Relative orders and slowly changing functions oriented growth analysis of composite entire functions

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ISSN: 2008-6822 (electronic)

http://dx.doi.org/10.22075/ijnaa.2015.259

Relative orders and slowly changing functions

oriented growth analysis of composite entire

functions

Sanjib Kumar Datta´ca,∗, Tanmay Biswasb, Sarmila Bhattacharyyac

aDepartment of Mathematics,University of Kalyani, Kalyani, Dist-Nadia, PIN- 741235, West Bengal, India

bRajbari, Rabindrapalli, R. N. Tagore Road, P.O. Krishnagar, P.S.- Kotwali, Dist-Nadia, PIN- 741101, West Bengal, India cJhorehat F. C. High School for Girls, P.O.- Jhorehat, P.S.- Sankrail, Dist-Howrah, PIN- 711302, West Bengal, India

(Communicated by M.B. Ghaemi)

Abstract

In the paper we establish some new results depending on the comparative growth properties of composition of entire functions using relativeL∗-order (relative L∗-lower order) as compared to their corresponding left and right factors where L≡L(r) is a slowly changing function.

Keywords: Entire function; maximum modulus; maximum term; composition; growth; relative L*-order ( relative L*-lower order); slowly changing function.

2010 MSC: Primary 30D30; Secondary 30D20.

1. Introduction, Definitions and Notations.

Let C denote the set of all finite complex numbers. Also let f be an entire function defined in the open complex plane C. The maximum term µ(r, f) of f =

∞ P n=0

anzn on |z| = r is defined by

µf (r) = max n≥0 (|an|r

n). On the other hand, maximum modulus M(r, f) of f on |z|=r is defined as

M(r, f) = max

|z|=r|f(z)|.The following notation:

log[k]x= loglog[k−1]x for k= 1,2,3, .... and log[0]x=x

is frequently used in the paper.

To start our paper we just recall the following definition :

Corresponding author

Email addresses: sanjib_kr_datta@yahoo.co.in(Sanjib Kumar Datta´c),

tanmaybiswas_math@rediffmail.com(Tanmay Biswas),bsarmila@gmail.com(Sarmila Bhattacharyya)

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Definition 1.1. The order ρf and lower order λf of an entire functionf are defined as

ρf = lim sup r→∞

log[2]Mf(r)

logr and λf = lim infr→∞

log[2]Mf (r)

logr .

Using the inequalities µ(r, f) ≤ M(r, f) ≤ R

R−rµ(R, f) ( cf.[15] ) , for 0 ≤ r < R one may verify that

ρf = lim sup r→∞

log[2]µf(r)

logr and λf = lim infr→∞

log[2]µf(r)

logr .

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 [12] defined it in the following way:

Definition 1.2. [12] A positive continuous functionL(r) is called a slowly changing function if for

ε(>0),

1

kε ≤

L(kr)

L(r) ≤k

ε f or rr(ε) and

uniformly for k(≥1).

Somasundaram and Thamizharasi [13] introduced the notions of L-order and L-lower order for entire functions. The more generalised concept forL-order and L-lower order for entire function are

L∗-order and L∗-lower order respectively. Their definitions are as follows:

Definition 1.3. [13] The L∗-order ρL∗

f and the L

-lower order λL∗

f of an entire function f are defined as

ρLf∗ = lim sup r→∞

log[2]Mf(r)

log [reL(r)] and λ L∗

f = lim infr→∞

log[2]Mf(r) log [reL(r)] .

If an entire functiong is non-constant thenMg(r) is strictly increasing and continuous. Its inverse Mg−1 : (|f(0)|,∞)→(0,∞) exists and is such that lim

s→∞M −1

g (s) = ∞.

Bernal [1] introduced the definition of relative order of an entire function f with respect to an entire function g , denoted byρg(f) as follows:

ρg(f) = inf{µ >0 :Mf(r)< Mg(rµ) for all r > r0(µ)>0}

= lim sup r→∞

logMg−1Mf(r)

logr .

