Başlık: On the growth analysis of wronskians in the light of some generalized growth indicatorsYazar(lar):DATTA, Sanjib Kumar; BISWAS, Tanmay; KAR, AnanyaCilt: 64 Sayı: 2 Sayfa: 001-034 DOI: 10.1501/Commua1_0000000730 Yayın Tarihi: 2015 PDF

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ON THE GROWTH ANALYSIS OF WRONSKIANS IN THE LIGHT OF SOME GENERALIZED GROWTH INDICATORS

SANJIB KUMAR DATTA, TANMAY BISWAS, AND ANANYA KAR

Abstract. In the paper we establish some new results depending on the com-parative growth properties of composite entire or meromorphic functions using m-th generalizedpL -order with rate p, m-th generalisedpL - type with rate

pand m-th generalisedpL -weak type with rate p and wronskians generated

by one of the factors where m and p are any two positive integers.

1. Introduction, Definitions and Notations.

Let f be an entire function de…ned in the open complex plane C: The maxi-mum modulus function M (r; f ) corresponding to f is de…ned on jzj = r as follows:

M (r; f ) = _jzj=rmax _{jf (z)j .}

When f is meromorphic, M (r; f ) cannot be de…ned as f is not analytic throughout the complex plane. In this situation, one may introduce another func-tion T (r; f ) known as Nevanlinna’s characteristic funcfunc-tion of f; playing the same role as maximum modulus function in the following manner:

T (r; f ) = N (r; f ) + m (r; f ) ; where N (r; f ) = r Z 0 n (t; f ) n (0; f ) t dt + n (0; f ) log r

is the pole-counting contribution, where n(r; f ) is the number of poles of f , including multiplicities, for jzj r. On the other hand, the function m (r; f ) known as the

Received by the editors: April 09, 2015; Accepted: July 12, 2015 . 2010 Mathematics Subject Classi…cation. 30D35, 30D30.

Key words and phrases. Transcendental entire function, transcendental meromorphic function, composition, growth, m-th generalisedpL - order with rate p; m-th generalisedpL - type with

rate p and m-th generalisedpL -weak type with rate p, wronskian, slowly changing function. c 2 0 1 5 A n ka ra U n ive rsity

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proximity function is de…ned as m (r; f ) = 1 2 2 Z 0 log+ f rei d ; where log+x = max (log x; 0) for all x> 0 :

In addition, we denote the order and lower order of growth of f by _f and f

respectively and they are de…ned as

f = lim sup r!1

log T (r; f )

log r and f = lim infr!1

log T (r; f ) log r . When f is entire, one can easily verify that

f = lim sup r!1

log[2]M (r; f )

log r and f = lim infr!1

log[2]M (r; f ) log r where log[k]x = log log[k 1]x for k = 1; 2; 3; :::: and log[0]x = x:

Somasundaram and Thamizharasi [12] introduced the notions of L-order and L-type for entire function where L L (r) is a positive continuous function increasing slowly i.e.,L (ar) _{L (r) as r ! 1 for every positive constant ‘a’.} The more generalized concept for L-order and L-type for entire and meromorphic functions are L -order and L -type. Their de…nitions are as follows:

De…nition 1. [12] The L -order L

f and the L -lower order L

f of an entire

function f are de…ned as

L f = lim sup r!1 log[2]M (r; f ) log reL(r) and L f = lim inf_r !1 log[2]M (r; f ) log reL(r) ;

When f is meromorphic, one can easily verify that

L f = lim sup r!1 log T (r; f ) log reL(r) and L f = lim inf_r !1 log T (r; f ) log reL(r) :

De…nition 2. [12] The L -type L

f of an entire function f is de…ned as L f = lim sup r!1 log M (r; f ) reL(r) Lf 0 < L_f _{< 1 :} For meromorphic f , L f = lim sup r!1 T (r; f ) reL(r) Lf 0 < L_f _{< 1 :}

In the line of Somasundaram and Thamizharasi [12] , for any two positive integers m and p, Datta and Biswas [3] introduced the following de…nition:

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De…nition 3. [3] The m-th generalized pL -order with rate p denoted by (m)(p) Lf

and the m-th generalizedpL -lower order with rate p denoted as(m)_(p) Lf of an entire

function f are de…ned in the following way:

(m) (p) L f = lim sup r!1 log[m+1]M (r; f ) log r exp[p]_{L (r)} and

(m) (p) L f = lim inf_r !1 log[m+1]M (r; f ) log r exp[p]_{L (r)} ;

where both m and p are positive integers.

When f is meromorphic, it can be easily veri…ed that

(m) (p) L f = lim sup r!1 log[m]T (r; f ) log r exp[p]_{L (r)} and

(m) (p) L f = lim inf r!1 log[m]T (r; f ) log r exp[p]_{L (r)} ;

These de…nitions extend the generalized L -order [m]L_f (respectively gen-eralized L -lower order [m]L_f ) of an entire or meromorphic function f for each integer m 2 since these correspond to the particular case [m]L_f =(m)₍₁₎ L

f (re-spectively [m]L_f =(m)₍₁₎ L_f ): Clearly(1)_(p) L f =(p) Lf (respectively (1) (p) L f =(p) Lf ) and(1)₍₁₎ L f = Lf (respectively (1) (1) L f = L f ).

In order to compare the relative growth of two entire or meromorphic func-tions having same non zero …nite generalizedpL -order with rate p, one may

intro-duce the de…nitions of generalisedpL -type with rate p and generalisedpL -lower

type with rate p of entire and meromorphic functions having …nite positive gener-alisedpL -order with rate p in the following manner:

De…nition 4. The m-th generalisedpL -type with rate p denoted by(m)_(p) Lf and

m-th generalisedpL -lower type with rate p of an entire function f denoted by(m)_(p) Lf

are respectively de…ned as follows:

(m) (p) L f = lim sup r!1 log[m]M (r; f ) r exp[p]_{L (r)} (m) (p) Lf and (m) (p) L f = lim inf_r !1 log[m]M (r; f ) r exp[p]_{L (r)} (m) (p) L f ; 0 < (m)_(p) L_f _{< 1;} where m and p are any two positive integers.

For meromorphic f , (m) (p) L f = lim sup r!1 log[m 1]T (r; f ) r exp[p]_{L (r)} (m) (p) L f and (m) (p) L f = lim inf_r !1 log[m 1]T (r; f ) r exp[p]_{L (r)} (m) (p) L f ; 0 < (m)_(p) L_f _{< 1;}

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If m = p = 1, then De…nition 4 becomes the classical one given in De…nition 2. If p = 1 and m is any positive integer, we get the de…nition of generalized L -type

[m]L

f (respectively generalized L -lower type [m]L

f ) and if m = 1 and p is any

positive integer; then (1)_(p) L_f =(p) Lf and (1) (p)

L

f =(p) Lf are respectively called

aspL -type with rate p andpL -lower type with rate p of an entire or meromorphic

function f .

Analogusly in order to determine the relative growth of two entire or mero-morphic functions having same non zero …nite generalized pL -lower order with

rate p one may introduce the de…nition of generalisedpL -weak type with rate p of

entire and meromorphic functions having …nite positivegeneralizedpL -lower order

with rate p in the following way:

De…nition 5. The m-th generalisedpL -weak type with rate p denoted by (m)(p) Lf

of an entire function f is de…ned as follows:

(m) (p) L f = lim inf_r !1 log[m]M (r; f ) r exp[p]_{L (r)} (m) (p) Lf ; 0 < (m)_(p) L_f _{< 1;} where both m and p are positive integers.

Also one may de…ne the growth indicator (m)_(p) L_f of an entire function f in the following manner : (m) (p) L f = lim sup r!1 log[m]M (r; f ) r exp[p]_{L (r)} (m) (p) L f ; 0 < (m)_(p) L_f _{< 1;} where m and p are any two positive integers.

