A Special Type Of Differential Polynomial And Its Comparative Growth Properties

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International Journal of Modern Engineering Research (IJMER)
Vol. 3, Issue. 5, Sep - Oct. 2013 pp-2606-2614
ISSN: 2249-6645

A Special Type Of Differential Polynomial And Its Comparative
Growth Properties
Sanjib Kumar Datta1, Ritam Biswas2
1

2

Department of Mathematics, University of Kalyani, Kalyani, Dist.- Nadia, PIN- 741235, West Bengal, India.
Murshidabad College of Engineering and Technology, Banjetia, Berhampore, P. O.-Cossimbazar Raj, PIN-742102, West
Bengal, India

Abstract: Some new results depending upon the comparative growth rates of composite entire and meromorphic function
and a special type of differential polynomial as considered by Bhooshnurmath and Prasad[3] and generated by one of the
factors of the composition are obtained in this paper.
AMS Subject Classification (2010) : 30D30, 30D35.

Keywords and phrases: Order (lower order), hyper order (hyper lower order), growth rate, entire function, meromorphic
function and special type of differential polynomial.

I.

INTRODUCTION, DEFINITIONS AND NOTATIONS.

Let ℂ be the set of all finite complex numbers. Also let f be a meromorphic function and g be an entire function defined on
ℂ . In the sequel we use the following two notations:
𝑖 log [𝑘] 𝑥 = log(log [𝑘−1] 𝑥) 𝑓𝑜𝑟 𝑘 = 1,2,3, … ; log [0] 𝑥 = 𝑥
and
𝑖𝑖 exp[𝑘] 𝑥 = exp 𝑒𝑥𝑝 𝑘 −1 𝑥 𝑓𝑜𝑟 𝑘 = 1,2,3, … ; exp[0] 𝑥 = 𝑥.
The following definitions are frequently used in this paper:
Definition 1 The order 𝜌 𝑓 and lower order 𝜆 𝑓 of a meromorphic function f are defined as
lim ⁡
sup log 𝑇(𝑟, 𝑓)
𝜌 𝑓 = 𝑟 →∞
log 𝑟
and
log 𝑇(𝑟, 𝑓)
inf
𝜆 𝑓 = lim𝑟⁡
.
→∞
log 𝑟
If f is entire, one can easily verify that
[2]
𝑀(𝑟, 𝑓)
lim ⁡
sup log
𝜌 𝑓 = 𝑟→∞
log 𝑟
and
log [2] 𝑀(𝑟, 𝑓)
⁡
inf
𝜆 𝑓 = lim𝑟→∞
.
log 𝑟
Definition 2 The hyper order 𝜌 𝑓 and hyper lower order 𝜆 𝑓 of a meromorphic function f are defined as follows
𝜌𝑓 =

lim ⁡
sup
𝑟 →∞

𝜆𝑓 =

lim ⁡
inf
𝑟→∞

𝜌𝑓 =

lim ⁡
sup
𝑟 →∞

log [2] 𝑇(𝑟, 𝑓)
log 𝑟

and
log [2] 𝑇(𝑟, 𝑓)
.
log 𝑟

If f is entire, then
log 𝑟

and
.
log 𝑟
Definition 3 The type 𝜍 𝑓 of a meromorphic function f is defined as follows
lim ⁡
sup 𝑇(𝑟, 𝑓)
𝜍 𝑓 = 𝑟→∞
, 0 < 𝜌 𝑓 < ∞.
𝑟𝜌𝑓
When f is entire, then
lim ⁡
sup log 𝑀(𝑟, 𝑓)
𝜍 𝑓 = 𝑟 →∞
, 0 < 𝜌 𝑓 < ∞.
𝑟𝜌𝑓
Definition 4 A function 𝜆 𝑓 (𝑟) is called a lower proximate order of a meromorphic function f of finite lower order 𝜆 𝑓 if
𝜆𝑓 =

(i)
(ii)

lim ⁡
inf
𝑟→∞

𝜆 𝑓 (𝑟) is non-negative and continuous for 𝑟 ≥ 𝑟0 , say
𝜆 𝑓 (𝑟) is differentiable for 𝑟 ≥ 𝑟0 except possibly at isolated points at which 𝜆′𝑓 (𝑟 + 0) and 𝜆′𝑓 (𝑟 − 0) exists,
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(iii)
lim 𝜆 𝑓 𝑟 = 𝜆 𝑓 ,
𝑟→∞

(iv)

lim 𝑟𝜆′𝑓 𝑟 log 𝑟 = 0
𝑟→∞

and

(v)
lim ⁡
inf
𝑟 →∞

𝑇(𝑟, 𝑓)