For g(z) = expz,the above definition coincides with the classical one (cf. [16]).

Similarly, one can define the relative lower order of an entire function f with respect to another entire function g denoted byλg(f) as follows :

λg(f) = lim inf r→∞

logM−1

g Mf(r)

logr .

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Definition 1.4. [5] The relative order ρg(f) and relative lower order λg(f) of an entire function f with respect to another entire function g are defined as follows:

ρg(f) = lim sup r→∞

logµ−1 g µf(r)

logr and λg(f) = lim infr→∞

logµ−1 g µf(r)

logr .

In the line of Somasundaram and Thamizharasi ( cf. [13] ), one can define the relative L∗-order

of an entire function in the following manner:

Definition 1.5. ([3],[4]) Therelative L∗-order of an entire functionf with respect to another entire function g , denoted by ρL∗

g (f) is defined in the following way

ρLg∗(f) = infnµ >0 :Mf(r)< Mg

reL(r) µ for all r > r0(µ)>0

o

= lim sup r→∞

logMg−1Mf (r) log [reL(r)] .

Analogusly, one may define therelative L∗-lower order of an entire functionf with respect to another entire function g denoted byλLg∗(f) as follows:

λLg∗(f) = lim inf r→∞

logM−1

g Mf(r) log [reL(r)] .

Datta, Biswas and Ali [6] also gave an alternative definition of relative L∗-order and relative

L∗-lower order of an entire function which are as follows:

Definition 1.6. [6] The relative L∗-order ρL∗

g (f) and the relative L∗-lower order λL

g (f) of an entire function f with respect to g are as follows:

ρLg∗(f) = lim sup r→∞

logµ−g1µf(r)

log [reL(r)] and λ L∗

g (f) = lim infr→∞

logµ−g1µf(r) log [reL(r)] .

For entire functions, the notions of thier growth indicators such as order (lower order), L-order

(L-lower order) and L∗-order (L∗-lower order) are classical in complex analysis and their respective generalized concepts are relative order (relative lower order), relative L-order (relative L-lower or-der) and relative L∗-order (relative L∗-lower order). During the past decades, several researchers have already been exploring their studies in the area of comparative growth properties of composite entire functions in different directions using the classical growth indicators. But at that time, the concepts of relative growth indicators of entire functions and as well as their technical advantages of not comparing with the growths of expz are not at all known to the researchers of this area. There-fore the studies of the growths of composite entire functions using their relative growth indicators are the prime concern of this paper. In fact, some light has already been thrown on such type of works in [7], [8], [9], [10] and [11]. In the paper we study some growth properties of maximum term and maximum modulus of composition of entire functions with respect to another entire function and compare the growth of their corresponding left and right factors on the basis ofrelative L∗-order

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2. Lemmas.

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

Lemma 2.1. [14] Let f and g be any two entire functions. Then for every α >1 and 0< r < R,

µf◦g(r)≤ α α−1µf

αR

R−rµg(R)

.

Lemma 2.2. [14] Let f and g be any two entire functions with g(0) = 0. Then for all sufficiently large values of r,

µf◦g(r)≥ 1 2µf

1 8µg

r

4

.

Lemma 2.3. [2] If f and g are any two entire functions then for all sufficiently large values of r,

Mf◦g(r)≤Mf(Mg(r)) .

Lemma 2.4. [2] If f and g are any two entire functions withg(0) = 0 then for all sufficiently large values of r,

Mf◦g(r)≥Mf

1 8Mg

r

2

.

Lemma 2.5. [5] If f be entire and α >1, 0< β < α, then for all sufficiently larger,

µf(αr)≥βµf(r) .