For meromorphic f , (m) (p) L f = lim sup r!1 log[m 1]T (r; f ) r exp[p]_{L (r)} (m) (p) Lf and (m) (p) L f = lim inf_r !1 log[m 1]T (r; f ) r exp[p]_{L (r)} (m) (p) L f ; 0 < (m)_(p) L_f _{< 1;} where both m and p are positive integers.

Particularly, when p = 1 and m is any positive integer, then(m)₍₁₎ L

f =

[m]L f

( respectively (m)₍₁₎ L_f = [m]L_f ) and m = 1 and p is any positive integer; then

(1) (p) L f =(p) Lf ( respectively (1) (p) L f =(p) Lf ). Clearly (1) (1) L f = Lf (respectively (1) (1) L f = Lf ).

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De…nition 6. A meromorphic function a a (z) is called small with respect to f if T (r; a) = S (r; f ) where S (r; f ) = o fT (r; f)g i.e., S(r;f )T (r;f ) ! 0 as r ! 1 :

De…nition 7. Let a1; a2; ::::ak be linearly independent meromorphic functions and

small with respect to f . We denote by L (f ) = W (a1; a2; ::::ak; f ) ; the Wronskian

determinant of a1; a2; ::::; ak; f i.e., L (f ) = a1 a2 : : : ak f a0₁ a0₂ : : : a0_k f0 : : : : : : : : : : : : : : : : : : : : : a(k)₁ a(k)₂ : : : a(k)_k f(k) .

De…nition 8. If a 2 C [ f1g,the quantity (a; f ) = 1 lim sup

r!1

N (r; a; f )

T (r; f ) = lim infr!1

m (r; a; f ) T (r; f ) is called the Nevanlinna de…ciency of the value ‘a’:

From the second fundamental theorem it follows that the set of values of a 2 C [ f1g for which (a; f) > 0 is countable and P

a6=1

(a; f ) + _{(1; f)} 2 (cf.[[7], p.43]). If in particular, P

a6=1 (a; f ) + (1; f) = 2, we say that f has the

maximum de…ciency sum.

Lakshminarasimhan [8] introduced the idea of the functions of L-bounded index. Later Lahiri and Bhattacharjee [10] worked on the entire functions of L-bounded index and of non uniform L-L-bounded index. Since the natural extension of a derivative is a di¤erential polynomial, in this paper we prove our results for a special type of linear di¤erential polynomials viz. the Wronskians. In the paper we establish some new results depending on the comparative growth properties of composite entire or meromorphic functions using generalised pL -order with rate

p, generalisedpL - type with rate p and generalisedpL -weak type with rate p and

wronskians generated by one of the factors. We use the standard notations and de…nitions in the theory of entire and meromorphic functions which are available in [7] and [13].

2. Lemmas.

In this section we present some lemmas which will be needed in the sequel. Lemma 1. [1]If f be meromorphic and g be entire then for all su¢ ciently large values of r,

T (r; f g) _{f1 + o (1)g} T (r; g)

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Lemma 2. [2]Let f be meromorphic and g be entire and suppose that 0 < <

g 1. Then for a sequence of values of r tending to in…nity,

T (r; f g) T (exp (r ) ; f ) :

Lemma 3. [9] Let g be an entire function with g < 1 and ai(i = 1; 2; 3; ; n;

n _{1) are entire functions satisfying T (r; a}i) = o fT (r; g)g. If n P i=1 (ai; g) = 1 then lim r!1 T (r;g) log M (r;g) = 1_.

Lemma 4. [11] Let f be a transcendental meromorphic function having the maxi-mum de…ciency sum . Then

lim

r!1

T (r; L (f ))

T (r; f ) = 1 + k k (1; f) :

Lemma 5. Let f be a transcendental meromorphic function having the maximum de…ciency sum and m and p are any two positive integers.Then the m-th generalized

pL -order with rate p (the m-th generalized pL -lower order with rate p) of L (f )

and that of f are same. Proof. By Lemma 4; lim

r!1

log[m]_{T (r;L(f ))}

log[m]_{T (r;f )} exists and is equal to 1 for m 1: Now (m) (p) L L(f ) = lim sup r!1 log[m]T (r; L(f )) log r exp[p]_{L (r)} = lim r!1 log[m]T (r; L(f ))

log[m]T (r; f ) lim supr!1

log[m]T (r; f ) log r exp[p]_{L (r)}

= (m)_(p) L_f :

In a similar manner,(m)_(p) L_{L(f )}=(m)_(p) L_f : This proves the lemma.

Lemma 6. Let f be a transcendental meromorphic function having the maximum de…ciency sum. Then

(i) (m)_(p) L_{L(f )}= 8 > < > : f1 + k k (1; f)g (m)(p) L f for m = 1 (m) (p) L f for m > 1 and (ii) (m)_(p) L_{L(f )}= 8 > < > : f1 + k k (1; f)g (m)(p) L f for m = 1 (m) (p) Lf for m > 1 :

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Proof. By Lemma 4 and Lemma 5 we get that (p) LL(f ) = lim sup r!1 T (r; L (f )) r exp[p]_{L (r)} (p) LL(f ) = lim r!1 T (r; L (f )) T (r; f ) lim supr!1 T (r; f ) r exp[p]_{L (r)} (p) Lf = _{f1 + k} _{k (1; f)g} (p) Lf :

Also for m > 1; lim

r!1

log[m 1]_{T (r;L(f ))}

log[m 1]_{T (r;f )} exists and is equal to 1 and therefore in view of Lemma 5 we obtain that

(m) (p) L L(f ) = lim sup r!1 log[m 1]T (r; L (f )) r exp[p]_{L (r)} (m) (p) L f = lim r!1 log[m 1]T (r; L (f ))

log[m 1]T (r; f ) lim supr!1

log[m 1]T (r; f ) r exp[p]_{L (r)} (m) (p) Lf = (m)_(p) L_f : In a similar manner, (m) (p) L L(f ) = f1 + k k (1; f)g (m) (p) L f for m = 1 and (m)_(p) L_{L(f )} = (m)_(p) L_f otherwise. Thus the lemma follows.

Lemma 7. Let f be a transcendental meromorphic function having the maximum de…ciency sum. Then

(i) (m)_(p) L_{L(f )}= 8 > < > : f1 + k k (1; f)g (m)(p) L f for m = 1 (m) (p) Lf for m > 1 and (ii) (m)_(p) L_{L(f )}= 8 > < > : f1 + k k (1; f)g (m)(p) L f for m = 1 (m) (p) L f for m > 1 :

We omit the proof of Lemma 7 because it can be carried out in the line of Lemma 6:

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3. Theorems.

