Sharp lower bounds for the first eigenvalue of Steklov-type eigenvalue problems on a compact surface

Let $\mathrm{\Omega }$ be a compact surface with smooth boundary and the geodesic curvature $k_g\ge c>0$ along $\mathrm{\partial }\mathrm{\Omega }$ for some constant $c\in \mathbb{R}$. We prove that, if the Gaussian curvature satisfies $K\ge -\alpha$ for a constant $\alpha \ge 0$, then the first eigenvalue ${\sigma }_1$ of the Steklov-type eigenvalue problem satisfies $${\sigma }_1+\frac{\alpha }{{\sigma }_1}\ge c\mathrm{.}$$ Moreover, equality holds if and only if $\mathrm{\Omega }$ is a Euclidean disk of radius $\frac{1}{c}$ and $\alpha =0$. Furthermore, we obtain a sharp lower bound for the first eigenvalue of the fourth-order Steklov-type eigenvalue problem on $\mathrm{\Omega }$.