= 1.
𝑟 𝜆 𝑓 (𝑟)
Definition 5 Let ‘a’ be a complex number, finite or infinite. The Nevanlinna’s deficiency and Valiron deficiency of ‘a’ with
respect to a meromorphic function f are defined as
𝑚(𝑟, 𝑎; 𝑓)
lim ⁡
sup 𝑁(𝑟, 𝑎; 𝑓)
inf
𝛿 𝑎; 𝑓 = 1 − 𝑟→∞
= lim ⁡
𝑟→∞
𝑇(𝑟, 𝑓)
𝑇(𝑟, 𝑓)
and
𝑁(𝑟, 𝑎; 𝑓) lim ⁡
sup 𝑚(𝑟, 𝑎; 𝑓)
inf
Δ 𝑎; 𝑓 = 1 − lim ⁡
= 𝑟→∞
.
𝑟→∞
𝑇(𝑟, 𝑓)
𝑇(𝑟, 𝑓)
Let f be a non-constant meromorphic function defined in the open complex plane ℂ. Also let n0j, n1j,…, nkj (k ≥ 1) be non𝑘
negative integers such that for each j, 𝑖=0 𝑛 𝑖𝑗 ≥ 1. We call
𝑀𝑗 𝑓 = 𝐴 𝑗 (𝑓) 𝑛 0𝑗 (𝑓 (1) ) 𝑛 1𝑗 … (𝑓 (𝑘) ) 𝑛 𝑘𝑗
where 𝑇 𝑟, 𝐴 𝑗 = 𝑆(𝑟, 𝑓), to be a differential monomial generated by f. The numbers
𝑘

𝛾 𝑀𝑗 =

𝑛 𝑖𝑗
𝑖=0

and
𝑘

Γ 𝑀𝑗 =

(𝑖 + 1)𝑛 𝑖𝑗
𝑖=0

are called the degree and weight of 𝑀𝑗 [𝑓] {cf. [4]} respectively. The expression
𝑠

𝑃 𝑓 =

𝑀𝑗 [𝑓]
𝑗 =1

is called a differential polynomial generated by f. The numbers
𝛾 𝑃 = max 𝛾 𝑀 𝑗
1≤𝑗 ≤𝑠

and
Γ 𝑃 = max Γ
1≤𝑗 ≤𝑠

𝑀𝑗

are respectively called the degree and weight of P[f] {cf. [4]}. Also we call the numbers
𝛾 𝑃 = min 𝛾 𝑀 𝑗
1≤𝑗 ≤𝑠

and k ( the order of the highest derivative of f) the lower degree and the order of P[f] respectively. If 𝛾 𝑃 = 𝛾 𝑃 , P[f] is called a
homogeneous differential polynomial.
Bhooshnurmath and Prasad [3] considered a special type of differential polynomial of the form 𝐹 = 𝑓 𝑛 𝑄[𝑓] where
Q[f] is a differential polynomial in f and n = 0, 1, 2,…. In this paper we intend to prove some improved results depending
upon the comparative growth properties of the composition of entire and meromorphic functions and a special type of
differential polynomial as mentioned above and generated by one of the factors of the composition. We do not explain the
standard notations and definitions in the theory of entire and meromorphic functions because those are available in [9] and
[5].

II.

LEMMAS.

In this section we present some lemmas which will be needed in the sequel.
Lemma 1 [1] If f is meromorphic and g is entire then for all sufficiently large values of r,
𝑇 𝑟, 𝑔
𝑇 𝑟, 𝑓𝑜 𝑔 ≤ 1 + 𝑜 1
𝑇 𝑀 𝑟, 𝑔 , 𝑓 .
log 𝑀 𝑟, 𝑔
Lemma 2 [2] Let f be meromorphic and g be entire and suppose that 0 < 𝜇 < 𝜌 𝑔 ≤ ∞. Then for a sequence of values of r
tending to infinity,
𝑇(𝑟, 𝑓𝑜 𝑔) ≥ 𝑇(exp( 𝑟 𝜇 ), 𝑓).
𝑛
Lemma 3 [3] Let 𝐹 = 𝑓 𝑄[𝑓] where Q[f] is a differential polynomial in f. If n ≥ 1 then 𝜌 𝐹 = 𝜌 𝑓 and 𝜆 𝐹 = 𝜆 𝑓 .
Lemma 4 Let 𝐹 = 𝑓 𝑛 𝑄[𝑓] where Q[f] is a differential polynomial in f. If n ≥ 1 then
𝑇(𝑟, 𝐹)
lim
= 1.
𝑟→∞ 𝑇(𝑟, 𝑓)
The proof of Lemma 4 directly follows from Lemma 3.
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In the line of Lemma 3 we may prove the following lemma:
Lemma 5 Let 𝐹 = 𝑓 𝑛 𝑄[𝑓] where Q[f] is a differential polynomial in f. If n ≥ 1 then 𝜌 𝐹 = 𝜌 𝑓 and 𝜆 𝐹 = 𝜆 𝑓 .
Lemma 6 For a meromorphic function f of finite lower order, lower proximate order exists.
The lemma can be proved in the line of Theorem 1 [7] and so the proof is omitted.
Lemma 7 Let f be a meromorphic function of finite lower order 𝜆 𝑓 . Then for 𝛿 > 0 the function 𝑟 𝜆 𝑓 +𝛿−𝜆 𝑓 (𝑟) is an
increasing function of r.
Proof. Since
𝑑 𝜆 +𝛿−𝜆 (𝑟)
𝑓
𝑟 𝑓
= 𝜆 𝑓 + 𝛿 − 𝜆 𝑓 𝑟 − 𝑟𝜆′𝑓 log 𝑟 𝑟 𝜆 𝑓 +𝛿−𝜆 𝑓 𝑟 −1 > 0
𝑑𝑟
for sufficiently large values of r, the lemma follows.
Lemma 8 [6] Let f be an entire function of finite lower order. If there exists entire functions a i (i= 1, 2, 3,…, n; n ≤ ∞)
satisfying 𝑇 𝑟, 𝑎 𝑖 = 𝑜{𝑇(𝑟, 𝑓)} and
𝑛

𝛿(𝑎 𝑖 , 𝑓) = 1,
𝑖=1

then
lim

𝑇(𝑟, 𝑓)
1
= .
𝑀(𝑟, 𝑓)
𝜋

𝑟→∞ log

III.

