An extremal property of entire functions with positive zeros (Q2639193)

Let \(f(z):=\prod^{\infty}_{k=1}E_ q(z/z_ k)\), \(z_ k\neq 0\), where \(E_ q\) denotes the usual Weierstraß primary factor and \(\sum^{\infty}_{k=1}| z_ k|^{-q-1}<\infty\). Let \(\hat f(z):=\prod^{\infty}_{k=1}Eq(z/| z_ k|)\) and define \[ u(re^{i\phi},f):=\sup_{\theta}\{\log | f(re^{i(\theta +\phi)})| +\log | f(re^{i(\theta -\phi)})| \}. \] The first main result of this interesting paper ``presents a rare instance when an inequality on primary factors is sharp for a range of \(\theta\), independent of r, and hence leads directly to extremal properties of \(\hat f\)''. In particular the authors show (cf. Theorem 1) that if \[ (*)\quad \frac{\pi}{2(q+1)}\leq \phi \leq \frac{\pi}{2q}\quad (q\geq 1)\quad (\frac{\pi}{2}\leq \phi \leq \pi,\quad if\quad q=0, \] then \(u(re^{i\phi},f)\leq u(re^{i\phi},\hat f)\). The first part of the second main result (cf. Theorem 2) states that if g is an entire function of nonintegral order \(\lambda\) and if \(\{r_ m\}\) is a sequence of Pólya peaks of order \(\lambda\) for the counting function N (for the zeros of g), then \[ \limsup_{m\to \infty}\frac{u(tr_ me^{i\phi},g)}{N(r_ m,0)}\leq \frac{2\pi \lambda t^{\lambda}}{\sin (\pi \lambda)}\cos ((\pi -\phi)\lambda), \] for \(\phi\) satisfying (*), uniformly in t in compact subsets of \(0<t<\infty\). Theorem 2 is sharp and extends a theorem of \textit{W. H. J. Fuchs} [Comment. Math. Helv. 53, 135- 141 (1978; Zbl 0369.30021)].