  While the critical nonlinearity $\int |u|^{2^*}$ for the Sobolev space $H^1$
in dimension $N>2$ lacks weak continuity at any point, Trudinger-Moser
nonlinearity $\int e^{4\pi u^2}$ in dimension $N=2$ is weakly continuous at any
point except zero. In the former case the lack of weak continuity can be
attributed to invariance with respect to actions of translations and dilations.
The Sobolev space $H_0^1$ of the unit disk $\mathbb D\subset\R^2$ possesses
transformations analogous to translations (M\"obius transformations) and
nonlinear dilations $r\mapsto r^s$. We present improvements of the
Trudinger-Moser inequality with sharper nonlinearities sharper than $\int
e^{4\pi u^2}$, that lack weak continuity at any point and possess (separately),
translation and dilation invariance. We show, however, that no nonlinearity of
the form $\int F(|x|,u(x))\mathrm{d}x$ is both dilation- and M\"obius
shift-invariant. The paper also gives a new, very short proof of the
conformal-invariant Trudinger-Moser inequality obtained recently by Mancini and
Sandeep and of a sharper version of Onofri-type inequality of Beckner.
