# Growth Rates of Meromorphic Functions Focusing Relative Order

Authors: Sanjib Kumar Datta; Tanmay Biswas; Sarmila Bhattacharyya
Journal: Chinese Journal of Mathematics
 (2014)

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## Abstract

A detailed study concerning some growth rates of composite entire and meromorphic functions on the basis of their relative orders (relative lower orders) with respect to entire functions has been made in this paper.


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## Body

## 1. Introduction, Definitions and Notations

Letf be a meromorphic and let g be an entire function defined in the open complex plane ℂ and Mg(r)=max⁡{|g(z)|:|z|=r}. If g is nonconstant, then Mg(r) is strictly increasing and continuous and its inverse Mg-1(r):(|g(0)|,∞)→(0,∞) exists and is such that lim⁡s→∞⁡Mg-1(s)=∞. We use the standard notations and definitions in the theory of entire and meromorphic functions those are available in [1, 2].To start our paper we just recall the following definitions.Definition 1.
The orderρf (the lower order λf) of an entire function f is defined as
(1)ρf=limsup⁡r→∞log⁡[2]Mf(r)log⁡r(λf=liminf⁡r→∞log⁡[2]Mf(r)log⁡r).
If f is a meromorphic function, one can easily verify that
(2)ρf=limsup⁡r→∞log⁡Tf(r)log⁡r(λf=liminf⁡r→∞log⁡Tf(r)log⁡r),
where Tf(r) is the Nevanlinna's characteristic function of f (cf. [1]).Bernal [3, 4] introduced the definition of relative order of an entire function f with respect to another entire function g denoted by ρg(f) as follows:
(3)ρg(f)=inf⁡{μ>0:Mf(r)<Mg(rμ)∀r>r0(μ)>0}=limsup⁡r→∞log⁡Mg-1Mf(r)log⁡r.The definition coincides with the classical one [5] if g(z)=exp⁡z.Similarly, one can define the relative lower order of an entire functionf with respect to another entire function g denoted by λg(f) as follows:
(4)λg(f)=liminf⁡r→∞log⁡Mg-1Mf(r)log⁡r.Extending this notion, Lahiri and Banerjee [6] introduced the definition of relative order of a meromorphic function with respect to an entire function in the following way.Definition 2 (see [6]).
Letf be any meromorphic function and let g be any entire function. The relative order of f with respect to g is defined as
(5)ρg(f)=inf⁡{μ>0:Tf(r)<Tg(rμ)∀larger}=limsup⁡r→∞log⁡Tg-1Tf(r)log⁡r.
Likewise, one can define the relative lower order of a meromorphic function f with respect to an entire function g denoted by λg(f) as follows:
(6)λg(f)=liminf⁡r→∞log⁡Tg-1Tf(r)log⁡r.
It is known (cf. [6]) that if g(z)=exp⁡z, then Definition 2 coincides with the classical definition of the order of a meromorphic function f.In this paper we wish to prove some results related to the growth rates of composite entire and meromorphic functions on the basis of relative order (relative lower order) of meromorphic functions with respect to an entire function.


## 2. Lemmas

In this section we present some lemmas which will be needed in the sequel.Lemma 1 (see [7]).
Letf be meromorphic and let g be entire and suppose that 0<μ<ρg≤∞. Then for a sequence of values of r tending to infinity,
(7)Tf∘g(r)≥Tf(exp⁡(rμ)).Lemma 2 (see [8]).
Letf be meromorphic and let g be entire such that 0<ρg<∞ and 0<λf. Then for a sequence of values of r tending to infinity,
(8)Tf∘g(r)>Tg(exp⁡(rμ)),
where 0<μ<ρg.Lemma 3 (see [9]).
Letf be a meromorphic function and let g be an entire function such that λg<μ<∞ and 0<λf≤ρf<∞. Then for a sequence of values of r tending to infinity,
(9)Tf∘g(r)<Tf(exp⁡(rμ)).Lemma 4 (see [9]).
Letf be a meromorphic function of finite order and let g be an entire function such that 0<λg<μ<∞. Then for a sequence of values of r tending to infinity,
(10)Tf∘g(r)<Tg(exp⁡(rμ)).


