In this paper, the author concerns two trace Trudinger-Moser inequalities and obtains the corresponding extrema] functions on a compact Riemann surface (Sigma, g) with smooth boundary partial derivative Sigma. Explicitly, let lambda(1)(partial derivative Sigma) = inf(u is an element of W1,2(Sigma,g),integral partial derivative Sigma udsg=0, u not equivalent to 0) integral(Sigma)(vertical bar del(g)u vertical bar(2)+u(2))dv(g)/integral(partial derivative Sigma)u(2)ds(g) and H = { u is an element of W-1,W-2(Sigma,g) : integral(Sigma)(vertical bar del(g)u vertical bar(2)+ u(2))dv(g)-alpha integral(partial derivative Sigma)u(2)ds(g) <= 1 and integral(partial derivative Sigma)uds(g) = 0}, where W-1,W-2(Sigma, g) denotes the usual Sobolev space and del(g) stands for the gradient operator. By the method of blow-up analysis, we obtain sup(u is an element of H)integral(partial derivative Sigma) e(pi u2) ds(g ){ < + infinity, 0 <= alpha < lambda 1 (partial derivative Sigma), = + infinity, alpha >= lambda(1) (partial derivative Sigma). Moreover, the author proves the above supremum is attained by a function u(alpha) is an element of H boolean AND C-infinity((Sigma) over bar) for any 0 <= alpha < lambda(1) (partial derivative Sigma). Further, he extends the result to the case of higher order eigenvalues. The results generalize those of [Li, Y. and Liu, P., Moser-Trudinger inequality on the boundary of compact Riemannian surface, Math. Z., 250, 2005, 363-386], [Yang, Y., Moser-Trudinger trace inequalities on a compact Riemannian surface with boundary, Pacific J. Math., 227, 2006, 177-200] and [Yang, Y., Extremal functions for TrudingerMoser inequalities of Adimurthi-Druet type in dimension two, J. Diff. Eq., 258, 2015, 3161-3193].
