# Growth analysis of composite entire and meromorphic functions in the light of their relative orders.

1. Introduction

Let f be meromorphic and g be an entire function defined in the open complex plane C. The maximum modulus function corresponding to entire g is defined as [M.sub.g](r) = max{[absolute value of (g(z))]: [absolute value of (z)] = r}. For meromorphic f, [M.sub.f](r) cannot be defined as f is not analytic. In this situation one may define another function [T.sub.f] (r) known as Nevanlinna's characteristic function of f, playing the same role as maximum modulus function in the following manner:

[T.sub.f] (r) = [N.sub.f] (r) + [m.sub.f] (r), (1)

where the function [N.sub.f] (r,a)([[bar.N].sub.f] (r,a)) known as counting function of a-points (distinct a-points) of meromorphic f is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

Moreover, we denote by [n.sub.f] (r,a)([[bar.n].sub.f] (r,a)) the number of apoints (distinct a-points) of f in [absolute value of (z)] [less than or equal to] r and an [infinity]-point is a pole of f. In many occasions [N.sub.f] (r, [infinity]) and [[bar.N].sub.f] (r, [infinity]) are denoted by [N.sub.f](r) and [N.sub.f](r), respectively

The function [m.sub.f] (r,[infinity]) alternatively denoted by [m.sub.j] (r) known as the proximity function of f is defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where [log.sup.+] x = max (log x,0) [for all]x [greater than or equal to] 0.

Also we may denote m(r, 1/(f - a)) by [m.sub.f] (r, a).

When f is an entire function, the Nevanlinna's characteristic function [T.sub.f] (r) of f is defined as

[T.sub.f] (r) = [m.sub.f] (r). (4)

Further, if f is a nonconstant entire function then [M.sub.f](r) and [T.sub.f](r) are both strictly increasing and continuous functions of r. Also their inverses [M.sup.-1.sub.f] (r) : ([absolute value of (f(0))], [infinity]) [right arrow] (0,([infinity]) and [T.sub.f.sup.-1] : ([T.sub.f](0), [infinity]) [right arrow] (0,([infinity]) exist, respectively,

and are such that [lim.sub.s[right arrow][infinity]] [M.sup.-1.sub.g] (s) = [infinity] and [lim.sub.s[right arrow][infinity]] [T.sup.-1.sub.f] (s) = [infinity].

In this connection we just recall the following definition which is relevant.

Definition 1 (see [1]). A nonconstant entire function f is said to have the Property (A) if, for any a > 1 and for all sufficiently large r, [[[M.sub.f] (r)].sup.2] < [M.sub.f] ([r.sup.[sigma]]) holds. For examples of functions with or without the Property (A), one may see [1].

However, for any two entire functions f and g, the ratio [M.sub.f](r)/[M.sub.g](r) as r [right arrow] [infinity] is called the growth of f with respect to g in terms of their maximum moduli. Similarly, when f and g are both meromorphic functions, the ratio [T.sub.f](r)/[T.sub.g] (r) as r [right arrow] [infinity] is called the growth of f with respect to g in terms of their Nevanlinna's characteristic functions. The notion of the growth indicators such as order and lower order of entire or meromorphic functions which are generally used in computational purpose is defined in terms of their growth with respect to the exponential function as the following.

Definition 2. The order pf (the lower order Xf) of an entire function f is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

where [log.sup.[k]]x = log([log.sup.[k-1]]v) for k = 1,2,3,... and [log.sup.[0]]x = x. Further, if f is a meromorphic function one can easily verify that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

Bernal [1, 2] introduced the definition of relative order of an entire function f with respect to another entire function g, denoted by pg(f) to avoid comparing growth just with exp z as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

The definition coincides with the classical one [3] if g(z) = exp z.

Similarly, one can define the relative lower order of an entire function f with respect to another entire function g denoted by [[lambda].sub.g] (f) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

Extending this notion, Lahiri and Banerjee [4]introduced the definition of relative order of a meromorphic function with respect to an entire function in the following way.

Definition 3 (see [4]). Let f be any meromorphic function and g be any entire function. The relative order of f with respect to g is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

Likewise, one can define the relative lower order of a meromorphic function f with respect to an entire function g denoted by [[lambda].sub.g](f) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)

It is known (cf. [4]) that if g(z) = exp z then Definition 3 coincides with the classical definition of the order of a meromorphic function f.

For entire and meromorphic functions, the notions of their growth indicators such as order and lower order are classical in complex analysis and during the past decades, several researchers have already been exploring their studies in the area of comparative growth properties of composite entire and meromorphic functions in different directions using the classical growth indicators. But at that time, the concepts of relative orders and relative lower orders of entire and meromorphic functions and their technical advantages of not comparing with the growths of exp z are not at all known to the researchers of this area. Therefore the studies of the growths of composite entire and meromorphic functions in the light of their relative orders and relative lower orders are the prime concern of this paper. In fact some light has already been thrown on such type of works by Datta et al. [5]. We do not explain the standard definitions and notations of the theory of entire and meromorphic functions as those are available in [6, 7].

