The growth of entire functions in the terms of generalized orders

  • T.Ya. Hlova Institute for Applied Problems of Mechanics and Mathematics, 3b Naukova str., 79060, Lviv, Ukraine
  • P.V. Filevych Stepan Gzhytskyi National University of Veterinary Medicine and Biotechnologies, 50 Pekarska str., 79000, Lviv, Ukraine
Keywords: entire function, maximum modulus, maximal term, central index, order, generalized order
Published online: 2012-06-28

Abstract


Let $\Phi$ be a convex function on $[x_0,+\infty)$ such that $\frac{\Phi(x)}x\to+\infty$, $x\to+\infty$, $\displaystyle f(z)=\sum_{n=0}^\infty a_nz^n$ is a transcendental entire function, let $M(r,f)$ be the maximum modulus of $f$ and let $$ \rho_\Phi(f)=\varlimsup_{r\to +\infty}\frac{\ln\ln M(r,f)}{\ln\Phi(\ln r)},\quad c_{\Phi}=\varlimsup_{x\to +\infty}\frac{\ln x}{\ln\Phi(x)},$$ $$d_{\Phi}=\varlimsup\limits_{x\to +\infty}\frac{\ln\ln\Phi'_+(x)}{\ln\Phi(x)}. $$ It is proved that for every transcendental entire function $f$ the generalized order $\rho_\Phi(f)$ is independent of the arguments of the coefficients $a_n$ (or defined by the sequence $(|a_n|)$) if and only if the inequality $d_{\Phi}\le c_{\Phi}$ holds.

How to Cite
(1)
HlovaT., FilevychP. The Growth of Entire Functions in the Terms of Generalized Orders. Carpathian Math. Publ. 2012, 4 (1), 28–35.