Two-sided estimates for the relative growth of functions and their derivatives

We provide an extended presentation of a talk given at the International mathematical conference on theory of functions dedicated to centenary of corresponding member of AS USSR A.F. Leont’ev. We propose a new method for obtaining uniform two-sided estimates for the fraction of the derivatives of two real functions on the base of the information of two-sided estimates for the functions themselves. At that, one of the functions possesses certain properties and serves as a reference for measuring a growth and introduces some scale. The other function, whose growth is compared with that of the reference function, is convex, increases unboundedly or decays to zero on a certain interval. The method is also applicable to some class of functions concave on an interval. We consider examples of applications of the obtained results to the behavior of entire functions.