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

Theorem 1. Let f be a transcendental meromorphic function with P

a6=1

(a; f ) + (1; f) = 2 and g be entire such that 0 <(m)(p)

L

f < 1 and (p) Lg < 1 where m

and p are any two positive integers. If exp[p 1]_{L (M (r; g)) = o} _{r exp}[p]_{L (r)}

as r ! 1 and for some positive <(p) Lg , then

lim inf

r!1

log[m]T (r; f g) log[m]T exp r exp[p]_{L (r)} (p) Lg

; L(f ) (p)

L g :

Proof. Since T (r; g) log+M (r; g) and by Lemma 1, we get for a sequence of values of r tending to in…nity that

log T (r; f g) _{log f1 + o (1)g + log T (M (r; g) ; f)} i:e:; log[m]T (r; f g) O (1) + log[m]T (M (r; g) ; f )

i:e:; log[m]T (r; f g) O (1) + (m)_(p) L_f + " h log M (r; g) + exp[p 1]L (M (r; g))i i:e:; log[m]T (r; f g) O (1) + (m)_(p) L_f + " " (p) Lg + " h r exp[p]L (r) i(p) Lg + exp[p 1]L (M (r; g)) # : (1)

Further in view of Lemma 5, we obtain for all su¢ ciently large values of r that log[m]T exphr exp[p]L (r)i(p)

L g ; L(f ) ! (m) (p) L L(f ) " "_h r exp[p]L (r)i(p) L g + exp[p 1] " L exphr exp[p]L (r)i(p) L g !##

i:e:; log[m]T exp h r exp[p]L (r) i(p) Lg ; L(f ) ! (m) (p) L f " h r exp[p]L (r)i(p) L g :

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Now from (1) and above it follows for a sequence of values of r tending to in…nity that

log[m]T (r; f g)

log[m]T exp r exp[p]_{L (r)} (p) Lg _{; L(f )}

O (1) + (m)_(p) L_f + " (p) Lg + " r exp[p]L (r) (p) Lg + exp[p 1]L (M (r; g)) (m) (p) L f " r exp[p]L (r) (p) Lg i:e:; log [m] T (r; f g) log[m]T exp r exp[p]_{L (r)} (p) Lg

; L(f ) O (1) (m) (p) L f " r exp[p]L (r) (p) Lg + (m) (p) L f + " " (p) Lg + " + exp[p 1]L(M (r;g)) [r exp[p]_L(r)_](p) Lg # (m) (p) L f " : (2)

As <(p) Lg and exp[p 1]L (M (r; g)) = o r exp[p]L (r) as r ! 1, we obtain

that lim r!1 exp[p 1]L (M (r; g)) r exp[p]_{L (r)} (p) Lg = 0 : (3)

Since " (> 0) is arbitrary, it follows from (2) and (3) that lim inf

r!1

; L(f ) (p)

L g :

Thus the theorem is established.

In the line of Theorem 1, the following two theorems can be carried out and therefore their proofs are omitted:

Theorem 2. Let f be a transcendental meromorphic function having the maximum de…ciency sum and g be entire with 0 < (m)_(p) L

lim inf

r!1

; L(f ) (p)

L g :

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a6=1

L f (m) (p) L f < 1 and(p) Lg < 1

where m and p are any two positive integers. If

exp[p 1]L (M (r; g)) = o hr exp[p]L (r)i _{as r ! 1} and for some positive <(p) Lg , then

lim sup

r!1

log[m]T (r; f g)

(m) (p) L f (p) Lg (m) (p) L f :

Remark 1. For p = 1, Theorem 3 reduces to Theorem 14 of [5].

Using the notion ofpL -lower type with rate p (p is any positive integer)

we may state the following theorem without its proof because it can be proved in the line of Theorem 3:

Theorem 4. Let f be a transcendental meromorphic function having maximum de-…ciency sum and g be entire with 0 <(m)_(p) L_f (m)_(p) L

f < 1 and(p) Lg < 1 where

m and p are any two positive integers. If exp[p 1]_{L (M (r; g)) = o} _{r exp}[p]_{L (r)}

lim inf

r!1

; L(f ) (m) (p) L f (p) Lg (m) (p) L f :

Now we state the following three theorems without their proofs as those can be carried out in the line of Theorem 1, Theorem 2 and Theorem 3 repectively. Theorem 5. Let f be meromorphic and g be transcendental entire with P

a6=1 (a; g) + _{(1; g) = 2;} (p) Lg > 0; (m) (p) L

f < 1 and(p) Lg < 1 where m and p are any

two positive integers. If exp[p 1]L (M (r; g)) = o r exp[p]L (r) _{as r ! 1 and} for some positive <(p) Lg , then

lim inf

r!1

log[m]T (r; f g)

log[m]T exp r exp[p]_{L (r)} (p) Lg _{; L(g)}

(m) (p) L f (p) Lg (p) Lg :

Theorem 6. Let f be meromorphic and g be transcendental entire having the max-imum de…ciency sum such that (p) Lg > 0;

(m)

(p) Lf < 1 and (p) Lg < 1 where m

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lim inf

r!1

; L(g)

(m)

(p) Lf (p) Lg (p) Lg

:

Theorem 7. Let f be meromorphic and g be transcendental entire with P

a6=1 (a; g) + _{(1; g) = 2;}(p) Lg > 0; (m) (p) L

f < 1 and (p) Lg < 1 where m and p are any

two positive integers. If exp[p 1]_{L (M (r; g)) = o} _{r exp}[p]_{L (r)} _{as r ! 1 and}

for some positive <(p) Lg , then

lim sup

r!1

; L(g) (m) (p) L f (p) Lg (p) Lg :

Remark 2. Theorem 7 improves Theorem 15 of Datta et. al. { cf. [5]}.

Theorem 8. Let f be meromorphic and g be transcendental entire having the max-imum de…ciency sum such that(p) Lg > 0;

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(p) Lf < 1 and (p) Lg < 1 where m

lim inf

r!1

; L(g) (m) (p) L f (p) Lg (p) Lg :

We omit the proof of Theorem 8 as it can easily be established in the line of Theorem 4.

Further we state the following two theorems which are based onpL -weak

type with rate p (p is any positive integer):

Theorem 9. Let f be a transcendental meromorphic function having the maximum de…ciency sum and g be entire with 0 < (m)_(p) L_f (m)_(p) L

f < 1 and (p) Lg < 1

where m and p are any two positive integers. If exp[p 1]L (M (r; g)) = o

h

r exp[p]L (r) i

lim inf

r!1

log[m]T (r; f g)

(m) (p) Lf (p) Lg (m) (p) L f :

Theorem 10. Let f be meromorphic and g be transcendental entire having the max-imum de…ciency sum such that(p) Lg > 0;

(m)

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and p are any two positive integers. If exp[p 1]L (M (r; g)) = o r exp[p]L (r) as r ! 1 and for some positive <(p) Lg , then

lim inf

r!1

; L(g)

(m)

:

The proofs of the above two theorems can be carried out in the line of Theorem 4 and Theorem 8 respectively and therefore their proofs are omitted.

Using the concept of the growth indicator [p] Lg (where p is any positive

integer) of an entire function g, we may state the subsequent six theorems without their proofs since those can be carried out in the line of Theorem 1, Theorem 2, Theorem 3, Theorem 5, Theorem 6 and Theorem 7 respectively.

a6=1

(a; f ) + (1; f) = 2 and g be entire such that 0 < (m)(p)

L

f < 1 and(p) Lg < 1 where m

and p are any two positive integers. If exp[p 1]L (M (r; g)) = o r exp[p]L (r) as r ! 1 and for some positive <(p) Lg , then

lim inf

r!1

log[m]T (r; f g)

log[m]T exp r exp[p]_{L (r)} (p) Lg _{; L(f )} (p) L g :

Theorem 12. Let f be a transcendental meromorphic function having the maxi-mum de…ciency sum and g be entire with 0 <(m)_(p) L

f < 1 and(p) Lg < 1 where m

lim inf

r!1

log[m]T (r; f g)

log[m]T exp r exp[p]_{L (r)} (p) Lg _{; L(f )} (p) L g :

a6=1

L f

(m)

(p) Lf < 1 and(p) Lg < 1

where m and p are any two positive integers. If

exp[p 1]L (M (r; g)) = o hr exp[p]L (r)i _{as r ! 1} and for some positive <(p) Lg , then

lim sup

r!1

; L(f ) (m) (p) Lf (p) Lg (m) (p) L f :

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Theorem 14. Let f be meromorphic and g be transcendental entire with P a6=1 (a; g) + _{(1; g) = 2;} (p) Lg > 0; (m) (p) L

f < 1 and (p) Lg < 1 where m and

p are any two positive integers. If exp[p 1]_{L (M (r; g)) = o} _{r exp}[p]_{L (r)} _as

r ! 1 and for some positive <(p) Lg , then

lim inf

r!1

; L(g) (m) (p) L f (p) Lg (p) Lg :

