%0 Journal Article
%T Growth Analysis of Composite Entire Functions Related to Slowly Changing Functions Oriented Relative Order and Relative Type
%A Sanjib Kumar Datta
%A Tanmay Biswas
%A Sarmila Bhattacharyya
%J Journal of Complex Analysis
%D 2014
%I Hindawi Publishing Corporation
%R 10.1155/2014/503719
%X Some results on comparative growth properties of maximum terms and maximum moduli of composite entire functions on the basis of relative -order and relative -type are proved in this paper. 1. Introduction, Definitions, and Notations We denote by the set of all finite complex numbers. Let be an entire function defined in the open complex plane . The maximum term of on is defined by and the maximum modulus of on is defined as . Let be a positive continuous function increasing slowly, that is, as for every positive constant . Singh and Barker [1] defined it in the following way. Definition 1 (see [1]). A positive continuous function is called a slowly changing function if, for £¿£¿(>0), and uniformly for £¿£¿(¡İ1). If, further, is differentiable, the above condition is equivalent to Somasundaram and Thamizharasi [2] introduced the notions of -order and -type for entire function where is a positive continuous function increasing slowly, that is, as for every positive constant ¡° ¡±. The more generalized concepts for -order and -type for entire functions are -order and -type. Their definitions are as follows. Definition 2 (see [2]). The -order and the -lower order of an entire function are defined as where for and . Using the inequalities £¿£¿{cf. [3]}, for£¿£¿ , one may verify that Definition 3 (see [2]). The -type of an entire function is defined as If an entire function is nonconstant then is strictly increasing and continuous and its inverse exists and is such that . Bernal [4] introduced the definition of relative order of an entire function with respect to an 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 an entire function denoted by as follows: Datta and Maji [6] gave an alternative definition of relative order and relative lower order in terms of maximum term of an entire function with respect to another entire in the following way. Definition 4 (see [6]). The relative order and relative lower order of an entire function with respect to an entire function are defined as follows: In the line of Somasundaram and Thamizharasi [2] and Bernal [4], one may define the relative -order of an entire function in the following manner. Definition 5 (see [7, 8]). The relative -order and relative -lower order of an entire function with respect to another entire function are defined as Datta et al. [9] also gave an alternative definition of -order and relative -lower order in terms of maximum term of an entire function which are as
%U http://www.hindawi.com/journals/jca/2014/503719/