Applications of Slowly Changing Functions in the Estimation of Growth Properties of Composite Entire Functions on the Basis of their Maximum Terms and Maximum Moduli

In the paper we prove some comparative growth properties of composite entire functions on the basis of their maximum terms and maximum moduli using generalised L∗-order and generalised L∗-lower order. Mathematics Subject Classification: 30D35, 30D30 238 Sanjib Kumar Datta, Tanmay Biswas and Azizul Haque

Let C be the set of all finite complex numbers and f be an entire function tions in the theory of entire functions which are available in [11].In the sequel we use the following notation : log [k] x = log log [k−1] x for k = 1, 2, 3, .... and log [0] x = x.
To start our paper we just recall the following definitions : Definition 1 The order ρ f and lower order λ f of an entire function f are defined as ρ f = lim sup r→∞ log [2] M(r, f ) log r and λ f = lim inf r→∞ log [2] M(r, f ) log r .
Extending this notion, Sato [6] defined the generalised order and generalised lower order of an entire function as follows : Definition 2 [6]Let m be an integer ≥ 2. The generalised order ρ For m = 2, Definition 2 reduces to Definition 1.If ρ f < ∞ then f is of finite order.Also ρ f = 0 means that f is of order zero.In this connection Datta and Biswas [2] gave the following definition : Definition 3 [2]Let f be an entire function of order zero.Then the quantities ρ * * f and λ * * f of f are defined by: Let L ≡ L (r) be a positive continuous function increasing slowly i.e., L (ar) ∼ L (r) as r → ∞ for every positive constant a. Singh and Barker [7] defined it in the following way: uniformly for k (≥ 1) .
If further, L (r) is differentiable, the above condition is equivalent to Somasundaram and Thamizharasi [8] introduced the notions of L-order (L-lower order ) for entire functions where L ≡ L (r) is a positive continuous function increasing slowly i.e.,L (ar) ∼ L (r) as r → ∞ for every positive constant 'a'.The more generalised concept for L-order ( L-lower order ) for entire function are L * -order ( L * -lower order ).Their definitions are as follows: Definition 5 [8]The L * -order ρ L * f and the L * -lower order λ L * f of an entire function f are defined as r) ] and λ L * f = lim inf r→∞ log [2] M (r, f ) log [re L(r) ] .
In the line of Sato [6] , Datta and Biswas [2] one can define the generalised L * -order ρ  r) ] and λ respectively.
Datta, Biswas and Hoque [3] reformulated Definition 6 in terms of the maximum terms of entire functions in the following way: Definition 7 [3] The growth indicators ρ r) ] and λ r) ] respectively where m be an integer ≥ 1.
Lakshminarasimhan [4] introduced the idea of the functions of Lbounded index.Later Lahiri and Bhattacharjee [5] worked on the entire functions of L-bounded index and of non uniform L-bounded index.In this paper we would like to investigate some growth properties of composite entire functions on the basis of their maximum terms and maximum moduli using generalised L * -order and generalised L * -lower order .

