CJM Chinese Journal of Mathematics 2314-8071 Hindawi Publishing Corporation 582082 10.1155/2014/582082 582082 Research Article Growth Rates of Meromorphic Functions Focusing Relative Order Datta Sanjib Kumar 1 Biswas Tanmay 2 Bhattacharyya Sarmila 3 Latushkin Y. Mallier R. 1 Department of Mathematics University of Kalyani Kalyani Nadia District West Bengal 741235 India klyuniv.ac.in 2 Rajbari Rabindrapalli R.N. Tagore Road P.O. Krishnagar P.S. Kotwali Nadia District West Bengal 741101 India 3 Jhorehat F. C. High School for Girls P.O. Jhorehat P.S. Sankrail Howrah District West Bengal 711302 India 2014 232014 2014 17 11 2013 15 01 2014 2 3 2014 2014 Copyright © 2014 Sanjib Kumar Datta et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction, Definitions and Notations

Let f be a meromorphic and let g be an entire function defined in the open complex plane 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 limsMg-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 order ρf (the lower order λf) of an entire function f is defined as (1)ρf=limsuprlogMf(r)logr(λf=liminfrlogMf(r)logr). If f is a meromorphic function, one can easily verify that (2)ρf=limsuprlogTf(r)logr(λf=liminfrlogTf(r)logr), where Tf(r) is the Nevanlinna's characteristic function of f (cf. ).

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}=limsuprlogMg-1Mf(r)logr.

The definition coincides with the classical one  if g(z)=expz.

Similarly, one can define the relative lower order of an entire function f with respect to another entire function g denoted by λg(f) as follows: (4)λg(f)=liminfrlogMg-1Mf(r)logr.

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

Definition 2 (see [<xref ref-type="bibr" rid="B7">6</xref>]).

Let f 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)    large  r}=limsuprlogTg-1Tf(r)logr. 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)=liminfrlogTg-1Tf(r)logr. It is known (cf. ) that if g(z)=expz, 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 [<xref ref-type="bibr" rid="B3">7</xref>]).

Let f 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)Tfg(r)Tf(exp(rμ)).

Lemma 2 (see [<xref ref-type="bibr" rid="B6">8</xref>]).

Let f 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)Tfg(r)>Tg(exp(rμ)), where 0<μ<ρg.

Lemma 3 (see [<xref ref-type="bibr" rid="B4">9</xref>]).

Let f be a meromorphic function and let g be an entire function such that λg<μ< and 0<λfρf<. Then for a sequence of values of r tending to infinity, (9)Tfg(r)<Tf(exp(rμ)).

Lemma 4 (see [<xref ref-type="bibr" rid="B4">9</xref>]).

Let f 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)Tfg(r)<Tg(exp(rμ)).

3. Theorems

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

Theorem 1.

Let f be a meromorphic function and let g, h be any two entire functions such that (11)(i)liminfrlogTh-1(r)(logr)α=A,arealnumber>0,(ii)liminfrlogTh-1Tf(exprμ)(logTh-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(fg)=.

Proof.

From (i) we have for all sufficiently large values of r that (13)logTh-1(r)(A-ε)(logr)α and from (ii) we obtain for all sufficiently large values of r that (14)logTh-1Tf(exprμ)(B-ε)(logTh-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)logTh-1Tfg(r)logTh-1Tf(exp(rμ)) that is, (16)logTh-1Tfg(r)(B-ε)(logTh-1(r))β+1 that is, (17)logTh-1Tfg(r)(B-ε)[(A-ε)(logr)α]β+1 that is, (18)logTh-1Tfg(r)(B-ε)(A-ε)β+1(logr)α(β+1) that is, (19)logTh-1Tfg(r)logr(B-ε)(A-ε)β+1(logr)α(β+1)logr that is, (20)limsuprlogTh-1Tfg(r)logrliminfr(B-ε)(A-ε)β+1(logr)α(β+1)logr. Since ε(>0) is arbitrary and α(β+1)>1, it follows from above that (21)ρh(fg)=. This proves the theorem.

In the line of Theorem 1 and with the help of Lemma 2 one may state the following theorem without its proof.

Theorem 2.

Let f 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)liminfrlogTh-1(r)(logr)α=A,arealnumber>0,(ii)liminfrlogTh-1Tg(exprμ)(logTh-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(fg)=.

Theorem 3.

Let f be a meromorphic function and let g, h be any two entire functions such that (24)(i)liminfrlogTh-1(exp(rμ))(logr)α=A,arealnumber>0,(ii)liminfrlog[logTh-1(Tf(exprμ))/logTh-1(exprμ)][logTh-1(exprμ)]β=B,arealnumber>0 for any α, β satisfying α>1, 0<β<1, αβ>1 and 0<μ<ρg. Then (25)ρh(fg)=.

