Isoperimetric inequalities for immersed closed spherical curves

Let $\alpha :{S^1} \to {S^2}$ be a ${C^2}$ immersion with length $L$ and total curvature $K$. If $\alpha$ is regularly homotopic to a circle traversed once then ${L^2} + {K^2} \geqslant 4{\pi ^2}$ with equality if and only if $\alpha$ is a circle traversed once. If $\alpha$ has nonnegative geodesic curvature and multiple points then $L + K \geqslant 4\pi$ with equality if and only if $\alpha$ is a great circle traversed twice.