3. Theorems.

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

Theorem 3.1. Let f and h be any two entire functions with 0< λL∗

h (f)≤ρL

h (f)<∞. Also let g

be an entire function withλL∗

g >0andg(0) = 0. Then for every positive constant A and real number x,

lim r→∞

logµ−h1µf◦g(r)

logµ−h1µf(rA)

1+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 2.2 and Lemma 2.5, we have for all sufficiently large values ofr that

µf◦g(r)≥µf

1 24µg

r

4

. (3.1)

Since µ−h1 is an increasing function, it follows from (3.1) for all sufficiently large values of r that

µ−h1µf◦g(r) ≥ µ−h1µf

1 24µg

r

4

i.e., logµ−h1µf◦g(r) ≥ logµ−h1µf

1 24µg

r

4

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i.e., logµ−h1µf◦g(r) ≥ O(1) + λL

h (f)−ε

log

1 24µg

r

4

+L

1 24µg

r

4

i.e., logµ−h1µf◦g(r) ≥ O(1) + λL

h (f)−ε

logµg r

4

+O(1) +L

1 24µg

r

4

i.e., logµ−h1µf◦g(r)

≥ O(1) + λLh∗(f)−εnr

4

eL(r)o

λL∗ g −ε

+L

1 24µg

r

4

. (3.2)

where we choose 0< ε <minλL∗

h (f), λL

g . Also for all sufficiently large values of r we get that

logµ−h1µf rA

≤ ρLh∗(f) +εlognrAeL(rA)o

i.e., logµ−h1µf rA

≤ ρLh∗(f) +ε

lognrAeL(rA)o

i.e., logµ−h1µf rA 1+x

≤ ρLh∗(f) +ε1+x Alogr+L rA1+x . (3.3)

Therefore from (3.2) and (3.3), it follows for all sufficiently large values ofr that

logµ−h1µf◦g(r)

logµ−h1µf(rA) 1+x

≥ O(1) + λ

L∗

h (f)−ε r 4

eL(r) λL

g −ε

+O(1) +L 241µg r4

(ρL∗

h (f) +ε) 1+x

(Alogr+L(rA))1+x . (3.4)

Since rλL

g −ε

logr1+x → ∞as r→ ∞, the theorem follows from (3.4).

In the line of Theorem 3.1, one may state the following theorem without proof :

Theorem 3.2. Let f and h be any two entire functions with 0< λLh∗(f)<∞or 0 < ρLh∗(f)<∞. Also let g be an entire function with λL∗

g > 0 and g(0) = 0.Then for every positive constant A and real number x,

lim sup r→∞

logµ−h1µf◦g(r)

logµ−h1µg(rA)

1+x =∞ .

Using the same technique of Theorem 3.1 and Theorem 3.2 and using Lemma 2.4 one may easily verify the following two theorems:

Theorem 3.3. Let f and h be any two entire functions with 0 < λL∗

h (f)≤ ρL

h (f) < ∞. Alos let g be an entire function with λLg∗ > 0 and g(0) = 0. Then for every positive constant A and real number x,

lim r→∞

logMh−1Mf◦g(r)

logµ−h1µf(rA)

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Theorem 3.4. Letf,g andhbe any three entire functions whereg is with non zeroL∗-lower order,

g(0) = 0 and either 0 < λL∗

h (f) < ∞ or 0 < ρL

h (f) < ∞.Then for every positive constant A and real number x,

lim sup r→∞

logMh−1Mf◦g(r)

logµ−h1µg(rA)

1+x =∞ .

Theorem 3.5. Let f and h be any two entire functions with 0 < λLh∗(f) ≤ ρLh∗(f) < ∞ and g be an entire function with non zero L∗-lower order and g(0) = 0. Then for any positive integer α and

β,

lim r→∞

log[2]µ−h1µf◦g(exp (exp (rα))) logµ−h1µf(exp (rβ)) +K(r, α;L)

=∞ ,

where K(r, α;L) =

  

0 if rβ =o{L(exp (exp (rα)))}

as r → ∞

L(exp (exp (rα))) otherwise .

