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 Jaket1 March 11th, 2018 06:23 AM

Starlike functions proof

Show the following regions are starlike:

The open right half plane $z\in\mathbb{C}:Re(z)>0$

The cut plane $\mathbb{C}=\mathbb{C}/(-\infty,0]$

This seems easy intuitively. However, how would I actually prove that these functions are starlike? Could I create a parameter and show that every arbitrary point from the starcentre is reached?

 Country Boy April 25th, 2018 03:52 AM

Well the first thing you should do is choose a point to be the "centre". Then, yes, it should be easy to write the equation of a straight line from that "centre" to any other point, (x, y), in the set.

 Country Boy April 25th, 2018 08:49 AM

A couple of points: a set (not a function) is "starlike" if there exist a point (not necessarily unique), $\displaystyle z= x_0+ y_0i$, such that the straight line from $\displaystyle z= x_0+ y_0i$ to any point, x+ iy, in the set, that lies entirely in the set. For both of these problems, I think I would choose that point to be z= 1+ 0i. What is the line from 1+ 0i to x+ yi? That will, of course, be a linear function of some parameter, t. Then show every point on that line segment is in the given set.

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