Theorem 15. Let f be meromorphic and g be transcendental entire having the max-imum de…ciency sum such that (p) Lg > 0;

(m) (p)

L

lim inf

r!1

; L(g)

(m)

:

Theorem 16. Let f be meromorphic and g be transcendental entire with P

a6=1

(a; g) + _{(1; g) = 2;} (p) Lg > 0; (m) (p)

L

f < 1 and (p) Lg < 1 where m and

p are any two positive integers. If exp[p 1]_{L (M (r; g)) = o} _{r exp}[p]_{L (r)} _as

r ! 1 and for some positive <(p) Lg , then

lim sup

r!1

; L(g)

(m)

:

Theorem 17. Let f be transcendental meromorphic function having the maximum de…ciency sum such that 0 <(m)_(p) L

f < g and (m) (p)

L

f > 0 where m and p are any

two positive integers and g be an entire function. If exp[p 1]L exp r exp[p]L (r) = o r exp[p]_{L (r)} _{as r ! 1 for any} _{> 0; then}

lim sup r!1 log[m]T r exp[p]_{L (r) ; f} _g log[m 1]T (r; L(f )) 8 > > > < > > > : (p) Lf f1+k k (1;f )g (p) Lf for m = 1 (m) (p) L f (m) (p) Lf for m > 1 :

(14)

Proof. From De…nition 4 and any arbitrary " (> 0) ; we obtain for all su¢ ciently large values of r that

log[m 1]T (r; L(f )) (m)_(p) L_{L(f )}+ " h r exp[p]L (r) i(m) (p) L L(f ) : (4)

Now in view of Lemma 5 and Lemma 6, it follows from (4) for all su¢ ciently large values of r that log[m 1]T (r; L(f )) 8 > > < > > : f1 + k k (1; f)g (m)(p) L f + " r exp[p]L (r) (m) (p) L f for m = 1 (m) (p) Lf + " r exp[p]L (r) (m) (p) L f for m > 1 : (5) As 0 < (m)_(p) L

f < g; we obtain in view of Lemma 2 for a sequence of values of r

tending to in…nity that

log[m]T r exp[p]L (r) ; f g log[m]T exp r exp[p]L (r) (m) (p) L f ; f !

i:e:; log[m]T r exp[p]L (r) ; f g

(m) (p) L f " "_h r exp[p]L (r) i(m) (p) L f

+ exp[p 1]L exp r exp[p]L (r) (m) (p) L f !# :

Therefore from (5) and above, it follows for a sequence of values of r tending to in…nity that log[m]T r exp[p]_{L (r) ; f} _g log[m 1]T (r; L(f )) 8 > > > > > > > > < > > > > > > > > : (m) (p) L f " " [r exp[p]_L(r)_](m)(p) L

f _+exp[p 1]_L _exp₍_{r exp}[p]_L(r)₎(m)(p) L f !# f1+k k (1;f )g (m)(p) L f +" [r exp[p]L(r)] (m) (p) Lf for m = 1 (m) (p) L f " " [r exp[p]L(r)] (m) (p) L f _+exp[p 1] L exp(r exp[p]L(r)) (m) (p) L f !# (m) (p) Lf +" [r exp[p]L(r)] (m) (p) Lf for m > 1:

(15)

Since lim

r!1

exp[p 1]_L _exp₍_{r exp}[p]_L(r)₎ (m) (p) Lf ! [r exp[p]_L(r)_] (m) (p) L f

= 0 as exp[p 1]L exp r exp[p]L (r) = o r exp[p]_{L (r)} _{(r ! 1) for any} _{> 0; we obtain from above that}

lim sup r!1 log[m]T r exp[p]L (r) ; f g log[m 1]T (r; L(f )) 8 > > > < > > > : (p) Lf f1+k k (1;f )g (p) Lf for m = 1 (m) (p) L f (m) (p) L f for m > 1 : Thus the theorem follows.

Now using the concept of the growth indicator(m)_(p) L_f (m and p are any two positive integers) of a meromorphic function f , we may state thefollowing theorem without its proof since it can be carried out in the line of Theorem 17.

a6=1

(a; f ) + (1; f) = 2; 0 < (m)(p) Lf < g and

(m)

(p) Lf > 0 where m and p are any two

positive integers and g be an entire function. If exp[p 1]L exp r exp[p]L (r) = o r exp[p]_{L (r)} _{as r ! 1 for any} _{> 0; then}

lim sup r!1 log[m]T r exp[p]_{L (r) ; f} _g log[m 1]T (r; L(f )) 8 > > > < > > > : (p) Lf f1+k k (1;f )g (p) Lf for m = 1 (m) (p) L f (m) (p) L f for m > 1 : Theorem 19. Let f be meromorphic function and g be entire function having the maximum de…ciency sum such that (i) (m)_(p) L

f < 1 and (ii) 0 < (p) Lg (p) Lg

< 1 where m and p are any two positive integers. Then (a) If exp[p 1]_{L (M (r; g)) = o fT (r; L(g))g then}

lim sup r!1 log[m]T (r; f g) T (r; L(g)) + exp[p 1]_{L (M (r; g))} (m) (p) L f (p) Lg (1 + k _{k (1; g))} (p) Lg

and (b) if T (r; L(g)) = o exp[p 1]_{L (M (r; g))} _then

lim sup r!1 log[m]T (r; f g) T (r; L(g)) + exp[p 1]_{L (M (r; g))} (m) (p) L f :

(16)

Proof. Since T (r; g) log+M (r; g) in view of Lemma 1, we obtain for all su¢ -ciently large values of r that

log T (r; f g) _{log f1 + o (1)g + log T (M (r; g) ; f)} i:e:; log[m]T (r; f g) o (1) + log[m]T (M (r; g) ; f )

i:e:; log[m]T (r; f g) o (1) + (m) (p) L f + " n log M (r; g) + exp[p 1]L (M (r; g)) o : (6)

Using the de…nition of(p)L -type, we obtain from (6) for all su¢ ciently large values

of r that log[m]T (r; f g) o (1) + (m)_(p) L_f + " (p) Lg + " h r exp[p]L (r)i(p) L g + (m)_(p) L_f + " exp[p 1]L (M (r; g)) : (7) Again from the de…nition of(p)L -lower type and in view of Lemma 5 and Lemma

6, we get for all su¢ ciently large values of r that T (r; L(g)) (p) LL(g) " h r exp[p]_{L (r)}i(p) L L(g) i:e:; T (r; L(g)) n (1 + k _{k (1; g))} (p) Lg " o h r exp[p]L (r) i(p) Lg i:e:; hr exp[p]L (r)i(p) L g T (r; L(g)) (1 + k _{k (1; g))} (p) Lg " : (8)

Now from (7) and (8) ; it follows for all su¢ ciently large values of r that log[m]T (r; f g) o (1) + (m)_(p) L_f + " exp[p 1]L (M (r; g)) + (m)_(p) L_f + " (p) Lg + " T (r; L(g)) (1 + k _{k (1; g))} (p) Lg " ie:; log [m]_{T (r; f} _g) T (r; L(g)) + exp[p 1]_{L (M (r; g))} o (1) T (r; L(g)) + exp[p 1]_{L (M (r; g))} + (m) (p) L f +" (p) Lg +" ((1+k k (1;g))(p) Lg ") 1 +exp[p_{T (r;L(g))}1]L(M (r;g)) + (m) (p) L f + " 1 + _exp[pT (r;L(g))1]_{L(M (r;g))} : (9)

If exp[p 1]_{L (M (r; g)) = o fT (r; L(g))g then from (9) we get that} lim sup r!1 log[m]T (r; f g) T (r; L(g)) + exp[p 1]_{L (M (r; g))} (m) (p) L f (p) Lg (1 + k _{k (1; g))} (p) Lg :

(17)

Thus the …rst part of theorem follows.

