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Analytic in the unit disc functions of bounded index. (Ukrainian. English, Russian summaries) Zbl 0783.30025
Let \(\ell(x)\) be a continuous positive function on \((1,\infty)\) and let \(f(z)\) be an analytic function in \(D=\{z: | z|<1\}\). The authors call \(f(z)\) a function of bounded \(\ell\)-index in \(D\) if it satisfies \[ {1\over n!}\bigl| f^{(n)}(z)\bigr| \ell^{-n}\left({1\over 1-| z|}\right)\leq\max\left\{{1\over k!}\bigl| f^{(k)}(z)\bigr| \ell^{-k}\left({1\over 1-| z|}\right):\;0\leq k\leq N\right\}, \] for some \(N\in\mathbb{Z}_ +\) and all \(n\in\mathbb{Z}_ +\) and \(z\in D\). Necessary and sufficient conditions for \(f(z)\) to be a function of bounded \(\ell\)-index in \(D\) and estimates of the growth of such functions are obtained provided \(\ell(x)\) satisfies some regularity restrictions. The results are analogues of the known theorems for entire functions of bounded \(\ell\)-index.

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