Let p,q be positive real numbers with p less than 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 "(x^2)/(p^2)+(y^2)/(q^2)=1" has a maximum curvature of q/(p^2) and is an arc belonging to the class H(p,q), but I cannot prove-and do not think- that q/(p^2) is actually a greatest lower bound for the whole class (except in the case p=q which is specifically excluded).