Again if T (r; L(g)) = o exp[p 1]_{L (M (r; g)) then from (9) it follows that}

lim sup r!1 log[m]T (r; f g) T (r; L(g)) + exp[p 1]_{L (M (r; g))} (m) (p) L f

which is the second part of the theorem.

Theorem 20. Let f be a meromorphic function and g be an entire function having the maximum de…ciency sum with (i) (m)_(p) L_f _{< 1 and (ii) 0 <} (p) Lg (p) Lg

lim inf r!1 log[m]T (r; f g) T (r; L(g)) + exp[p 1]_{L (M (r; g))} (m) (p) L f (p) Lg (1 + k _{k (1; g))} (p) Lg

lim inf r!1 log[m]T (r; f g) T (r; L(g)) + exp[p 1]_{L (M (r; g))} (m) (p) L f :

We omit the proof of the above theorem as it can be carried out in the line of Theorem 19.

Using the concept of the growth indicator(p) Lg and(p) Lg (p is any positive

integer) of an entire function g, we may state the subsequent two theorems without their proofs since those can be carried out in the line of Theorem 19 and Theorem 20 respectively.

Theorem 21. Let f be a meromorphic function and g be an entire function having the maximum de…ciency sum such that (i)(m)_(p) L

f < 1 and (ii) 0 <(p) Lg (p) Lg

Theorem 22. Let f be a meromorphic function and g be an entire function having the maximum de…ciency sum with (i) (m)_(p) L_f _{< 1 and (ii) 0 <} (p) Lg (p) Lg

(18)

and (b) if T (r; L(g)) = o exp[p 1]_{L (M (r; g))} _then lim inf r!1 log[m]T (r; f g) T (r; L(g)) + exp[p 1]_{L (M (r; g))} (m) (p) L f :

Now we state the following four theorems under some di¤erent conditions which can also be carried out using the same technique of Theorem 19 and therefore their proofs are omitted.

f < 1; (ii) (p) Lg < 1 and (iii) (p) Lg > 0 where m and p are any two positive integers. Then

(a) If exp[p 1]_{L (M (r; g)) = o fT (r; L(g))g then}

lim sup r!1 log[m]T (r; f g) T (r; L(g)) + exp[p 1]_{L (M (r; g))} (m) (p) Lf (p) Lg (1 + k _{k (1; g))} (p) Lg

and (b) if T (r; L(g)) = o exp[p 1]L (M (r; g)) then lim sup r!1 log[m]T (r; f g) T (r; L(g)) + exp[p 1]_{L (M (r; g))} (m) (p) L f :

Theorem 24. Let f be a meromorphic function and g be an entire function having the maximum de…ciency sum with (i)(m)_(p) L_f _{< 1; (ii)}(p) Lg < 1 and (iii)(p) Lg

> 0 where m and p are any two positive integers. Then (a) If exp[p 1]_{L (M (r; g)) = o fT (r; L(g))g then}

f < 1; (ii) (p) Lg > 0 and (iii) (p) Lg < 1 where m and p are any two positive integers. Then

(19)

Theorem 26. Let f be a meromorphic function and g be an entire function having the maximum de…ciency sum with (i)(m)_(p) L_f _{< 1; (ii)}(p) Lg > 0 and (iii)(p) Lg

The following theorem can also be carried out in the line of Theorem 19 and therefore its proof is omitted.

Theorem 27. Let f be a meromorphic function with (m)_(p) L_f _{< 1 where m and p} are any two positive integers. Also let g be an entire function with P

a6=1

(a; g) + (1; g) = 2 and also satisfy any one of the following conditiona:

(i) 0 <(p) Lg < 1; (ii) 0 <(p) Lg < 1; (iii) 0 < (p) Lg < 1; or (iv) 0 <(p) Lg

< 1. Then

lim inf r!1 log[m]T (r; f g) T (r; L(g)) + exp[p 1]_{L (M (r; g))} (m) (p) Lf (1 + k _{k (1; g))} and (b) if T (r; L(g)) = o exp[p 1]_{L (M (r; g))} _then

Remark 3. Theorem 27 extends Theorem 26 of Datta et. al. { cf. [5]}. Theorem 28. Let f be a transcendental meromorphic function with P

a6=1

(a; f ) + (1; f) = 2 and g be entire such that (i)(m)(p)

L

f =(p) Lg ; (ii) 0 < (p) Lg < 1

and (iii)(m)_(p) L

(20)

(a) If exp[p 1]L (M (r; g)) = o n log[m 1]T (r; L(f )) o then lim sup r!1 log[m]T (r; f g) log[m 1]T (r; L(f )) + exp[p 1]_{L (M (r; g))} 8 > > > < > > > : (p) Lf (p) Lg (1+k k (1;f )) (p) Lf for m = 1 (m) (p) L f (p) Lg (m) (p) L f for m > 1 and (b) if log[m 1]T (r; L(f )) = o exp[p 1]L (M (r; g)) then

lim sup r!1 log[m]T (r; f g) log[m 1]T (r; L(f )) + exp[p 1]_{L (M (r; g))} (m) (p) L f :

Proof. In view of condition (ii) we obtain from (7) for all su¢ ciently large values of r that log[m]T (r; f g) o (1) + (m)_(p) L_f + " (p) Lg + " h r exp[p]L (r)i (m) (p) L f + (m)_(p) L_f + " exp[p 1]L (M (r; g)) : (10) Again from the de…nition of(p)L -lower type and in view of Lemma 5, we get for

all su¢ ciently large values of r that

log[m 1]T (r; L(f )) (m)_(p) L_{L(f )} " hr exp[p]L (r)i (m) (p) L L(f ) i:e:; log[m 1]T (r; L(f )) (m)_(p) L_{L(f )} " h r exp[p]L (r) i(m) (p) L f i:e:; hr exp[p]L (r)i (m) (p) L f log[m 1]T (r; L(f )) (m) (p) L L(f ) " : (11)

Now from (10) and (11) ; it follows for all su¢ ciently large values of r that log[m]T (r; f g) o (1) + (m)_(p) L_f + " (p) Lg + " log[m 1]T (r; L(f )) (m) (p) LL(f ) " + (m)_(p) L_f + " exp[p 1]L (M (r; g)) ie:; log [m]_{T (r; f} _g) log[m 1]T (r; L(f )) + exp[p 1]_{L (M (r; g))} o (1) log[m 1]T (r; L(f )) + exp[p 1]_{L (M (r; g))}

(21)

+ (m) (p) L f +" (p) Lg +" (m) (p) LL(f ) " 1 +exp[p 1]L(M (r;g)) log[m 1]_{T (r;L(f ))} + (m) (p) L f + " 1 +_explog[m[p 1]1]_{L(M (r;g))}T (r;L(f )) : (12) If exp[p 1]L (M (r; g)) = o n log[m 1]T (r; L(f )) o

then from (12) we get that

lim sup r!1 log[m]T (r; f g) log[m 1]T (r; L(f )) + exp[p 1]_{L (M (r; g))} (m) (p) Lf + " (p) Lg + " (m) (p) LL(f ) " :

Since " (> 0) is arbitrary, it follows from above that lim sup r!1 log[m]T (r; f g) log[m 1]T (r; L(f )) + exp[p 1]_{L (M (r; g))} (m) (p) L f (p) Lg (m) (p) LL(f ) : Now in view of Lemma 6, we get from above that

lim sup r!1 log[m]T (r; f g) log[m 1]T (r; L(f )) + exp[p 1]_{L (M (r; g))} 8 > > > < > > > : (p) Lf (p) Lg (1+k k (1;f )) (p) Lf for m = 1 (m) (p) L f (p) Lg (m) (p) L f for m > 1 : Thus the …rst part of the theorem follows.

