Real Slice of Mandelbrot set
I'm trying to prove that the real slice of a Mandelbrot set is M n R =[-2,1/4]
I've tried researching it on the internet but everything I find seems to just state this and not prove it.
The hint I've been given to start is is to let
$\displaystyle c$\in$$(1/4, $\displaystyle $\infty$$) proving that for each $\displaystyle $z$$\in$R$ that $\displaystyle f$_{\text{c}}$(z)-z $\geq$ c -$1/4
where $\displaystyle f$_{\text{c}}$(z)= z^2+c$
Any help would be greatly appreciated
Re: Real Slice of Mandelbrot set
What you have written makes no sense. For any real number c, fc(z)- z is a complex number. You cannot say it is "larger than c- 1/4, a real number.
Perhaps you meant $\displaystyle \left|f_c(z)- z\right|\ge c- \frac{1}{4}$?