Proof . Taking x= 0 and A = 1 in Theorem 3.1 we obtain for all sufficiently large values ofr and for K >1

logµ−h1µf◦g(r) > Klogµ−h1µf(r) i.e., µ−h1µf◦g(r) >

µ−h1µf(r) K

i.e., µ−h1µf◦g(r) > µ−h1µf(r) . (3.5)

Therefore from (3.5), we get for all sufficiently large values ofr that

logµ−h1µf◦g(exp (exp (rα)))>logµ−h1µf (exp (exp (rα)))

i.e., logµ−h1µf◦g(exp (exp (rα))) > λLh∗(f)−ε

.log{exp (exp (rα)).expL(exp (exp (rα)))}

i.e., logµ−h1µf◦g(exp (exp (rα)))

> λLh∗(f)−ε.{(exp (rα)) +L(exp (exp (rα)))}

i.e., logµ−h1µf◦g(exp (exp (rα)))

> λLh∗(f)−ε.

(exp (rα))

1 + L(exp (exp (r α))) (exp (rα))

i.e., log[2]µ−h1µf◦g(exp (exp (rα))) > O(1) + log exp (rα)

+ log

1 + L(exp (exp (r α))) (exp (rα))

i.e., log[2]µ−h1µf◦g(exp (exp (rα))) > O(1) + rα + log

1 + L(exp (exp (r α))) (exp (rα))

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i.e., log[2]µ−h1µf◦g(exp (exp (rα))) > O(1) +rα+L(exp (exp (rα)))

−log [exp{L(exp (exp (rα)))}] + log

1 + L(exp (exp (r α))) exp (µrα)

i.e., log[2]µ−h1µf◦g(exp (exp (rα)))> O(1) +rα+L(exp (exp (rα)))

+ log

1

exp{L(exp (exp (rα)))}+

L(exp (exp (rα)))

exp{L(exp (exp (rα)))}.exp (rα)

i.e., log[2]µ−h1µf◦g(exp (exp (rα))) > O(1) +r(α−β).rβ+L(exp (exp (rα))). (3.6)

Again we have for all sufficiently large values of r that

logµ−h1µf exp rβ

≤ ρLh∗(f) +εlognexp rβeL(exp(rβ))o

logµ−h1µf exp rβ

≤ ρLh∗(f) +εlognexp rβeL(exp(rβ))o

i.e., logµ−h1µf exp rβ

≤ ρLh∗(f) +ε log exp rβ

+L exp rβ i.e., logµ−h1µf exp rβ

≤ ρLh∗(f) +ε rβ+L exp rβ

i.e., logµ −1

h µf exp rβ

− ρL∗

h (f) +ε

L exp rβ

(ρL∗

h (f) +ε)

≤rβ. (3.7)

Now from (3.6) and (3.7), it follows for all sufficiently large values of r that

log[2]µ−h1µf◦g(exp (exp (rα)))

≥O(1) +

r(α−β) ρL∗

h (f) +ε

logµ−h1µf exp rβ

− ρLh∗(f) +εL exp rβ

+L(exp (exp (rα))) (3.8)

i.e., log

[2]

µ−h1µf◦g(exp (exp (rα))) logµ−h1µf(exp (rβ))

≥ L(exp (exp (r

α))) +O(1) logµ−h1µf(exp (rβ))

+ r

(α−β)

ρL∗

h (f) +ε (

1− ρ

L∗

h (f) +ε

L exp rβ

logµ−h1µf (exp (rβ)) )

. (3.9)

Again from (3.8) we get for all sufficiently large values of r that

log[2]µ−h1µf◦g(exp (exp (rα))) logµ−h1µf(exp (rβ)) +L(exp (exp (rα)))

≥ O(1) +r

(α−β)L exp rβ

logµ−h1µf(exp (rβ)) +L(exp (exp (rα)))

+

r(α−β)

ρL∗ h (f)+ε

logµ−h1µf exp rβ

logµ−h1µf(exp (rβ)) +L(exp (exp (rα)))

+ L(exp (exp (r

α)))

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i.e., log

[2]µ−1

h µf◦g(exp (exp (rα))) logµ−h1µf(exp (rβ)) +L(exp (exp (rα)))

O(1)+r(α−β)L(exp(rβ)) L(exp(exp(rα)))

logµ−1h µf(exp(rβ)) L(exp(exp(rα))) + 1

+

r(α−β)

ρL∗ h (f)+ε

1 + L(exp(exp(rα))) logµ−1h µf(exp(rβ))

+ 1

1 + logµ

−1

h µf(exp(rβ)) L(exp(exp(rα)))

. (3.10)

Case I. If rβ =o{L(exp (exp (rα)))} then it follows from (3.9) that

lim r→∞

log[2]µ−h1µf◦g(exp (exp (rα))) logµ−h1µf (exp (rβ))

=∞ .