Again if log[m 1]T (r; L(f )) = o exp[p 1]L (M (r; g)) then from (12) it follows that lim sup r!1 log[m]T (r; f g) log[m 1]T (r; L(f )) + exp[p 1]_{L (M (r; g))} (m) (p) L f + " :

As " (> 0) is arbitrary, we obtain from above that lim sup r!1 log[m]T (r; f g) log[m 1]T (r; L(f )) + exp[p 1]_{L (M (r; g))} (m) (p) L f :

Thus the second part of the theorem is established.

Theorem 29. Let f be a transcendental meromorphic function having the maxi-mum de…ciency sum and g be entire with (i) (m)_(p) Lf < 1; (ii)

(m) (p)

L

f = (p) Lg ;

(iii)(p) Lg < 1 and (iv) (m) (p)

L

f > 0 where m and p are any two positive integers:

(22)

(a) If exp[p 1]L (M (r; g)) = o n log[m 1]T (r; L(f )) o then lim inf r!1 log[m]T (r; f g) log[m 1]T (r; L(f )) + exp[p 1]_{L (M (r; g))} 8 > > > < > > > : (p) Lf (p) Lg (1+k k (1;f )) (p) Lf for m = 1 (m) (p) L f (p) Lg (m) (p) Lf for m > 1

and (b) if log[m 1]T (r; L(f )) = o exp[p 1]L (M (r; g)) then

lim inf r!1 log[m]T (r; f g) log[m 1]T (r; L(f )) + exp[p 1]_{L (M (r; g))} (m) (p) L f :

a6=1

(a; f ) + (1; f) = 2 and g be entire such that (i)(m)(p) Lf =(p) Lg ; (ii) 0 < (p) Lg < 1

and (iii)(m)_(p) L

f > 0 where m and p are any two positive integers: Then

(a) If exp[p 1]_{L (M (r; g)) = o}n_log[m 1]_{T (r; L(f ))}o _then

lim inf r!1 log[m]T (r; f g) log[m 1]T (r; L(f )) + exp[p 1]_{L (M (r; g))} 8 > > > < > > > : (p) Lf (p) Lg (1+k k (1;f )) (p) Lf for m = 1 (m) (p) L f (p) Lg (m) (p) L f for m > 1

and (b) if log[m 1]T (r; L(f )) = o exp[p 1]_{L (M (r; g))} _then

a6=1

L

f =(p) Lg ; (ii) 0 < (p) Lg < 1

and (iii)(m)_(p) L

(23)

(a) If exp[p 1]L (M (r; g)) = o n log[m 1]T (r; L(f )) o then lim inf r!1 log[m]T (r; f g) log[m 1]T (r; L(f )) + exp[p 1]_{L (M (r; g))} 8 > > > < > > > : (p) Lf (p) Lg (1+k k (1;f )) (p) Lf for m = 1 (m) (p) L f (p) Lg (m) (p) Lf for m > 1 and (b) if log[m 1]T (r; L(f )) = o exp[p 1]_{L (M (r; g))} _then

We omit the proof of the above three theorems as those can be carried out in the line of Theorem 28.

Remark 4. For p = 1, Theorem 30 reduces to Theorem 27 of [5]. Similarly using the concept of the growth indicator (m)_(p) L

f and (p) Lg we

may state the subsequent four theorems without their proofs since those can be carried out in the line of Theorem 28, Theorem 29, Theorem 30 and Theorem 31 respectively.

a6=1

L f < 1; (ii) (m) (p) L f =(p) Lg ; (iii) (p) Lg < 1 and (iv) (m)

(p) Lf > 0 where m and p are any two positive integers: Then

(a) If exp[p 1]L (M (r; g)) = o n log[m 1]T (r; L(f )) o then lim sup r!1 log[m]T (r; f g) log[m 1]T (r; L(f )) + exp[p 1]_{L (M (r; g))} 8 > > > < > > > : (p) Lf (p) Lg (1+k k (1;f )) (p) Lf for m = 1 (m) (p) L f (p) Lg (m) (p) Lf for m > 1 and (b) if log[m 1]T (r; L(f )) = o exp[p 1]_{L (M (r; g))} _then

(24)

Theorem 33. Let f be a transcendental meromorphic function having the maxi-mum de…ciency sum and g be entire with (i)(m)_(p) L_f =(p) Lg ; (ii) 0 <(p) Lg < 1

and (iii)(m)_(p) L

a6=1

(25)

(a) If exp[p 1]L (M (r; g)) = o n log[m 1]T (r; L(f )) o then lim inf r!1 log[m]T (r; f g) log[m 1]T (r; L(f )) + exp[p 1]_{L (M (r; g))} 8 > > > < > > > : (p) Lf (p) Lg (1+k k (1;f )) (p) Lf for m = 1 (m) (p) L f (p) Lg (m) (p) L f for m > 1 and (b) if log[m 1]T (r; L(f )) = o exp[p 1]_{L (M (r; g))} _then

Remark 5. Theorem 35 extends Theorem 1 of Datta et. al. { cf. [4]}.

Analogously we state the following four theorems under some di¤erent con-ditions which can also be carried out using the same technique of Theorem 28 and therefore their proofs are omitted.

a6=1

(a) If exp[p 1]L (M (r; g)) = o n log[m 1]T (r; L(f )) o then lim sup r!1 log[m]T (r; f g) log[m 1]T (r; L(f )) + exp[p 1]_{L (M (r; g))} 8 > > > < > > > : (p) Lf (p) Lg (1+k k (1;f )) (p) Lf for m = 1 (m) (p) L f (p) Lg (m) (p) L f for m > 1 and (b) if log[m 1]T (r; L(f )) = o exp[p 1]L (M (r; g)) then

Remark 6. Theorem 36 extends Theorem 3 of Datta et. al. { cf. [4]}.

Theorem 37. Let f be a transcendental meromorphic function having the maxi-mum de…ciency sum and g be entire with (i) (m)_(p) L_f =(p) Lg ; (ii) 0 < (p) Lg < 1

(26)

and (iii)(m)_(p) L

Remark 7. Theorem 37 extends Theorem 2 of Datta et. al. { cf. [4]}. Theorem 38. Let f be a transcendental meromorphic function with P

a6=1

L

f =(p) Lg ; (ii) 0 < (p) Lg < 1

and (iii)(m)_(p) L

(a) If exp[p 1]L (M (r; g)) = o n log[m 1]T (r; L(f )) o then lim sup r!1 log[m]T (r; f g) log[m 1]T (r; L(f )) + exp[p 1]_{L (M (r; g))} 8 > > > < > > > : (p) Lf (p) Lg (1+k k (1;f )) (p) Lf for m = 1 (m) (p) L f (p) Lg (m) (p) Lf for m > 1 and (b) if log[m 1]T (r; L(f )) = o exp[p 1]_{L (M (r; g))} _then

Theorem 39. Let f be a transcendental meromorphic function having the maxi-mum de…ciency sum and g be entire with (i)(m)_(p) L

f =(p) Lg ; (ii) 0 <(p) Lg < 1

and (iii)(m)_(p) L

(27)

