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 \[ σ_1 + α{σ_1} ≥ c. \] Moreover, equality holds if and only if $Ω$ is a Euclidean disk of radius 1/c and $α= 0$. Furthermore, we obtain a sharp lower bound for the first eigenvalue of the fourth-order Steklov-type eigenvalue problem on $Ω$.