%0 Journal Article
%T Growth Analysis of Wronskians in terms of Slowly Changing Functions
%A Sanjib Kumar Datta
%A Tanmay Biswas
%A Sultan Ali
%J Journal of Complex Analysis
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/395067
%X In the paper we establish some new results depending on the comparative growth properties of composite entire or meromorphic functions using generalised -order and generalised -type and Wronskians generated by one of the factors. 1. Introduction, Definitions, and Notations We denote by the set of all finite complex numbers. Let be a meromorphic function defined on . We use the standard notations and definitions in the theory of entire and meromorphic functions which are available in [1] and [2]. In the sequel we use the following notation: for . and . The following definitions are well known. Definition 1. A meromorphic function is called small with respect to if . Definition 2. Let be linearly independent meromorphic functions and small with respect to . We denote by the Wronskian determinant of , that is, Definition 3. If , the quantity is called the Nevanlinna deficiency of the value ¡° .¡± From the second fundamental theorem it follows that the set of values of , the quaintity for which is countable and (cf. [1, ]). If, in particular, , we say that has the maximum deficiency sum. Let be a positive continuous function increasing slowly, that is, as for every positive constant . Singh and Barker [3] defined it in the following way. Definition 4 (see [3]). A positive continuous function is called a slowly changing function if, for , uniformly for . If further, is differentiable, the above condition is equivalent to Somasundaram and Thamizharasi [4] introduced the notions of -order and -type for entire functions. The more generalised concept for -order and -type for entire and meromorphic functions are -order and -type, respectively. In the line of Somasundaram and Thamizharasi [4], for any positive integer one may define the generalised -order (generalised -lower order and generalised -type in the following manner. Definition 5. The generalised -order and the generalised -lower order of an entire function are defined as When is meromorphic, it can be easily verified that Definition 6. The generalised -type of an entire function is defined as follows: For meromorphic , For , we may get the classical cases {cf. [4]} of Definitions 5 and 6, respectively. Lakshminarasimhan [5] introduced the idea of the functions of L-bounded index. Later Lahiri and Bhattacharjee [6] worked on the entire functions of L-bounded index and of nonuniform L-bounded index. Since the natural extension of a derivative is a differential polynomial, in this paper we prove our results for a special type of linear differential polynomials, namely, the Wronskians. In the paper we
%U http://www.hindawi.com/journals/jca/2013/395067/