=\Omega_2^+\cup\Omega_2^-, \end{align*} $$

and in turn

(1.7) $$ \begin{align} & \Omega_1^+=\left\lbrace \omega_1 \colon Re\left(\omega_1 e^{i\alpha_1}\right)<-A^+_1\right\rbrace, \\\nonumber & \Omega_1^-=\left\lbrace \omega_1 \colon Re\left(\omega_1 e^{-i\alpha_1}\right)<-A^-_1\right\rbrace, \\\nonumber & \Omega_2^+=\left\lbrace \omega_2 \colon Re\left(\omega_2 e^{i\alpha_2}\right)<-A^+_2\right\rbrace, \\\nonumber & \Omega_2^-=\left\lbrace \omega_2 \colon Re\left(\omega_2 e^{-i\alpha_2}\right)<-A^-_2\right\rbrace \end{align} $$

(see Figure 1). The function m is defined on $\Omega _1\times \Omega _2$ by the following four formulas:

(1.8) $$ \begin{align} & m\left(\omega_1,\omega_2\right)=\frac{-1}{4\pi^2} \int_{e^{i\alpha_2}[0,+\infty)}\int_{e^{i\alpha_1}[0,+\infty)} f\left(\zeta_1,\zeta_2\right) e^{\omega_1\zeta_1+\omega_2\zeta_2} d\zeta_1d\zeta_2 ,\\\nonumber &\quad\quad\quad\quad \text{for } \omega_1\in\Omega_1^+ ,\; \omega_2\in\Omega_2^+, \\\nonumber & m\left(\omega_1,\omega_2\right)=\frac{-1}{4\pi^2} \int_{e^{i\alpha_2}[0,+\infty)}\int_{e^{-i\alpha_1}[0,+\infty)} f\left(\zeta_1,\zeta_2\right) e^{\omega_1\zeta_1+\omega_2\zeta_2} d\zeta_1d\zeta_2 ,\\\nonumber & \quad\quad\quad\quad\text{for } \omega_1\in\Omega_1^+ ,\; \omega_2\in\Omega_2^-, \\\nonumber & m\left(\omega_1,\omega_2\right)=\frac{-1}{4\pi^2} \int_{e^{-i\alpha_2}[0,+\infty)}\int_{e^{i\alpha_1}[0,+\infty)} f\left(\zeta_1,\zeta_2\right) e^{\omega_1\zeta_1+\omega_2\zeta_2} d\zeta_1d\zeta_2 ,\\\nonumber &\quad\quad\quad\quad \text{for } \omega_1\in\Omega_1^- ,\; \omega_2\in\Omega_2^+, \\\nonumber & m\left(\omega_1,\omega_2\right)=\frac{-1}{4\pi^2} \int_{e^{-i\alpha_2}[0,+\infty)}\int_{e^{-i\alpha_1}[0,+\infty)} f\left(\zeta_1,\zeta_2\right) e^{\omega_1\zeta_1+\omega_2\zeta_2} d\zeta_1d\zeta_2 ,\\\nonumber & \quad\quad\quad\quad\text{for } \omega_1\in\Omega_1^- ,\; \omega_2\in\Omega_2^-. \end{align} $$

Figure 1: The set $\Omega $ .

Definition 1.6 For a class of functions $Exp\left (\alpha _1,\alpha _2\right ),$ denote by $\Gamma _1$ and $\Gamma _2$ the curves given by the following parametrizations:

(1.9) $$ \begin{align} & \gamma_1\colon\mathbb{R}\rightarrow\mathbb{C},\nonumber \\ & \gamma_1(t)=p_1-ie^{i\alpha_1}|t|,\quad\text{for }t\in (-\infty,0],\nonumber \\ & \gamma_1(t)=p_1+ie^{-i\alpha_1}|t|,\quad\text{for }t\in (0,+\infty] \end{align} $$

(see Figure 2), and the real number $p_1$ appearing in parametrization (1.9) is chosen in such a way that it satisfies inequality

(1.10) $$ \begin{align} p_1\cos(\alpha_1)<-h_1 \end{align} $$

and correspondingly the curve $\Gamma _2$ is parameterized by

(1.11) $$ \begin{align} & \gamma_2\colon\mathbb{R}\rightarrow\mathbb{C},\nonumber \\ & \gamma_2(t)=p_2-ie^{i\alpha_2}|t|,\quad\text{for }t\in (-\infty,0],\nonumber \\ & \gamma_2(t)=p_2+ie^{-i\alpha_2}|t|,\quad\text{for }t\in (0,+\infty], \end{align} $$

and the real number $p_2$ appearing in parametrization (1.9) is chosen in such a way that it satisfies inequality

(1.12) $$ \begin{align} p_2\cos(\alpha_2)<-h_2. \end{align} $$

Figure 2: The curve $\Gamma _1$ .

