International Journal of Pure and Applied Mathematics

International Journal of Pure and Applied Mathematics ————————————————————————– Volume 52 No. 1 2009, 143-153

ON THE GROWTH OF ENTIRE ALGEBROIDAL FUNCTIONS Sanjib Kumar Datta1 § , Arindam Jha2 1 Department

of Mathematics University of North Bengal Raja Rammohunpur, District Darjeeling PIN Code 734013, West Bengal, INDIA e-mail: sanjib kr [email protected] 2 Tarangapur N.K. High School Post: Tarangapur, District Uttar Dinajpur PIN Code 733129, West Bengal, INDIA e-mails: d [email protected], [email protected] Abstract: In this paper we study the comperative growth properties of entire algebroidal functions. AMS Subject Classification: 30D35, 30D30 Key Words: entire algebroidal function, composition, growth, order, lower order, lower proximate order

1. Introduction, Definitions and Notations Let F be a k-valued function defined by the following irreducible equation fk F k + fk−1 F k−1 + fk−2 F k−2 + · · · + f0 = 0, where fk 6≡ 0 and all fi (i = 0, 1, 2, · · · , k) are entire functions having no common zeros. If at least one of the fi (i = 0, 1, 2, · · · , k) is transcendental then F is called a k-valued algebroidal function. Further, if fk ≡ 1 then F is called a k-valued entire algebroidal function. In the paper we establish some results on the growth properties of composition of two k-valued entire algebroidal function. We do not explain the standard notations and definitions of the theory of entire functions as those are available in [5]. Received:

March 27, 2009

§ Correspondence address: 25, School Road,

c 2009 Academic Publications

Kalianibash, Barrackpore, P.O.: Nonachandanpukur, Dist.: 24 Parganas (North), P.S.: Titagarh, PIN Code: 743102, West Bengal, INDIA

144

S.K. Datta, A. Jha

Before starting our discussion we just mention the following well known definitions: Definition 1. The order ρf and lower order λf of an entire function f is defined as log[2] M (r, f ) log[2] M (r, f ) and λf = lim inf , ρf = lim sup r→∞ log r log r r→∞ where log[k] x = log log[k−1] x for k = 1, 2, 3, · · · and log[0] x = x.

Since T (r, f ) and log M (r, f ) are mutually replaceble in the formula for the order and lower order of f then ρf = lim sup r→∞

log T (r, f ) log T (r, f ) and λf = lim inf . r→∞ log r log r

Definition 2. A function λf (r) is called a lower proximate order of an entire function f if: (i) λf (r) is non negative and continuous for r ≥ r0 say; (ii) λf (r) is differentiable for r ≥ r0 except possibly at isolated points at which λ′f (r − 0) and λ′f (r + 0) exist; (iii) lim λf (r) = λf < ∞; r→∞

(iv) lim rλ′f (r) log r = 0; r→∞

(r,f ) = 1. (v) lim inf log λMf (r) r→∞

r

2. Some Lemmas In this section we present some lemmas which will be needed in the sequel. 2+ε ε

Lemma 1. (see [3]) Let f and g be two entire functions. If M (r, g) > |g (0)| for any ε (> 0) then T (r, f ◦ g) < (1 + ε) T (M (r, g) , f ) , In particular if g (0) = 0, then T (r, f ◦ g) < T (M (r, g) , f )

for all r > 0.

Lemma 2. (see [2]) Let g be an entire function with λg < ∞ and ai (i = 1, 2, 3, · · · , n; n ≤ ∞) are entire functions satisfying T (r, ai ) = o {T (r, g)}.

ON THE GROWTH OF ENTIRE ALGEBROIDAL FUNCTIONS

If

n P

T (r,g) r→∞ log M (r,g)

δ (ai , g) = 1 then lim

i=1

145

= π1 .

Lemma 3. Let f be an entire function. Then for δ (> 0) the function is ultimately an increasing function of r.

r λf +δ−λf (r)

Proof. Since d λf +δ−λf (r) r = λf + δ − λf (r) − rλ′f (r) log r r λf +δ−λf (r)−1 > 0, dr for all sufficiently large values of r, the lemma is proved.

