Radial segments and conformal mapping of an annulus onto domains bounded by a circle and a k-circle

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 ${\displaystyle l\left({\phi }_1\right)+l\left({\phi }_2\right)}$ for fixed ${\displaystyle {\phi }_1}$ and ${\displaystyle {\phi }_2}$ ${\displaystyle (0\le {\phi }_1\le {\phi }_2\le 2\pi )}$.