This is a simple question I asked in math.SE last month but unfortunately no one gives any comment. So I decided to try some luck here.

 
Given a smooth arc (part of an ellipse actually) on the complex plane by 
   > $z=\cos t + 0.5 i \sin t,\; t\in[\pi/10,\pi/5] $ , 

and a non-analytic function 
$f(z) = \text{Re } z $  defined on the arc.
Obviously, $f(z) = g(t) := \cos t.$  
Suppose we compute the "derivatives" of $f$ on the arc recursively by   
$f'(z) = g'(t)/z'(t),\quad$
$f''(z) = \dfrac{df'(z)}{dt}\dfrac{1}{z'(t)},\quad$
$f'''(z) = \dfrac{df''(z)}{dt}\dfrac{1}{z'(t)},\quad \dots$

Is there an estimate on the upper bound of magnitude of $n^\text{th}$ order derivative of  $f$ ? 
For example, can we show something like 

> $|f^{(n)}(z)|\leq C n! r^n $, where $C$ and $r$ are positive constants independent of $n$  ?

Note that in the case above the form of $f$ is really simple. 
If $f$ is more complicated, for  example, $f\circ z(t) := \frac{|z'(t)|}{z'(t)}$,
what can we say about $|f^{(n)}(z)|$ ?

**Update:** According To Fedor's answer, function $f$ in first example actually coincides with an analytic function on the arc. I need to modify the example later so that no analytic function coincides with $f$ on the arc. (I did not verify that $f$ in the second example is genuinely non-analytic yet.)
Moreover, if you can construct such a function $f$, please let me know and I will update the example. Thank you!