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ISSN 2074-1871 Уфимский математический журнал. Том 8. № 2 (2016). С. 97-105.
SOME RESULTS ON GENERALIZED RELATIVE ORDERS OF
MEROMORPHIC FUNCTIONS
S.K. DATTA, T. BISWAS, P. DAS
Abstract. In this paper we discuss some growth rates of compositions of entire and meromorphic functions on the base of generalized relative order and generalized relative lower order of meromorphic functions with respect to entire functions.
Keywords: meromorphic function, entire function, generalized relative order, generalized relative lower order, composition, growth.
Mathematics Subject Classification: 30D35, 30D30, 30D20
1. Introduction, Definitions and Notations
Let f be an entire function defined in the complex plane C. The maximum modulus function corresponding to entire function f is defined as Mf (r) = max||/(z)| : |z| = r}. If f is non-constant, it has the following property:
Property (A) [2] : A non-constant entire function f is said to have Property (A) if for any a > 1 and for all sufficiently large values of r, [Mf (r)]2 ^ Mf (ra) holds. For examples of functions with or without the Property (A), one may see [2].
When f is meromorphic, Mf (r) can not be defined as f is not analytic. In this case one may define another function Tf (r) known as Nevanlinna's Characteristic function of f, playing the same role as maximum modulus function, in the following manner:
Tf (r) = Nf (r) + mf (r), where functions Nf(r) and mf(r) are defined as follows. We first introduce function
Nf (r, a) ^Nf (r, aknown as the counting function of a-points (distinct a-points) of
meromorphic f as
Nf (r, a) = j (t,a) - (°,a) dt + nf О a) log r, 0
Nf (r, a) = j (t,a) - (°,a) dt + nf О a) log r к о
moreover, we denote by nf (r,a) (nf (r,a)^ the number of a-points (distinct a-points) of f in
|z| ^ r and an ro -point is a pole of f. In many situations, Nf (r, ro) and Nf (r, ro) are denoted
by Nf (r) and Nf (r), respectively.
S.K. Datta, T. Biswas, P. Das, Some results focusing generalized relative orders of MEROMORPHIC functions.
© S.K. Datta, T. Biswas, P. Das 2016. Поступила 22 июля 2015 г.
Function m,f (r, to) alternatively denoted by m/ (r) known as the proximity function of f is defined as
mf (r) = 1 j l°g+ |/ (re%d) | dd, where 0
l°g+ x = max (logx, 0) for all x ^ 0.
Also we may denote m (r, j—^ by m/ (r, a).
If f is entire function, then the Nevanlinna's Characteristic function Tf (r) of f is defined as
Tf (r) = mf (r).
For any two entire functions / and g, the ratio as r ^ to is called the growth of
/ with respect to g in terms of their maximum moduli. Also the ratio Tp-j^ as r ^ to is called the growth of f with respect to g in terms of the Nevanlinna's Characteristic functions, when and are both meromorphic functions. Accordingly, the study of comparative growth properties of entire and meromorphic functions which is one of the prominent branches of the value distribution theory of entire and meromorphic functions is the prime concern of the paper. We do not explain the standard definitions and notations in the theory of entire and meromorphic functions as those are available in [12] and [15]. In the sequel the following two notations are used:
x = log (l°g[fc-1'x
and
l°g[fc'x = log (l°g[fc-1] x) for fc=1, 2, 3, ••• ; log[0] x = x
exp[fc] x = exp (exp[fc_1] x) for k = l, 2, 3, ■ ■ ■ ; exp[0] x = x.
Taking this into account, the generalized order (respectively, generalized lower order) of an entire function f as introduced by Sato [14] is given by:
p« = limsup '°g"'M' W , = l.msup'og^),
J l°gl°gMexp * (f) l°gf
( respectively, Aj = liminf-——^J ( ) N = liminf g ^ ( )
) •
\ ' 1 loglogMjXp^ (r) logr
where I ^ l.
When f is meromorphic function, one can easily verify that
m l°g[f-1' Tf (r) l°g['"1'T/ (r) l°g[f-1' Tf (r)
p[/ = limsup----— = limsup---T7T- = limsu^---,
J log Texpz (r) log l°gr + 0(1)
( • , ^[г' V • Pl°g['-1'Tf (r) v . Pl°g['-1'Tf (r)
respectively, AV = liming-——--- = liminf--
y j r^rc logTexp3 (r) r^ logr + 0(1)
where I ^ 1.