Lemmas.
In this section we present some lemmas which will be needed in the sequel.
Lemma 1 [9] Let f and g be any two entire functions with g(0) = 0. Then for all sufficiently large values of r, Lemma 2 [1] If f and g are any two entire functions then for all sufficiently large values of r, 3 Theorems.
In this section we present the main results of the paper.
Theorem 1 Let f and g be any two entire functions such that 0 < λ Then for every constant A and real number x, Proof.If x is such that 1 + x ≤ 0, then the theorem is obvious.So we suppose that 1 + x > 0. Now in view of Lemma 1, we get for all sufficiently large values of r that where we choose 0 < ε < min λ . Also for all sufficiently large values of r, we obtain that Therefore from ( 1) and ( 2) it follows for all sufficiently large values of r that log [m] Thus the theorem follows from (3).
In the line of Theorem 1, we may establish the following theorem for the right factor of the composite entire function : Theorem 2 Let f and g be any two entire functions with 0 < λ Then for every constant A and real number x, The proof is omitted.
Theorem 3 Let f and g be any two entire functions such that 0 < λ Then for any two positive integers α and β, Proof.Taking x = 0 and A = 1 in Theorem 1, we obtain for K > 1 and for all sufficiently large values of r that Therefore from (4) we get for all sufficiently large values of r that ( Again we have for all sufficiently large values of r that Now from ( 5) and ( 6) it follows for all sufficiently large values of r that log [m+1] μ (exp (exp (r α )) , f • g) i.e., log [m] μ (exp (r β ) , f) Again from (7) we get for all sufficiently large values of r that Case II.If r β = o {L (exp (exp (r α )))} then two sub cases may arise: Sub case (a).If L (exp (exp (r α ))) = o log [m] μ exp r β , f , then we get from (9) that lim inf and we obtain from ( 9) that lim inf Combining Case I and Case II we obtain that where K (r, α; L) = 0 if r μ = o {L (exp (exp (r α )))} as r → ∞ L (exp (exp (r α ))) otherwise .This proves the theorem.
Theorem 4 Let f and g be any two entire functions with 0 < λ Then for any two positive integers α and β, The proof is omitted because it can be carried out in the line of Theorem 3.

Remark 1 In view of Lemma 2 , the results analogous to Theorem 1, Theorem 2, Theorem 3 and Theorem 4 can also be derived in terms of maximum moduli of composite entire functions.
Theorem 5 Let f and g be any two entire functions such that 0 r) as r → ∞ and for some α < λ Proof.In view of Lemma 2 and taking R = βr in the inequality μ (r, f [10] } , we have for all sufficiently large values of r that i.e., log [m] Also we obtain for all sufficiently large values of r that Now from ( 11) and ( 12) we get for all sufficiently large values of r that log , we can choose ε (> 0) in such a way that Case I. Let L (μ (βr, g)) = o r α e αL(r) as r → ∞ and for some α < λ , we can choose ε (> 0) in such a way that Since L (μ (βr, g)) = o r α e αL(r) as r → ∞ we get on using (15) that L (μ (βr, g)) r α e αL(r) → 0 as r → ∞ i.e., L (μ (βr, g)) Now in view of (13), ( 14) and ( 16) we obtain that lim r→∞ log [m] μ(r, f • g) Case II.If L (μ (βr, g)) = o r α e αL(r) as r → ∞ and for some α < λ [m]L * f then we get from (13) that for a sequence of values of r tending to infinity, Now using (14) it follows from (18) that Combining ( 17) and ( 19) we obtain that r) as r → ∞ and for some α < λ Thus the theorem is established.
The following theorem can be carried out in the line of Theorem 5 and therefore its proof is omitted : Theorem 6 Let f and g be any two entire functions with r) as r → ∞ and for some α < ρ Replacing maximum term by maximum modulus in Theorem 5 and Theorem 6 we respectively get Theorem 7 and Theorem 8 and therefore their proofs are omitted.
We omit the proof of Corollary 1 because it can be carried out in the line of Theorem 7.
Corollary 2 Let f and g be any two entire functions with ρ [m]L * f < ∞ and 0 < ρ L * g < ∞ where m ≥ 1.Then for any β > 1, (a) if L (M (r, g)) = o log [2] M (r, g) then lim inf r→∞ log [m+1] M (r, f • g) log [2] M (r, g) + L (M (r, g)) ≤ 1 defined in C. The maximum term μ (r, f ) of f = ∞ n=0 a n z n on |z| = r is defined by μ (r, f ) = maxn≥0 |a n | r n and the maximum modulus M (r, f ) of f on |z| = r is defined as M (r, f ) = max |z|=r |f (z)| .We use the standard notations and defini- f of an entire function f are defined byρ [m] f = lim sup r→∞ log [m] M (r, f ) log r and λ [m] f = lim inf r→∞ log [m] M (r, f ) log rrespectively.

Definition 6
[m]L * f and generalised L * -lower orderλ [m]L * f of an entire function f in the following manner : Let m be an integer ≥ 1.The generalised L * -order ρ [m]L * f and generalised L * -lower order λ [m]L * f of an entire function f are defined as