ELEMENTARY PROOF OF A THEOREM OF P. MONTEL ON ENTIRE FUNCTIONS A. V. JATEGAONKAR 1. Introduction Let/(z) be an analytic function, regular in a simply-connected domain D. A complex number a is called a completely p-multiple value off(z) in D, p being an integer > 1, if all a-points of f(z) in D have multiplicities divisible by p. f(z) is said to be of rank (p, q, r) in D, where 1 1 < 1, if there exist three distinct values a, b and c such that a is completely ^-multiple in D, b is completely ^-multiple in D, and c is completely r-multiple in D. P. Mcntel [1,2] has proved, using Schwarz functions, the following theorem: A non-constant meromorphic function cannot be of rank (p, q,r). In the case of entire functions, he has remarked [2] that it would be interesting to give an elementary proof of the theorem. We shall in fact show how this problem can be reduced very simply to the Pi card theorem in the form that an equation exp[/(z)]+exp[0(z)] = l, where/(z) and g(z) are non-constant entire functions, cannot hold. We also prove an analogue of Montel's theorem for regular functions in unit circle and of order p>\. These results, except for Theorem 4, are all well-known consequences from Nevanlinna theory [3]. However, it is hoped that the elementary proofs of this paper will be of some interest. 2. We have THEOREM 1. A non-constant entire function cannot have a completely p-multiple value a and a completely q-multiple value b, ivhere a^b and p q LEMMA 1. Let F(z) and G(z) be non-constant entire functions. Then, if p is an integer, and p> 2 the equation }v=l (1) cannot hold. Received 20 May, 1963; revised 18 February, 1964. [JOUKNAL LONDON MATH. SOC, 40 (1965), 166-170]