These definitions extend the definitions of order pf and lower order A/ of an entire and
[2'
meromorphic function f since for I = 2, these correspond to the particular case py = pf and A/' = Af.
Given a non-constant entire function g defined in the complex plane C, its maximum modulus function Mg (r) and Nevanlinna's Characteristic function Tg (r) are both strictly increasing and continuous functions of r. Also their inverses M~1 (r) : (lg (0)| , ro) ^ (0, ro) and T~1 : (Tg (0) , ro) ^ (0, ro) exist and are such that lim M~1 (s) = ro and limT"1 (s) = ro.
S^TO S^TO
Extending the idea of relative order of entire functions as established by Bernal [1], [2], Lahiri and Banerjee [13] introduced the definition of relative order of a meromorphic function f with respect to another entire function g, denoted by pg (f) to avoid comparing growth just with expz as follows:
Pg ( f) = inf [n > 0 : Tf (r) < Tg (r^) for all sufficiently large r}
v log Tfl~% (r) = lim sup-f- .
r^TO log r
The definition coincides with the classical one if g (z) = exp z, cf. [14].
Likewise, one can define the relative lower order of a meromorphic function f with respect to an entire function g denoted by Xg (f) as follows :
, s logT" 1Tf (r)
\g (f) = lim inf g l f () .
r^TO log r
Further, Banerjee and Jana [4] gave a more generalized concept of relative order of a mero-morphic function with respect to an entire function in the following way:
Definition 1.1. [4] If I ^ 1 is a positive integer, then the l-th generalized relative order of a meromorphic function f with respect to an entire function g denoted by p( f) is defined by
log^T- 1Tf (r) r^TO log r
[/1 l PlJ (/) = Um sup-
Likewise one can define the generalized relative lower order of a meromorphic function f with respect to an entire function g denoted by A^ (f) as
y r^TO log r
For entire and meromrophic functions, the notions of their growth indicators such as order is 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 compositions of entire and meromorphic functions in different directions using the classical growth indicators. But at that time, the concepts of relative orders andconsequently, the generalized relative orders of entire and meromorphic functions with respect to another entire function and as well as their technical advantages of not comparing with the growths of exp were not at all known to the researchers of this area. Therefore the growth of compositions of entire and meromorphic functions needs to be modified on the basis of their relative order some of which has been explored in [5], [6], [7], [8], [9], [10] and [11]. In this paper we establish some newly developed results related to the growth rates of composite entire and meromorphic functions on the basis of their generalized relative orders (respectively, generalized relative lower orders).
2. Lemma
In this section we present a lemma which will be needed in the sequel.
Lemma 2.1. [3] Let f be meromorphic and g be entire and suppose that 0 < ^ < pg ^ ro. Then for a sequence of values of r tending to infinity,
Tfo9(r) >Tf (exp (r")).
3. Theorems In this section we present the main results of the paper.
Theorem 3.1. Let f be a meromorphic function and g,h be any two entire functions such that
l°g[l't -1 (r)
( i) liminf—-—— = A, a is a real number > 0
(log r)
and
r--\ 1- • fl°g[l]Tf-1Tf (exp r") . .
(xi) liminf-f+^ = B, a is a real number > 0
V't-1
for any a, /, p satisfying 0 < a < 1, ¡3 > 0, a (/ + 1) > 1 and 0 < p < pg ^ to. Then
Ph ( f ° 9) =
where I is any positive integer.
Proof. From (i) we have for all sufficiently large values of r that
log^T-1 (r) Z (A - e) (logrf (3.1)
and from ( ) we obtain for all sufficiently large values of that
log"1 T-lTf (exp r") Z (B - e) (log"1 T-1 (r))^+1 . (3.2)
Also T-1 (r) is an increasing function of r and it follows from (3.1), (3.2) and Lemma 2.1 for a sequence of values of r tending to infinity that
logN T-1Tfog(r) Z log"1 T-1Tf (exp (r*)) , log"1 T-1Tfo9(r) Z (B - s) flogW T-1 (r)'
log"' T-1Tfog (r) J (B - s) [(A - e) (l°gr)a]^+1,
log"] T-1Tfo9(r) J (B - e)(A - ef+1 (l°gr)a(f+1),
l°g"'T-1T/ofl (r) ^ (B - g) (A - e)f+1 (l°gr)"(f+1), l°g l°g
'^UP'O^tM:) j liminf(B - £)(A - .
r^^ l°g r l°g r
Since e > 0 is arbitrary and a (/ + 1) > 1, it follows from above that
Ph ( f ° 9> = to.
