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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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