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On a problem of Erdős. (English) Zbl 0573.30032

Entire functions are considered, bounded outside a set of finite measure. Thus the maximum modulus M(r) has a strong growth. In fact, the integral towards \(\infty\) of r/ln ln M(r) must converge. The author now continues to show that the condition is best possible in the following sense. Let \(\psi\) (r) be a positive non-decreasing function such that the integral towards \(\infty\) of r/\(\psi\) (r) is convergent. Then there exists an entire function as above for which \(\ln^+\ln^+M(r)\leq \psi (r)\) for \(r\geq 0\). The method of proof is first to construct an appropriate subharmonic function and then an entire function of similar growth.
Reviewer: B.Kjellberg

MSC:

30D20 Entire functions of one complex variable (general theory)
30D15 Special classes of entire functions of one complex variable and growth estimates
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