Let f(z) be a conformal mapping of an annulus A(R) = 1 < |z| < R and let f(A(R)) be a ring domain bounded by a circle and a k-circle. If R(φ) = w : arg w = φ, and l(φ) - 1 is the linear measure of f(A(R)) ∩ R(φ), then we determine the sharp lower bound of $l\left({\phi }_1\right)+l\left({\phi }_2\right)$ for fixed ${\phi }_1$ and ${\phi }_2$$(0\le ...$