I am given $h(x)$ a positive continuous function on the real line. I want to show that there exists a non0zero entire function $f(z)$ such that $0\leq |f(x)|<|h(x)|$ for all real value $x$. I am trying to use only Runge's theorem, meaning without using Mergelyan's theorem.
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Has $f$ to be continuos? And what does it mean for you entire function? – Federico Fallucca Jun 15 '22 at 13:22
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@FedericoFallucca Holomorphic at every point in $\Bbb C$ – FShrike Jun 15 '22 at 15:08
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$0\le f(x)\lt h(x)$ has no meaning unless $f$ is always real valued when $x$ is real. Did you mean instead $0\le|f(x)|\le h(x)$? In which case the answer is false when $h(x)$ is a bounded function – FShrike Jun 15 '22 at 15:09