Asymptotic bounds for the modulus of an entire function of noninteger order in terms of a majorant of its zero counting function
Let \(f\) be an entire function of non-integer order \(\rho\), set \(p =\lfloor \rho \rfloor\) and \(M_0(r) =\log (1+r)\). For \(p\geq 1\) let \[ M_p(r) = \max_{\varphi}\left(\frac{1}{2} \log \left(1-2r\cos \varphi +r^2\right)+ \sum_{k=1}^p\frac{r^k \cos k\varphi}{k}\right) \quad\text{and}\quad S(\rho) =\int_0^\infty r^{-\rho}d M_p(r). \] Define \(M(r,f) = \max_{\varphi} |f(r e^{i\varphi})|\) and denote by \(\rho(r)\) a refined (proximate) order in the sense of Valiron. Further, let \(l(r)= r^{\rho(r) -\rho }\) and \( w(r) = \frac{rl^{'}(r)}{l(r)}\). The author declares (without proofs) estimates involving the above quantities under some additional conditions for \(\rho(r)\). If \(n(r) \leq r^{\rho(r)}\) then \(\log M(r,f) \leq (S(\rho)+(S^{'}(\rho) + o(1))w(r))r^{\rho(r)}\) as \(r\to\infty\); \(\log M(r,f) \leq (S(\rho)+b(\rho)w(r))r^{\rho(r)}\), \(r\geq r_0\), where \(b(\rho)\) are different a priori defined functions. The author discusses the function \(S(\rho)\), proving the following two inequalities: \[ S(\rho) = 2^{1-\rho} \rho \left( \frac{1}{2-\rho} + \frac{1}{\rho - 1} \right) +\sum_{k=0}^\infty \frac{1}{2^{k+\rho}(k+\rho)},\quad \rho\in (1;2), \] \[ 2\log(p+1) - 2{.}2 < S(\rho) - \frac{1}{\{\rho\}} - \frac{1}{1 -\{\rho\}} < 2\log p +\frac{5}{3}, \quad \rho > 2,\{\rho\} = \rho - \lfloor \rho \rfloor. \] An asymptotic formula for \(S(\rho)\) when \(\rho \to \infty\) is also given. Other results are also presented in the article.
scientific article; zbMATH DE number 5998362
scientific article