Case II. rβ 6=o{L(exp (exp (rα)))} then the following two sub cases may arise:

Sub case (a). IfL(exp (exp (rα))) =o

logµ−h1µf exp rβ , then we get from (3.10) that

lim r→∞

log[2]µ−h1µf◦g(exp (exp (rα))) logµ−h1µf(exp (rβ)) +L(exp (exp (rα)))

=∞ .

Sub case (b). If L(exp (exp (rα)))logµ−1

h µf exp r β

then

lim r→∞

L(exp (exp (rα))) logµ−h1µf(exp (rβ))

= 1

and we obtain from (3.10) that

lim inf r→∞

log[2]µ−h1µf◦g(exp (exp (rα))) logµ−h1µf(exp (rβ)) +L(exp (exp (rα)))

=∞ .

Combining Case I and Case II we obtain that

lim r→∞

log[2]µ−h1µf◦g(exp (exp (rα))) logµ−h1µf(exp (rβ)) +L(exp (exp (rα)))

=∞ ,

where K(r, α;L) =

  

0 if rµ=o{L(exp (exp (rα)))} as r→ ∞

L(exp (exp (rα))) otherwise . This proves the theorem.

Theorem 3.6. Let f, g and h be any three entire functions where g is an entire function with non zero L∗-lower order,g(0) = 0, λLh∗(f)>0 and ρLh∗(g)<∞. Then for any positive integerα and β,

lim r→∞

log[2]µ−h1µf◦g(exp (exp (rα))) logµ−h1µg(exp (rβ)) +K(r, α;L)

=∞ ,

where K(r, α;L) =

  

0 if rβ =o{L(exp (exp (rα)))} as r→ ∞

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The proof is omitted because it can be carried out in the line of Theorem 3.5.

In the line of Theorem 3.5 and Theorem 3.6 and with the help of Theorem 3.3 one may easily establish the following two theorems:

Theorem 3.7. Let f and h be any two entire functions with 0 < λL∗

h (f) ≤ ρL

h (f) <∞ and g be an entire function with non zero L∗-lower order and g(0) = 0. Then for any positive integer α and

β,

lim r→∞

log[2]Mh−1Mf◦g(exp (exp (rα))) logMh−1Mf(exp (rβ)) +K(r, α;L)

=∞ ,

where K(r, α;L) =

  

0 if rβ =o{L(exp (exp (rα)))}

as r→ ∞

L(exp (exp (rα))) otherwise .

Theorem 3.8. Let f, g and h be any three entire functions where g is an entire function with non zero L∗-lower order,g(0) = 0, λLh∗(f)>0 and ρLh∗(g)<∞. Then for any positive integerα and β,

lim r→∞

log[2]Mh−1Mf◦g(exp (exp (rα))) logMh−1Mg(exp (rβ)) +K(r, α;L)

=∞ ,

where K(r, α;L) =

  

0 if rβ =o{L(exp (exp (rα)))} as r→ ∞

L(exp (exp (rα))) otherwise .

Theorem 3.9. Let f, g andh be any three entire functions such that ρLg∗ < λhL∗(f)≤ρLh∗(f)<∞.

Then for any β >1,

lim r→∞

logµ−h1µf◦g(r) µ−h1µf(r)·K(r, g;L)

= 0 ,

where K(r, g;L) =

  

1 if L(µg(βr)) =o

eαL(r) as r → ∞

and for some α < λL∗ h (f) L(µg(βr)) otherwise.

