Sequences of some meromorphic function spaces

Abstract

Our goal in this paper is to introduce some new sequences of some meromorphic function spaces, which will be called $b_q$ and $q_{K}$-sequences. Our study is motivated by the theories of normal, $Q^{\#}_K$ and meromorphic Besov functions. For a non-normal function $f$ the sequences of points $\{a_n\}$ and $\{b_n\}$ for which $$\lim_{n\rightarrow \infty}(1-|a_n|^2)f^{\#}(a_n)=+\infty\,\,\,\mbox{and} $$ $$ \lim_{n\rightarrow\infty}\iint_\Delta \bigl(f^{\#}(z)\bigr)^q (1-|z|^2)^{q-2}(1-|\varphi_{a_n}(z)|^2)^s dA(z)=+\infty\;$$ or $$ \lim_{n\rightarrow\infty}\iint_\Delta \bigl(f^{\#}(z)\bigr)^2 K(z,a_n)dA(z)=+\infty\;$$ are considered and compared with each other. Finally, non-normal meromorphic functions are described in terms of the distribution of the values of these meromorphic functions.