Theorem 1.3 For a function $f \in Exp\left (\alpha _1,\alpha _2\right ) $ , the following Fourier inversion formula holds:

(1.13) $$ \begin{align} f\left(z_1,z_2\right)=\int_{\Gamma_1}\int_{\Gamma_2}m\left(\omega_1,\omega_2\right) e^{-\omega_1 z_1-\omega_2z_2} d\omega_2d\omega,\quad \text{for }z_1\in \Delta_{\alpha_1},\;z_2\in \Delta_{\alpha_2}. \end{align} $$

Theorem 1.3 is proved in Section 2.

We complement Theorem 1.1 by the following remarks:

Remark 1.4. Theorem 1.1 is stated for functions of two complex variables. The corresponding theorem for functions of n complex variables also holds.

Remark 1.5. Theorem 1.1 is sharp, that is, there exists a function f for whom the assumptions of Theorem 1.1 are satisfied, and the inequality (1.6) is an equality.

Remark 1.5 is proved in Section 4.

Remark 1.6. Integral representation (1.13) in Theorem 1.3 could be compared to multivariate integrals of Mellin–Barnes type [8, 20, 26].

1.2 Indicators of functions in one complex variable

We say that an entire function $f\colon \mathbb {C}\rightarrow \mathbb {C}$ is of exponential type if

(1.14) $$ \begin{align} \limsup_{z \to \infty} \frac{\ln^+|f(z)|}{|z|}<+\infty. \end{align} $$

The notion of indicator of an entire function of exponential type was introduced by Phragmen and Lindeleph [3, 21] as follows:

(1.15) $$ \begin{align} h_{f}(\alpha):=\limsup_{r \to \infty} \frac{\ln\left|f\left(re^{i\alpha}\right)\right|}{r}, \;\; \alpha \in \mathbb{R}. \end{align} $$

The indicator $h_f(\alpha )$ describes the growth of function f along the ray $e^{i\alpha }[0,+\infty ).$ It follows from the definition (1.15) that the indicator $h_{f}(\alpha )$ is a real-valued $2\pi $ -periodic function. It also follows that the indicator of the product of two functions does not exceed the sum of the indicators of the factors,

$$ \begin{align*} h_{fg}(\alpha)\leq h_{f}(\alpha)+h_{g}(\alpha), \end{align*} $$

and that the indicator of the sum of two functions does not exceed the larger of the two indicators,

$$ \begin{align*} h_{f+g}(\alpha)\leq \max\left(h_{f}(\alpha),h_{g}(\alpha)\right). \end{align*} $$

One of the main properties of the indicator $h_{f}(\alpha )$ is its trigonometric convexity [4, 15]: if $\alpha _1<\alpha <\alpha _2$ and $\alpha _2-\alpha _1<\pi ,$ then the following inequality holds:

(1.16) $$ \begin{align} h_{f}(\alpha)\sin{(\alpha_2-\alpha_1)}\leq h_{f}(\alpha_1)\sin{(\alpha_2-\alpha)}+h_{f}(\alpha_2)\sin{(\alpha-\alpha_1)}. \end{align} $$

The following property follows from the trigonometric convexity (1.16) [15]: if the indicator is bounded on an open interval, then it is continuous. The latter claim does not hold for a closed interval.

The notion of indicator is known to be an important tool in some methods regarding to finding analytic continuation [1, 5, 6, 22]. In particular, the indicator appears in problems relating to analytic continuation of power series via interpolation of coefficients. It also plays a role in problems relating to localization of singularities of power series [2].

Let

$$ \begin{align*} f(z)=\sum_{k=0}^\infty {\frac{a_k}{k!}z^k} \end{align*} $$

be the power-series representation of the entire function $f.$ Consider its Borel transform defined by the Laurent series in the following way:

(1.17) $$ \begin{align} g(\omega)=\sum_{k=0}^\infty {a_k \omega^{-k-1}}. \end{align} $$

The interrelation between the set of singularities of g and the indicator of f is described by Polya’s theorem [14, 23].