THEOREMS.

In this section we present the main results of the paper.
Theorem 1 Let f be a meromorphic function and g be an entire function satisfying
𝑖 𝜆 𝑓 , 𝜆 𝑔 are both finite and
𝑖𝑖 𝑓𝑜𝑟 𝑛 ≥ 1, 𝐺 = 𝑔 𝑛 𝑄[𝑔]. Then
log 𝑇(𝑟, 𝑓𝑜 𝑔)
lim ⁡
inf
≤ 3. 𝜌 𝑓 . 2 𝜆 𝑔 .
𝑟→∞
𝑇(𝑟, 𝐺)
Proof. If 𝜌 𝑓 = ∞, the result is obvious. So we suppose that 𝜌 𝑓 < ∞. Since 𝑇 𝑟, 𝑔 ≤ log + 𝑀 𝑟, 𝑔 , in view of Lemma 1 we
get for all sufficiently large values of r that
𝑇 𝑟, 𝑓𝑜 𝑔 ≤ 1 + 𝑜 1 𝑇(𝑀 𝑟, 𝑔 , 𝑓)
i.e.,
log 𝑇(𝑟, 𝑓𝑜 𝑔) ≤ log{1 + 𝑜(1)} + log 𝑇(𝑀 𝑟, 𝑔 , 𝑓)
i.e.,
log 𝑇(𝑟, 𝑓𝑜 𝑔) ≤ 𝑜 1 + 𝜌 𝑓 + 𝜀 log 𝑀(𝑟, 𝑔)
i.e.,
log 𝑀(𝑟, 𝑔)
lim ⁡
inf
⁡
inf
≤ 𝜌 𝑓 + 𝜀 lim𝑟→∞
.
𝑟 →∞
𝑇(𝑟, 𝑔)
𝑇(𝑟, 𝑔)
Since 𝜀(> 0) is arbitrary, it follows that
lim ⁡
inf
⁡
inf
≤ 𝜌 𝑓 . lim𝑟→∞
.
1
𝑟→∞
𝑇(𝑟, 𝑔)
𝑇(𝑟, 𝑔)
As by condition (v) of Definition 4
𝑇(𝑟, 𝑔)
lim ⁡
inf
= 1,
𝑟 →∞
𝑟 𝜆 𝑔 (𝑟)
so for given 𝜀(0 < 𝜀 < 1) we get for a sequence of values of r tending to infinity that
𝑇 𝑟, 𝑔 ≤ 1 + 𝜀 𝑟 𝜆 𝑔 𝑟
(2)
and for all sufficiently large values of r,
𝑇 𝑟, 𝑔 > 1 − 𝜀 𝑟 𝜆 𝑔 𝑟
(3)
Since
log 𝑀(𝑟, 𝑔) ≤ 3𝑇(2𝑟, 𝑔)
{cf. [5]}, we have by (2), for a sequence of values of r tending to infinity,
log 𝑀(𝑟, 𝑔) ≤ 3𝑇 2𝑟, 𝑔 ≤ 3 1 + 𝜀 2𝑟 𝜆 𝑔 2𝑟 .
(4)
Combining (3) and (4), we obtain for a sequence of values of r tending to infinity that
log 𝑀(𝑟, 𝑔) 3(1 + 𝜀) (2𝑟) 𝜆 𝑔 (2𝑟)
≤
.
.
𝑇(𝑟, 𝑔)
(1 − 𝜀)
𝑟 𝜆 𝑔 (𝑟)
Now for any 𝛿(> 0), for a sequence of values of r tending to infinity we obtain that
log 𝑀(𝑟, 𝑔) 3(1 + 𝜀)
(2𝑟) 𝜆 𝑔 +𝛿
1
≤
.
. 𝜆 (𝑟)
𝜆 𝑔 +𝛿−𝜆 𝑔 (2𝑟)
𝑇(𝑟, 𝑔)
(1 − 𝜀) (2𝑟)
𝑟 𝑔
i.e.,
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log 𝑀(𝑟, 𝑔) 3(1 + 𝜀) 𝜆 +𝛿
≤
.2 𝑔
(5)
𝑇(𝑟, 𝑔)
(1 − 𝜀)
is an increasing function of r by Lemma 7. Since 𝜀(> 0) and 𝛿(> 0) are arbitrary, it follows from (5)

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because 𝑟 𝜆 𝑔 +𝛿−𝜆 𝑔 (𝑟)
that

lim ⁡
inf
𝑟→∞

≤ 3. 2 𝜆 𝑔 .
𝑇(𝑟, 𝑔)

(6)

Thus from (1) and (6) we obtain that
lim ⁡
inf
𝑟 →∞

≤ 3. 𝜌 𝑓 . 2 𝜆 𝑔 .
𝑇(𝑟, 𝑔)

(7)