2. Lemmas

In this section we present some lemmas which will be needed in the sequel.

Lemma 1 (see [8]). Let f be meromorphic and g be entire; then, for all sufficiently large values of r,

[T.sub.fog](r) [less than or equal to] {1 + o(1)} [T.sub.g](r)/log [M.sub.g](r) [T.sub.f] ([M.sub.g](r)). (11)

Lemma 2 (see [9]). Let f be meromorphic and g be entire and suppose that 0 < [mu] < [[rho].sub.g] < [[lambda].sub.f]. Then, for a sequence of values of r tending to infinity,

[T.sub.fog] (r) [greater than or equal to] [T.sub.f] (exp([r.sup.[mu]])), (12)

Lemma 3 (see [10]). Let f be meromorphic and g be entire such that 0 < pg < rn and 0 < Aj. Then, for a sequence of values of r tending to infinity,

[T.sub.fog] (r) > [T.sub.g] (exp([r.sup.[mu]])), (13)

where 0 < [mu] < [[rho].sub.g].

Lemma 4 (see [11]). Let f be an entire function which satisfies the Property (A), [beta] > 0, [delta] > 1, and [alpha] > 2. Then

[beta][T.sub.f] (r) < [T.sub.f] ([ar.sup.[delta]]). (14)

3. Theorems

In this section we present the main results of the paper.

Theorem 1. Let f be a meromorphic function and h be an entire function with 0 < [[lambda].sub.h] (f) [less than or equal to] [[rho].sub.h] (f) < [infinity] and let g be an entire function with finite order. Ifh satisfies the Property (A), then, for every positive constant g and each [alpha] [member of] (-[infinity], [infinity]),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

Proof. Let us suppose that [beta] > 2 and [delta] > 1. If 1 + [alpha] [less than or equal to] 0, then the theorem is obvious. We consider l + [alpha] > 0.

Since [T.sup.-1.sub.h] (r) is an increasing function of r, it follows from Lemma 1,Lemma4, and the inequality [T.sub.g] (r) [less than or equal to] log [M.sub.g] (r) {cf. [6]} for all sufficiently large values of r that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

Again for all sufficiently large values of r we get that

log [T.sup.-1.sub.h] [T.sub.f] (exp [r.sup.[mu]]) [greater than or equal to] ([[lambda].sub.h] (f) - [epsilon]) [r.sup.[mu]]. (17)

Hence for all sufficiently large values of r we obtain from (16) and (17) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (18)

where we choose 0 < [epsilon] < min {[[lambda].sub.h] (f), ([mu]/(1 + [alpha])) - [[rho].sub.g]}.

So from (18) we obtain that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (19)

This proves the theorem.

Remark 2. In Theorem 1 if we take the condition 0 < [[rho].sub.h](f) < rn instead of 0 < [[lambda].sub.h] (f) [less than or equal to] [[rho].sub.h] (f) < [infinity], the theorem remains true with "limit inferior" in place of "limit"

In view of Theorem 1 the following theorem can be carried out.

Theorem 3. Let f be a meromorphic function and let g, h be any two entire functions where g is of finite order and [[lambda].sub.h] (g) > 0, [[rho].sub.h] (f) < [infinity]. If h satisfies the Property (A), then, for every positive constant g and each a e (-rn, rn),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (20)

The proof is omitted.

Remark 4. In Theorem 3 if we take the condition [[rho].sub.h](g) > 0 instead of [[lambda].sub.h](g) > 0, the theorem remains true with "limit" replaced by "limit inferior".

Theorem 5. Let f be a meromorphic function and let g, h be any two entire functions such that 0 < [[lambda].sub.h](f) < [[rho].sub.h](f) < [infinity]i and [[lambda].sub.g] < g < [infinity]. Also suppose that h satisfies the Property (A). Then, for a sequence of values of r tending to infinity,

[T.sup.-1.sub.h][T.sub.fog] (r) < [T.sup.-1.sub.h][T.sub.f] (exp [r.sup.[mu]]). (21)

Proof. Let us consider [delta] > 1. Since [T.sup.-1.sub.h] (r) is an increasing function of r, it follows from Lemma 1 that, for a sequence of values of r tending to infinity,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (22)

Now from (17) and 22), it follows for a sequence of values of r tending to infinity that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (23)

As [[lambda].sub.g] < [mu] we can choose e (> 0) in such a way that

[[lambda].sub.g] + [epsilon] < [mu] < [[rho].sub.g]. (24)

Thus from (23) and 24) we obtain that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (25)

Now from (25), we obtain for a sequence of values of r tending to infinity and also for K > 1

[T.sup.-1.sub.h] [T.sub.f] (exp [r.sup.[mu]]) > [T.sup.-1.sub.h] [T.sub.fog] (r). (26)

Thus the theorem follows. ?

