Measures of transcendency for entire functions (Q1432731)

Let \(\varDelta_r(a):=\{z\in\mathbb C: | z-a| \leq r\}\), \(\varDelta_r:=\varDelta_r(0)\), \(\varDelta:=\varDelta_1\). For a transcendental entire function \(f\in\mathcal O(\mathbb C)\), let \(M(r):=\max_{| z| =r}| f(z)| \), \(m(r):=\max\{\log^+M(r),\log^+r\}\), \(e_n:=\log\sup\{\| P\| _{\varDelta^2}: P\in\mathcal P_n,\; \| P_f\| _{\varDelta}\leq1\}\), where \(\mathcal P_n\) denotes the set of all complex polynomials of two complex variables of degree at most \(n\) and \(P_f(z):=P(z,f(z))\). For \(\theta\in[0,2\pi]\) and \(r\geq2\) let \(D(\theta,r):=\{z\in\mathbb C: 2\leq| z| \leq r,\;f(z)=e^{i\theta}\}\), \(d_n(\theta,r):=\min\{1,\text{diam}_n(D(\theta,r))\}\), where the \(n\)th diameter \(\text{diam}_n(G)\) of a set \(G\subset\mathbb C\) is defined by the formula \(\text{diam}_n(G):= \{\sum_{j=1}^kr_j: k\leq n,\;G\subset\bigcup_{j=1}^k\varDelta_{r_j}(a_j)\}\). Let, moreover, \(m_n(r):=\log\sup\{\| P_f\| _{\varDelta_r}: P\in\mathcal P_n,\;\| P_f\| _\varDelta\leq1\}\). The authors study the problem how the above growth characteristics of \(f\) measure the transcendency of \(f\). In particular, they prove the following estimates: \(m_n(r)\leq\frac{e_n+nm(t)}{\log t}\log r\), \(1\leq r\leq t\). If \(r\geq2\) and \(d_n(\theta,r)\geq a\) on a set \(E\subset\partial\varDelta\) of length \(\ell\), then \(e_n\leq n m(er)\log r+n\log(er)(\log\frac{36er}a+\frac{4\pi}\ell)\).