  The main results of this paper concern sharp constant of the Trudinger-Moser
inequality in $\mathbb{R}^{2}$ for Aharonov-Bohm magnetic fields. This is a
borderline case of the Hardy type inequalities for Aharonov-Bohm magnetic
fields in $\mathbb{R}^2$ studied by A. Laptev and T. Weidl. As an application,
we obtain the exact asymptotic estimates on best constants of magnetic
Hardy-Sobolev inequalities. In order to achieve our goal, we introduce a new
operator $T_{a}$ on the unit circle $\mathbb{S}^{1}$ and give the asymptotic
estimates of the heat kernel $e^{tT_{a}}$ via the Poisson summation formula.
Finally, we show that such Trudinger-Moser inequalities in the unit ball
$\mathbb{B}^{2}$ can be improved via subtraction of an additional Hardy term to
derive a Hardy-Trudinger-Moser inequality.