3. Theorems In this section we present the main results of the paper. Theorem 1. such that

Let F and G be two k-valued entire algebroidal functions F k + fk−1 F k−1 + fk−2 F k−2 + · · · + f0 = 0

and Gk + gk−1 Gk−1 + gk−2 Gk−2 + · · · + g0 = 0 , where fi (i = 0, 1, 2, · · · , k − 1) and gi (i = 0, 1, 2, · · · , k − 1) are entire functions having no common zeros. Also let 0 < λfi ≤ ρfi < ∞, 0 < λgi ≤ ρgi < ∞, 0 < ρF < ∞ and 0 < ρG < ∞ for i = 0, 1, 2, · · · , k − 1. Then lim inf r→∞

k−1 log[2] T (r, F ◦ G) Y log[2] T (r, fi ◦ gi ) ρG lim inf ≤ r→∞ log T (r, F ) log T (r, gi ) ρF i=0

k−1 log[2] T (r, fi ◦ gi ) log[2] T (r, F ◦ G) Y lim sup . ≤ lim sup log T (r, F ) log T (r, gi ) r→∞ r→∞ i=0

Proof. We know that for r > 0 (Niino and Yang[4]) 1 r 1 M , gi + o (1) , fi . (1) T (r, fi ◦ gi ) ≥ log M 3 8 4 Since λfi and λgi are the lower orders of fi and gi respectively, for given εi and for all large values of r we get log M (r, fi ) > r λfi −εi and log M (r, gi ) > r λgi −εi ,

146

S.K. Datta, A. Jha

where 0 < εi < min {λfi , λgi }. So from (1) we get for all large values of r, λfi −εi λfi −εi 1 1 r 1 1 r ≥ M , gi + o (1) M , gi , T (r, fi ◦ gi ) ≥ 3 8 4 3 9 4 r λg −εi i , log T (r, fi ◦ gi ) ≥ O (1) + (λfi − εi ) 4 log[2] T (r, fi ◦ gi ) ≥ O (1) + (λgi − εi ) log r .

(2)

Also for a sequence of values of r tending to infinity, log T (r, gi ) < (λgi + εi ) log r.

(3)

So from (2) and (3) it follows for a sequence of values of r tending to infinity, log[2] T (r, fi ◦ gi ) O (1) + (λgi − εi ) log r > . log T (r, gi ) (λgi + εi ) log r Since εi (> 0) is arbitrary we obtain from (4) that

(4)

log[2] T (r, fi ◦ gi ) ≥ 1. log T (r, gi ) r→∞ Now taking products for i = 0, 1, 2, · · · , k − 1 it follows from (5) that lim sup

k−1 Y i=0

lim sup r→∞

(5)

log[2] T (r, fi ◦ gi ) ≥ 1. log T (r, gi )

(6)

Again in a similar manner, for a sequence of values of r tending to infinity, log[2] T (r, F ◦ G) O (1) + (ρG − ε) log r > , log T (r, F ) (ρF + ε) log r log[2] T (r, F ◦ G) ρG ≥ . log T (r, F ) ρF r→∞ Thus from (6) and (7) we obtain, lim sup

k−1 log[2] T (r, fi ◦ gi ) ρG log[2] T (r, F ◦ g) Y lim sup ≥ . lim sup log T (r, F ) log T (r, gi ) ρF r→∞ r→∞

(7)

(8)

i=0

Now by Lemma 1 we get for all sufficiently large values of r,

log T (r, fi ◦ gi ) ≤ O (1) + log T (M (r, gi ) , fi ) , log T (r, fi ◦ gi ) ≤ O (1) + (ρfi + εi ) log M (r, gi ) .

(9)

Also for a sequence of values of r tending to infinity, log M (r, gi ) < r λgi +εi .

(10)


147

So from (9) and (10) it follows for a sequence of values of r tending to infinity, log T (r, fi ◦ gi ) < O (1) + (ρfi + εi ) r λgi +εi , log[2] T (r, fi ◦ gi ) < O (1) + (λgi + εi ) log r.

(11)

Also for all large values of r and for 0 < εi < λgi , log T (r, gi ) > (λgi − εi ) log r.

(12)

Now combining (11) and (12) we get for a sequence of values of r tending to infinity, log[2] T (r, fi ◦ gi ) O (1) + (λgi + εi ) log r < , log T (r, gi ) (λgi − εi ) log r λg + εi log[2] T (r, fi ◦ gi ) ≤ i . r→∞ log T (r, gi ) λgi − εi As εi (> 0) is arbitrary it follows that lim inf

log[2] T (r, fi ◦ gi ) ≤ 1. r→∞ log T (r, gi ) Now taking products for i = 0, 1, 2, · · · , k − 1 we get from (14) that lim inf

k−1 Y i=0

lim inf r→∞

log[2] T (r, fi ◦ gi ) ≤ 1. log T (r, gi )

(13)

(14)

(15)

In a like manner for a sequence of values of r tending to infinity, log[2] T (r, F ◦ G) O (1) + (ρG + ε) log r < . log T (r, F ) (ρF − ε) log r Since εi (> 0) is arbitrary we obtain from (16) that ρG log[2] T (r, F ◦ G) ≤ . r→∞ log T (r, F ) ρF Now from (15) and (17) it follows that lim inf

lim inf r→∞

k−1 ρG log[2] T (r, fi ◦ gi ) log[2] T (r, F ◦ G) Y lim inf ≤ . r→∞ log T (r, F ) log T (r, gi ) ρF

(16)

(17)

(18)

i=0

Thus the theorem follows from (8) and (18).