## 3. Theorems

In this section we present the main results of the paper.Theorem 1.
Letf be a meromorphic function and let g, h be any two entire functions such that
(11)(i)liminf⁡r→∞log⁡Th-1(r)(log⁡r)α=A,arealnumber>0,(ii)liminf⁡r→∞log⁡Th-1Tf(exp⁡rμ)(log⁡Th-1(r))β+1=B,arealnumber>0,
for any α,β,μ satisfying 0<α<1, β>0, α(β+1)>1, and 0<μ<ρg≤∞. Then
(12)ρh(f∘g)=∞.Proof.
From(i) we have for all sufficiently large values of r that
(13)log⁡Th-1(r)≥(A-ε)(log⁡r)α
and from (ii) we obtain for all sufficiently large values of r that
(14)log⁡Th-1Tf(exp⁡rμ)≥(B-ε)(log⁡Th-1(r))β+1.
Also Th-1(r) is an increasing function of r it follows from (13) and (14) and Lemma 1 for a sequence of values of r tending to infinity that
(15)log⁡Th-1Tf∘g(r)≥log⁡Th-1Tf(exp⁡(rμ))
that is,
(16)log⁡Th-1Tf∘g(r)≥(B-ε)(log⁡Th-1(r))β+1
that is,
(17)log⁡Th-1Tf∘g(r)≥(B-ε)[(A-ε)(log⁡r)α]β+1
that is,
(18)log⁡Th-1Tf∘g(r)≥(B-ε)(A-ε)β+1(log⁡r)α(β+1)
that is,
(19)log⁡Th-1Tf∘g(r)log⁡r≥(B-ε)(A-ε)β+1(log⁡r)α(β+1)log⁡r
that is,
(20)limsup⁡r→∞log⁡Th-1Tf∘g(r)log⁡r≥liminf⁡r→∞(B-ε)(A-ε)β+1(log⁡r)α(β+1)log⁡r.
Since ε(>0) is arbitrary and α(β+1)>1, it follows from above that
(21)ρh(f∘g)=∞.
This proves the theorem.In the line of Theorem1 and with the help of Lemma 2 one may state the following theorem without its proof.Theorem 2.
Letf be a meromorphic function and let g, h be any two entire functions such that 0<ρg<∞ and 0<λf. Further suppose that
(22)(i)liminf⁡r→∞log⁡Th-1(r)(log⁡r)α=A,arealnumber>0,(ii)liminf⁡r→∞log⁡Th-1Tg(exp⁡rμ)(log⁡Th-1(r))β+1=B,arealnumber>0
for any α, β, μ satisfying 0<α<1, β>0, α(β+1)>1 and 0<μ<ρg. Then
(23)ρh(f∘g)=∞.Theorem 3.
Letf be a meromorphic function and let g, h be any two entire functions such that
(24)(i)liminf⁡r→∞log⁡Th-1(exp⁡(rμ))(log⁡[2]r)α=A,arealnumber>0,(ii)liminf⁡r→∞log⁡[log⁡Th-1(Tf(exp⁡rμ))/log⁡Th-1(exp⁡rμ)][log⁡Th-1(exp⁡rμ)]β=B,arealnumber>0
for any α, β satisfying α>1, 0<β<1, αβ>1 and 0<μ<ρg≤∞. Then
(25)ρh(f∘g)=∞.Proof.
From(i) we have for all sufficiently large values of r that
(26)log⁡Th-1(exp⁡(rμ))≥((A-ε)log⁡[2]r)α
and from (ii) we obtain for all sufficiently large values of r that
(27)log⁡[log⁡Th-1(Tf(exp⁡rμ))log⁡Th-1(exp⁡rμ)]≥(B-ε)[log⁡Th-1(exp⁡rμ)]βthatis,log⁡Th-1(Tf(exp⁡rμ))log⁡Th-1(exp⁡rμ)≥exp⁡[(B-ε)[log⁡Th-1(exp⁡rμ)]β].
As Th-1(r) is an increasing function of r, it follows from (26) and (27) and Lemma 1 for a sequence of values of r tending to infinity that
(28)log⁡Th-1Tf∘g(r)log⁡r≥log⁡Th-1Tf(exp⁡(rμ))log⁡rthatis,111111111111111111111111111111111111111111log⁡Th-1Tf∘g(r)log⁡r≥log⁡Th-1Tf(exp⁡(rμ))log⁡Th-1(exp⁡(rμ))·log⁡Th-1(exp⁡(rμ))log⁡rthatis,111111111111111111111111111111111111111111log⁡Th-1Tf∘g(r)log⁡r≥exp⁡[(B-ε)[log⁡Th-1(exp⁡rμ)]β]·(A-ε)(log⁡[2]r)αlog⁡rthatis,11111111111111111111111111111111111111111log⁡Th-1Tf∘g(r)log⁡r≥exp⁡[(B-ε)(A-ε)β(log⁡[2]r)αβ]·(A-ε)(log⁡[2]r)αlog⁡rthatis,1111111111111111111111111111111111111111log⁡Th-1Tf∘g(r)log⁡r≥exp⁡[(B-ε)(A-ε)β(log⁡[2]r)αβ-1log⁡[2]r]·(A-ε)(log⁡[2]r)αlog⁡rthatis,1111111111111111111111111111111111111111log⁡Th-1Tf∘g(r)log⁡r≥(log⁡r)(B-ε)(A-ε)β(log⁡[2]r)αβ-1·(A-ε)(log⁡[2]r)αlog⁡rthatis,1111111111111111111111111111111111111111limsup⁡r→∞log⁡Th-1Tf∘g(r)log⁡r≥liminf⁡r→∞(log⁡r)(B-ε)(A-ε)β(log⁡[2]r)αβ-1·(A-ε)(log⁡[2]r)αlog⁡r.
