Can smooth LTB models mimicking the cosmological constant for the
  luminosity distance also satisfy the age constraint?
  The central smoothness of the functions defining a LTB solution plays a
crucial role in their ability to mimick the effects of the cosmological
constant. So far attention has been focused on $C^{1}$ models while in this
paper we approach it a more general way, investigating the implications of
higher order central smoothness conditions for LTB models reproducing the
luminosity distance of a $\Lambda CDM$ Universe. Our analysis is based on a low
red-shift expansion, and extends previous investigations by including also the
constraint coming from the age of the Universe and re-expressing the equations
for the solution of the inversion problem in a manifestly dimensionless form
which makes evident the freedom to accommodate any value of $H_0$ as well,
correcting some wrong claims that the observed value of $H_0$ would be enough
to rule out LTB models.
  Higher order smoothness conditions strongly limit the number of possible
solutions respect to the first order condition. Neither a $C^{1}$ or a $C^{i}$
LTB model can both satisfy the age constraint and mimick the cosmological
constant for the luminosity distance. One difference is in the case in which
the age constraint is not included and the bang function is zero, in which
there is a unique solution for $C^1$ models but no solution for the $C^{i}$
case. Another difference is in the case in which the age constraint is not
included and the bang function is not zero, in which the solution is
undetermined for both $C^1$ and $C^{i}$ models, but the latter ones have much
less residual parametric freedom.
  Our results imply that any LTB model able to fit luminosity distance data and
satisfy the age constraint is either not mimicking exactly the $\Lambda CDM$
red-shift space observables theoretical predictions or it is not $C^{\infty}$
smooth.