In the line of Theorem 5, we may state the following theorem without its proof.

Theorem 6. Let g and h be any two entire functions with [[lambda].sub.h] (g) > 0 and let f be a meromorphic function with finite relative order with respect to h. Also suppose that [[lambda].sub.g] < [mu] < x and h satisfies the Property (A). Then, for a sequence of values of r tending to infinity,

[T.sup.-1.sub.h] [T.sub.fog] (r) < [T.sup.-1.sub.h] [T.sub.g] (exp [r.sup.[mu]]). (27)

As an application of Theorem 5 and Lemma 2, we may state the following theorem.

Theorem 7. Let f be a meromorphic function and let g, h be any two entire functions such that 0 < [[lambda].sub.h](f) [less than or equal to] [[rho].sub.h](f) < [infinity] and [[lambda].sub.g] < g < [[rho].sub.g]. If h satisfies the Property (A), then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (28)

The proof is omitted.

Similary in view of Theorem 6 and Lemma 3, the following theorem can be carried out.

Theorem 8. Let f be a meromorphic function and let g, h be any two entire functions with 0 < [[lambda].sub.h](f) < [[rho].sub.h](f) < [infinity], 0 < [[lambda].sub.h](g) [less than or equal to] [[rho].sub.h] (g) < [infinity], and 0 < [[lambda].sub.g] < g < [[rho].sub.g] < [infinity]. Moreover h satisfies the Property (A). Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (29)

The proof is omitted.

Theorem 9. Let f be a meromorphic function and let h, g be any two entire functions with [[lambda].sub.h](f) > 0 and 0 < [[rho].sub.h] (g) < [infinity]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (30)

where 0 < [mu] < [[rho].sub.g].

Proof. Let 0 < [mu] < [mu] < [[rho].sub.g]. As [T.sup.-1.sub.h] (r) is an increasing function of r, it follows from Lemma 2 for a sequence of values of r tending to infinity that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (31)

that is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Again for all sufficiently large values of r we get that

log [T.sup.-1.sub.h] [T.sub.g] (exp [r.sup.[mu]]) [less than or equal to] ([[rho].sub.h] (g) + [epsilon]) [r.sup.[mu]]. (32)

So combining (31) and (32), we obtain for a sequence of values of r tending to infinity that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (33)

Since [mu] < [[mu].sup./] (it follows from 33) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (34)

Hence the theorem follows.

Corollary 10. Under the assumptions of Theorem 9,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (35)

Proof. In view of Theorem 9, we get for a sequence of values of r tending to infinity that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (36)

from which the corollary follows. ?

Theorem 11. Let f be a meromorphic function and let h, g be any two entire functions such that (i) 0 < [[rho].sub.h](g) < [infinity], (ii) [[lambda].sub.h](f) > 0, and (iii) [[lambda].sub.h](f o g) > 0. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (37)

where 0 < [mu] < [[rho].sub.g].

Proof. From the definition of relative order and relative lower order, we obtain for arbitrary positive e and for all sufficiently large values of r that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (38)

Therefore, from (38), it follows for all sufficientlylarge values of r that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (39)

that is, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus the theorem follows from (34) and 39).

Similarly, one may state the following theorems and corollary without their proofs as those can be carried out in the line of Theorems 9 and 11 and Corollary 10, respectively

Theorem 12. Let f be a meromorphic function and h be an entire function with 0 < [[lambda].sub.h] (f) [less than or equal to] [[rho].sub.h] (f) < [infinity]. Then, for any entire function g,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (40)

where 0 < [mu] < [[rho].sub.g].

Theorem 13. Let f be a meromorphic function and let h, g be any two entire functions such that (i) 0 < [[lambda].sub.h] (f) [less than or equal to] [[rho].sub.h] (f) < [infinity] and (ii) [[lambda].sub.h] (f o g) > 0. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (41)

where 0 < [mu] < [[rho].sub.g].

Corollary 14. Under the assumptions of Theorem 12,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (42)

Theorem 15. Let f be a meromorphic function and let h be an entire function with 0 < [[lambda].sub.h](f) < [[rho].sub.h](f) < x. Then, for any entire function g,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (43)

where 0 < g < [[rho].sub.g] and B > 0.