Sinceε(>0) is arbitrary and α>1, αβ>1, the theorem follows from above.Theorem 4.
Letf be a meromorphic function and let g, h be any two entire functions such that 0<ρg<∞ and 0<λf. Further suppose that
(29)(i)liminf⁡r→∞log⁡Th-1(exp⁡(rμ))(log⁡[2]r)α=A,arealnumber>0,(ii)liminf⁡r→∞log⁡[log⁡Th-1(Tg(exp⁡rμ))/log⁡Th-1(exp⁡rμ)][log⁡Th-1(exp⁡rμ)]β=B,arealnumber>0
for any α,β satisfying α>1, 0<β<1, αβ>1, and 0<μ<ρg. Then
(30)ρh(f∘g)=∞.We omit the proof of Theorem4 as it can be carried out in the line of Theorem 3 and with the help of Lemma 2.Theorem 5.
Letf be a meromorphic function and let g, h be any two entire functions such that 0<ρg≤∞ and λh(f)>0. Then
(31)ρh(f∘g)=∞.Proof.
Suppose0<μ<ρg≤∞.
AsTh-1(r) is an increasing function of r, we get from Lemma 1 for a sequence of values of r tending to infinity that
(32)log⁡Th-1Tf∘g(r)≥log⁡Th-1Tf(exp⁡(rμ))thatis,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiilog⁡Th-1Tf∘g(r)≥(λh(f)-ε)rμthatis,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiilog⁡Th-1Tf∘g(r)log⁡r≥(λh(f)-ε)rμlog⁡rthatis,limsup⁡r→∞log⁡Th-1Tf∘g(r)log⁡r≥lim⁡inf⁡r→∞(λh(f)-ε)rμlog⁡rthatis,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiρh(f∘g)=∞.
Thus the theorem follows.In the line of Theorem5, one can easily prove the following theorem.Theorem 6.
Letf be a meromorphic function and let g, h be any two entire functions such that 0<min⁡{λf,ρg} and λh(g)>0. Then
(33)ρh(f∘g)=∞.Theorem 7.
Letf be a meromorphic function and let g, h be any two entire functions such that 0<ρg≤∞ and λh(f)>0. Then
(34)limsup⁡r→∞log⁡Th-1Tf∘g(r)log⁡Th-1Tf(r)=∞.Proof.
In view of Theorem5, we obtain that
(35)limsup⁡r→∞log⁡Th-1Tf∘g(r)log⁡Th-1Tf(r)≥limsup⁡r→∞log⁡Th-1Tf∘g(r)log⁡r·liminf⁡r→∞log⁡rlog⁡Th-1Tf(r)thatis,limsup⁡r→∞log⁡Th-1Tf∘g(r)log⁡Th-1Tf(r)≥ρh(f∘g)·1ρh(f)thatis,limsup⁡r→∞log⁡Th-1Tf∘g(r)log⁡Th-1Tf(r)=∞.
Thus the theorem follows.Theorem 8.
Letf be a meromorphic function and let g, h be any two entire functions such that 0<min⁡{λf,ρg} and λh(g)>0. Then
(36)limsup⁡r→∞log⁡Th-1Tf∘g(r)log⁡Th-1Tg(r)=∞.The proof of Theorem8 is omitted as it can be carried out in the line of Theorem 7 and in view of Theorem 6.Theorem 9.
Letf be a meromorphic function and let h be an entire functions such that 0<λh(f)≤ρh(f)<∞. Also let g be an entire function with nonzero order. Then for every positive constant A and every real number α,
(37)limsup⁡r→∞log⁡Th-1Tf∘g(r){log⁡Th-1Tf(rA)}1+α=∞.Proof.
Ifα is such that 1+α≤0, then the theorem is trivial. So we suppose that 1+α>0.
SinceTh-1(r) is an increasing function of r, we get from Lemma 1 for a sequence of values of r tending to infinity that
(38)log⁡Th-1Tf∘g(r)≥log⁡Th-1Tf(exp⁡(rμ))thatis,log⁡Th-1Tf∘g(r)≥(λh(f)-ε)rμ,
where we choose 0<μ<ρg≤∞.
Again from the definition ofρh(f), it follows for all sufficiently large values of r that
(39)log⁡Th-1Tf(rA)≤(ρh(f)+ε)Alog⁡rthatis,{log⁡Th-1Tf(rA)}1+α≤(ρh(f)+ε)1+αA1+α(log⁡r)1+α.
Now from (38) and (39), it follows for a sequence of values of r tending to infinity that