Proof . Taking R = βr in Lemma 2.1 and in view of Lemma 2.5 we have for all sufficiently large values of r that

µf◦g(r) ≤

α α−1

µf

αβ

(β−1)µg(βr)

i.e., µf◦g(r) ≤ µf

2α2β

(α−1) (β−1)µg(βr)

.

Since µ−h1 is an increasing function, it follows from above for all sufficiently large values ofr that

µ−h1µf◦g(r)≤µ−h1µf

2α2β

(α−1) (β−1)µg(βr)

i.e., logµ−h1µf◦g(r)≤logµ−h1µf

2α2β

(α−1) (β−1)µg(βr)

i.e., logµ−h1µf◦g(r)≤ ρL

h (f) +ε

"

logµg(βr)·e L

2α2β

(α−1)(β−1)µg(βr)

+O(1)

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i.e., logµ−h1µf◦g(r)≤ ρL

h (f) +ε

[logµg(βr) +L(µg(βr)) +O(1)] (3.11)

i.e., logµ−h1µf◦g(r)≤ ρL

h (f) +ε

βreL(βr) (ρ L∗ g +ε)

+L(µg(βr)) +O(1)

i.e., logµ−h1µf◦g(r)≤ ρL

h (f) +ε

βreL(r) (ρ L∗ g +ε)

+L(µg(βr))

. (3.12)

Also we obtain for all sufficiently large values ofr that

logµ−h1µf (r)≥ λL

h (f)−ε

logreL(r) i.e., logµ−h1µf (r)≥ λL

h (f)−ε

logreL(r)

i.e., µ−h1µf (r)≥

reL(r)(λL

h (f)−ε)

. (3.13)

Now from (3.12) and (3.13) we get for all sufficiently large values ofr that

logµ−h1µf◦g(r) µ−h1µf(r)

ρL∗

h (f) +ε

βreL(r) (ρL

g +ε)

+L(µg(βr))

[reL(r)](λL

h (f)−ε)

. (3.14)

Since ρL∗ g < λL

h (f), we can choose ε(>0) in such a way that

ρLg∗+ε < λLh∗(f)−ε . (3.15)

Case I. LetL(µg(βr)) =o

rαeαL(r) as r→ ∞ and for some α < λLh∗(f).

As α < λL∗

h (f) we can choose ε(>0) in such a way that

α < λLh∗(f)−ε . (3.16)

Since L(µg(βr)) = o

rαeαL(r) asr → ∞we get on using (3.16) that

L(µg(βr))

eαL(r) →0 as r → ∞

i.e., L(µg(βr))

[reL(r)](λL

h (f)−ε)

→0 as r → ∞. (3.17)

Now in view of (3.14), (3.15) and (3.17) we obtain that

lim r→∞

logµ−h1µf◦g(r) µ−h1µf(r)

= 0 . (3.18)

Case II. If L(µg(βr))6= o

eαL(r) as r → ∞ and for some α < λL∗

h (f) then we get from (3.14) that for a sequence of values ofr tending to infinity that

logµ−h1µf◦g(r) µ−h1µf(r).L(µg(βr))

≤ (ρL ∗

h (f)+ε){reL(r)}( ρLg∗+ε)

[reL(r)](λL ∗

h (f)−ε)·L(µ(βr,g))

+ (ρ

L∗ h (f)+ε)

[reL(r)](λL ∗

h (f)−ε)

. (3.19)

Now using (3.15) it follows from (3.19) that

lim r→∞

logµ−h1µf◦g(r) µ−h1µf (r)·L(µg(βr))

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Combining (3.18) and (3.20) we obtain that

lim r→∞

logµ−h1µf◦g(r) µ−h1µf (r)·L(µg(βr))

= 0 ,

where K(r, g;L) =

  

1 ifL(µg(βr)) =o

rαeαL(r) as r→ ∞

and for some α < λLh∗(f)

L(µg(βr)) otherwise.

Thus the theorem is established.

