Growth Analysis of Wronskians in terms of Slowly Changing Functions

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