Theorem 1.7 (Polya)

Let f be an entire function of exponential type. Denote by $K\subset \mathbb {C}$ the convex set whose support function

$$ \begin{align*} k(\theta)=\sup_{\omega\in K}Re\left(\omega e^{-i\theta}\right) \end{align*} $$

is determined by f’s indicator as follows:

(1.18) $$ \begin{align} k(-\theta)=h_f(\theta). \end{align} $$

Then f can be restored by

(1.19) $$ \begin{align} f(z)=\frac{1}{2\pi i}\int_{\Gamma}g(\omega)e^{z\omega}d\omega, \end{align} $$

where $\Gamma $ is a closed contour containing the set $K,$ and g (1.17) is the Borel transform of $f.$ Additionally, K is the smallest convex set such that g is analytic in $\mathbb {C}\setminus K.$

1.3 Indicators of functions in several complex variables

The works of Ronkin [23], Lelon [13], Levin [15], Ivanov [10], Kiselman [11], and others are dedicated to exploring multidimensional analogues of the indicator function. We say that $f\left (\vec z\right )=f(z_1,\ldots ,z_n)$ is an n-valued entire function of exponential type if f is holomorphic in $\mathbb {C}^n, $ i.e.,

$$ \begin{align*} f\left(\vec z\right) \in \mathcal O (\mathbb C^n), \end{align*} $$

and if there exist constants $k, \sigma _1,\ldots ,\sigma _n$ such that

(1.20) $$ \begin{align} \left|f\left(\vec z\right)\right|\leq ke^{\sigma_1|z_1|+\cdots+\sigma_n|z_n|},\quad \text{for }\vec z\in\mathbb{C}^n. \end{align} $$

In the case of several complex variables, different characteristics of an analytic function’s growth in directions have been introduced. For example, introduce the radial indicator of the function f as follows [13, 23]:

$$ \begin{align*}L_r(z,f)=\limsup_{t \to \infty}\frac{\ln|f(tz)|}{t}, \;\; z\in\mathbb{C}^n\end{align*} $$

and correspondingly introduce regularization of the function $L(z,f)$ as follows:

(1.21) $$ \begin{align} L^{\ast}(z,f)=\limsup_{z' \to z}L(z,f). \end{align} $$

We will refer to $L^{\ast }(z,f)$ as regularized radial indicator of the function f. Just as in the one-dimensional case, the function $L^{\ast }(z,f)$ is semi-continuous from above. Note that the regularized radial indicator is a plurisubharmonic function in $\mathbb {C}^n.$ Consequently, more information about its properties may be found in works related to plurisubharmonic functions in potential theory [11] and their different generalizations [12, 16].

Another characteristic of an analytic functions growth was introduced by Ivanov; namely Ivanov [10] has introduced the set $T_{f}\left (\vec \theta \right )$ as follows:

(1.22) $$ \begin{align} T_{f}\left(\vec \theta\right)= \{& \vec \nu\in \mathbb R^n: \ln{\left|f\left(\vec re^{i\vec \theta}\right)\right|}\leq \nu_1 r_1+\cdots+\nu_nr_n+C_{\vec \nu,\vec \theta}, \\\nonumber & \text{ for some } C_{\vec \nu,\vec\theta}, \text{ for all }\vec r\in \mathbb R^n_+ \}; \end{align} $$

here, $\vec re^{i\vec \theta }$ is the vector $(r_1e^{i\theta _1},\ldots ,r_ne^{i\theta _n}).$ The set $T_{f}\left (\vec \theta \right )$ implicitly reflects the notion of an indicator of an entire function.

For example, for the closure of the set $T_\varphi \left (\theta _1,\theta _2\right )$ defined for the function $\varphi (\zeta _1,\zeta _2)=\cos (\zeta _1\zeta _2)^{\frac {1}{2}}$ of exponential type, we have the following one [18]:

$$ \begin{align*} \overline{{T}_{\varphi}\left(\theta_1,\theta_2\right)}=\left\lbrace \vec \nu\in\mathbb R^2: \nu_1\nu_2=\frac{1}{4}\left|\sin\left(\frac{\theta_1+\theta_2}{2}\right)\right|{}^2,\;\;\nu_1\geq0, \;\; \nu_2\geq0 \right\rbrace. \end{align*} $$

There exist many more analogues of indicator for functions in several complex variables. However, for none of them, a property resembling trigonometric convexity was obtained.

Remark 1.8. Following our main result (Theorem 1.1), we now formulate trigonometric convexity for multidimensional indicator after Ivanov.

