The growth of entire functions in the terms of generalized orders

• Main Authors: Hlova T.Ya., Filevych P.V.
• Format: Article
• Language: English
• Published: Vasyl Stefanyk Precarpathian National University 2012-01-01
• Series: Karpatsʹkì Matematičnì Publìkacìï
• Online Access: http://journals.pu.if.ua/index.php/cmp/article/view/73/62

Let $Phi$ be a convex function on $[x_0,+infty)$ such that$frac{Phi(x)}xo+infty$, $xo+infty$, $f(z)=sum_{n=0}^infty a_nz^n$--- a transcendental entire function, let $M(r,f)$ be the maximum modulus of$f$ and let$$ho_Phi(f)=varlimsup_{ro +infty}frac{lnln M(r,f)}{lnPhi(ln r)},quad c_{Phi}=varlimsup_{xo +infty}frac{ln x}{lnPhi(x)},quad d_{Phi}=varlimsuplimits_{xo +infty}frac{lnlnPhi'_+(x)}{lnPhi(x)}.$$It is proved that for every transcendental entire function $f$ thegeneralized order $ho_Phi(f)$ is independent on the arguments of thecoefficients $a_n$ (or defined by the sequence $(|a_n|)$) if and only if theinequality $d_{Phi}le c_{Phi}$ holds.