We consider finding the best possible constants $C = C(n,\alpha ,\beta ,p,q)$ such that $\| f \|_p \leqq C(b - a)^{n + 1/p -1/q} \| {f^{(n)} } \|_q $ under several differen sets of boundary conditions. Specifically in one case, f is required to have a zeros at $a $ and $\beta $ zeros at b where $n \leqq \alpha + \beta \leqq 2n$, $\alpha \leqq n$ and $\beta \leqq n$. In the other case, f has $\alpha $ zeros at $a $, $\beta $ zeros at b and $\int _a^b f(x)x^r (x - a)^\alpha (x - \beta )^\beta dx = 0$, $r = 0, \cdots ,n - 1 - \alpha - \beta $, where $0 \leqq \alpha $, $0 \leqq \beta $ and $\alpha + \beta < n$. The various problems are related and the numbers C are calculated exactly in some cases. Finally, we show how some of these can be applied to disconjugacy problems in differential equations.