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

$$ \begin{align*} \left(C_1,\dots,C_n\right)\in \overline {T_f\left(\theta_1,\dots,\theta_n\right)}, \end{align*} $$

where the constants $C_1,\dots ,C_n$ are determined from the following formulas:

$$ \begin{align*} C_j\sin\left(2\alpha_j\right)=A^+_j\sin\left(\theta_j+\alpha_j\right)+A^-_j\sin\left(\alpha_j-\theta_j\right), \;\; j=1,\ldots,n. \end{align*} $$

2 Two-dimensional sectorial Fourier inversion formula

2.1 Two-dimensional concatenated Fourier transform

Due to (1.7) for any pair of complex numbers $\left (\omega _1,\omega _2\right )\subset \Omega _1^+\times \Omega _2^+$ , we can pick $\varepsilon>0$ so that the following inequalities are satisfied:

(2.1) $$ \begin{align} & Re\left(\omega_1 e^{i\alpha_1}\right)+\varepsilon<-A^+_1, \\\nonumber & Re\left(\omega_2 e^{i\alpha_2}\right)+\varepsilon<-A^+_2. \end{align} $$

Then we have the following estimate on the function m defined by the first formula in (1.8):

(2.2) $$ \begin{align} & \left|m\left(\omega_1,\omega_2\right)\right| \overset{({1.8})}{\leq}\\&\nonumber \leq \frac{1}{4\pi^2} \int_{e^{i\alpha_1}[0,+\infty)} \int_{e^{i\alpha_2}[0,+\infty)} \left|f\left(\zeta_1,\zeta_2\right)\right| \left|e^{\omega_1\zeta_1+\omega_2\zeta_2}\right| \left|d\zeta_2\right|\left|d\zeta_1\right| \leq \\\nonumber & \overset{({1.5})}{\leq} \frac{k_{\varepsilon}}{4\pi^2} \left[ \int_{e^{i\alpha_1}[0,+\infty)} e^{ \left( A^+_1+\varepsilon+Re\left(\omega_1 e^{i\alpha_1}\right) \right) \left|\zeta_1\right| } \left|d\zeta_1\right| \right] \cdot \\\nonumber & \cdot \left[ \int_{e^{i\alpha_2}[0,+\infty)} e^{ \left( A^+_2+\varepsilon+Re\left(\omega_2e^{i\alpha_2}\right) \right)\left|\zeta_2\right| } \left|d\zeta_2\right|\right] = \\\nonumber & \overset{({2.1})}{=} \frac{k_{\varepsilon}}{4\pi^2}\cdot \frac{-1}{A^+_1+\varepsilon+Re\left(\omega e^{i\alpha_1}\right)}\cdot\frac{-1}{A^+_2+\varepsilon+Re\left(\omega_2 e^{i\alpha_2}\right)}<+\infty. \end{align} $$

Due to the estimate (2.2), the first of four double integrals in (1.8) is absolutely convergent. And that double integral determines a function that is analytic in two complex variables on the set $\Omega _1^+\times \Omega _2^+.$ The same claims are true for the other three double integrals in the definition (1.8).

We now address the formal ambiguity in the definition (1.8) of function $m,$ arising from the fact that some of the four domains that appear in (1.8) might intersect. Due to Lemma 4.1 in [25] (a lemma that uses the exponential estimate (1.1) on the function f and that is based on application of the Phragmen–Lindelöf maximum principle), the first and the second double integrals in (1.8) are equal on the intersection of their corresponding domains: $\Omega _1^+\times \Omega _2^+$ and $\Omega _1^+\times \Omega _2^-.$ Due to the estimate (2.2), Fubini’s theorem applies to each of the four double integrals in (1.8). By changing the order of integration in the first and third double integrals and referring Lemma 4.1 in [25], we see that the first and the third double integrals are equal on the intersection of their corresponding domains.

As for the first and the fourth double integrals, the intersection of their corresponding domains lies within the intersection of any two of the four domains. Hence in there, the first integral equals the second integral, and in turn, the second integral equals the fourth integral.

Consequently, any two of the four definitions of the function m in (1.8) are equivalent on the intersection of their corresponding domains. And, the function m defined by formulas (1.8) is analytic on $\Omega _1\times \Omega _2$ in each of its variables.