Now in view of (7) and Lemma 3, we get that
log 𝑇(𝑟, 𝑓𝑜 𝑔) lim ⁡ log 𝑇(𝑟, 𝑓𝑜 𝑔)
𝑇(𝑟, 𝑔)
lim ⁡
inf
inf
= 𝑟→∞
. lim
𝑟 →∞
𝑟→∞ 𝑇(𝑟, 𝐺)
𝑇(𝑟, 𝐺)
𝑇(𝑟, 𝑔)
≤ 3. 𝜌 𝑓 . 2 𝜆 𝑔 .
This proves the theorem.
Theorem 2 Let f be meromorphic and g be entire such that 𝜌 𝑓 < ∞, 𝜆 𝑔 < ∞ and for 𝑛 ≥ 1, 𝐺 = 𝑔 𝑛 𝑄 𝑔 . Then
log [2] 𝑇(𝑟, 𝑓𝑜 𝑔)
lim ⁡
inf
≤ 1.
𝑟 →∞
log 𝑇(𝑟, 𝐺)
Proof. Since
𝑇 𝑟, 𝑔 ≤ log + 𝑀 𝑟, 𝑔 ,
in view of Lemma 1, we get for all sufficiently large values of r that
log 𝑇(𝑟, 𝑓𝑜 𝑔) ≤ log 𝑇(𝑀 𝑟, 𝑔 , 𝑓) + log{1 + 𝑜(1)}
i.e.,
log 𝑇(𝑟, 𝑓𝑜 𝑔) ≤ 𝜌 𝑓 + 𝜀 log 𝑀(𝑟, 𝑔) + 𝑜(1)
i.e.,
log [2] 𝑇 𝑟, 𝑓𝑜 𝑔 ≤ log [2] 𝑀 𝑟, 𝑔 + 𝑂 1 .
(8)
It is well known that for any entire function g,
log 𝑀(𝑟, 𝑔) ≤ 3𝑇 2𝑟, 𝑔
𝑐𝑓. 5 .
Then for 0 < 𝜀 < 1 and 𝛿 > 0 , for a sequence of values of r tending to
infinity it follows from (5) that
log 2 𝑀 𝑟, 𝑔 ≤ log 𝑇 𝑟, 𝑔 + 𝑂 1 .
9
Now combining (8) and (9), we obtain for a sequence of values of r tending to infinity that
log 2 𝑇(𝑟, 𝑓𝑜 𝑔) ≤ log 𝑇 𝑟, 𝑔 + 𝑂(1)
i.e.,
log 2 𝑇(𝑟, 𝑓𝑜 𝑔)
≤ 1.
(10)
log 𝑇 𝑟, 𝑔
As by Lemma 4,
log 𝑇(𝑟, 𝑔)
lim
𝑟→∞ log 𝑇(𝑟, 𝐺)
exists and is equal to 1, then from (10) we get that
log [2] 𝑇(𝑟, 𝑓𝑜 𝑔) lim ⁡ log [2] 𝑇(𝑟, 𝑓𝑜 𝑔)
lim ⁡
inf
inf
= 𝑟→∞
. lim
𝑟→∞
𝑟→∞ log 𝑇(𝑟, 𝐺)
≤ 1.1 = 1.
Thus the theorem is established.
Remark 1 The condition 𝜌 𝑓 < ∞ is essential in Theorem 2 which is evident from the following example.
Example 1 Let 𝑓 = exp[2] 𝑧 and 𝑔 = exp 𝑧. Then 𝑓𝑜 𝑔 = exp[3] 𝑧 and for 𝑛 ≥ 1, 𝐺 = 𝑔 𝑛 𝑄 𝑔 . Taking 𝑛 = 1, 𝐴 𝑗 = 1, 𝑛0𝑗 =
1 and 𝑛1𝑗 = ⋯ = 𝑛 𝑘𝑗 = 0; we see that 𝐺 = exp 2𝑧 . Now we have
𝜌𝑓 =

lim ⁡
sup
𝑟 →∞

=
log 𝑟

lim ⁡
sup
𝑟 →∞

log [2] (exp[2] 𝑟)
=∞
log 𝑟

and
𝜆𝑔 =

lim ⁡
inf
𝑟→∞

log [2] 𝑀(𝑟, 𝑔)
=
log 𝑟

lim ⁡
inf
𝑟→∞

log 2 (exp 𝑟)
= 1.
log 𝑟

Again from the inequality
𝑇(𝑟, 𝑓) ≤ log + 𝑀(𝑟, 𝑓) ≤ 3𝑇(2𝑟, 𝑓)
{cf. p.18, [5]} we obtain that
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𝑇 𝑟, 𝐺 ≤ log 𝑀 𝑟, 𝐺 = log(exp 2𝑟)

i.e.,
log 𝑇(𝑟, 𝐺) ≤ log 𝑟 + 𝑂(1)
and
1
𝑟
1
𝑟
𝑇 𝑟, 𝑓𝑜 𝑔 ≥ log 𝑀 , 𝑓𝑜 𝑔 = exp[2] ( )
3
2
3
2
i.e.,
log [2] 𝑇 𝑟, 𝑓𝑜 𝑔 ≥

𝑟
+ 𝑂 1 .
2

Combining the above two inequalities, we have

𝑟
+ 𝑂(1)
2
≥
.
log 𝑟 + 𝑂(1)

