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On the growth of entire functions of exponential type along the imaginary axis. (Russian) Zbl 0671.30025
Let \(\Lambda =\{\lambda_ n\}\) and \(\Gamma\) be sequences of complex numbers with finite upper density, with \(\Gamma\) in the right-hand half plane. The following three properties are equivalent for entire functions g (\(\not\equiv 0)\). (1) For every g that vanishes on \(\Gamma\) and every \(\epsilon >0\) there is an entire function \(f_{\epsilon}\) of exponential type that vanishes on \(\Lambda\) and has the property \[ \log | f_{\epsilon}(iy)| \leq \log | g(iy)| +\epsilon | y|, \] for all y lying outside a set of finite Lebesgue measure on R. (2) For every \(\epsilon >0\) there is a constant M such that for \(0<r<R<\infty\), \[ \ell_{\Lambda}(r,R)\leq \ell_{\Gamma}(r,R)+\epsilon \log (R/r)+M_{\epsilon}, \] where \(\ell\) denotes logarithmic measure. (3) For some entire function g (\(\not\equiv 0)\) of exponential type that vanishes on \(\Gamma\) and has no other zero in the right-hand half plane, for every \(\epsilon >0\) there is an entire function \(f_{\epsilon}\) of exponential type that vanishes on \(\Lambda\) and satisfies the inequality in (1) for \(y\in R\setminus E_{\epsilon}\), where the Lebesgue measure of \(E_{\epsilon}\) is finite.
Reviewer: R.P.Boas

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