A Trudinger-Moser inequality with mean value zero on a compact Riemann surface with boundary

In this paper, on a compact Riemann surface $(\Sigma, g)$ with smooth boundary $\partial\Sigma$, we concern a Trudinger-Moser inequality with mean value zero. To be exact, let $\lambda_1(\Sigma)$ denotes the first eigenvalue of the Laplace-Beltrami operator with respect to the zero mean value condition and $\mathcal{ S }= \left\{ u \in W^{1,2} (\Sigma, g) : \|\nabla_g u\|_2^2 \leq 1\right.$ and $\left.\int_\Sigma u \,dv_g = 0 \right \},$ where $W^{1,2}(\Sigma, g)$ is the usual Sobolev space, $\|\cdot\|_2$ denotes the standard $L^2$-norm and $\nabla_{g}$ represent the gradient. By the method of blow-up analysis, we obtain \begin{eqnarray*} \sup_{u \in \mathcal{S}} \int_{\Sigma} e^{ 2\pi u^{2} \left(1+\alpha\|u\|_2^{2}\right) }d v_{g} <+\infty, \ \forall \ 0 \leq\alpha<\lambda_1(\Sigma); \end{eqnarray*} when $\alpha \geq\lambda_1(\Sigma)$, the supremum is infinite. Moreover, we prove the supremum is attained by a function $u_{\alpha} \in C^\infty\left(\overline{\Sigma}\right)\cap \mathcal {S}$ for sufficiently small $\alpha> 0$. Based on the similar work in the Euclidean space, which was accomplished by Lu-Yang \cite{Lu-Yang}, we strengthen the result of Yang \cite{Yang2006IJM}.