On the Pólya conjecture concerning the maximum and minimum of the modulus of an entire function of finite order given by a lacunary power series (Q753985)

Let f be an entire function of finite order with power series representation \[ (*)\quad f(z)=\sum^{\infty}_{k=0}a_ kz^{n_ k}\text{ where } k=o(n_ k)\quad (k\to \infty). \] If \(M(r,f)=\max_{| z| =r}| f(z)|\) and \(L(r,f)=\min_{| z| =r}| f(z)|\), then \textit{W. H. J. Fuchs} [Illinois J. Math. 7, 661-667 (1963; Zbl 0113.287)] proved a conjecture of Pólya that for \(0<\epsilon <1\), \[ (**)\quad \ln L(r,f)>(1-\epsilon)\ln M(r,f) \] for all r outside a set of zero logarithmic density. The result was extended by the reviewer [Proc. Lond. Math. Soc., III. Ser. 21, 525-539 (1970; Zbl 0206.088)] to functions of finite lower order for r on a set of infinite logarithmic measure; the latter was extended by \textit{W. K. Hayman} [ibid. 24, 590-624 (1972; Zbl 0239.30035)] to r outside a set of zero lower logarithmic density. In the paper under review, the author proves that in order for every entire function of the form (*) of finite order (finite lower order) to have the relation (**) satisfied outside a set of zero lower logarithmic density, it is necessary and sufficient that \[ \liminf_{t\to \infty}((\ln t)^{-1}\sum_{n_ k\leq t}(n_ k)^{-1})=0. \] The sufficiency proofs are a synthesis of the work of Hayman and Sons mentioned above. The necessity parts already appear in \textit{J. M. Anderson\textit{and \textit{K. G. Binmore}}} [Glasgow Math. J. 12, 89-97 (1971; Zbl 0237.30006)] and \textit{J. Kühn} [Mitt. Math. Sem. Giessen 75, 1-50 (1967; Zbl 0158.316)].