BLOCH AND NORMAL FUNCTIONS

This work is a general and detailed study of the theories of normal meromorphic functions and of Bloch functions on the unit disk. After a brief account of their historical development, we start by treating the problem of characterizing normal meromorphic functions and Bloch functions. A necessary and sufficient condition for a meromorphic function to be normal (resp. Bloch in the analytic case), is given in terms of the Lipschitz condition with respect to the hyperbolic and the chordal (resp. euclidean) distances. A convergence theorem for normal (resp. Bloch) function is a straightforward consequence. After observing that exponentials of Bloch functions are normal, it is shown that not all analytic functions g on the unit disk satisfying the condition that "exp (g)" is normal for all in C'' are Bloch functions.;With the general problem in mind of characterizing the normal functions of bounded characteristic, we treat the case of the quotient of Blaschke products. We extend the condition for normality of a quotient of Blaschke products whose zeros form interpolating sequences, given by J. A. Cima and P. Colwell Proc. A.M.S. 19 (1968), 796-798 , to the case of zeros and poles of bounded multiplicities. It is also shown that a similar characterization does not apply in the case of unbounded multiplicities.;P. Lappan proved that the product of a normal unbounded analytic function (resp. a normal meromorphic function with infinitely many poles) and a Blaschke product need not be normal. The meromorphic functions which preserve normality under multiplication by a normal function are precisely those which are bounded and bounded away from zero near the unit circumference.