Theorem 3.10. Letf,g andh be any three entire functions withρL∗ g < ρL

h (f)<∞.Then for any β >1,

lim inf r→∞

logµ−h1µf◦g(r) µ−h1µf(r)·K(r, g;L)

= 0 ,

where K(r, g;L) =

  

1 ifL(µg(βr)) =o

eαL(r) as r→ ∞ and for some α < ρL∗

h (f) L(µg(βr)) otherwise.

The proof of Theorem 3.10 is omitted because it can be carried out in the line of Theorem 3.9.

Theorem 3.11. Letf,gandhbe any three entire functions such thatρLg∗ < λhL∗(f)≤ρLh∗(f)<∞.

Then

lim r→∞

logMh−1Mf◦g(r) Mh−1Mf (r)·K(r, g;L)

= 0 ,

where K(r, g;L) =

  

1 ifL(Mg(r)) = o

eαL(r) asr → ∞ and for some α < λL∗

h (f) L(Mg(r)) otherwise.

Theorem 3.12. Letf,g and h be any three entire functions with ρL∗ g < ρL

h (f)<∞. Then

lim inf r→∞

logMh−1Mf◦g(r) Mh−1Mf(r)·K(r, g;L)

= 0 ,

where K(r, g;L) =

  

1 ifL(Mg(r)) = o

eαL(r) asr → ∞ and for some α < ρLh∗(f)

L(Mg(r)) otherwise.

We omit the proof of Theorem 3.11 and Theorem 3.12 as those can be carried out with the help of Lemma 2.3 and in the line of Theorem 3.9 and Theorem 3.10.

Theorem 3.13. Letf, g andh be any three entire functions such that ρL∗

h (f)<∞, λL

h (g)>0and ρLg∗ <∞. Then for any β >1,

(a) if L(µg(βr)) =o

logµ−h1µg(r) then

lim sup r→∞

log[2]µ−h1µf◦g(r) logµ−h1µg(r) +L(µg(βr))

≤ ρ

L∗ g λL∗

h (g)

and (b) if logµ−h1µg(r) =o{L(µg(βr))} then

lim r→∞

log[2]µ−h1µf◦g(r) logµ−h1µg(r) +L(µg(βr))

(12)

Proof . Taking logn1 + O(1)+L(µg(βr))

logµg(βr) o

∼ O(1)+L(µg(βr))

logµg(βr) we have from (3.11) for all sufficiently large

values of r that

logµ−h1µf◦g(r)≤ ρL

h (f) +ε

.logµg(βr)

1 + O(1) +L(µg(βr)) logµg(βr)

i.e., log[2]µ−h1µf◦g(r) ≤ log ρL

h (f) +ε

+ log[2]µg(βr) + log

1 + O(1) +L(µg(βr)) logµg(βr)

i.e., log[2]µ−h1µf◦g(r)≤log ρL

h (f) +ε

+ ρLg∗+εlogβreL(βr)

+ log

1 + O(1) +L(µg(βr)) logµg(βr)

i.e., log[2]µ−h1µf◦g(r)≤log ρL

h (f) +ε

+ ρLg∗+εlogβreL(r)

+ log

1 + O(1) +L(µg(βr)) logµg(βr)

i.e., log[2]µ−h1µf◦g(r)≤O(1) + ρL

g +ε

{logβr+L(r)}+ O(1) +L(µg(βr)) logµg(βr)

i.e., log[2]µ−h1µf◦g(r)≤O(1) + ρL

g +ε

{logr+L(r)}

+ ρLg∗+εlogβ+ O(1) +L(µg(βr)) logµg(βr)

. (3.21)

Again from the definition of relative L∗-lower order of an entire function with respect to another entire function in term of their maximum terms we have for all sufficiently large values of r that

logµ−h1µg(r)≥ λL

h (g)−ε

logreL(r) i.e., logµ−h1µg(r)≥ λL

h (g)−ε

logreL(r) i.e., logµ−h1µg(r)≥ λL

h (g)−ε

[logr+L(r)]

i.e., logr+L(r)≤ logµ

−1 h µg(r) (λL∗

h (g)−ε)

. (3.22)

Hence from (3.21) and (3.22), it follows for all sufficiently large values ofr that

log[2]µ−h1µf◦g(r)

≤ O(1) + ρ

L∗ g +ε λL∗

h (g)−ε !