Remark 2.1. Thanks to the estimate (2.2), the function m is bounded on any subset of $\Omega _1^+\times \Omega _2^+$ that is bounded away from its boundary $\partial \left (\Omega _1^+\times \Omega _2^+\right ).$ Similarly, one can prove that the function m is bounded on any subset of $\Omega _1\times \Omega _2$ that is bounded away from the boundary $\partial \left (\Omega _1\times \Omega _2\right ).$

2.2 Verification of the sectorial Fourier inversion formula

Due to the condition (1.1), for any $z_2\in \Delta _{\alpha _2}$ , the Fourier inversion formula (see [25, Theorem 1.2]) applies to the function $z_1\rightarrow f\left (z_1,z_2\right )$ : that is, we have

(2.3) $$ \begin{align} f(z_1,z_2)=\int_{\Gamma_1}g\left(\omega_1,z_2\right)e^{-\omega_1 z_1}d\omega_1 ,\quad\text{for } z_2\in \Delta_{\alpha_2}, \end{align} $$

where the curve $\Gamma _1$ is defined by (1.9), and the function $\omega _1\rightarrow g\left (\omega _1,z_2\right )$ is well defined by the following formulas:

(2.4) $$ \begin{align} & g\left(\omega_1,z_2\right)=\frac{1}{2\pi i} \int_{e^{-i\alpha_1}[0,+\infty)} f\left(\zeta_1,z_2\right) e^{\omega_1 \zeta_1} d\zeta_1 ,\quad \text{for } Re\left(\omega_1 e^{-i\alpha_1}\right)<-h_1, \\\nonumber & g\left(\omega_1,z_2\right)=\frac{1}{2\pi i} \int_{e^{i\alpha_1}[0,+\infty)} f\left(\zeta_1,z_2\right) e^{\omega_1 \zeta_1} d\zeta_1 ,\quad \text{for } Re\left(\omega_1 e^{i\alpha_1}\right)<-h_1. \end{align} $$

We now derive the estimate (2.6) for function $g\left (\omega _1,z_2\right ),$ that is uniform in its first variable $\omega _1$ for $\omega _1\in \Gamma _1$ and that depends exponentially on its second variable $z_2.$

Due to inequality (1.10) imposed on the real number $p_1,$ we can pick $\varepsilon>0$ such that

(2.5) $$ \begin{align} h_1+\varepsilon+p_1\cos\left(\alpha_1\right)<0. \end{align} $$

Let $\omega _1\in \Gamma _1.$ Then, due to parameterization (1.9), we have $\omega _1=\gamma _1(t),\; t\in \mathbb {R}.$ Specifically, assume that $ t\geq 0, $ so that the second formula of the two formulas (2.4) holds for $g\left (\gamma _1(t),z_2\right ).$ We estimate

(2.6) $$ \begin{align} & \left|g\left(\gamma_1(t),z_2\right)\right| \overset{({2.4})}{\leq} \frac{1}{2\pi} \int_{e^{i\alpha_1}[0,+\infty)} \left|f\left(\zeta_1,z_2\right)\right| \cdot \left|e^{\gamma_1(t) \zeta_1}\right| \left|d\zeta_1\right|\leq \\\nonumber & \overset{({1.1})}{\leq} \frac{1}{2\pi} \int_{e^{i\alpha_1}[0,+\infty)} k_{\varepsilon}e^{\left(h_1+\varepsilon\right)\left|\zeta_1\right| +\left(h_2+\varepsilon\right)\left|z_2\right|} \cdot e^{Re\left(\gamma_1(t) \zeta_1\right)} \left|d\zeta_1\right|= \\\nonumber & = \frac{k_{\varepsilon}e^{\left(h_2+\varepsilon\right)\left|z_2\right|}}{2\pi} \int_{e^{i\alpha_1}[0,+\infty)} e^{\left(h_1+\varepsilon+Re\left(\gamma_1(t) e^{i\alpha_1}\right)\right)\left|\zeta_1\right|} \left|d\zeta_1\right|= \\\nonumber & \overset{({1.9})}{=} \frac{k_{\varepsilon}e^{\left(h_2+\varepsilon\right)\left|z_2\right|}}{2\pi} \int_{e^{i\alpha_1}[0,+\infty)} e^{\left(h_1+\varepsilon+p_1\cos\left(\alpha_1\right)\right)\left|\zeta_1\right|} \left|d\zeta_1\right|= \\\nonumber & \overset{({2.5})}{=} \left[ \frac{k_{\varepsilon}}{2\pi} \cdot \frac{-1}{h_1+\varepsilon+p_1\cos\left(\alpha_1\right)} \right] \cdot e^{\left(h_2+\varepsilon\right)\left|z_2\right|}. \end{align} $$

And, we would get the same estimate on $g\left (\gamma _1(t),z_2\right )$ if we assumed that $t<0.$ Due to the estimate (2.6), for any $-\infty <t<+\infty ,$ the Fourier inversion formula (see [25, Theorem 1.2]) applies to the function $z_2\rightarrow g\left (\gamma _1(t),z_2\right ),$ and we have

(2.7) $$ \begin{align} g\left(\gamma_1(t),z_2\right)=\int_{\Gamma_2} m\left(\gamma_1(t),\omega_2\right)e^{-\omega_2 z_2}d\omega_2 ,\quad\text{for } z_2\in \Delta_{\alpha_2}, \end{align} $$

where the curve $\Gamma _2$ is parameterized by (1.11), and for any $\infty <t<+\infty ,$ the function $\omega _2\rightarrow m\left (\gamma _1(t),\omega _2\right )$ is well defined by the formulas (1.8).