Therefore
lim ⁡
inf
𝑟→∞

= ∞,

which is contrary to Theorem 2.
Theorem 3 Let f and g be any two entire functions such that 𝜌 𝑔 < 𝜆 𝑓 ≤ 𝜌 𝑓 < ∞ and for 𝑛 ≥ 1, 𝐹 = 𝑓 𝑛 𝑄[𝑓] and
𝐺 = 𝑔 𝑛 𝑄 𝑔 . Also there exist entire functions ai (i = 1, 2,…, n; n ≤ ∞) with
𝑖 𝑇 𝑟, 𝑎 𝑖 = 𝑜 𝑇 𝑟, 𝑔
𝑎𝑠 𝑟 → ∞ 𝑓𝑜𝑟 𝑖 = 1, 2, … , 𝑛
and
𝑛

𝑖𝑖

𝛿 𝑎 𝑖 ; 𝑔 = 1.
𝑖=1

Then
{log 𝑇(𝑟, 𝑓𝑜 𝑔)}2
= 0.
𝑟→∞ 𝑇 𝑟, 𝐹 𝑇(𝑟, 𝐺)

lim
Proof. In view of the inequality

𝑇(𝑟, 𝑔) ≤ log + 𝑀(𝑟, 𝑔)
and Lemma 1, we obtain for all sufficiently large values of r that
𝑇 𝑟, 𝑓𝑜 𝑔 ≤ {1 + 𝑜 1 }𝑇(𝑀 𝑟, 𝑔 , 𝑓)
i.e.,
log 𝑇(𝑟, 𝑓𝑜 𝑔) ≤ log{1 + 𝑜(1)} + log 𝑇(𝑀 𝑟, 𝑔 , 𝑓)
i.e.,
log 𝑇(𝑟, 𝑓𝑜 𝑔) ≤ 𝑜 1 + 𝜌 𝑓 + 𝜀 log 𝑀(𝑟, 𝑔)
i.e.,
log 𝑇(𝑟, 𝑓𝑜 𝑔) ≤ 𝑜 1 + 𝜌 𝑓 + 𝜀 𝑟 (𝜌 𝑔 +𝜀) .
Again in view of Lemma 3, we get for all sufficiently large values of r that
log 𝑇 𝑟, 𝐹 > 𝜆 𝐹 − 𝜀 log 𝑟
i.e.,
log 𝑇 𝑟, 𝐹 > 𝜆 𝑓 − 𝜀 log 𝑟
i.e.,
𝑇 𝑟, 𝐹 > 𝑟 𝜆 𝑓 −𝜀 .
Now combining (11) and (12), it follows for all sufficiently large values of r that
𝑜 1 + (𝜌 𝑓 + 𝜀)𝑟 (𝜌 𝑔 +𝜀)
≤
.
𝑇(𝑟, 𝐹)
𝑟 𝜆 𝑓 −𝜀
Since 𝜌 𝑔 < 𝜆 𝑓 , we can choose 𝜀(> 0) in such a way that
𝜌 𝑔 + 𝜀 < 𝜆 𝑓 − 𝜀.
So in view of (13) and (14), it follows that
lim
= 0.
𝑟→∞
𝑇(𝑟, 𝐹)
Again from Lemma 4 and Lemma 8, we get for all sufficiently large values of r that
𝑜 1 + 𝜌 𝑓 + 𝜀 log 𝑀(𝑟, 𝑔)
≤
𝑇(𝑟, 𝐺)
𝑇(𝑟, 𝐺)
i.e.,
lim ⁡
sup log 𝑇(𝑟, 𝑓𝑜 𝑔)
lim ⁡
sup log 𝑀(𝑟, 𝑔)
≤ (𝜌 𝑓 + 𝜀) 𝑟 →∞
𝑟→∞
𝑇(𝑟, 𝐺)
𝑇(𝑟, 𝐺)
i.e.,
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(12)
(13)
(14)
(15)

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𝑇(𝑟, 𝑔)
lim ⁡
sup log 𝑇(𝑟, 𝑓𝑜 𝑔)
lim sup log 𝑀 𝑟, 𝑔
≤ 𝜌 𝑓 + 𝜀 𝑟→∞
. lim
𝑟→∞
𝑟→∞ 𝑇(𝑟, 𝐺)
𝑇(𝑟, 𝐺)
𝑇 𝑟, 𝑔

i.e.,
lim ⁡
sup
𝑟 →∞

≤
𝑇(𝑟, 𝐺)

𝜌 𝑓 + 𝜀 . 𝜋.

Since 𝜀(> 0) is arbitrary, it follows from above that
lim ⁡
sup
𝑟→∞

≤ 𝜌 𝑓 . 𝜋.
𝑇(𝑟, 𝐺)

(16)

Combining (15) and (16), we obtain that
𝑇 𝑟, 𝐹 𝑇(𝑟, 𝐺)
log 𝑇(𝑟, 𝑓𝑜 𝑔) lim ⁡ log 𝑇(𝑟, 𝑓𝑜 𝑔)
sup
= lim
. 𝑟 →∞
𝑟→∞
𝑇(𝑟, 𝐹)
𝑇(𝑟, 𝐺)
lim ⁡
sup
𝑟→∞

