The meromorphic functions of completely regular growth on the upper half-plane

K. G. Malyutin ; M. V. Kabanko

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 30 (2020) no. 3, pp. 396-409

Résumé

A strictly positive continuous unbounded increasing function $\gamma(r)$ on the half-axis $[0,+\infty)$ is called growth function. Let the growth function $\gamma(r)$ satisfies the condition $\gamma(2r)\leq M\gamma(r)$ for some $M>0$ and for all $r>0$. In the paper, the class $JM(\gamma(r))^o$ of meromorphic functions of completely regular growth on the upper half-plane with respect to the growth function $\gamma$ is considered. The criterion for the meromorphic function $f$ to belong to the space $JM(\gamma(r))^o$ is obtained. The definition of the indicator of function from the space $JM(\gamma(r))^o$ is introduced. It is proved that the indicator belongs to the space $\mathbf{L}^p[0,\pi]$ for all $p>1$.

Détail

Keywords: just meromorphic function, complete measure, function of growth, function of completely regular growth, conjugate series, indicator.
Mots-clés : Fourier coefficients