Redshift at the beginning of current expansion epoch? The $q$ parameter is at present -ve which implies accelerated expansion , hence we can approximate our universe now to be dominated by dark energy since dark energy provides exponential growth $~e^{\alpha t}$
It is therefore after matter-dark energy equality time inflation starts e.g $$\epsilon(t)_d=\epsilon(t)_m,$$
$$\frac{\epsilon_{0m}}{a^3}=\epsilon_{0d},$$
$$a^3=\frac{\epsilon_{0m}}{\epsilon_{0d}},$$
$$a^3=\frac{\Omega_{m0}(.27)}{\Omega_{d0}(.73)},$$
with $a_e=0.71,$
then $$z=0.3908.$$
Can we say that at this redshift the inflation period of the universe starts and is continuing till now when z=0?
 A: Yes, you are right about the idea. In the calculation process, you can change your $\Omega_m$ and $\Omega_{\Lambda}$ values with respect to the 2018 Planck data 
$\Omega_m=0.3111$ and $\Omega_{\Lambda}=0.6889$ 
Hence we get
$$a(t)=0.76$$
so 
$$z=0.3157$$
I am not exactly sure about the inflation part. The dark energy is dominant for the last 4 billion years however I think its still early to say that the inflation period is started. I think we can still use the scale factor related to the matter-lambda universe, 
$$a(t)=(\Omega_m/\Omega_{\Lambda})^{1/3}sinh^{2/3}(t/t_{\Lambda})$$
for $t_{\Lambda}=\frac {2} {3H_0\sqrt {\Omega_{\Lambda}}}$
A: Inflation is just a model, and we don't have any proof that it actually happened. It is nowhere near as solidly tested by observation as other aspects of our cosmological models. Inflationary theories use crude models of the potential of the field that causes the inflation (Mexican hat potentials), and these crude models actually tend to have a hard time producing reasonable descriptions of the exit from inflation. In fact, the inability to get the exit to work is one of the main unsolved problems with inflationary theories at the present time.
So basically we have no idea whether inflation occurred, and if so, when it ended.
However, if inflation did occur, then it has to have ended before big bang nucleosynthesis started, so $z$ would be many orders of magnitude lower than 1.
