Quasiadditive properties and bilipschitz conditions (Q1271587)

Let \({\mathcal H}\) be the class of all positive nondecreasing functions \(f\) on \((0,+\infty)\), and let \[ S_f(x,y)= \bigl(f(x) +f(y) \bigr)/f(x+y). \] A function \(f\in {\mathcal H}\) is quasiadditive if \[ \lambda_f= \inf_{x,y>0} S_f(x,y)>0. \] A function \(f\in {\mathcal H}\) is quasimultiplicative if there exist positive constants \(a\) and \(b\) such that \[ a\leq {f(x)f(y) \over f(xy)}\leq b \] for all \(x,y\in (0,+\infty)\). In the paper the authors present a series of properties of quasiadditive and quasimultiplicative functions. Various Hölderian growth estimates are obtained and functional inequalities are proved for these functions.