This proves the theorem. □
Theorem 3.2. Let f be a meromorphic function and g, h be any two entire functions such that
l°g[i] T-1 (exp (r">) ( i) liminf-y~-ra-= A, a is a real number > 0
™ (l°g[2'r) and _
"logt'l T-1(Tf (expr»)>
log
(ii) liminf
log™ T-1(exp rv)
log11 T- (exp r^)
= B, is a real number > 0
for any a, [ satisfying a > 1, 0 < [ < 1, a[ > 1 and 0 < p < pg ^ ro. Then
Ph (/ ° 9) = ro,
where I is any integer with I ^ 1.
Proof. From (i) we have for all sufficiently large values of r that
log"] T-1 (exp (r»)) ^ ((A - e) log[2] r and from (ii) we obtain for all sufficiently large values of r that
(3.3)
log
log"^-1 (Tf (exp r»))
log[l]T-1 (exp r») log''1 T-1 (Tf (exp r")) log"] T-1 (exp r^)
£ (B - e) \log[l1T-1 (exp r")
>
exp (B - e) |~log"] T-1 (exp rß)
(3.4)
Also Th 1 (r) is an increasing function of r and it follows from(3.3), (3.4) and Lemma 2.1 for a sequence of values of r tending to infinity that
log"lT"1T/ofl(r) > log^T"1^ (exp (r"))
log
log"1 T-1Tf 0g (r)
log
log"1 T-%oa (r)
log
log"1 T-1Tf og (r)
log
log"1 T-1Tf og (r)
log
log"1 T-%oa (r)
log
1
>
log[l]T-1 (exp (r»)) £ exp
exp
exp
(B - e) log"] T-1 (exp rß) (B - e) (A - e)ß (log[2] ryß
log
ß] (A - e) (log[2] r)'
log
( A - ) log[2]
(B - e) (A - e)ß (log[2] ryß 1 log[2] r
log
(A - e) (log[2] r J
log
log
£ (log r)(B-£)(Ä-£)ß(log[2]r)
aß-1
(A - e) (log[2] ry
log|i^ £ liminf (logr)^Ä^W--—
logr logr
lim sup -
log
ß-i (A - e)(log[2]r)'
Since e > 0 is arbitrary and a > 1, aß > 1, it completes the proof.
□
Theorem 3.3. Let f be a meromorphic function and g, h be any two entire functions such that 0 < pg ^ ro and A"1 (/) > 0, where I is any positive integer. Then
p[h ( f ° 9) = x
Proof. Suppose 0 < p < pg ^ x.
As T-1 (r) is an increasing function of r, we get from Lemma 2.1 for a sequence of values of r tending to infinity that
log"1 T-1Tfog(r) ^ log"1 T-1Tf (exp (r")), log[l]T-1Tfo9 (r) > (\®(f) - e) r*
ß
ß
a
log^'r,-1T,rv(г) ^ (\W (/) ,
log r log r '
у log^-^T,ofl(r) ^ .. f (А?Ш - 0 limsup—2——J ^ liminf v '
log Г r^x log Г
Ph ( f ° 9) = ж
Thus, the theorem follows. □
Theorem 3.4. Let f be a meromorphic function and g, h be any two entire functions such
И h
that 0 < pg ^ to and A"' (/) > 0, where I is any positive integer. Then
r log^T"1 Tf oa (r) limsup m - / = ж. ™ log[i] T- Tf (r)
Proof. In view of Theorem 3.3, we obtain that
lim sup
Г^Х
lim sup
Г^Х
lim sup
log"
log
log"
r^x log log"
log
T-lTfoa (r) JogMT"lTfoä (rï f bgr
,, ^ limsup---liminf- m .
T- Tf (r) r^x logr ™ logViT-1Tf(r)
Th'lTf0a(r) >P[H(f ° 9) 1
T-1Tf o9 (r)
l]T-1Tf (r)
.
Thus, the theorem follows. □
Theorem 3.5. Let f be a meromorphic function and h be an entire function such that 0 < А"' (/) ^ p"] (/) < ж. Also let д be an entire function with a nonzero order. Then for every positive constant A and every real number a
y log"] T"1 Tf o9 (r) limsup--———1+- = ж,
rWT-1T/ (г V
where I is any positive integer.