Proof.

From (i) we have for all sufficiently large values of r that (26)logTh-1(exp(rμ))((A-ε)logr)α and from (ii) we obtain for all sufficiently large values of r that (27)log[logTh-1(Tf(exprμ))logTh-1(exprμ)](B-ε)[logTh-1(exprμ)]βthat  is,logTh-1(Tf(exprμ))logTh-1(exprμ)exp[(B-ε)[logTh-1(exprμ)]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)logTh-1Tfg(r)logrlogTh-1Tf(exp(rμ))logrthat  is,111111111111111111111111111111111111111111logTh-1Tfg(r)logrlogTh-1Tf(exp(rμ))logTh-1(exp(rμ))·logTh-1(exp(rμ))logrthat  is,111111111111111111111111111111111111111111logTh-1Tfg(r)logrexp[(B-ε)[logTh-1(exprμ)]β]·(A-ε)(logr)αlogrthat  is,11111111111111111111111111111111111111111logTh-1Tfg(r)logrexp[(B-ε)(A-ε)β(logr)αβ]·(A-ε)(logr)αlogrthat  is,1111111111111111111111111111111111111111logTh-1Tfg(r)logrexp[(B-ε)(A-ε)β(logr)αβ-1logr]·(A-ε)(logr)αlogrthat  is,1111111111111111111111111111111111111111logTh-1Tfg(r)logr(logr)(B-ε)(A-ε)β(logr)αβ-1·(A-ε)(logr)αlogrthat  is,1111111111111111111111111111111111111111limsuprlogTh-1Tfg(r)logrliminfr(logr)(B-ε)(A-ε)β(logr)αβ-1·(A-ε)(logr)αlogr.

Since ε(>0) is arbitrary and α>1, αβ>1, the theorem follows from above.

Theorem 4.

Let f 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)liminfrlogTh-1(exp(rμ))(logr)α=A,arealnumber>0,(ii)liminfrlog[logTh-1(Tg(exprμ))/logTh-1(exprμ)][logTh-1(exprμ)]β=B,arealnumber>0 for any α,β satisfying α>1, 0<β<1, αβ>1, and 0<μ<ρg. Then (30)ρh(fg)=.

We omit the proof of Theorem 4 as it can be carried out in the line of Theorem 3 and with the help of Lemma 2.

Theorem 5.

Let f be a meromorphic function and let g, h be any two entire functions such that 0<ρg and λh(f)>0. Then (31)ρh(fg)=.

Proof.

Suppose 0<μ<ρg.

As Th-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)logTh-1Tfg(r)logTh-1Tf(exp(rμ))that  is,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiilogTh-1Tfg(r)(λh(f)-ε)rμthat  is,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiilogTh-1Tfg(r)logr(λh(f)-ε)rμlogrthat  is,limsuprlogTh-1Tfg(r)logrliminfr(λh(f)-ε)rμlogrthat  is,iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiρh(fg)=. Thus the theorem follows.

In the line of Theorem 5, one can easily prove the following theorem.

Theorem 6.

Let f 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(fg)=.

Theorem 7.

Let f be a meromorphic function and let g, h be any two entire functions such that 0<ρg and λh(f)>0. Then (34)limsuprlogTh-1Tfg(r)logTh-1Tf(r)=.

Proof.

In view of Theorem 5, we obtain that (35)limsuprlogTh-1Tfg(r)logTh-1Tf(r)limsuprlogTh-1Tfg(r)logr·liminfrlogrlogTh-1Tf(r)that  is,limsuprlogTh-1Tfg(r)logTh-1Tf(r)ρh(fg)·1ρh(f)that  is,limsuprlogTh-1Tfg(r)logTh-1Tf(r)=. Thus the theorem follows.

Theorem 8.

Let f 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)limsuprlogTh-1Tfg(r)logTh-1Tg(r)=.

The proof of Theorem 8 is omitted as it can be carried out in the line of Theorem 7 and in view of Theorem 6.

Theorem 9.

Let f 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)limsuprlogTh-1Tfg(r){logTh-1Tf(rA)}1+α=.

Proof.

If α is such that 1+α0, then the theorem is trivial. So we suppose that 1+α>0.

Since Th-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)logTh-1Tfg(r)logTh-1Tf(exp(rμ))that  is,logTh-1Tfg(r)(λh(f)-ε)rμ, where we choose 0<μ<ρg.

