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On entire functions of infinite order

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On entire functions of infinite order COMPOSITIO MATHEMATICA MANSOORAHMAD On entire functions of infinite order Compositio Mathematica, tome 13 (1956-1958), p. 159-172. <http://www.numdam.org/item?id=CM_1956-1958__13__159_0> © Foundation Compositio Mathematica, 1956-1958, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ On Entire Functions of Infinite Order by Mansoor Ahmad 1. Introduction. The purpose of this paper is to extend to a class of entire functions of infinite order some theorems on entire functions of finite order. Theorems 1 and 2 are formal analogues of two theorems [1] and [2] of Shah. Theorems 3, 4 and 5 are new; but they are closely connected with some theorems [3] of Shah. Theorem 6 is an analogue of a theorem of Lindelôf [4]. 2. DEFINITIONS. We define the k-th order and the k-th lower order of an entire or meromorphic function as and Similarly, we define the k-th order and the k-th lower order of the zeros of f(z) as and where T(r), n(r) have their usual meanings and llx = log x, l2x = log log x, and so on. 3. LEMMA (i) If x(x) is a positive function continuous almost every where in every interval (rp, r); and if then 160 where LEMMA (ii) If X(x) and e(r) are the same functions as before; and if then PROOF. If f(x) and g(x) are two positive functions which tend to infinity with x; and if each of the functions is differentiable almost every where in every interval (ro, r), such that their derivatives f’(x) and g’(x) have a definite finite value at every p

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