does there exist a holomorphic function which has a multiple root whose fixed-point of the corresponding Newton map is not indifferent?

Question

Does there exist an analytic function $f(t)$ which has a root $\alpha$ with multiplicity > 1 but which the multiplier $|N'_f(t)|$ of the Newton map $N_f(t)=t-f(t)/f'(t)$ is not equal to 1?

The multiplier of a fixed point $\alpha$ of a map $f (x)$ where $f (\alpha) = \alpha$ is equal to the absolute value of the derivative of the map evaluated at the point $\alpha$. \begin{equation} \lambda_f (\alpha) = | \dot{f} (\alpha) | \end{equation} If $\lambda_f (\alpha) < 1$ then $\alpha$ is a said to be an attractive fixed-point of the map $f (x)$. If $\lambda_f (\alpha) = 1$ then $\alpha$ is an indifferent fixed point, and if $\lambda_f (\alpha) > 1$ then $\alpha$ is a repelling fixed-point. When $\lambda_f (\alpha) = 0$ the fixed-point $\alpha$ is said to be super-attractive

Answer 1

$f(z)$ is analytic and has a zero of multiplicity $m$ at $z=a$ means that for some function $h(z)$ analytic around $z=a$ : $$\log f(z) = h(z)+ m\log (z-a), \quad \frac{f'(z)}{f(z)} = h'(z)+\frac{m}{z-a}$$ $$N_f(z) =z- \frac{1}{h'(z)+\frac{m}{z-a}},\quad N_f'(z)= 1+\frac{h''(z)-\frac{m}{(z-a)^2}}{(h'(z)+\frac{m}{z-a})^2}$$ $$ \boxed{N_f'(a) = 1-\frac{1}{m}}$$

Answer 2

Let $g(t)=f(t)/f'(t)$ where $f(t)=tanh(t-2)^2*tanh(t-4)$ then $g(t)$ has the same roots of $f(t)$ but with only simple roots. Here $g(t)={\frac { \left( \tanh \left( t-2 \right) \right) ^{2}\tanh \left( t-4 \right) }{2\,\tanh \left( t-2 \right) \tanh \left( t-4 \right) \left( 1- \left( \tanh \left( t-2 \right) \right) ^{2} \right) + \left( \tanh \left( t-2 \right) \right) ^{2} \left( 1- \left( \tanh \left( t-4 \right) \right) ^{2} \right) }} $ then the multiplier of the root of $g$ at $t=2$ is equal to $1/2$ so that the multiplicity of $f(2)$ is $m=1/(1-1/2)=1/(1/2)=2$ , a formula which cannot be applied to $f$ directly due to its multiple root being an indifferent fixed-point of $N_f(x)$

Answer 3

Let $f(t)=tanh(t-2)^2*tanh(t-4)$ which has a double root at $t=2$ then the multiplier of the fixed-point at 2 is equal to 1/2 so its multiplicity is 1/(1-1/2)=2