Proof. If a be such that 1 + a ^ 0 then the theorem is trivial. So we suppose that 1 + a > 0. Since T-1 (r) is an increasing function of r, we get from Lemma 2.1 for a sequence of values of r tending to infinity that
log"] T"1 Tf og (r) ^ log"] T-1 Tf (exp (r* ))
log[i] T-1 Tfoa(r) > (а£?(/) - e) r», (3.5)
where we choose 0 < p < pg ^ to. Again from the definition of p"' (/), it follows for all sufficiently large values of r that
log[i] T-1Tf (И) ^ (pj?(/)+e)Alogr {log"] T-1Tf (ГА)}1+а ^ (pj? (/) + e)1+a A1+a (logr)1+a . (3.6)
Now from (3.5) and (3.6), it follows for a sequence of values of r tending to infinity that
log« T-1T,„(r) (A,H (/) - e) r»
{log["T-1T/ m}1+" ' (p-'(/) + e)1+°A1+" (logr)1+°
Since 7:——>■ co as r —> oo, the theorem follows from above. □
1 + a
(log r)
Theorem 3.6. Let f be a meromorphic function and g be an entire function with a non-zero order. Also let h and k be any two entire function such that 0 < A"1 (/) and pk (g) < ro, where I is any positive integer. Then for every positive constant A and every real number a,
log[fc] T-1Tfog (r)
limsup-2--—J y = ro.
™ {logT-%(rA)}l+a
We omit the proof of Theorem 3.6 since it follows the same line as for Theorem 3.5.
Theorem 3.7. Let f be a meromorphic function and g, h be any two entire functions such that p"] ( f) < ro and A"] (f ° g) = ro. Then for every p (> 0),
lim'og^-^ = ro,
™ log[l]T-1Tf (r^)
where I is any positive integer.
Proof. We argue by contradiction and assume that there exist a constant [ such that for a sequence of values of tending to infinity,
logW T-1Tfo9(r) ^ 3 • log"! T-1Tf (r*). (3.7)
Again from the definition of p"] ( f), it follows for all sufficiently large values of r that
log"] T-lTf (r>) ^ (/) + e)plogr . (3.8)
Now combining (3.7) and (3.8), we have for a sequence of values of r tending to infinity that
log"1 T-lTfog(r) ^ [ • (pf1 (/)+ e)p • logr i.e., A£(f ° g) ^ [ •p (p^ (/) + e) ,
which contradicts the condition A"] (f ° g) = ro. Hence, for all sufficiently large values of r we get that
log"]T"1T/ofl (r) >3 • log"]T"1T/(r")
that completes the proof. □
Remark 3.1. Theorem 3.7 is also valid with "limit superior" instead of "limit" if Aj] (f ° g) = ro is replaced by p"] (f ° g) = ro and the other conditions remain the same.
Corollary 1. Under the assumptions of Theorem 3.7 and Remark 3.1, lim T-11T/(r] = ro and limsup(r} = ro
respectively.
K
Proof. By Theorem 3.7, we obtain for all sufficiently large values of r and for K > 1 that
log"' T-1Tfog (r) ^ K log"lT"1T/(r")
i.e., log[i-1]T"1T/ofl(r) ^ {log[i-1]T"1T/(r^
from which the first part of the corollary follows. Similarly, using Remark 3.1, we obtain the second part of the corollary. □
In the same way one may state the following theorem and corollaries; their proofs follows the same line as for Remark 3.1, Theorem 3.7 and Corollary 1, respectively.
Theorem 3.8. If f be a meromorphic function and g, h be any two entire functions such that p"' (g) < oo and p"' (f o g) = oo, then for every p (> 0),
™ log''' T- Ty(r*),
where I is any positive integer.
Corollary 2. Theorem 3.7 is also valid with "limit " instead of "limit superior" if p"' ( f o g) = oo is replaced by A"' (f o g) = oo and the other conditions remain the same.
Corollary 3. Under the assumptions of Theorem 3.7 and Corollary 2,
limsupTh (r) =oo and lim Tfe ^^ = oo r^JT-% (r») ^xT-% (r»)
respectively.
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Sanjib Kumar Datta,
Department of Mathematics, University of Kalyani, P.O.-Kalyani, Dist-Nadia, PIN- 741235, West Bengal, India. E-mail: [email protected]
Tanmay Biswas,
Rajbari, Rabindrapalli, R. N. Tagore Road, P.O.-Krishnagar, Dist-Nadia, PIN- 741101, West Bengal, India. E-mail: [email protected]
Pranab Das,
Department of Mathematics, University of Kalyani, P.O.-Kalyani, Dist-Nadia, PIN- 741235, West Bengal, India. E-mail: [email protected]