·logµ−h1µg(r) + ρL

g +ε

logβ+O(1) +L(µg(βr)) logµg(βr)

i.e., log

[2]

µ−h1µf◦g(r) logµ−h1µg(r) +L(µg(βr))

≤ O(1) + ρ

L∗ g +ε

logβ

logµ−h1µg(r) +L(µg(βr))

+ ρ

L∗ g +ε λL∗

h (g)−ε !

· logµ

−1 h µg(r)

logµ−h1µg(r) +L(µg(βr))

+ O(1) +L(µg(βr))

logµ−h1µg(r) +L(µg(βr))

(13)

i.e.,

log[2]µ−h1µf◦g(r) logµ−h1µg(r) +L(µg(βr))

O(1)+(ρL∗

g +ε)logβ L(µg(βr))

logµ−1h µg(r) L(µg(βr)) + 1

+

ρL∗

g +ε λL∗

h (g)−ε

1 + L(µg(βr))

logµ−1h µg(r)

+h 1

1 + logµ

−1

h µg(r) L(µg(βr))

i

logµg(βr)

. (3.23)

Since L(µg(βr)) = o

logµ−h1µg(r) asr → ∞and ε(>0) is arbitrary we obtain from (3.23) that

lim sup r→∞

log[2]µ−h1µf◦g(r) logµ−h1µg(r) +L(µg(βr))

≤ ρ

L∗

g λL∗

h (g)

. (3.24)

Again if logµ−h1µg(r) = o{L(µg(βr))} then from (3.23) we get that

lim r→∞

log[2]µ−h1µf◦g(r) logµ−h1µg(r) +L(µg(βr))

= 0 . (3.25)

Thus from (3.24) and (3.25), the theorem is established.

Corollary 3.14. Let f, g and h be any three entire functions with ρLh∗(f) < ∞, ρLh∗(g) > 0 and

ρL∗

g <∞. Then for any β >1, (a) if L(µg(βr)) = o

logµ−h1µg(r) then

lim inf r→∞

log[2]µ−h1µf◦g(r) logµ−h1µg(r) +L(µg(βr))

≤ ρ

L∗

g ρL∗

h (g)

and (b) if logµ−h1µg(r) = o{L(µg(βr))} then

lim inf r→∞

log[2]µ−h1µf◦g(r) logµ−h1µg(r) +L(µg(βr))

= 0 .

We omit the proof of Corollary 3.14 because it can be carried out in the line of Theorem 3.13.

Theorem 3.15. Let f, g and h be any three entire functions such that ρL∗

h (f) < ∞, λL

h (g) > 0 and ρLg∗ <∞. Then

(a) if L(Mg(r)) =o

logMh−1Mg(r) then

lim sup r→∞

log[2]Mh−1Mf◦g(r) logMh−1Mg(r) +L(Mg(r))

≤ ρ

L∗ g λL∗

h (g)

and (b) if logMh−1Mg(r) =o{L(Mg(r))}then

lim r→∞

log[2]Mh−1Mf◦g(r) logMh−1Mg(r) +L(Mg(r))

(14)

Corollary 3.16. Let f, g and h be any three entire functions with ρLh∗(f) < ∞, ρLh∗(g) > 0 and

ρL∗ g <∞.

(a) If L(Mg(r)) = o

logMh−1Mg(r) then

lim inf r→∞

log[2]Mh−1Mf◦g(r) logMh−1Mg(r) +L(Mg(r))

≤ ρ

L∗

g ρL∗

h (g)

and (b) if logMh−1Mg(r) =o{L(Mg(r))}then

lim inf r→∞

log[2]Mh−1Mf◦g(r) logMh−1Mg(r) +L(Mg(r))

= 0 .

We omit the proof of Theorem 3.15 and Corollary 3.16 because those can be respectively carried out in the line of Theorem 3.13 and Corollary 3.14 and with the help of Lemma 2.3.

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References