Theorem 2. Let F and G be two k-valued entire algebroidal functions with F k + fk−1 F k−1 + fk−2 F k−2 + · · · + f0 = 0 and Gk + gk−1 Gk−1 + gk−2 Gk−2 + · · · + g0 = 0, where fi (i = 0, 1, 2, · · · , k − 1) and gi (i = 0, 1, 2, · · · , k − 1) are entire functions having no common zeros. Also let ρfi and λgi are both finite

148

S.K. Datta, A. Jha

for i = 0, 1, 2, · · · , k − 1. Assume that aij (i = 0, 1, 2, · · · , k − 1, j = 1, 2, 3, · · · , n; n ≤ ∞) n P are entire functions satisfying T (r, aij ) = o {T (r, gi )}. If δ (aij , gi ) = 1 for j=1

i = 0, 1, 2, · · · , k − 1 then

lim sup r→∞

k−1 Y i=0

k−1

Y log T (r, fi ◦ gi ) ρfi . ≤ πk log T (r, gi ) i=0

Proof. Taking products for i = 0, 1, 2, · · · , k − 1, from (9) we obtain for all large values of r and for arbitrary εi (> 0) that k−1 Y i=0

lim sup r→∞

k−1 Y log T (r, fi ◦ gi ) log M (r, gi ) (ρfi + εi ) ≤ + O (1) , T (r, gi ) T (r, gi )

k−1 Y i=0

i=0

log T (r, fi ◦ gi ) T (r, gi )

log M (r, gi ) T (r, gi )

(ρfi + εi ) lim sup

log M (r, gi ) . T (r, gi )

r→∞

≤

k−1 Y i=0

k−1 Y

(ρfi + εi )

≤ lim sup

i=0

r→∞

(19)

Since εi (> 0) is arbitrary, in view of Lemma 2 it follows from (19) that lim sup r→∞

This proves the theorem.

k−1 Y i=0

k−1

Y log T (r, fi ◦ gi ) ρfi . ≤ πk log T (r, gi )

(20)

i=0

Theorem 3. Let F and G be two k-valued entire algebroidal functions such that F k + fk−1 F k−1 + fk−2 F k−2 + · · · + f0 = 0 and Gk + gk−1 Gk−1 + gk−2 Gk−2 + · · · + g0 = 0, where fi and gi are entire functions having no common zeros for i = 0, 1, 2, · · · , k − 1. Also let ρF ◦G < ∞ and λG > 0. Then for any positive constant A, # " k−1 k−1 ρF ◦G k Y log[2] M (r, F ◦ G) Y log T (r, fi ◦ gi ) ≤ ρfi . π lim sup T (r, gi ) AλG r→∞ log[2] M (r A , G) i=0 i=0 Proof. From the definition of order and lower order we get for all sufficiently large values of r and for arbitrary positive ε, log[2] M (r, F ◦ G) ≤ (ρF ◦G + ε) log r

(21)

log[2] M r A , G ≥ A (λG − ε) log r.

(22)

and


149

Now from (21) and (22) it follows for all large values of r, log[2] M (r, F ◦ G) log lim sup

[2]

M (r A , G)

≤

(ρF ◦G + ε) , A (λG − ε)

log[2] M (r, F ◦ G) [2]

(r A , G)

≤

log M As ε (> 0) is arbitrary we get from above that r→∞

lim sup

log[2] M (r, F ◦ G) [2]

(r A , G)

(ρF ◦G + ε) . A (λG − ε)

≤

ρF ◦G . AλG

log M Now in view of (20) and (23) we obtain that " # k−1 log[2] M (r, F ◦ G) Y log T (r, fi ◦ gi ) lim sup T (r, gi ) r→∞ log[2] M (r A , G) r→∞

(23)

i=0

≤ lim sup

≤

log

[2]

M (r, F ◦ G)

[2]

r→∞

log

ρF ◦G k π AλG

k−1 Y

M (r A , G)

lim sup r→∞

k−1 Y i=0

log T (r, fi ◦ gi ) T (r, gi )