(a) If exp[p 1]L (M (r; g)) = o n log[m 1]T (r; L(f )) o then lim inf r!1 log[m]T (r; f g) log[m 1]T (r; L(f )) + exp[p 1]_{L (M (r; g))} 8 > > > < > > > : (p) Lf (p) Lg (1+k k (1;f )) (p) Lf for m = 1 (m) (p) L f (p) Lg (m) (p) L f for m > 1 and (b) if log[m 1]T (r; L(f )) = o exp[p 1]_{L (M (r; g))} _then

a6=1 (a; f ) + (1; f) = 2 and 0 < (m)(p) L f (m) (p) L

f < g where m and p are any two positive

integers. Also let g be an entire function. Then lim sup r!1 log[m]T (r; f g) log[m]T (exp r ; L(f )) (m) (p) L f (m) (p) Lf where 0 < < g 1:

Proof. In view of Lemma 2, we obtain for a sequence of values of r tending to in…nity that

log[m]T (r; f g) log[m]T (exp r ; f ) i:e:; log[m]T (r; f g) (m)_(p) L_f "

h

r + exp[p 1]L (exp r ) i

: (13) Also for any arbitrary " (> 0) ; it follows from De…nition 3 and in view of Lemma 5 for all su¢ ciently large values of r that

log[m]T (exp r ; L(f )) (m)_(p) L_{L(f )}+ " hr + exp[p 1]L (exp r )i i:e:; log[m]T (exp r ; L(f ))

(m) (p) L f + " h r + exp[p 1]L (exp r ) i : (14)

Now from (13) and (14) ; we get for a sequence of values of r tending to in…nity that log[m]T (r; f g) log[m]T (exp r ; L(f )) (m) (p) L f " r + exp[p 1]L (exp r ) (m) (p) Lf + " r + exp[p 1]L (exp r ) :

(28)

Since " (> 0) is arbitrary, it follows from above that lim sup r!1 log[m]T (r; f g) log[m]T (exp r ; L(f )) (m) (p) L f (m) (p) Lf : Thus the theorem follows.

Theorem 41. Let f be meromorphic with (m)_(p) L

f < 1 and g be transcendental

entire with …nite lower order and P

a6=1 (a; g) + (1; g) = 2. Also let there exists

entire functions ai(i = 1; 2; 3; ; q; q 1) such that T (r; ai) = o fT (r; g)g and q P i=1 (ai; g) = 1. If exp[p 1]L (M (r; g)) = o fT (r; L(g))g then lim sup r!1 log[m]T (r; f g) T (r; L(g)) (m) (p) Lf (1 + k _{k (1; g))} ; otherwise lim r!1 log[m]T (r; f g) T (r; L(g)) exp[p 1]_{L (M (r; g))} = 0

where m and p are any two positive integers.

Proof. From (6) we get for all su¢ ciently large values of r that log[m]T (r; f g)

T (r; L(g)) :

(m)

(p) Lf + " log M (r; g) + exp[p 1]L (M (r; g)) + o(1)

T (r; L(g)) (15) i:e:; log [m] T (r; f g) T (r; L(g)) +O(1) (m) (p) L f + " log M (r; g) T (r; L(g)) + exp[p 1]_{L (M (r; g))} T (r; L(g)) i:e:; log [m]_{T (r; f} _g) T (r; L(g)) +O(1) (m) (p) L f + " log M (r; g) T (r; L(g)) T (r; g) T (r; L(g))+ exp[p 1]L (M (r; g)) T (r; L(g)) : (16)

Case I.Let exp[p 1]_{L (M (r; g)) = o fT (r; L(g))g. Then}

lim

r!1

exp[p 1]_{L (M (r; g))}

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Now combining (17) and (16) and in view of Lemma 3 and Lemma 4, it follows that lim sup r!1 log[m]T (r; f g) T (r; L(g)) (m) (p) L f (1 + k _{k (1; g))} : (18)

Case II.Let exp[p 1]_{L (M (r; g)) 6= o fT (r; L(g))g : Then from (15) we get for all}

su¢ ciently large values of r that log[m]T (r; f g) T (r; L(g)) exp[p 1]_{L (M (r; g))} (m) (p) L f + " log M (r; g) T (r; L(g)) exp[p 1]_{L (M (r; g))}+ O(1) T (r; L(g)) : i:e:; lim sup

r!1

log[m]T (r; f g)

T (r; L(g)) exp[p 1]_{L (M (r; g))} = 0 :

Thus combining Case I and Case II the theorem follows.

In the line of Theorem 41 the following theorem can be proved:

Theorem 42. Let f be meromorphic with (m)_(p) L_f _{< 1 where m} 1 and g be transcendental entire with …nite lower order and P

a6=1 (a; g) + (1; g) = 2. Also

let there exists entire functions ai(i = 1; 2; 3; ; q; q 1) such that T (r; ai) =

o fT (r; g)g and q P i=1 (ai; g) = 1. If exp[p 1]L (M (r; g)) = o fT (r; L(g))g then lim inf r!1 log[m]T (r; f g) T (r; L(g)) (m) (p) L f (1 + k _{k (1; g))} ; otherwise lim inf r!1 log[m]T (r; f g) T (r; L(g)) exp[p 1]_{L (M (r; g))} = 0

where m and p are any two positive integers.

Remark 8. Theorem 40 and Theorem 41 respectively extend Theorem 3:4 and Theorem 3:3 of [6].

Theorem 43. Let f be a transcendental meromorphic function having the maxi-mum de…ciency sum and g be entire such that 0 <(m)_(p) L_f (m)_(p) L

f < 1 where m

and p are any two positive integers. Then for any positive real number A; lim sup r!1 log[m+1]T exp rA ; f g log[m]T (exp (r ) ; L(f )) + K (r; g; L) = 1 ; where 0 < < _g and K (r; g; L) = 0 if r = o exp [p 1]_{L exp exp} _rA _{as r ! 1}

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Proof. Let 0 < < 0 _<

g. Using De…nition 3 we obtain in view of Lemma 2 for

a sequence of values of r tending to in…nity that

log[m]T exp rA ; f g log[m]T exp exp rA 0

; f i:e:; log[m]T exp rA ; f g

(m) (p) L f " n exp rA 0

+ exp[p 1]L exp exp rA

0 o

i:e:; log[m]T exp rA ; f g

(m) (p) L f " 8 < : exp r A 0 0 @1 + exp [p 1]_{L exp exp r}A 0 (exp (rA₎₎ 0 1 A 9 = ; i:e:; log[m+1]T exp rA ; f g O (1) + 0rA

+ log 8 < :1 +

exp[p 1]_{L exp exp r}A 0

(exp (rA₎₎ 0

9 = ; i:e:; log[m+1]T exp rA ; f g O (1) + 0rA

+ log " 1 + exp [p 1]_{L exp exp} 0_rA exp ( 0_rA₎ #

i:e:; log[m+1]T exp rA ; f g O (1) + 0rA+ exp[p 1]L exp exp rA loghexp[p] L exp exp rA i

+ log " 1 + exp [p 1]_{L exp exp} 0_rA exp ( 0_rA₎ #

i:e:; log[m+1]T exp rA ; f g O (1) + 0rA+ exp[p 1]L exp exp rA

+ log 1

exp[p]_{fL (exp (exp ( r}A_)))g

+ exp

[p 1]_{L exp exp} 0_rA

exp[p]_{fL (exp (exp ( r}A_{)))g exp (} 0_rA₎

#

i:e:; log[m+1]T exp rA ; f g O (1) + 0r(A ) r

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Again in view of Lemma 5, we have for all su¢ ciently large values of r that log[m]T (exp (r ) ; L(f )) (m)_(p) L_{L(f )}+ " log

h

exp (r ) exp[p]L (exp (r )) i i:e:; log[m]T (exp (r ) ; L(f )) (m)_(p) Lf + "

h r + exp[p 1]L (exp (r )) i i:e:; log[m]T (exp (r ) ; L(f )) (m)_(p) L f + " exp[p 1]L (exp (r )) (m) (p) Lf + " r : (20)

