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

Let Ω be a compact surface with smooth boundary and the geodesic curvature kg c > 0 along Ω for some constant c R. We prove that, if the Gaussian curvature satisfies K -α for a constant α 0, then the first eigenvalue σ₁ of the Steklov-type eigenvalue problem satisfies \ σ₁ + ασ₁ c. \ Moreover, equality holds if and only if Ω is a Euclidean disk of radius 1c and α= 0. Furthermore, we obtain a sharp lower bound for the first eigenvalue of the fourth-order Steklov-type eigenvalue problem on Ω.