ρfi .

i=0

Thus the theorem is established. Theorem 4. Let F and G be two k-valued entire algebroidal functions with F k + fk−1 F k−1 + fk−2 F k−2 + · · · + f0 = 0 and Gk + gk−1 Gk−1 + gk−2 Gk−2 + · · · + g0 = 0, where fi and gi are entire functions for i = 0, 1, 2, · · · , k − 1, having no common zeros. Also let 0 < λfi ◦gi ≤ ρfi ◦gi < ∞ and 0 < λgi ≤ ρgi < ∞. Then for any positive constant A, k−1 k−1 Y log[2] M (r, fi ◦ gi ) 1 Y ρfi ◦gi lim sup ≤ Ak λg i r→∞ log[2] M (r A , gi ) i=0

i=0

and

lim inf r→∞

k−1 Y i=0

k−1 1 Y λfi ◦gi ≥ k . A ρgi log[2] M (r A , gi ) i=0

log[2] M (r, fi ◦ gi )

Proof. For arbitrary positive εi and for all sufficiently large values of r we obtain log[2] M (r, fi ◦ gi ) ≥ (λfi ◦gi − εi ) log r

(24)

log[2] M r A , gi ≤ A (ρgi + εi ) log r.

(25)

and

150

S.K. Datta, A. Jha

Taking products for i = 0, 1, 2, · · · , k − 1 from (24) and (25) we get for all large values of r k−1 Y (λf ◦g − εi ) Y log[2] M (r, fi ◦ gi ) k−1 i i ≥ . (26) [2] A A (ρ gi + εi ) i=0 i=0 log M (r , gi ) As εi (> 0) is arbitrary we obtain from (26) that lim inf r→∞

k−1 Y

log[2] M (r, fi ◦ gi ) log[2] M (r A , gi )

i=0

≥

k−1 1 Y λfi ◦gi . Ak ρgi

(27)

i=0

Again for all large values of r,

log[2] M (r, fi ◦ gi ) ≤ (ρfi ◦gi + εi ) log r

(28)

log[2] M r A , gi ≥ A (λgi − εi ) log r.

(29)

and Now taking products for i = 0, 1, 2, · · · , k − 1 , it follows from (28) and (29) for all large values of r k−1 Y (ρf ◦g + εi ) Y log[2] M (r, fi ◦ gi ) k−1 i i ≤ . (30) [2] A (λgi − εi ) log M (r A , gi ) i=0

i=0

As εi (> 0) is arbitrary it follows from (30) that lim sup r→∞

k−1 Y

k−1 1 Y ρfi ◦gi . ≤ k A λgi log[2] M (r A , gi ) i=0


i=0

(31)


Theorem 5. Let F and G be two k-valued entire algebroidal functions such that F k + fk−1 F k−1 + fk−2 F k−2 + · · · + f0 = 0 and Gk + gk−1 Gk−1 + gk−2 Gk−2 + · · · + g0 = 0, where fi and gi are entire functions having no common zeros for i = 0, 1, 2, · · · , k − 1. Also let λgi > 0 and ρfi ◦gi < ∞ then ) ( k−1 k−1 k−1 Y log[2] M (r, fi ◦ gi ) 1 Y λfi ◦gi 1 Y ρfi ◦gi , . lim inf ≤ min r→∞ log[2] M (r A , g ) Ak ρgi Ak λg i i i=0

i=0

i=0

Proof. For a sequence of values of r tending to infinity, log[2] M (r, fi ◦ gi ) ≤ (λfi ◦gi + εi ) log r

(32)

and for all large values of r, log[2] M r A , gi ≥ A (λgi − εi ) log r.

(33)

So for a sequence of values of r tending to infinity, it follows from (32) and (33)


151

that log[2] M (r, fi ◦ gi ) [2]

(r A , gi )

≤

(λfi ◦gi + εi ) . A (λgi − εi )

log M As εi (> 0) is arbitrary, we get from (34) that lim inf r→∞

log[2] M (r, fi ◦ gi ) [2]

(r A , gi )

≤

λfi ◦gi . Aλgi

log M Now taking products for i = 0, 1, 2, · · · , k − 1 we obtain from (35), k−1 1 Y λfi ◦gi . lim inf ≤ k r→∞ log [2] M (r A , g ) A λgi i i=0 i=0

k−1 Y


Also for a sequence of values of r tending to infinity, log[2] M r A , gi ≥ A (ρgi − εi ) log r.

