  Let $\Omega$ be a smooth bounded domain in $\mathbf R^2$ and
$\lambda^{\mathsf N} (\Omega)$ the first non-zero Neumann eigenvalue of the
operator $-\Delta$ on $\Omega$. In this paper, for any $\gamma \in [0,
\lambda^{\mathsf N} (\Omega) )$, we establish the following improved
Moser-Trudinger inequality \[ \sup_{u} \int_{\Omega} e^{2\pi u^2} dx < +\infty
\] for arbitrary functions $u$ in $H^1(\Omega)$ satisfying $\int_\Omega u dx
=0$ and $\|\nabla u\|_2^2 -\alpha \|u\|_2^2 \leqslant 1$. Furthermore, this
supremum is attained by some function $u^*\in H^1(\Omega)$. This strengthens
the results of Chang and Yang (J. Differential Geom. 27 (1988) 259-296) and of
Lu and Yang (Nonlinear Anal. 70 (2009) 2992-3001).
