Let $p$ and $q$ be positive real numbers with $p \leq q$. Suppose that $H(p,q)$ is the class of all convex arcs $c$ in the Cartesian $x-y$ plane which satisfy the following conditions:

(1)The $y$-axis is an axis of symmetry of $c$.

(2)The points $(-p,0)$ and $(p,0)$ are the end-points of $c$ and the point $(0,q)$ is also a point of $c$.

(3)The curvature of $c$ is defined and continuous at each point of $c$ and never changes sign (which can always be taken to be non-negative).

QUESTION:Given $p$ and $q$ is there a formula for the greatest lower bound of the maximum curvature that an arc $c$ belonging to the class $H(p,q)$ can have? Is there a particular arc in this class which actually attains this greatest lower bound?

The (upper half of the) ellipse whose equation is $\frac{x^2}{p^2}+\frac{y^2}{q^2}=1$ has a maximum curvature of $\frac{q}{p^2}$ and is an arc belonging to the class $H(p,q)$, but I cannot prove-and do not think- that $\frac{q}{p^2}$ is actually a greatest lower bound for the whole class (except in the case $p=q$ which is specifically excluded).