On Iterated Entire Functions with ( )th Order

We prove some results relating to the comparative growth properties of iterated entire functions using (p, q)th order ((p, q)th lower order). 1. Introduction, Definitions, and Notations Let be an entire function defined in the open complex plane . The maximum term , the maximum modulus and Nevanlinna’s characteristic function of on are, respectively, defined as , and where for all . We do not explain the standard definitions and notations in the theory of entire function as those are available in [1]. In the sequel the following two notations are used: To start our paper we just recall the following definitions. Definition 1. The order and lower order of an entire function are defined as follows: Definition 2 (see [2]). Let be an integer . The generalised order and generalised lower order of an entire function are defined as When , Definition 2 coincides with Definition 1. Definition 3. A function is called a generalised proximate order of a meromorphic function relative to if(i) is nonnegative and continuous for ,(ii) is differentiable for except possibly at isolated points at which and exist,(iii) , (iv) ,(v) . The existence of such a proximate order is proved by Lahiri [3]. Similarly one can define the generalised lower proximate order of in the following way. Definition 4. A function is defined as a generalised lower proximate order of a meromorphic function relative to if(i) is nonnegative and continuous for ,(ii) is differentiable for except possibly at isolated points at which and exist,(iii) , (iv) ,(v) . Definitions 3 and 4 are both valid for entire . Juneja et al. [4] defined the th order and th lower order of an entire function , respectively, as follows: where are positive integers with . For and , we respectively denote and by and . Since for , {cf. [5]} It is easy to see that According to Lahiri and Banerjee [6] if and are entire functions, then the iteration of with respect to is defined as follows: , according to the fact that？？ ？？is odd or even, and so Clearly all and are entire functions. In this paper we would like to investigate some growth properties of iterated entire functions on the basis of their maximum terms, th order and th lower order where ,？？ are positive integers with . 2. Lemmas In this section we present some lemmas which will be needed in the sequel. Lemma 5 (see [7]). If and are any two entire functions then for all sufficiently large values of , Lemma 6. Let and be any two entire functions such that and where ,？？ ,？？ , and are any four positive integers with and . Then for any even number and for all sufficiently