≤ 0. 𝜋. 𝜌 𝑓 = 0,
i.e.,
= 0.
𝑟→∞ 𝑇 𝑟, 𝐹 𝑇(𝑟, 𝐺)

lim

Theorem 4 Let f and g be any two entire functions satisfying the following conditions: 𝑖 𝜆 𝑓 > 0 𝑖𝑖 𝜌 𝑓 <
∞ 𝑖𝑖𝑖 0 < 𝜆 𝑔 ≤ 𝜌 𝑔 and also let for 𝑛 ≥ 1, 𝐹 = 𝑓 𝑛 𝑄 𝑓 . Then
[2]
𝜆𝑔 𝜌𝑔
𝑇(𝑟, 𝑓𝑜 𝑔)
lim ⁡
sup log
≥ max{ , }.
𝑟 →∞
[2] 𝑇(𝑟, 𝐹)
log
𝜆𝑓 𝜌𝑓
Proof. We know that for r > 0 {cf. [8]} and for all sufficiently large values of r,
1
1
𝑟
𝑇 𝑟, 𝑓𝑜 𝑔 ≥ log 𝑀
𝑀 , 𝑔 + 𝑜 1 , 𝑓 .
(17)
3
8
4
Since 𝜆 𝑓 and 𝜆 𝑔 are the lower orders of f and g respectively then for given 𝜀(> 0) and for all sufficiently large values of r
we obtain that
log 𝑀 𝑟, 𝑓 > 𝑟 𝜆 𝑓 −𝜀
and
log 𝑀 𝑟, 𝑔 > 𝑟 𝜆 𝑔 −𝜀
where 0 < 𝜀 < min 𝜆 𝑓 , 𝜆 𝑔 . So from (17) we have for all sufficiently large values of r,
𝜆 𝑓 −𝜀
1 1
𝑟
𝑇(𝑟, 𝑓𝑜 𝑔) ≥
𝑀 , 𝑔 + 𝑜 1
3 8
4
i.e.,
1 1
𝑟
𝑇(𝑟, 𝑓𝑜 𝑔) ≥ { 𝑀( , 𝑔)} 𝜆 𝑓 −𝜀
3 9 4
i.e.,
𝑟
log 𝑇(𝑟, 𝑓𝑜 𝑔) ≥ 𝑂 1 + 𝜆 𝑓 − 𝜀 log 𝑀( , 𝑔)
4
i.e.,
𝑟
log 𝑇(𝑟, 𝑓𝑜 𝑔) ≥ 𝑂 1 + 𝜆 𝑓 − 𝜀 ( ) 𝜆 𝑔 −𝜀
4
i.e.,
log 2 𝑇 𝑟, 𝑓𝑜 𝑔 ≥ 𝑂 1 + 𝜆 𝑔 − 𝜀 log 𝑟 .
18
Again in view of Lemma 1, we get for a sequence of values r tending to infinity that
log 2 𝑇 𝑟, 𝐹 ≤ 𝜆 𝐹 + 𝜀 log 𝑟
i.e.,
log 2 𝑇 𝑟, 𝐹 ≤ 𝜆 𝑓 + 𝜀 log 𝑟 .
(19)
Combining (18) and (19), it follows for a sequence of values of r tending to infinity that
𝑂 1 + 𝜆 𝑔 − 𝜀 log 𝑟
≥
.
[2] 𝑇(𝑟, 𝐹)
log
𝜆 𝑓 + 𝜀 log 𝑟
Since 𝜀(> 0) is arbitrary, we obtain that
𝜆𝑔
≥ .
[2] 𝑇(𝑟, 𝐹)
log
𝜆𝑓
Again from (17), we get for a sequence of values of r tending to infinity that
lim ⁡
sup
𝑟 →∞

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𝑟 𝜌 −𝜀
𝑔
log 𝑇(𝑟, 𝑓𝑜 𝑔) ≥ 𝑂 1 + (𝜆 𝑓 − 𝜀)( )
4

i.e.,
log 2 𝑇 𝑟, 𝑓𝑜 𝑔 ≥ 𝑂 1 + 𝜌 𝑔 − 𝜀 log 𝑟 .
Also in view of Lemma 5, for all sufficiently large values of r we have
log [2] 𝑇 𝑟, 𝐹 ≤ 𝜌 𝐹 + 𝜀 log 𝑟
i.e.,

(21)

log [2] 𝑇 𝑟, 𝐹 ≤ 𝜌 𝑓 + 𝜀 log 𝑟 .
Now from (21) and (22), it follows for a sequence of values of r tending to infinity that
𝑂 1 + 𝜌 𝑔 − 𝜀 log 𝑟
≥
.
[2] 𝑇(𝑟, 𝐹)
log
𝜌 + 𝜀 log 𝑟

(22)