Again from the definition of ρh(f), it follows for all sufficiently large values of r that (39)logTh-1Tf(rA)(ρh(f)+ε)Alogrthat  is,{logTh-1Tf(rA)}1+α(ρh(f)+ε)1+αA1+α(logr)1+α. Now from (38) and (39), it follows for a sequence of values of r tending to infinity that (40)logTh-1Tfg(r){logTh-1Tf(rA)}1+α(λh(f)-ε)rμ(ρh(f)+ε)1+αA1+α(logr)1+α. Since rμ/(logr)1+α as r, the theorem follows from above.

In the line of Theorem 9 and with the help of Lemma 2, one may state the following theorem without its proof.

Theorem 10.

Let f 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)limsuprlogTh-1Tfg(r){logTh-1Tf(rA)}1+α=.

Theorem 11.

Let f 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)limsuprlogTh-1Tfg(r){logTh-1Tg(rA)}1+α=.

Theorem 12.

Let f 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)limsuprlogTh-1Tfg(r){logTh-1Tg(rA)}1+α=.

We omit the proof of Theorems 11 and 12 as those can be carried out in the line of Theorems 9 and 10, respectively.

Theorem 13.

Let f 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)liminfr{logTh-1Tfg(r)}1+αlogTh-1Tf(exp(rμ))=0if  μ>λg.

Proof.

If 1+α0, then the theorem is obvious. We consider 1+α>0.

Since Th-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)logTh-1Tfg(r)<logTh-1Tf(exp(rμ))that  is,logTh-1Tfg(r)<(ρh(f)+ε)rμ. Again for all sufficiently large values of r we get that (46)logTh-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){logTh-1Tfg(r)}1+αlogTh-1Tf(exp(rμ))(ρh(f)+ε)1+αrμ(1+α)(λh(f)-ε)rμ. So from (47) we obtain that (48)liminfr{logTh-1Tfg(r)}1+αlogTh-1Tf(exp(rμ))=0. This proves the theorem.

Theorem 14.

Let f 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)liminfr{logTh-1Tfg(r)}1+αlogTh-1Tg(exp(rμ))=0if  μ>λg.

The proof is omitted as it can be carried out in the line of Theorem 13.

Theorem 15.

Let f 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)liminfr{logTh-1Tfg(r)}1+αlogTh-1Tf(exp(rμ))=0if  μ>λg.

Theorem 16.

Let f 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)liminfr{logTh-1Tfg(r)}1+αlogTh-1Tg(exp(rμ))=0if  μ>λg.

We omit the proofs of Theorems 15 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.

Let f be a meromorphic function and let g, h be any two entire functions such that ρh(f)< and λh(fg)=. Then for every μ(>0), (52)limrlogTh-1Tfg(r)logTh-1Tf(rμ)=.

Proof.

If possible, let there exist a constant β such that for a sequence of values of r tending to infinity, (53)logTh-1Tfg(r)β·logTh-1Tf(rμ). Again from the definition of ρh(f), it follows for all sufficiently large values of r that (54)logTh-1Tf(rμ)(ρh(f)+ε)μlogr. Now combining (53) and (54), we have for a sequence of values of r tending to infinity that (55)logTh-1Tfg(r)β·(ρh(f)+ε)μ·logrthat  is,λh(fg)β·μ(ρh(f)+ε), which contradicts the condition λh(fg)=.

So for all sufficiently large values of r we get that (56)logTh-1Tfg(r)β·logTh-1Tf(rμ), from which the theorem follows.

Remark 18.

Theorem 16 is also valid with “limit superior” instead of “limit” if λh(fg)= is replaced by ρh(fg)= and the other conditions remain the same.

Corollary 19.

Under the assumptions of Theorem 16 and Remark 18, (57)limrTh-1Tfg(r)Th-1Tf(rμ)=,limsuprTh-1Tfg(r)Th-1Tf(rμ)=, respectively hold.

Proof.

By Theorem 16, we obtain for all sufficiently large values of r and for K>1 that (58)logTh-1Tfg(r)KlogTh-1Tf(rμ)that  is,Th-1Tfg(r){Th-1Tf(rμ)}K, from which the first part of the corollary follows.

Similarly using Remark 18, 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 Remark 18, Theorem 16, and Corollary 19, respectively.

Theorem 20.

If f is a meromorphic function and g, h are any two entire functions such that ρh(g)< and ρh(fg)=, then for every μ(>0), (59)limsuprlogTh-1Tfg(r)logTh-1Tg(rμ)=.

Corollary 21.

Theorem 17 is also valid with “limit” instead of “limit superior” if ρh(fg)= is replaced by λh(fg)= and the other conditions remain the same.

Corollary 22.

Under the assumptions of Theorem 17 and Corollary 21, (60)limsuprTh-1Tfg(r)Th-1Tg(rμ)=,limrTh-1Tfg(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.

Conflict of Interests

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

Acknowledgment

The authors are thankful to the referee for his/her valuable suggestions towards the improvement of the paper.

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