Remark 2.2. Note that due to the uniform bound (2.6), the curve $\Gamma _2,$ involved in formula (2.7), is defined by formula (1.11) in such a way that it does not depend on the choice of $\omega _1=\gamma _1(t).$

By combining the Fourier inversion formulas (2.3) and (2.7), we obtain representation (1.13) of function f as two consecutive integrals. Due to (2.2), we estimate Fubini’s theorem applies to the consecutive integrals in the representation (1.13) of function $f,$ and we can consider those consecutive integrals as a double integral.

3 Estimates for two-dimensional indicator after Ivanov

3.1 A calculation relating to Figure 3

We now justify that the length $|[0,c]|$ of the interval $[0,c]$ constructed in Figure 3 indeed equals $C_1.$ Indeed, we observe three triangles determined by their vertices $0,q_1,a;\ 0,q_1,b,$ and $0,q_1,c,$ to see that

(3.1) $$ \begin{align} & Re\left(q_1 e^{i\alpha_1}\right)=-|[0,a]|, \\\nonumber & Re\left(q_1 e^{-i\alpha_1}\right)=-|[0,b]|, \\\nonumber & Re\left(q_1 e^{i\theta_1}\right)=-|[0,c]|. \end{align} $$

Denote $x=Re\left (q_1 \right ),y=Im\left (q_1\right ).$ Then we can paraphrase (3.1) as

$$ \begin{align*}& x\cos\left(\alpha_1\right)-y\sin\left(\alpha_1\right)=-A^+_1, \\ & x\cos\left(\alpha_1\right)+y\sin\left(\alpha_1\right)=-A^-_1, \\ & x\cos\left(\theta_1\right)-y\sin\left(\theta_1\right)=-|[0,c]|, \end{align*} $$

so that

$$ \begin{align*} |[0,c]|=\frac{A^+_1}{2} \left[\frac{\sin\left(\theta_1\right)}{\sin\left(\alpha_1\right)}+\frac{\cos\left(\theta_1\right)}{\cos\left(\alpha_1\right)}\right]+\frac{A^-_1}{2}\left[\frac{\cos\left(\theta_1\right)}{\cos\left(\alpha_1\right)}-\frac{\sin\left(\theta_1\right)}{\sin\left(\alpha_1\right)}\right]. \end{align*} $$

Consequently, due to (1.3), we have $|[0,c]|=C_1.$

Figure 3: Construction of $C_1$ .

3.2 An auxiliary estimate

We now estimate the integral

(3.2) $$ \begin{align} \int_{\Lambda_1}e^{-Re\left(\omega_1 z_1\right)}\left|d\omega_1\right|, \quad \text{for }\arg\left(z_1\right)=\theta_1, \end{align} $$

where the curve $\Lambda _1$ is defined by Figure 4. Due to the choice (1.3) of the constant $C_1$ and parametrization (1.9) of the curve $\Gamma _1,$ the curve $\Lambda _1$ is a union of three subcurves: the finite segment $\overline {\Lambda _1\setminus \Gamma _1}$ and two infinite segments $\Lambda _1^+$ and $\Lambda _1^-$ correspondingly parameterized by

(3.3) $$ \begin{align} & \gamma_1(t),\quad t\geq t_+>0, \\\nonumber & \gamma_1(t),\quad t\leq t_-<0, \end{align} $$

where $t_+$ and $t_-$ are determined by $s_{1}^{-}=\gamma _1(t_-),s_{1}^{+}=\gamma _1(t_+).$

Figure 4: Construction of $\Lambda $ .

Figure 5: Construction of $\Lambda _1$ .

Step 1.

Note that due to the choice (1.3) of the constant $C_1,$

(3.4) $$ \begin{align}& Re\left(\omega_1 z_1\right)=Re\left(\omega_1 e^{i\theta_1}\right)\left|z_1\right| = -\left(C_1+\delta\right)\left|z_1\right| ,\\&\nonumber \text{for }\omega_1\in\bar{\Lambda_1\setminus \Gamma_1},\;\arg\left(z_1\right)=\theta_1. \end{align} $$

And, consequently,

(3.5) $$ \begin{align} \int_{\Lambda_1\setminus\Gamma_1} e^{-Re\left(\omega_1 z_1\right)} \left|d\omega\right| = e^{\left(C_1+\delta\right)\left|z_1\right|} \left|\Lambda_1\setminus\Gamma_1\right|. \end{align} $$

Step 2.

