Because you are using the principal branch of log, you take all the arguments of the numbers sqrt(z) from , therefore there is no z whose square root sqrt(z) lies on the negative real axis.
So log(sqrt(z)) analicity region is
The region of analyticity of log(sqrt(z)). (using principle branch)
I know the domain of log(z) is the complex plane minus the origin and the negative x-axis (C\{x+iy | x =< 0, y=0}). I'm not sure what domain of sqrt(z) produces this as an image, though. I know z --> sqrt(z) cuts angles in half... sooo would the domain be everything but the positive x-axis? I'm a little confused
So you get more idea of whats happening see this applet:
Math 132 Applet 4
choose sqrt as a function and select principal branch.
You will see how there isnt a single point z whose image sqrt lies on the negative real line.