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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.

##### MSC:
 30D15 Special classes of entire functions of one complex variable and growth estimates 30D99 Entire and meromorphic functions of one complex variable, and related topics
##### Keywords:
local behavior; $$\ell$$-index