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The famous Yang-Yau inequality provides an upper bound for the firsteigenvalue of the Laplacian on an orientable Riemannian surface solely in termsof its genus $\gamma$ and the area. Its proof relies on the existence ofholomorhic maps to $\mathbb{CP}^1$ of low degree. Very recently, A.~Ros wasable to use certain holomorphic maps to $\mathbb{CP}^2$ in order to give aquantitative improvement of the Yang-Yau inequality for $\gamma=3$. In thepresent paper, we generalize Ros' argument to make use of holomorphic maps to$\mathbb{CP}^n$ for any $n>0$. As an application, we obtain a quantitativeimprovement of the Yang-Yau inequality for all genera $\gamma>3$ except for$\gamma = 4,6,8,10,14$.

The first eigenvalue of the Laplacian on orientable surfaces