Growth Rates of Meromorphic Functions Focusing Relative Order

A detailed study concerning some growth rates of composite entire and meromorphic functions on the basis of their relative orders (relative lower orders) with respect to entire functions has been made in this paper. 1. Introduction, Definitions and Notations Let be a meromorphic and let be an entire function defined in the open complex plane and . If is nonconstant, then is strictly increasing and continuous and its inverse exists and is such that . We use the standard notations and definitions in the theory of entire and meromorphic functions those are available in [1, 2]. To start our paper we just recall the following definitions. Definition 1. The order (the lower order ) of an entire function is defined as If is a meromorphic function, one can easily verify that where is the Nevanlinna's characteristic function of (cf. [1]). Bernal [3, 4] introduced the definition of relative order of an entire function with respect to another entire function denoted by as follows: The definition coincides with the classical one [5] if . Similarly, one can define the relative lower order of an entire function with respect to another entire function denoted by as follows: Extending this notion, Lahiri and Banerjee [6] introduced the definition of relative order of a meromorphic function with respect to an entire function in the following way. Definition 2 (see [6]). Let be any meromorphic function and let be any entire function. The relative order of with respect to is defined as Likewise, one can define the relative lower order of a meromorphic function with respect to an entire function denoted by as follows: It is known (cf. [6]) that if , then Definition 2 coincides with the classical definition of the order of a meromorphic function . In this paper we wish to prove some results related to the growth rates of composite entire and meromorphic functions on the basis of relative order (relative lower order) of meromorphic functions with respect to an entire function. 2. Lemmas In this section we present some lemmas which will be needed in the sequel. Lemma 1 (see [7]). Let be meromorphic and let be entire and suppose that . Then for a sequence of values of tending to infinity, Lemma 2 (see [8]). Let be meromorphic and let be entire such that and . Then for a sequence of values of tending to infinity, where . Lemma 3 (see [9]). Let be a meromorphic function and let be an entire function such that and . Then for a sequence of values of tending to infinity, Lemma 4 (see [9]). Let be a meromorphic function of finite order and let be an entire function such that . Then