We rewrite

(3.6) $$ \begin{align} \int_{\Lambda_1^+}e^{-Re\left(\omega_1 z_1\right)}\left|d\omega_1\right| \overset{({3.3}),({1.9})}{=} \int_{t_+}^{+\infty}e^{-Re\left(\gamma_1(t)z_1\right)}dt. \end{align} $$

Note that

$$ \begin{align*} \left|\theta_1\right|{<}\alpha_1{<}\pi/2. \end{align*} $$

Consequently,

$$ \begin{align*} Re\left(ie^{-i\alpha_1}z_1\right)>0,\quad \text{for }\arg\left(z_1\right)=\theta_1, \end{align*} $$

so that the following integral is convergent and equals

(3.7) $$ \begin{align} \int_{t_+}^{+\infty}e^{-Re\left(\gamma_1(t)z_1\right)}dt = \frac{e^{-Re\left(\gamma_1\left(t_+\right)z_1\right)}}{Re\left(ie^{-i\alpha_1}z_1\right)}, \quad \text{for }\arg\left(z_1\right)=\theta_1. \end{align} $$

Note that $\gamma _1(t_+)\in \overline {\Lambda _1\setminus \Gamma _1}.$ Consequently, due to (3.4), we have

(3.8) $$ \begin{align} Re\left(\gamma_1\left(t_+\right)z_1\right)=-\left(C_1+\delta\right)\left|z_1\right|, \quad \text{for }\arg\left(z_1\right)=\theta_1. \end{align} $$

By combining (3.6)–(3.8), we get

(3.9) $$ \begin{align} \int_{\Lambda_{1}^+}e^{-Re\left(\omega_1 z_1\right)}\left|d\omega_1\right|\leq \frac{e^{\left(C_1+\delta\right)\left|z_1\right|}}{Re\left(ie^{-i\alpha_1}z_1\right)}, \quad \text{for }\arg\left(z_1\right)=\theta_1. \end{align} $$

Similarly,

(3.10) $$ \begin{align} \int_{\Lambda_1^-}e^{-Re\left(\omega_1 z_1\right)}\left|d\omega_1\right|\leq \frac{e^{\left(C_1+\delta\right)\left|z_1\right|}}{Re\left(-ie^{i\alpha_1}z_1\right)}, \quad \text{for }\arg\left(z_1\right)=\theta_1. \end{align} $$

Step 3.

By combining the estimates (3.5),(3.9), and (3.10), we get

(3.11) $$ \begin{align} \int_{\Lambda_1}e^{-Re\left(\omega_1 z_1\right)}\left|d\omega_1\right|\leq k_{\alpha_1,\theta_1}\frac{e^{\left(C_1+\delta\right)\left|z_1\right|}}{\left|z_1\right|}, \quad \text{for }\arg\left(z_1\right)=\theta_1, \end{align} $$

where the number $k_{\alpha _1,\theta _1},$ while depending on $\alpha $ and $\theta ,$ does not depend on $|z|.$ We remark that the estimate (3.11) that we have obtained is similar to the one in Lemma 3.1 in [25].

3.3 Proof of trigonometric convexity

As the function m is analytic on the set $\Omega _1\times \Omega _2,$ by consecutive applications of the Cauchy integral theorem in variables $z_1$ and $z_2,$ we can rewrite (1.13) as

(3.12) $$ \begin{align} f\left(z_1,z_2\right) = \int_{\Lambda_1}\int_{\Lambda_2} m\left(\omega_1,\omega_2\right) e^{-\omega_1 z_1-\omega_2z_2} d\omega_2d\omega_1, \end{align} $$

where the curve $\Lambda _1$ is constructed by Figure 4. Due to the construction of the curve $\Lambda _1,$ it is bounded away from the boundary $\partial \Omega _1.$ Similarly, the curve $\Lambda _2$ is bounded away from the boundary $\partial \Omega _2.$ Consequently, due to Remark 2.1, the function m is uniformly bounded (by some constant $k_{\delta }(m)$ ) on the Cartesian product $\Lambda _1\times \Lambda _2.$ We estimate

