On the growth of a composition of entire functions
Journal Title: Карпатські математичні публікації - Year 2017, Vol 9, Issue 2
Abstract
Let $\gamma$ be a positive continuous on $[0,\,+\infty)$ function increasing to $+\infty$ and $f$ and $g$ be arbitrary entire functions of positive lower order and finite order. In order to $$ \lim\limits_{r\to+\infty} \dfrac{\ln\ln\,M_{f(g)}(r)}{\ln\ln\,M_f(\exp\{\gamma(r)\})}=+\infty, \quad M_f(r)=\max\{|f(z)|:\,|z|=r\}, $$ it is necessary and sufficient $(\ln\,\gamma(r))/(\ln\,r)\to 0$ as $r\to+\infty$. This statement is an answer to the question posed by A.P. Singh and M.S. Baloria in 1991. Also in order to $$ \lim\limits_{r\to+\infty}\dfrac{\ln\ln\,M_F(r)} {\ln\ln\,M_f(\exp\{\gamma(r)\})}=0,\quad F(z)=f(g(z)), $$ it is necessary and sufficient $(\ln\,\gamma(r))/(\ln\,r)\to \infty$ as $r\to+\infty$.
Authors and Affiliations
M. Sheremeta
Representation of spectra of algebras of block-symmetric analytic functions of bounded type
The paper contains a description of a symmetric convolution of the algebra of block-symmetric analytic functions of bounded type on $\ell_1$-sum of the space $\mathbb{C}^2$. We show that the specrum of such algebra does...
Some analytic properties of the Weyl function of a closed linear relation
Let L and L0, where L is an expansion of L0, be closed linear relations (multivalued operators) in a Hilbert space H. In terms of abstract boundary operators (i.e. in the form which in the case of differential operators...
On the crossings number of a hyperplane by a stable random process
The numbers of crossings of a hyperplane by discrete approximations for trajectories of an α-stable random process (with 1<α<2) and some processes related to it are investigated. We consider an α-stable process is killed...
Lateral continuity and orthogonally additive operators
We generalize the notion of a laterally convergent net from increasing nets to general ones and study the corresponding lateral continuity of maps. The main result asserts that, the lateral continuity of an orthogonally...
Properties of composite positive continuous functions in Cn
The properties of positive continuous functions with Qnb and Q are investigated. We prove that some composite functions with Q belong to class Qnb. A relation between functions with these classes are established.