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.

翻译：对单变量函数在有限区间（称为信任域）上构造多项式上界或下界具有重要应用价值。经典方法通过约束信任域内k阶导数的取值范围生成k次多项式界，但这种方法得到的界并非最优。本文针对多种一维函数，推导出尖锐的多项式上下界。进一步证明：当信任域宽度趋近零时，k次尖锐界在渐近意义上至少比经典方法获得的界紧致k+1倍。最后探讨了这些尖锐界在极大极小化优化等场景中的应用。