Abstract :
We consider positive functionsh=h(x) defined forx∈R+0. Conditions for the existence of a power seriesN(x)=∑ cnxn,cn⩾0, with the propertyd1⩽h(x)/N(x)⩽d2, x⩾0,for some constantsd1, d2∈R+, are investigated in [J. Clunie and T. Kövari,Canad. J. Math.20(1968), 7–20; P. Erdős and T. Kövari,Acta Math. Acad. Sci. Hung.7(1956), 305–316; U. Schmid,Complex Variables18(1992), 187–192; U. Schmid, J.Approx. Theory83(1995), 342–346]. In this paper, methods are discussed which allow for a given functionhthe construction of the coefficientscn,n∈N0, for the above defined power seriesNand to find suitable constantsd1andd2. We also study the power seriesH(x)=∑ xn/un, where we setun=sup{xn/h(x), x⩾0}, forn∈N0, and the relation betweenhandHconcerning the above stated inequalities.