(40)log⁡Th-1Tf∘g(r){log⁡Th-1Tf(rA)}1+α⩾(λh(f)-ε)rμ(ρh(f)+ε)1+αA1+α(log⁡r)1+α.
Since rμ/(log⁡r)1+α→∞ as r→∞, the theorem follows from above.In the line of Theorem9 and with the help of Lemma 2, one may state the following theorem without its proof.Theorem 10.
Letf be a meromorphic function with nonzero finite lower order and let g be an entire function with nonzero finite order. Also let h be an entire function such that ρh(f)<∞ and λh(g)>0. Then for every positive constant A and every real number α,
(41)limsup⁡r→∞log⁡Th-1Tf∘g(r){log⁡Th-1Tf(rA)}1+α=∞.Theorem 11.
Letf be a meromorphic function and let g be an entire function with nonzero order. Also let h be an entire function such that 0<λh(f) and ρh(g)<∞. Then for every positive constant A and every real number α,
(42)limsup⁡r→∞log⁡Th-1Tf∘g(r){log⁡Th-1Tg(rA)}1+α=∞.Theorem 12.
Letf be a meromorphic function with nonzero finite lower order and let g be an entire function with nonzero finite order. Also let h be an entire function such that 0<λh(g)≤ρh(g)<∞. Then for every positive constant A and every real number α,
(43)limsup⁡r→∞log⁡Th-1Tf∘g(r){log⁡Th-1Tg(rA)}1+α=∞.We omit the proof of Theorems11 and 12 as those can be carried out in the line of Theorems 9 and 10, respectively.Theorem 13.
Letf be a meromorphic function with nonzero finite order and lower order. Also let g, h be any two entire functions such that 0<λh(f)≤ρh(f)<∞. Then for every positive constant μ and each α∈(-∞,∞),
(44)liminf⁡r→∞{log⁡Th-1Tf∘g(r)}1+αlog⁡Th-1Tf(exp⁡(rμ))=0ifμ>λg.Proof.
If1+α≤0, then the theorem is obvious. We consider 1+α>0.
SinceTh-1(r) is an increasing function of r, it follows from Lemma 3 for a sequence of values of r tending to infinity that
(45)log⁡Th-1Tf∘g(r)<log⁡Th-1Tf(exp⁡(rμ))thatis,log⁡Th-1Tf∘g(r)<(ρh(f)+ε)rμ.
Again for all sufficiently large values of r we get that
(46)log⁡Th-1Tf(exp⁡(rμ))≥(λh(f)-ε)rμ.
Hence for a sequence of values of r tending to infinity, we obtain from (45) and (46) that
(47){log⁡Th-1Tf∘g(r)}1+αlog⁡Th-1Tf(exp⁡(rμ))≤(ρh(f)+ε)1+αrμ(1+α)(λh(f)-ε)rμ.
So from (47) we obtain that
(48)liminf⁡r→∞{log⁡Th-1Tf∘g(r)}1+αlog⁡Th-1Tf(exp⁡(rμ))=0.
This proves the theorem.Theorem 14.
Letf be a meromorphic function with nonzero finite order and lower order. Also let g, h be any two entire functions such that ρh(f)<∞ and λh(g)>0. Then for every positive constant μ and each α∈(-∞,∞),
(49)liminf⁡r→∞{log⁡Th-1Tf∘g(r)}1+αlog⁡Th-1Tg(exp⁡(rμ))=0ifμ>λg.The proof is omitted as it can be carried out in the line of Theorem13.Theorem 15.
Letf be a meromorphic function with finite order and let g be an entire function with nonzero finite lower order. Also let h be another entire function such that λh(f)>0 and ρh(g)<∞. Then for every positive constant μ and each α∈(-∞,∞),
(50)liminf⁡r→∞{log⁡Th-1Tf∘g(r)}1+αlog⁡Th-1Tf(exp⁡(rμ))=0ifμ>λg.Theorem 16.
Letf be a meromorphic function with finite order and let g be an entire function with nonzero finite lower order. Also let h be another entire function such that 0<λh(g)≤ρh(g)<∞. Then for every positive constant μ and each α∈(-∞,∞),
