Lahiri, B. K.; Banerjee, Dibyendu A note on relative order of entire functions. (English) Zbl 1089.30022 Bull. Calcutta Math. Soc. 97, No. 3, 201-206 (2005). 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. Reviewer: Anatoly Filip Grishin (Khar’kov) Cited in 2 Documents MSC: 30D15 Special classes of entire functions of one complex variable and growth estimates Keywords:order; relative order of Bernal; function of regular growth; exponent of convergence PDFBibTeX XMLCite \textit{B. K. Lahiri} and \textit{D. Banerjee}, Bull. Calcutta Math. Soc. 97, No. 3, 201--206 (2005; Zbl 1089.30022)