Does modified Morlet wavelet function satisfy the admissibility condition?
No, it doesn't (at least not in theory). You know that the admissibility condition is

$$\int_{-\infty}^{\infty}\frac{|\Psi(\omega)|^2}{|\omega|}d\omega&lt;\infty$$

where $\Psi(\omega)$ is the Fourier transform of the mother wavelet $\psi(t)$. If $\Psi(\omega)$ decays sufficiently fast (which is the case for the Morlet wavelet) then the admissibility condition reduces to $\Psi(0)=0$, i.e. $\psi(t)$ must satisfy a zero-mean condition. However, this condition is not satisfied for the Morlet wavelet. So the Morlet wavelet is not admissible, but in practice this is no problem because $\Psi(0)$ is very small.

[1] M. Vetterli, J. Kovačević: Wavelets and Subband Coding