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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).

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).

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

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).