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 ${\phi }_2$ $\left(0\le {\phi }_1\le {\phi }_2\le 2\pi \right)$.

We give some explicit values of the constants $C_1$ and $C_2$ in the inequality $C_1/sin\left(\frac{\pi }{p}\right)\le {\left|P\right|}_p\le C_2/sin\left(\frac{\pi }{p}\right)$ where ${\left|P\right|}_p$ denotes the norm of the Bergman projection on the $L^p$ space.

We study the existence of nodal solutions of the $m$-point boundary value problem $$u^{\text{'}\text{'}}+f\left(u\right)=0,0<t<1,u^{\text{'}}\left(0\right)=0,u\left(1\right)=\sum _{i=1}^{m-2}{\alpha }_{i}u\left({\eta }_i\right)$$ where ${\eta }_i\in \mathbb{Q}$ $(i=1,2,\cdots ,m-2)$ with $0<{\eta }_1<{\eta }_2<\cdots <{\eta }_{m-2}<1$, and ${\alpha }_i\in ℝ$ $(i=1,2,\cdots ,m-2)$ with ${\alpha }_i>0$ and $0<{\sum }_{i=1}^{m-2}{\alpha }_i<1$. We give conditions on the ratio $f\left(s\right)/s$ at infinity and zero that guarantee the existence of nodal solutions. The proofs of the main results are based on bifurcation techniques.