  It is often useful to have polynomial upper or lower bounds on a
one-dimensional function that are valid over a finite interval, called a trust
region. A classical way to produce polynomial bounds of degree $k$ involves
bounding the range of the $k$th derivative over the trust region, but this
produces suboptimal bounds. We improve on this by deriving sharp polynomial
upper and lower bounds for a wide variety of one-dimensional functions. We
further show that sharp bounds of degree $k$ are at least $k+1$ times tighter
than those produced by the classical method, asymptotically as the width of the
trust region approaches zero. We discuss how these sharp bounds can be used in
majorization-minimization optimization, among other applications.
