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 𝐿𝐿 ∗ -order and generalised 𝐿𝐿 ∗ -type and Wronskians generated by one of the factors.


1� �ntroduction, De�nitions, and �otations
We denote by ℂ the set of all �nite complex numbers. Let be a meromorphic function de�ned on ℂ. We use the standard notations and de�nitions in the theory of entire and meromorphic functions which are available in [1] and [2]. In the sequel we use the following notation: log [ log(log [ for . and log [0 . e following de�nitions are well known.

De�nition 1. A meromorphic function
( is called small with respect to if ( ( .
is called the �evanlinna de�ciency of the value � . " From the second fundamental theorem it follows that the set of values of ℂ , the quaintity for which ( ; 0 is countable and ∑ ( ; ( ; (cf. [1, page 4 ]). If, in particular, ∑ ( ; ( ; , we say that has the maximum de�ciency sum.
Let ( be a positive continuous function increasing slowly, that is, ( ( as for every positive constant . Singh and Barker [3] de�ned it in the following way.

Journal of Complex Analysis
If further, is differentiable, the above condition is equivalent to Somasundaram and amizharasi [4] introduced the notions of -order and -type for entire functions. e more generalised concept for -order and -type for entire and meromorphic functions are * -order and * -type, respectively. In the line of Somasundaram and amizharasi [4], for any positive integer one may de�ne the generalised * -order [ * (generalised * -lower order [ * and generalised * -type [ * in the following manner.
�e�nition �. e generalised * -order [ * and the generalised * -lower order [ * of an entire function are de�ned as When is meromorphic, it can be easily veri�ed that �e�nition �. e generalised * -type [ * of an entire function is de�ned as follows� For meromorphic , For = , we may get the classical cases {cf.
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 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.

Lemmas
In this section we present some lemmas which will be needed in the sequel.
Lemma 7 (see [7]). If be meromorphic and be entire then for all sufficiently large values of , Lemma 8 (see [8]). Let be meromorphic and entire and suppose that 0 < < ≤ . en for a sequence of values of tending to in�nity, Lemma 9 (see [9]). Let be a transcendental meromorphic function having the ma�imum de�ciency sum. en Lemma 10 (see [10]). If be a transcendental meromorphic function having the ma�imum de�ciency sum. en the generalised * -order (generalised * -lower order) of and that of are the same.

Lemma 11. Let be a transcendental meromorphic function having the ma�imum de�ciency sum. en
Proof. By Lemmas 9 and 10 we get that * Also by Lemma 9, lim log [ log [ exists and is equal to for . erefore us the lemma follows.
Journal of Complex Analysis 3

Theorems
In this section we present the main results of the paper. where 0 < < and Proof. Let 0 < < ′ < . �sing the de�nition of generalised * -lower order we obtain in view of Lemma Now from (17) and (18) and we obtain from (21) is is a contradiction. is proves the theorem.
Remark 17. eorem 16 is also valid with "limit superior" instead of "limit" if [ * ∞ is replaced by [ * ∞ and the other conditions remain the same. We omit the proof of eorem 22 because it can be carried out in the line of eorem 16.
Remark 23. eorem 22 is also valid with "limit superior" instead of "limit" if [ * is replaced by [ * and the other conditions remain the same.
In the line of Corollary 18 we may easily verify the following.
As ( ) is arbitrary, we obtain from the above that us the second part of eorem 27 follows.