Sharp bounds for the type of an entire function of order less than 1 whose zeros are located on a ray and have given averaged densities (Q1761080)

Let \(\sigma_\rho(f)=\limsup _{r\to+\infty} r^{-\rho} \ln M(r,f)\), where \(M(r,f)=\max\{|f(z)|: |z|=r\} \), denote the value of type of an entire function \(f\) of order \(\rho\). For its zero sequence \(\Lambda=\Lambda_f=(\lambda_n)\), the counting function and Nevanlinna's counting function are defined by \[ n_\Lambda(r)=\sum_{|\lambda_n|\leq r} 1\quad\text{and}\quad N_\Lambda(r)=\int_0^r \frac{n_\Lambda(t)}{t} dt, \quad\Lambda\not\ni 0, \] respectively. The averaged upper (lower) density of the sequence \(\Lambda\) is defined to be \(D^*_\rho(\Lambda)=\limsup_{r\to+\infty} r^{-\rho} N_\Lambda(r)\) (\(d^*_\rho(\Lambda)=\liminf_{r\to+\infty} r^{-\rho} N_\Lambda(r)\)). It is well known [\textit{B. Ya. Levin}, The distribution of the zeros of entire functions (Russian). Moskau: Staatsverlag für technisch-theoretische Literatur (1956; Zbl 0111.07401)] that for entire functions of order \(\rho\in (0,1)\), \[ D^*_\rho(\Lambda_f)\leq \sigma_\rho(f)\leq \frac{\pi \rho}{\sin\pi \rho}D^*_\rho(\Lambda_f). \eqno(1) \] The lower bound follows from Jensen's formula and is valid for any positive \(\rho\). For entire functions of order \(\rho\in (0,1)\) with all zeros on a ray we have \[ \ln M(r,f)=r\int_0^\infty \frac{n_\Lambda(t)}{t(r+t)}dt =r\int_0^\infty \frac{N_\Lambda(t)}{(r+t)^2}dt \] [\textit{A. A. Goldberg} and \textit{I. V. Ostrovskii}, Value distribution of meromorphic functions. Transl. from the Russian by Mikhail Ostrovskii. With an appendix by Alexandre Eremenko and James K. Langley. Providence, RI: American Mathematical Society (AMS) (2008; Zbl 1152.30026)]. In this case it was plausible that the lower bound in (1) can be improved. The main results establish sharp lower bounds for \(\sigma_\rho(f)\) in terms of densities of \(\Lambda_f\) in the case \(\rho\in (0,1)\). Given \(\theta\in [0,1]\), let \(a_1\), \(a_2\) be the roots of the equation \(a\ln \frac ea=\theta (2)\) satisfying \(0\leq a_1\leq 1\leq a_2\leq e\). Theorem 1. Let \(f\) be an entire function of order \(\rho\in(0,1)\) with positive zeros, \(D_\rho^*(f)=D^*>0\), and \(d_\rho^*(f)\geq d^*>0\). Then the following bound holds: \[ \sigma_\rho(f)\geq C^*\Bigl(\frac{d^*}{D^*}, \rho\Bigr) D^*, \eqno (3) \] where \[ C^*(\theta, \rho)=\rho \Bigl(\frac{\pi \theta}{\sin \pi \rho} +\max_{a>0}\int_{aa_1^{\frac{1}{\rho}}}^{aa_2^{\frac{1}{\rho}}} \frac{a^{-\rho}-\theta \tau^{-\rho}}{\tau+1}d\tau \Bigr), \] \(a_j \), \(j\in \{1,2\}\), are the roots of the equation (2) with \(\theta=d^*/D^*\). For any \(\rho\in(0,1)\) there exists an entire function \(f\) with positive zeros such that the equality in (3) is attained. Without assumptions on the lower density \((d^*=0)\) the author obtains the following: Theorem 2. Let \(f\) be an entire function of order \(\rho\in(0,1)\) with positive zeros, \(D_\rho^*(f)=D^*>0\) and \(d_\rho^*(f)\geq d^*>0\). Then the sharp bound \[ \sigma_\rho(f)\geq \rho e C(\rho)D^*_\rho(f) \eqno (4) \] holds, where \(C(\rho)=\max_{a>0} a^{-\rho} \ln (1+a)\). Since \(C(\rho)> (\rho e)^{-1}\) [\textit{A. Yu. Popov}, Vestn. Mosk. Univ., Ser. I 2005, No. 1, 31--36 (2005); translation in Mosc. Univ. Math. Bull. 60, No. 1, 32--36 (2005; Zbl 1101.30009)], (4) implies that the lower bound in (1) cannot be attained in the class of entire functions of order \(\rho\in (0,1)\), and with the zeros on a ray. The author also establishes some relations between the constants \(C^*(\theta, \rho)\) and \(C(\rho)\), \(\theta\in [0,1]\), \(\rho\in (0,1)\) (Theorem 3), and finds an asymptotic for \(C^*(\theta, \rho)\) as \(\rho\to 0\), \(\theta\in (0,1)\) (Theorem 4).