Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 8 (2017), pp. 4099–4109 © Research India Publications http://www.ripublication.com/gjpam.htm Some growth of composite entire and meromorphic functions with finite iterated order Samten Tamang Department of Mathematics, The University of Burdwan, Burdwan, Pin - 713104, West Bengal, India. Nityagopal Biswas Department of Mathematics, University of Kalyani, Kalyani, Dist. Nadia, PIN - 741235, West Bengal, India. Abstract The object of this paper is to investigate the growth properties of composite entire and meromorphic functions with finite iterated order. We have established some new results which are the improvement and extensions of the earliar results. AMS subject classification: 30D35, 30D30. Keywords: Entire function, Meromorphic function, Composition, Finite iterated order. 1. Introduction, Definitions and Notations We will assume that the reader is familiar with the standard notations and the fundamental results employed in the theory of entire and meromorphic functions, see [18] and [5]. Let f be a entire function with M (r, f ) = max |z|=r |f (z)|. A well known theorem of Polya [13] asserts that: If f and g are entire functions, then the composite function f ◦ g is of infinite order unless (a) f is of finite order and g is a polynomial or (b) f is of order zero and g is of finite order. For the two transcendental entire functions f (z) and g (z), Clunie [4] showed that lim r →∞ T (r, f ◦ g) T (r, f ) =∞ and lim r →∞ T (r, f ◦ g) T (r, g) =∞ .