Let \(\rho> 1\), \(p= [\rho]\), \(\mathbb{C}_ += \{z: \text{Im }z> 0\}\), \([\rho,\infty)^ +\) be the class of functions \(f(z)\) holomorphic in \(\mathbb{C}_ +\) such that \(\varlimsup| z|^{-\rho}\ln| f(z)|0\)). Furthermore, \(D= \{a_ n,q_ n\}\), \(n=1,2,\dots,a_ n\in \mathbb{C}_ +\), \(q_ n\in \mathbb{N}\) is the divisor, \(\mu\) be the measure \[ \mu(G)= \sum_{a_ n\in G} q_ n\sin\arg a_ n,\qquad \mu(r)= \sum_{| a_ n|\leq r} q_ n\sin\arg a_ n,\qquad \text{and} \] \[ E(z)= E_ D(z)= \prod_{| a_ n|\leq 1}\left({z- a_ n\over z-\bar a_ n}\right)^{q_ n} \prod_{| a_ n|>1} E^{q_ n}_ p(z,a_ n), \] where \[ E_ p(z,a_ n)= {(1-{z\over a_ n})e^{{z\over a_ n}+\cdots+ {z^ p\over pa^ p_ n}}\over (1-{z\over \bar a_ n}) e^{{z\over \bar a_ n}+\cdots+ {z^ p\over p\bar a^ p_ n}}} \] is the Nevanlinna primary factor, \(\Lambda_ n= \min(1,\text{Im }a_ n)\). Problem I. For given \(D= \{a_ n,q_ n\}\) and \(b_{n,k}\in \mathbb{C}\), \(k= 1,2,\dots, q_ n\), \(n= 1,2,\dots\) find \(f\in [\rho,\infty)^ +\) such that \(f^{(k-1)}(a_ n)= b_{n,k}\). Let \(\rho\) be nonintegral. The author proves that problem I is solvable for every sequence \(b_{n,k}\) such that \[ \varlimsup_{n\to\infty} | a_ n|^{-\rho}\ln \max_{1\leq k\leq q_ n} {\Lambda^{k-1}_ n| b_{n,k}|\over (k-1)!}< \infty \] if and only if \[ \varlimsup_{r\to\infty} r^{- \rho}\mu(r)<\infty,\;\varlimsup_{n\to\infty} | a_ n|^{- \rho}\ln{q_ n!\over | E^{(q_ n)}(a_ n)|\Lambda^{q_ n}_ n}< \infty. \] He also gives a criterion for the solvability of problem I in terms of the measure \(\mu\). The more difficult case of \(\rho\) being an integer is also investigated. Let \(\rho(r)\) be a proximate order of Valiron. The author studies the class \([\rho(r),\infty)^ +\). We formulate the results for the case \(\rho(r)\equiv \rho\). An interpolation problem for analogous classes \([\rho(r),\infty)\) of entire functions was previously studied.