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A note on relative order of entire functions. (English) Zbl 1089.30022

Let \(f\) and \(g\not\equiv c\) be entire functions. Bernal (1988) defined relative order of \(f\) with respect to \(g\) by the following \[ \rho_g(f) = \limsup _{r\to \infty} \frac{\log G^{-1}(F(r))}{\log r}, \] where \(F(r)=\max_{\varphi}| f(r e^{i\varphi})| ,\quad \) \(G(r) = \max_{\varphi}| g(r e^{i\varphi})| .\) The authors define relative lower order: \[ \lambda_g(f) = \liminf _{r\to \infty} \frac{\log G^{-1}(F(r))}{\log r}. \] The function \(f\) is said to be of regular relative growth with respect to \(g\) if \(\rho_f(g)=\lambda_f(g).\) The authors prove the following Theorem. If \(f\) be of the regular growth and of regular relative growth with respect to \(g\) and \(\rho_g(f) = \rho(f) >0\) then \(g\) is of regular growth of order one. Conversely if \(g\) is of regular growth of order one then \(\rho_g(f) = \rho(f)\) for every entire function \(f\) with regular relative growth. One can see that the theorem follows from the equation \[ \frac{\log G^{-1}(F(r))}{\log r} = \frac{\log G^{-1}(F(r))} {\log \log F(r)} \frac{\log \log F(r)}{\log r}= \]
\[ \frac{\log R}{\log \log G(R)} \frac{\log\log F(r)}{\log r},\quad R = G^{-1}(F(r)). \] The authors (1999) introduced the relative exponent of the convergence of \(f\) with respect to \(g\) \((\rho_1(f,g) ).\) One can see that \(\rho_1(f,g)\) is the exponent of convergence of zeros of the meromorphic function \(\frac{f(z)}{g(z)}.\) The authors prove the inequality \(\rho_1(f,g) \leq \rho_g(f)\) if \(\rho(g) \leq 1.\) There are other propositions in the paper.

MSC:

30D15 Special classes of entire functions of one complex variable and growth estimates
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