×

A necessary condition for all the zeros of an entire function of exponential type to lie in a curvilinear half-plane. (English. Russian original) Zbl 0863.30038

Sb. Math. 186, No. 9, 1353-1362 (1995); translation from Mat. Sb. 186, No. 9, 125-134 (1995).
Let \(F(z)\) be an entire function of exponential type such that the following integral \[ \int_\mathbb{R} {\ln \bigl |F(x) \bigr|\over 1+ x^2}dx \] converges. In this paper the author has received the following Theorem. Let all the zeros of \(F(z)\) lie in the curvilinear half-plane \[ y\leq h \bigl(|x |\bigr)\;\biggl(y \geq h \bigl(|x |\bigr) \biggr),\;x\in \mathbb{R},\;z= x+ iy, \] where \(h(t)\) is bounded on each segment and positive on some semi-axis \(t>A\) function such that for each \(\lambda>0\), \[ {h (\lambda t) \over h(t)} \to \lambda^\alpha, \quad t \to\infty,\;\alpha\in [0,1). \] If the indicator diagram of \(F(z)\) is \([-ia, ia]\), \(a >0\) then \[ \limsup_{y\to\infty} {\ln \bigl|F(-iy)/F(iy) \bigr|\over h(y)} \leq {2 \alpha \over \cos (\pi \alpha/2)} \]
\[ \left( \liminf_{y\to\infty} {\ln \bigl |F(-iy)/F(iy) \bigr|\over h(y)} \geq {2\alpha \over \cos (\pi \alpha/2)} \right). \]

MSC:

30D15 Special classes of entire functions of one complex variable and growth estimates
30D20 Entire functions of one complex variable (general theory)
PDFBibTeX XMLCite
Full Text: DOI