Proof. Let 0 < [[mu].sup./] < [[rho].sub.g]. As [T.sup.-1.sub.h] (t) is an increasing function of r, it follows from 31) for a sequence of values of r tending to infinity that

[log.sup.[2]] [T.sup.-1.sub.h] [T.sub.fog] (r) [greater than or equal to] O(1) + [[mu].sup./] log r. (44)

So for a sequence of values of r tending to infinity we get from above that

[log.sup.[2]] [T.sup.-1.sub.h] [T.sub.fog] (exp [r.sup.B]) [greater than or equal to] O(1) + [[mu].sup./] log [r.sup.B]. (45)

Again we have for all sufficiently large values of r that

[log.sup.[2]] [T.sup.-1.sub.h] [T.sub.f] (exp [r.sup.[mu]]) [greater than or equal to] O(1) + [mu] log r. (46)

Now combining (45) and (46), we obtain for a sequence of values of r tending to infinity that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (47)

from which the theorem follows.

In view of Theorem 15 the following theorem can be carried out.

Theorem 16. Let f be a meromorphic function and let h, g be any two entire functions with [[lambda].sub.h](f) > 0 and 0 < [[rho].sub.h](g) < [infinity]. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (48)

where 0 < [mu] < [[rho].sub.g] and B > 0.

The proof is omitted.

Theorem 17. Let l be an entire function satisfying the Property (A) and let h be a meromorphic function such that [[lambda].sub.1] (h) > 0. Also let g and k be any two entire functions with finite nonzero order such that [[rho].sub.g] < [[rho].sub.k]. Then, for every meromorphic function f with 0 < [[rho].sub.l] (f) < [infinity],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (49)

Proof. Since [[rho].sub.g] < [[rho].sub.k], we can choose [epsilon] (>0) in such a way that

[[rho].sub.g] + [epsilon] < [mu] < [[rho].sub.k] - [epsilon]. (50)

As [T.sup.-1.sub.l] (r) is an increasing function of r, it follows from Lemma 2 for a sequence of values of r tending to infinity that

log [T.sup.-1.sub.l] [T.sub.hok] (r) [greater than or equal to] log [T.sup.-1.sub.l] [T.sub.h] (exp [r.sup.[mu]]),

where 0 < [mu] < [[rho].sub.k] [less than or equal to] [infinity]; (51)

that is, log [T.sup.-1.sub.l] [T.sub.hok] (r) [greater than or equal to] ([[lambda].sub.l] (h) - [epsilon]) [r.sup.[mu]].

Now from the definition of relative order of f with respect to l we have for arbitrary positive e and for all sufficiently large values of r that

log [T.sup.-1.sub.l] [T.sub.f] (r) [less than or equal to] ([[rho].sub.l] (f) + [epsilon]) log r. (52)

Now for any [delta] > 1, we get from (16), (51), (52), and in view of (50) for a sequence of values of r tending to infinity that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (53)

that is, log [T.sup.-1.sub.l][T.sub.hok](r)/log [T.sup.-1.sub.l][T.sub.hok](r) + log [T.sup.-1.sub.l][T.sub.hok](r) = [infinity],

which proves the theorem.

In the line of Theorem 17 the following theorem can be carried out.

Theorem 18. Let l be an entire function satisfying the Property (A) and let h be a meromorphic function such that Afh) > 0. Also let g and k be any two entire functions with finite nonzero order and also [[rho].sub.g] < [[rho].sub.k]. Then, for every meromorphic function f with 0 < [[rho].sub.l] (f) < [infinity],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (54)

4. Conclusion

Actually this paper deals with the extension of the works on the growth properties of composite entire and meromorphic functions on the basis of their relative orders and relative lower orders. These theories can also be modified by the treatment of the notions of generalized relative orders (generalized relative lower orders) and (p,q)th relative orders ((p,q)th relative lower orders). Moreover, some extensions of the same type may be done in the light of slowly changing functions.

http://dx.doi.org/ 10.1155/2014/538327

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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[11] S. K. Datta, T. Biswas, and C. Biswas, "Measure of growth ratios of composite entire and meromorphic functions with a focus on relative order," International Journal of Mathematical Sciences and Engineering Applications, vol. 8, no. 4, pp. 207-218, 2014.

Sanjib Kumar Datta, (1) Tanmay Biswas, (2) and Chinmay Biswas (3)

(1) Department of Mathematics, University of Kalyani, Kalyani, Nadia District, West Bengal 741235, India

(2) Rajbari, Rabindra Pally, R. N. Tagore Road, Krishnagar, Nadia District, West Bengal 741101, India

(3) Taraknagar Jamuna Sundari High School, Taraknagar, Hanskhali, Nadia District, West Bengal 741502, India

Correspondence should be addressed to Sanjib Kumar Datta; sanjib_kr_datta@yahoo.co.in

Received 29 June 2014; Accepted 20 August 2014; Published 29 October 2014