Let f(z)=z+.... be regular and bounded in the unit circle |f(z)<=M and n points z1, z2, z3, .... zn, some of which may be superposed, be assigned within the unit circle. What conditions are necessary and sufficient, in order that, choosing a suitable value , there should exist a function f(z) which assumes at the points z1, z2, ...., zn?-We will first give an answer for this problem. Next, using the above results, we will give a new proof for an interesting theorem of J. Dieudonne on the

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... lency of a limited function. And finally we will state a theorem analogous to Dieudonne's. II. First we will state a lemma Lemma. Suppose that f(z)=z+.... is regular and bounded in the vni t circle : f(z) <=M. Suppose that f(z) assumes a given value ( 0) at the given points z1, z2, ....,zn within the circle. For the existence of , such a function f(z), it is necessary and sufficicnt that, putting t= /M,