𝑓

As 𝜀(0 < 𝜀 < 𝜌 𝑔 ) is arbitrary, we obtain from above that
[2]
𝜌𝑔
lim ⁡
sup log
≥ .
𝑟 →∞
[2] 𝑇(𝑟, 𝐹)
log⁡
𝜌𝑓
Therefore from (20) and (23), we get that
[2]
𝜆𝑔 𝜌𝑔
lim ⁡
sup log
≥ max{ , }.
𝑟 →∞
log [2] 𝑇(𝑟, 𝐹)
𝜆𝑓 𝜌𝑓
Thus the theorem is established.
Theorem 5 Let f be meromorphic and g be entire such that 𝑖 0 < 𝜆 𝑓 < 𝜌 𝑓 , 𝑖𝑖 𝜌 𝑔 < ∞,
𝑖𝑣 𝑓𝑜𝑟 𝑛 ≥ 1, 𝐹 = 𝑓 𝑛 𝑄 𝑓 . Then
𝜆𝑔 𝜌𝑔
lim ⁡
inf
≤ min{ , }.
𝑟 →∞
𝜆𝑓 𝜌 𝑓
Proof. In view of Lemma 1 and the inequality
𝑇 𝑟, 𝑔 ≤ log + 𝑀 𝑟, 𝑔 ,
we obtain for all sufficiently large values of r that
log 𝑇 𝑟, 𝑓𝑜 𝑔 ≤ 𝑜 1 + 𝜌 𝑓 + 𝜀 log 𝑀(𝑟, 𝑔).
Also for a sequence of values of r tending to infinity,
log 𝑀(𝑟, 𝑔) ≤ 𝑟 𝜆 𝑔 +𝜀 .
Combining (24) and (25), it follows for a sequence of values of r tending to infinity that
log 𝑇 𝑟, 𝑓𝑜 𝑔 ≤ 𝑜 1 + (𝜌 𝑓 + 𝜀)𝑟 𝜆 𝑔 +𝜀
i.e.,
log 𝑇 𝑟, 𝑓𝑜 𝑔 ≤ 𝑟 𝜆 𝑔 +𝜀 {𝑜 1 + (𝜌 𝑓 + 𝜀)}
i.e.,
log [2] 𝑇 𝑟, 𝑓𝑜 𝑔 ≤ 𝑂 1 + 𝜆 𝑔 + 𝜀 log 𝑟 .
Again in view of Lemma 5, we obtain for all sufficiently large values of r that
log 2 𝑇 𝑟, 𝐹 > 𝜆 𝐹 − 𝜀 log 𝑟

(23)

𝑖𝑖𝑖 𝜌 𝑓 < ∞ and

(24)
(25)

(26)

i.e.,
log 2 𝑇 𝑟, 𝐹 > 𝜆 𝑓 − 𝜀 log 𝑟 .
Now from (26) and (27), we get for a sequence of values of r tending to infinity that
𝑂 1 + 𝜆 𝑔 + 𝜀 log 𝑟
log [2] 𝑇 𝑟, 𝑓𝑜 𝑔
≤
.
2 𝑇 𝑟, 𝐹
log
𝜆 𝑓 − 𝜀 log 𝑟

(27)

As 𝜀(> 0) is arbitrary, it follows that
𝜆𝑔
log [2] 𝑇 𝑟, 𝑓𝑜 𝑔
≤ .
log 2 𝑇 𝑟, 𝐹
𝜆𝑓
In view of Lemma 1, we obtain for all sufficiently large values of r that
log 2 𝑇 𝑟, 𝑓𝑜 𝑔 ≤ 𝑂 1 + 𝜌 𝑔 + 𝜀 log 𝑟 .
Also by Remark 1, it follows for a sequence of values of r tending to infinity that
log 2 𝑇 𝑟, 𝐹 > 𝜌 𝐹 − 𝜀 log 𝑟
i.e.,
lim ⁡
inf
𝑟 →∞

log 2 𝑇 𝑟, 𝐹 > 𝜌 𝑓 − 𝜀 log 𝑟 .
Combining (29) and (30), we get for a sequence of values of r tending to infinity that

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(29)

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𝑂 1 + 𝜌 𝑔 + 𝜀 log 𝑟
≤
.
𝜌 − 𝜀 log 𝑟
𝑓

Since 𝜀(> 0) is arbitrary, it follows from above that
lim ⁡
inf
𝑟 →∞

𝜌𝑔
≤ .
[2] 𝑇(𝑟, 𝐹)
log
𝜌𝑓

(31)

Now from (28) and (31), we get that
lim ⁡
inf
𝑟 →∞

𝜆𝑔 𝜌𝑔
≤ min{ , }.
[2] 𝑇(𝑟, 𝐹)
log
𝜆𝑓 𝜌 𝑓

The following theorem is a natural consequence of Theorem 4 and Theorem 5:
Theorem 6 Let f and g be any two entire functions such that
𝑖 0 < 𝜆 𝑓 < 𝜌 𝑓 < ∞, 𝑖𝑖 0 < 𝜆 𝑓 ≤ 𝜌 𝑓 < ∞, 𝑖𝑖𝑖 0 < 𝜆 𝑔 ≤ 𝜌 𝑔 < ∞ and
𝑖𝑣 𝑓𝑜𝑟 𝑛 ≥ 1, 𝐹 = 𝑓 𝑛 𝑄 𝑓 . Then
𝜆𝑔 𝜌𝑔
𝜆𝑔 𝜌𝑔
lim ⁡
inf
≤ min{ , } ≤ max{ , } ≤
𝑟→∞
[2] 𝑇(𝑟, 𝐹)
log
𝜆𝑓 𝜌𝑓
𝜆𝑓 𝜌𝑓

lim ⁡
sup
𝑟→∞

.

