A CRITERION FOR THE NORMALITY OF A FAMILY OF MEROMORPHIC FUNCTIONS

Let D be a domain in the complex plane. A family I of meromorphic functions on D is said to be normal if every sequence of functions in F contains a subsequence which converges uniformly on compact subsets of D to function / which is mero-morphic or identically-, th € convergence being with respect to the spherical metric 46:ldwllQ+ lwl'). A well known criterion of Marty (cf. [], p. 226) asserts that g is normal if, and only if, for each compact subset KcD therc is constant C * such that l.f'Q)l = cKt+lf (z)|'z) for all f(F and all zQK. Although this criterion is necessary and sufficientn it does not tell whether the family of functions in D satisfying, for example, I f'l ettt is normal or not. The purpose of the present note is to prove the following strengthen-ing of the sufficiency part of Marty's criterion: Theorem. Let I be afamily of meromarphicfunctions on D with the prop-erty that for each compact set KcD there is a monotone increasing fimction h* such that v,e)r h*qf(zii') for all f(F and all z€K. Then F is normal. Proof. Since the property of being normal is.a local one, it suffices to consider the case when lf'(z\l=n(lf(z)l) in a domain /. Since this inequality holds a fortiori for any larger h, we may assume that h is continuous and that h(t)-l*/2. For a differentiable curve y on the Riemann sphere C we define the length of y b