Estimate of the variation of an entire function for the shifts of its zeros (Q1360769)

Let \(N\in\mathcal O(\mathbb C)\), \(\{\lambda_1,\lambda_2,\dots\}:=N^{-1}(0)\), \(m_{j}:=\text{ord}_{\lambda_{j}}N\), \(j=1,2,\dots\). Assume that \(N(0)\neq0\) and let \(N(\lambda)=g(\lambda)\prod_{j=1}^\infty (1-\lambda/\lambda_{j})^{m_{j}}\exp p_{j}(\lambda/\lambda_{j})\) be the Weierstrass decomposition of \(N\). Fix \(d_{j}\in\mathbb C\), \(\rho_{j}>0\), and \(r_{j}>0\) such that \(\lambda_{j}+d_{j}\neq0\), \(|d_{j}|+\rho_{j}\leq r_{j}\), and \(\sum_{j=1}^\infty m_{j}|d_{j}|r_{j}<+\infty\). Put \(N_{d}(\lambda):=g(\lambda)\prod_{j=1}^\infty (1-\lambda/(\lambda_{j}+d_{j}))^{m_{j}} \exp p_{j}(\lambda/\lambda_{j})\). The authors announce the following results. (1) There exists \(C>0\) such that \(|\log|N_{d}(\lambda)|-\log|N(\lambda)||\leq C\) for \(\lambda\in\mathbb C\setminus\bigcup_{j=1}^\infty B(\lambda_{j},r_{j})\), where \(B(\lambda_0,r):=\{\lambda\in\mathbb C\: |\lambda-\lambda_0|<r\}\). (2) If, moreover, the discs \(B(\lambda_{j},r_{j})\), \(j=1,2,\dots\), are pairwise disjoint, then there exist \(c_1\), \(c_2>0\) such that \[ c_1(\rho_{j}/(r_{j}+|d_{j}|)^{m_{j}}\min_{\Gamma_{j}}|N|\leq\min_{\gamma_{j}}|N_{d}|\leq\max_{\gamma_{j}}|N_{d}|\leq c_2(\rho_{j}/(r_{j}-|d_{j}|)^{m_{j}}\max_{\Gamma_{j}}|N|, \] where \(\Gamma_{j}:=\partial B(\lambda_{j},r_{j})\), \(\gamma_{j}:=\partial B(\lambda_{j}+d_{j},\rho_{j})\), \(j=1,2,\dots\). As an application the authors present an approximation theorem for subharmonic functions.