Theorem 7 Let f be meromorphic and g be entire satisfying 0 < 𝜆 𝑓 ≤ 𝜌 𝑓 < ∞,𝜌 𝑔 > 0 and also let for 𝑛 ≥ 1,
𝐹 = 𝑓 𝑛 𝑄[𝑓]. Then
[2]
𝑇(exp 𝑟 𝜌 𝑔 , 𝑓𝑜 𝑔)
lim ⁡
sup log
= ∞,
𝑟 →∞
log [2] 𝑇(exp 𝑟 𝜇 , 𝐹)
where 0 < 𝜇 < 𝜌 𝑔 .
Proof. Let 0 < 𝜇 ′ < 𝜌 𝑔 . Then in view of Lemma 2, we get for a sequence of values of r tending to infinity that
log 𝑇(𝑟, 𝑓𝑜 𝑔) ≥ log 𝑇(exp(𝑟 𝜇 ) , 𝑓)
i.e.,
′
log 𝑇(𝑟, 𝑓𝑜 𝑔) ≥ 𝜆 𝑓 − 𝜀 log{exp(𝑟 𝜇 )}
i.e.,
′
log 𝑇(𝑟, 𝑓𝑜 𝑔) ≥ 𝜆 𝑓 − 𝜀 𝑟 𝜇
i.e.,
log [2] 𝑇 𝑟, 𝑓𝑜 𝑔 ≥ 𝑂 1 + 𝜇 ′ log 𝑟 .
(32)
Again in view of Lemma 3, we have for all sufficiently large values of r,
log 𝑇(exp 𝑟 𝜇 , 𝐹) ≤ 𝜌 𝐹 + 𝜀 log{exp(𝑟 𝜇 )}
i.e.,
log 𝑇(exp 𝑟 𝜇 , 𝐹) ≤ 𝜌 𝑓 + 𝜀 𝑟 𝜇
i.e.,
log 𝑇(exp 𝑟 𝜇 , 𝐹) ≤ 𝑂 1 + 𝜇 log 𝑟 .
(33)
Now combining (32) and (33), we obtain for a sequence of values of r tending to infinity that
log [2] 𝑇(exp 𝑟 𝜌 𝑔 , 𝑓𝑜 𝑔)
𝑂 1 + 𝜇′ 𝑟 𝜌 𝑔
≥
log [2] 𝑇(exp 𝑟 𝜇 , 𝐹)
𝑂 1 + 𝜇 log 𝑟
from which the theorem follows.
Remark 2 The condition 𝜌 𝑔 > 0 is necessary in Theorem 7 as we see in the following example.
Example 2 Let 𝑓 = exp 𝑧 , 𝑔 = 𝑧 and 𝜇 = 1 > 0 . Then 𝑓𝑜 𝑔 = exp 𝑧 and for 𝑛 ≥ 1, 𝐹 = 𝑓 𝑛 𝑄 𝑓 . Taking
𝑛 = 1, 𝐴 𝑗 = 1, 𝑛0𝑗 = 1 and 𝑛1𝑗 = ⋯ = 𝑛 𝑘𝑗 = 0; we see that 𝐹 = exp 2𝑧. Then
= 1,
log 𝑟
⁡
inf
𝜆 𝑓 = lim𝑟→∞
=1
log 𝑟

𝜌𝑓 =

lim ⁡
sup
𝑟→∞

𝜌𝑔 =

lim ⁡
sup
𝑟→∞

and

Also we get that

log [2] 𝑀(𝑟, 𝑔)
= 0.
log 𝑟

𝑟
𝑇 𝑟, 𝑓 = .
𝜋
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Therefore
𝑇 exp 𝑟 𝜌 𝑔 , 𝑓𝑜 𝑔 =

𝑒
𝜋

and
𝑇 exp 𝑟 𝜇 , 𝐹 =

2 exp 𝑟
.
𝜋

So from the above two expressions we obtain that
log [2] 𝑇(exp 𝑟 𝜌 𝑔 , 𝑓𝑜 𝑔)
𝑂(1)
=
[2] 𝑇(exp 𝑟 𝜇 , 𝐹)
log
log 𝑟 + 𝑂(1)
i.e.,
[2]
𝑇(exp 𝑟 𝜌 𝑔 , 𝑓𝑜 𝑔)
lim ⁡
sup log
= 0,
𝑟 →∞
[2] 𝑇(exp 𝑟 𝜇 , 𝐹)
log
which contradicts Theorem 7.

REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]

Bergweiler, W.: On the Nevanlinna Characteristic of a composite function, Complex Variables, Vol. 10 (1988), pp. 225-236.
Bergweiler, W.: On the growth rate of composite meromorphic functions, Complex Variables, Vol. 14 (1990), pp. 187-196.
Bhooshnurmath, S. S. and Prasad, K. S. L. N.: The value distribution of some differential polynomials, Bull. Cal. Math. Soc., Vol.
101, No. 1 (2009), pp. 55- 62.
Doeringer, W.: Exceptional values of differential polynomials, Pacific J. Math., Vol. 98, No. 1 (1982), pp. 55-62.
Hayman, W. K.: Meromorphic functions, The Clarendon Press, Oxford, 1964.
Lin, Q. and Dai, C.: On a conjecture of Shah concerning small functions, Kexue Tongbao (English Ed.), Vol. 31, No. 4 (1986),
pp. 220-224.
Lahiri, I.: Generalised proximate order of meromorphic functions, Mat. Vensik, Vol. 41 (1989), pp. 9-16.
Niino, K. and Yang, C. C.: Some growth relationships on factors of two composite entire functions, Factorization Theory of
Meromorphic Functions and Related Topics, Marcel Dekker, Inc. (New York and Basel), (1982), pp. 95- 99.
Valiron, G.:
Lectures on the general theory of integral functions, Chelsea Publishing Company, 1949.

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