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An extension of B. Ya. Levin’s theorem on proximate orders for entire functions. (English) Zbl 0768.30017

The author proves the following result related to the concept of generalized proximate order: Let \(\beta(r)\) be defined in \([a,\infty)\) with \(a>0\) and let \(\beta(r)\) be strictly increasing, differentiable, tending to \(\infty\) with \(r\). Then there exists for each continuous \(\alpha(r)>0\) with \(\limsup_{r\to\infty} (\log \alpha(r))/\beta(r)=\rho(\alpha,\beta)\) a generalized proximate order \(\rho(r)\) such that for some \(c>0\) and all \(r>r_ 0\) holds \(\alpha(r)\leq c\exp(\rho (r)\beta(r))\) and for some sequence \(r_ n\to\infty\) \(\alpha(r_ n)=c \exp(\rho(r_ n)\beta(r_ n))\). A similar result with respect to the generalized lower proximation order is proved.
Reviewer: J.Winkler (Berlin)

MSC:

30D15 Special classes of entire functions of one complex variable and growth estimates
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