(51)liminf⁡r→∞{log⁡Th-1Tf∘g(r)}1+αlog⁡Th-1Tg(exp⁡(rμ))=0ifμ>λg.We omit the proofs of Theorems15 and 16 as those can be carried out in the line of Theorems 13 and 14, respectively, and with the help of Lemma 4.Theorem 17.
Letf be a meromorphic function and let g, h be any two entire functions such that ρh(f)<∞ and λh(f∘g)=∞. Then for every μ(>0),
(52)lim⁡r→∞log⁡Th-1Tf∘g(r)log⁡Th-1Tf(rμ)=∞.Proof.
If possible, let there exist a constantβ such that for a sequence of values of r tending to infinity,
(53)log⁡Th-1Tf∘g(r)≤β·log⁡Th-1Tf(rμ).
Again from the definition of ρh(f), it follows for all sufficiently large values of r that
(54)log⁡Th-1Tf(rμ)≤(ρh(f)+ε)μlog⁡r.
Now combining (53) and (54), we have for a sequence of values of r tending to infinity that
(55)log⁡Th-1Tf∘g(r)≤β·(ρh(f)+ε)μ·log⁡rthatis,λh(f∘g)≤β·μ(ρh(f)+ε),
which contradicts the condition λh(f∘g)=∞.
So for all sufficiently large values ofr we get that
(56)log⁡Th-1Tf∘g(r)≥β·log⁡Th-1Tf(rμ),
from which the theorem follows.Remark 18.
Theorem16 is also valid with “limit superior” instead of “limit” if λh(f∘g)=∞ is replaced by ρh(f∘g)=∞ and the other conditions remain the same.Corollary 19.
Under the assumptions of Theorem16 and Remark 18,
(57)lim⁡r→∞Th-1Tf∘g(r)Th-1Tf(rμ)=∞,lim⁡sup⁡r→∞Th-1Tf∘g(r)Th-1Tf(rμ)=∞,
respectively hold.Proof.
By Theorem16, we obtain for all sufficiently large values of r and for K>1 that
(58)log⁡Th-1Tf∘g(r)≥Klog⁡Th-1Tf(rμ)thatis,Th-1Tf∘g(r)≥{Th-1Tf(rμ)}K,
from which the first part of the corollary follows.
Similarly using Remark18, we obtain the second part of the corollary.Analogously one may state the following theorem and corollaries without their proofs as those can be carried out in the line of Remark18, Theorem 16, and Corollary 19, respectively.Theorem 20.
Iff is a meromorphic function and g, h are any two entire functions such that ρh(g)<∞ and ρh(f∘g)=∞, then for every μ(>0),
(59)limsup⁡r→∞log⁡Th-1Tf∘g(r)log⁡Th-1Tg(rμ)=∞.Corollary 21.
Theorem17 is also valid with “limit” instead of “limit superior” if ρh(f∘g)=∞ is replaced by λh(f∘g)=∞ and the other conditions remain the same.Corollary 22.
Under the assumptions of Theorem17 and Corollary 21,
(60)limsup⁡r→∞Th-1Tf∘g(r)Th-1Tg(rμ)=∞,lim⁡r→∞Th-1Tf∘g(r)Th-1Tg(rμ)=∞,
respectively hold.


## 4. Conclusion

The notion of order and lower order which are the main tools to study the comparative growth properties of entire and meromorphic functions is very much classical in complex analysis. On the basis of the order and lower order of entire or meromorphic functions, several researchers have already explored their works on the area of comparative growth rates of composite entire and meromorphic functions in different directions. In fact, the main aim of this paper is actually to generalize the growth estimates of composite entire and meromorphic functions on the basis of their relative orders and relative lower orders with respect to an entire function. Moreover, the treatment of these notions may also be done by taking the help of slowly changing functions in order to study some different growth properties of composite entire and meromorphic functions.



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