(3.13) $$ \begin{align} & \left| f\left(z_1,z_2\right)\right| \overset{({1.13})}{\leq} \int_{\Lambda_1}\int_{\Lambda_2} \left| m\left(\omega_1,\omega_2\right) \right| e^{-Re(\omega_1 z_1)-Re\left(\omega_2z_2\right)} \left|d\omega_2\right|\left|d\omega_1\right|\leq \\\nonumber & \overset{({2.3})}{\leq} k_{\delta}(m) \left[\int_{\Lambda_1} e^{-Re(\omega_1 z_1)} \left|d\omega_1\right| \right]\cdot \left[ \int_{\Lambda_2} e^{-Re\left(\omega_2z_2\right)} \left|d\omega_2\right|\right]\leq \\\nonumber & \overset{({3.11})}{\leq} k_{\delta}(m) k_{\alpha_1,\theta_1}k_{\alpha_2,\theta_2}\frac{e^{\left(C_1+\delta\right)\left|z_1\right|}}{\left|z_1\right|}\frac{e^{\left(C_2+\delta\right)\left|z_2\right|}}{\left|z_2\right|} , \\\nonumber & \quad\text{for } \arg\left(z_1\right)=\theta_1, \; \arg\left(z_2\right)=\theta_2. \end{align} $$

Additionally, due to the estimate (1.5), the function f is bounded in: the intersection of $\Omega _1\times \Omega _2,$ and a vicinity of $0.$ By combining this fact with the estimate (3.13), we obtain the following estimate on the function f:

$$ \begin{align*} & \left| f\left(z_1,z_2\right)\right| \leq k_{\delta,\theta_1,\theta_2,\alpha_1,\alpha_2} e^{(C_1+\delta)\left|z_1\right|+\left(C_2+\delta\right)\left|z_2\right|}, \\\nonumber & \text{for } \arg\left(z_1\right)=\theta_1, \; \arg\left(z_2\right)=\theta_2, \end{align*} $$

where the constant $k_{\delta ,\theta _1,\theta _2,\alpha _1,\alpha _2},$ while depending on $\delta ,\theta _1,\theta _2,\alpha _1,\alpha _2,$ does not depend on $\left |z_1\right |$ or $\left |z_2\right |.$

Figure 6: $T_f\left (\theta _1,\theta _2\right )$ .

4 Proof of sharpness

Consider the entire function $f(z_1,z_2)=e^{z_1+z_2}.$ The function f is of exponential type,

$$ \begin{align*} \left|f\left(z_1,z_2\right)\right|\leq e^{\left|z_1\right|\cos \theta_1+\left|z_2\right|\cos \theta_2},\quad\text{for }z_1,z_2\in\mathbb{C}. \end{align*} $$

For our choice of the function $f,$ the set $T_f(\theta _1, \theta _2)$ defined by (1.2) equals

(4.1) $$ \begin{align} T_f\left(\theta_1, \theta_2\right) = \left\lbrace(\nu_1,\nu_2) : \nu_1\geq\cos \theta_1, \nu_2\geq\cos \theta_2 \right\rbrace \end{align} $$

(see Figure 6). In particular, for $\alpha _1=\alpha _2=\pm \frac {\pi }{4}$ , we have

$$ \begin{align*} T_f\left(\pm\frac{\pi}{4}, \pm\frac{\pi}{4}\right) = \left\lbrace(\nu_1,\nu_2) \colon \nu_1\geq \frac{\sqrt{2}}{2}, \nu_2\geq\frac{\sqrt{2}}{2}\right\rbrace. \end{align*} $$

Thus,

$$ \begin{align*} A^+_1=A^+_2=A^-_1=A^-_2=\frac{\sqrt{2}}{2}. \end{align*} $$

Take $\theta _1=\theta _2 = 0.$ By (1.3), we evaluate

$$ \begin{align*} C_1=C_2=\frac{\sqrt{2}}{4} \frac{2}{\sqrt{2}}+\frac{\sqrt{2}}{4} \frac{2}{\sqrt{2}} = 1. \end{align*} $$

Thus, according to Theorem 1.1, the following inequality holds:

$$ \begin{align*} f(z_1, z_2) \leq k_{\varepsilon} e^{(1+\varepsilon)\left|z_1\right|+(1+\varepsilon)\left|z_2\right|},\quad\text{for }\arg z_1 = \arg z_2 = 0. \end{align*} $$

On the other hand, by (4.1),

$$ \begin{align*} T_f(0, 0)=\left\lbrace(\nu_1,\nu_2) : \nu_1\geq 1, \nu_2\geq 1 \right\rbrace. \end{align*} $$

That is, in this case, Theorem 1.1 is sharp.