(34)

(35)

(36)

(37)

From (28) and (37) it follows for a sequence of values of r tending to infinity, log[2] M (r, fi ◦ gi ) [2]

(r A , gi )

≤

(ρfi ◦gi + εi ) . A (ρgi − εi )

log M Since εi (> 0) is arbitrary we obtain from (38) that lim inf r→∞

log[2] M (r, fi ◦ gi ) [2]

(r A , gi )

≤

ρfi ◦gi . Aρgi

log M Taking products for i = 0, 1, 2, · · · , k − 1 it follows from (39) that k−1 1 Y ρfi ◦gi ≤ k . lim inf r→∞ log [2] M (r A , g ) A ρg i i i=0 i=0

k−1 Y


(38)

(39)

(40)


Theorem 6. If F and G be two k-valued entire algebroidal functions with F k + fk−1 F k−1 + fk−2 F k−2 + · · · + f0 = 0 and Gk + gk−1 Gk−1 + gk−2 Gk−2 + · · · + g0 = 0, where fi and gi are entire functions having no common zeros and ρfi ,λgi are both finite for i = 0, 1, 2, · · · , k − 1, Then ! k−1 P k−1 k−1 λgi Y Y log T (r, fi ◦ gi ) i=0 k . ρfi .2 lim inf ≤3 r→∞ T (r, gi ) i=0

i=0

Proof. If at least one of ρfi for i = 0, 1, 2, · · · , k − 1 is infinity then the result is obvious. So we suppose that ρfi < ∞ for all i = 0, 1, 2, · · · , k − 1. For any two entire functions fi and gi , the following two inequalities are well known, cf. [1], p. 18: T (r, fi ) ≤ log+ M (r, fi ) ≤ 3T (2r, fi )

(41)

152

S.K. Datta, A. Jha

and M (r, fi ◦ gi ) ≤ M (M (r, gi ) , fi ) .

(42)

For εi (> 0) we get from (41) and (42) for all large values of r that T (r, f ◦ g ) ≤ log M (M (r, g ) , f ) ≤ {M (r, g )}(ρfi +εi ) , i

i

i

i

i

log T (r, fi ◦ gi ) log M (r, gi ) ≤ (ρfi + εi ) . T (r, gi ) T (r, gi ) Since εi (> 0) is arbitrary, we obtain from above that log T (r, fi ◦ gi ) log M (r, gi ) lim inf ≤ ρfi lim inf . r→∞ r→∞ T (r, gi ) T (r, gi )

(43)

i) Since lim inf Tλ(r,g (r) = 1, for given εi (0 < εi < 1) we get for a sequence of values r→∞ r gi of r tending to infinity

T (r, gi ) < (1 + εi ) r λgi (r) .

(44)

Also for all large values of r T (r, gi ) > (1 − εi ) r λgi (r) .

(45)

Therefore, for a sequence of values of r tending to infinity we get for any δi (> 0) log M (r, gi ) T (r, gi )

≤

1 3 (1 + ε) (2r)λgi +δi . . λ g (1 − ε) (2r)λgi +δi −λgi (2r) r i (r)

≤

3 (1 + ε) λgi +δi , .2 (1 − ε)

because r λf +δ−λf (r) is ultimately an increasing function of r by Lemma 3. Since εi (> 0) and δi (> 0) are arbitrary, we get from above that λg log M (r, gi ) lim inf (46) ≤ 3.2 i . r→∞ T (r, gi ) Therefore from (43) and (46), it follows that λg log T (r, fi ◦ gi ) lim inf (47) ≤ ρfi .3.2 i . r→∞ T (r, gi ) Now taking products for i = 0, 1, 2, · · · , k − 1 we get from (47) that ! k−1 P k−1 k−1 λg Y Y i log T (r, fi ◦ gi ) i=0 k . ≤3 ρfi .2 lim inf r→∞ T (r, gi ) i=0

This proves the theorem.

i=0


153

References [1] W.K. Hayman, Meromorphic Functions, The Clarendon Press, Oxford (1964). [2] Q. Lin, C. Dai, On a conjecture of Shah concerning small functions, Kexue Tongbao, English Edition, 31, No. 4 (1986), 220-224. [3] K. Niino, N. Suita, Growth of a composite function of entire functions, Kodai Math. J., 3 (1980), 374-379. [4] K. Niino, C.C. Yang, Some growth relationships on the factors of two composite entire functions, Factorization Theory of Meromorphic Fumctions and Related Topics, Marcel Dekker, New York-Basel, (1982), 95-99. [5] G. Valiron, Lectures on the general theory of integral functions, Chelsea Publishing Company (1949).

154