# entire function which maps _Horizontal line_ to _Horizontal line_

I know $Aut(\mathbb{C})=\{f(z): f(z)=az+b\}$ for some constant $a,b$,$a\in\mathbb{R}$, my question is if an entire function which maps Horizontal line to Horizontal line then it is of the form $az+b$? or in general $f$ maps any lines to to lines then is it of the form $az+b$?

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An entire function $f$ that maps horizontal lines to horizontal lines must have a derivative $f'$ that is real and $\ne0$ at all points $z\in\Bbb C$. It follows that $f'$ has to be a real nonzero constant. Therefore the most general $f$ satisfying your description is of the form $$f(z)=\sigma z+ c,\qquad \sigma\in\dot{\mathbb R},\quad c\in \Bbb C\ .$$

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Any function that maps horizontal lines to horizontal lines must be of the form

$$f(x + \mathbf{i} y) = u(x,y) + \mathbf{i} v(y)$$

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why ?add detail please. – La Belle Noiseuse Mar 22 '13 at 17:06
.... A horizontal line is completely determined by its imaginary part. – Hurkyl Mar 23 '13 at 3:53