On the interpolation in some classes of holomorphic in the unit disk functions

Abstract

There is considered an interpolation problem $f(\lambda_n )=b_n$ in the class of holomorphic in the unit disk $U(0;1)=\{z\in\mathbb{C}\colon |z|<1\}$functions of finite $\eta$-type, i.e such that $\displaystyle (\exists A>0)(\forall z\in U(0;1))\colon \quad |f(z)|\leq\exp\Big(A\eta\Big(\frac A{1-|z|}\Big)\Big),$ where $\eta\colon [1;+\infty)\to [0;+\infty)$ is an increasing convex function with respect to $\ln{t}$ and $\ln{t}=o\left(\eta ( t)\right)$ $(t\to+\infty)$.There were received sufficient conditions of the interpolation problem solvability in terms of the counting functions $\displaystyle N(r)=\int\nolimits_{0}^{r}\frac{\left(n(t)-1\right)^+}{t}dt$ and $\displaystyle N_{\lambda_n} (r)=\int\nolimits_{0}^{r}{\frac{{{(n}_{\lambda_n}\left(t\right)-1)}^+}{t}dt}$. Earlier, in 2004, necessary conditions were obtained (Ukr. Math. J., {\bf 56} (2004), \No 3) in these terms.For the moderate growth of $f$ (when the majorant $\eta=\psi$ satisfies the condition $\psi\left(2x\right)=O\left(\psi\left(x\right)\right),\ x\rightarrow+\infty$) that problem was solved in J. Math. Anal. Appl., {\bf 414} (2014), \No 1.In this paper, we remove any restrictions on the growth of $\eta$ and construct an interpolation function $f$ such that$\displaystyle (\exists A'>0)(\forall z\in U(0;1))\colon \quad |{f}(z)|\leq\exp\Big(\frac{A'}{(1-|z|)^{3/2}}\eta\Big(\frac{A'}{1-|z|}\Big)\Big)$.