Now from (19) and (20) ; it follows for a sequence of values of r tending to in…nity that log[m+1]T exp rA ; f g O (1) + 0 @ 0r(A ) (m) (p) L f + " 1

Ahlog[m]T (exp (r ) ; L(f )) (m)_(p) L_f + " exp[p 1]L (exp (r ))i

+ exp[p 1]L exp exp rA (21)

i:e:; log

[m+1]_{T exp r}A _{; f} _g

log[m]T (exp (r ) ; L(f ))

exp[p 1]_{L exp exp} _rA _{+ O (1)}

log T (exp (r ) ; L(f )) + 0_r(A ) (m) (p) Lf + " 8 < :1 (m) (p) Lf + " exp[p 1]L (exp (r )) log[m]T (exp (r ) ; L(f )) 9 = ; : (22)

Again from (21) we get for a sequence of values of r tending to in…nity that log[m+1]T exp rA _{; f} _g

log[m]T (exp (r ) ; L(f )) + exp[p 1]_{L (exp (exp ( r}A₎₎₎

O (1) 0_r(A ) _exp[p 1]_{L (exp (r ))}

+ 0_r(A ) (m) (p) L f +" log[m]T (exp (r ) ; L(f ))

+ exp

[p 1]_{L exp exp} _rA

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i:e:; log

[m+1]_{T exp r}A _{; f} _g

log[m]T (exp (r ) ; L(f )) + exp[p 1]_{L (exp (exp ( r}A₎₎₎ O(1) 0_r(A )_exp[p 1]_{L(exp(r ))}

exp[p 1]_{L(exp(exp( r}A₎₎₎ log[m]_{T (exp(r );L(f ))} exp[p 1]_{L(exp(exp( r}A₎₎₎+ 1 + 0_r(A ) (m) (p) L f +" log[m]T (exp (r ) ; L(f )) 1 +exp[p 1]L(exp(exp( rA))) log[m]_{T (exp(r );L(f ))} + 1

1 + _explog[p[m]1]T (exp(r );L(f ))_{L(exp(exp( r}A₎₎₎

: (23)

Case I.If r = o exp[p 1]_{L exp exp} _rA _{then it follows from (22) that}

lim sup

r!1

log[m+1]T exp rA _{; f} _g

log[m]T (exp (r ) ; L(f )) = 1 :

Case II. r _{6= o exp}[p 1]_{L exp exp} _rA _{then the following two sub cases}

may arise:

Sub case (a). If exp[p 1]L exp exp rA = o n

log[m]T (exp (r ) ; L(f )) o

, then we get from (23) that

lim sup

r!1

log[m+1]T exp rA ; f g

log[m]T (exp (r ) ; L(f )) + L (exp (exp ( rA₎₎₎ = 1 :

Sub case (b). If exp[p 1]_{L exp exp} _rA _log[m]_{T (exp (r ) ; L(f )) then}

lim

r!1

exp[p 1]_{L exp exp} _rA

log[m]T (exp (r ) ; L(f )) = 1 and we obtain from (23) that

lim sup

r!1

log[m+1]T exp rA ; f g

log[m]T (exp (r ) ; L(f )) + L (exp (exp ( rA₎₎₎ = 1 :

Combining Case I and Case II we obtain that lim sup r!1 log[m+1]T exp rA _{; f} _g log[m]T (exp (r ) ; L(f )) + K (r; g; L) = 1 ; where K (r; g; L) = 0 if r = o exp [p 1]_{L exp exp} _rA _{as r ! 1}

exp[p 1]L exp exp rA otherwise . This proves the theorem.

Theorem 44. Let f be a meromorphic function and g be transcendental entire such that (m)_(p) L_f > 0;(p) Lg < 1 and

P

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are any two positive integers: Then for any positive real number A, lim sup r!1 log[m+1]T exp rA _{; f} _g log T (exp (r ) ; L(g)) + K (r; f ; L) = 1 ; where 0 < < _g and K (r; f ; L) = 0 if r = o exp [p 1]_{L exp exp} _rA _{as r ! 1}

exp[p 1]_{L exp exp} _rA _{otherwise .}

The proof is omitted because it can be carried out in the line of Theorem 43. Remark 9. Theorem 43 and Theorem 44 are respectively improve Theorem 12 and Theorem 13 of Datta et. al. { cf. [5]}.

References

[1] W. Bergweiler : On the Nevanlinna Characteristic of a composite function, Complex Vari-ables, Vol. 10 (1988), pp.225-236.

[2] W. Bergweiler : On the growth rate of composite meromorphic functions, Complex Variables, Vol. 14 (1990) pp.187-196.

[3] S. K. Datta and A. Biswas : Some generalized growth properties of composite entire func-tions involving their maximum terms on the basis of slowly changing funcfunc-tions, International Journal of Mathematical Analysis, Vol. 5, No.22 (2011), pp. 1093-1101.

[4] S. K. Datta, T. Biswas and S. Ali : Some results on wronskians using slowly changing functions, News Bull. Cal. Math. Soc., Vol. 36, No. 7-9 (2013). pp. 8-24.

[5] S. K. Datta, T. Biswas and S. Ali : Growth analysis of wronskians in terms of slowly changing functions, Journal of Complex Analysis, Vol. 2013, Article ID 395067, 9 pages, http://dx.doi.org/10.1155/2013/395067.

[6] S. K. Datta, T. Biswas and S. Ali : Growth rates of wronskians generated by complex valued functions, Gulf Journal of Mathematics, Vol. 2, Issue 1 (2014), pp. 58-74.

[7] W. K. Hayman : Meromorphic Functions, The Clarendon Press, Oxford, 1964.

[8] T.V. Lakshminarasimhan : A note on entire functions of bounded index, J. Indian Math. Soc. Vol. 38 (1974), pp. 43-49.

[9] Q. Lin and C. Dai : On a conjecture of Shah concerning small functions, Kexue Tong bao (English Ed.), Vol.31, No. 4, (1986), pp. 220-224.

[10] I. Lahiri and N. R. Bhattacharjee : Functions of bounded index and of non-uniform L-bounded index, Indian J. Math. Vol. 40, No. 1 (1998), pp. 43-57.

[11] I. Lahiri and A. Banerjee : Value distribution of a Wronskian , Portugaliae Mathematica , Vol.. 61, No. Fasc.2 Nova Série (2004), pp. 161-175.

[12] D. Somasundaram and R. Thamizharasi : A note on the entire functions of L-bounded index and L-type, Indian J. Pure Appl.Math., Vol..19, No. 3 (March 1988), pp. 284-293.

[13] G. Valiron : Lectures on the general theory of integral functions, Chelsea Publishing Com-pany, 1949.

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Address : Sanjib Kumar Datta, Department of Mathematics,University of Kalyani, P.O. Kalyani, Dist-Nadia,PIN- 741235, West Bengal, India.

E-mail : sanjib_kr_datta@yahoo.co.in

Address : Tanmay Biswas, Rajbari, Rabindrapalli, R. N. Tagore Road, P.O. Krishnagar,Dist-Nadia,PIN- 741101, West Bengal, India.

E-mail : tanmaybiswas_math@rediffmail.com

Address : Ananya Kar, Taherpur Girls’High School, P.O.- Taherpur, Dist-Nadia PIN- 741159, West Bengal, India.

E-mail : ananyakaronline@gmail.com

0_Ba¸_{sl¬k: Baz¬ genelle¸}_stirilmi¸_{s büyüme belirteçleri ¬¸}_s¬¼_{g¬nda wronskiyenlerin büyüme analizi}

ü-zerine.

Anahtar Kelimeler: Transendental tam fonksiyon, transendental meromor…k fonksiyon, bile¸ske, büyüme, m-y.nci genelleme, p oranl¬pL - basama¼g¬,