indicator function of an entire function of finite exponential type?

Let $\Phi(z)$ be an entire function of finite exponential type. The indicator function of $\Phi(z)$ is defined as $$h_{\Phi}(\theta)=\overline{\lim_{r\rightarrow\infty}}\frac{\ln|\Phi(re^{i\theta})|}{r}\;,~\theta\in\mathbb{R}\;.$$ The indicator function of $\Phi(z)$ essentially characterizes the growth of $\Phi(z)$. I need to compute the indicator function of the function $$F(z)=\sum_{j=1}^{m}e^{\lambda_jz}f_{j}(z)\;,$$ where $f_{j}(z)$ are real polynomials, and $\lambda_1<\lambda_2<\cdots <\lambda_m$. Any step-by-step help is appreciated. In addition, I would like some ideas on how to deal with the particular function $$F(z)=(a_3z^3+a_2z^2+a_1z+a_0)e^{z\tau}+(b_2z^2+b_1z+b_0)\;,$$ where $a_j,b_j\in\mathbb{R}$, and $\tau\in\mathbb{R}^{+}$. I appreciate the monumental work of B. Ya. Levin, but I don't see how he computes the indicator functions - in a clear step-by-step process.

• For $\cos \theta > 0$, write $$F(z) = e^{\lambda_m z} \sum_{j=1}^m e^{(\lambda_j - \lambda_m)z} f_j(z).$$ – Daniel Fischer May 9 '15 at 18:26

It is not so simple to obtain the answer. Up to Levin, B. Ya. Lectures on entire functions. In collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko. Translated from the Russian manuscript by Tkachenko. Translations of Mathematical Monographs, 150. American Mathematical Society, Providence, RI, 1996. xvi+248 pp. ISBN: 0-8218-0282-8 MR1400006 (97j:30001) (see here), the indicator diagram of $F(z):=\sum_{j=1}^{j=m} e^{\lambda_jz}f_{j}(z)$, where $\lambda_1,\dots,\lambda_m$ are complex numbers, and $f_{j}(z), \,j=1,\dots,m,$ are polynomials, coincides with the convex hull of the set $\{\overline{\lambda_1},\dots,\overline{\lambda_m} \}$. In the case of positive lambdas, this is the segment $I:=[\lambda_1,\lambda_m].$ Next, the indicator $h_F(\theta)$ equals the supporting function of $I$ (see the definitions in the cited link) which is $$\begin{cases}\lambda_m\cos\theta,\, \cos \theta \ge 0;\\ \lambda_1\cos\theta,\, \cos \theta < 0.\end{cases}$$
• Addition. For the function $$F(z):=(a_3z^3+a_2z^2+a_1z+a_0)e^{z\tau}+(b_2z^2+b_1z+b_0)$$ we have $\log|F(re^{i\theta})| = r \tau \cos\theta)(1+o(1))$ as $r \to \infty$ uniformly in $\{|\theta| < \pi/2- \delta|\} \cup\{|\theta| >\pi/2 +\delta\}$ for arbitrary $\delta >0$. This implies $h(\theta)=\tau\cos\theta.$ – user64494 May 10 '15 at 5:44