Some Results on the Growth Properties of Entire Functions Satifying Second Order Linear Differential Equations

Transcription

1 Int Journal of Math Analysis, Vol 3, 2009, no 1, 1-14 Some Results on the Growth Properties of Entire Functions Satifying Second Order Linear Differential Equations Sanjib Kumar Datta Department of Mathematics, University of North Bengal Darjeeling, Pin , West Bengal, India sanjib kr Tanmay Biswas Department of Mathematics, University of North Bengal Darjeeling, Pin , West Bengal, India Tanmaybiswas Abstract In this paper we study the comparative growth properties of composite entire functions which satisfy second order linear differential equations Mathematics Subject Classification: 30D35, 30D30 Keywords: Entire function, meromorphic function, second order linear differential equation, proximate lower order, composition, growth, sharing of values 1 Introduction, Definitions and Notations We denote by C the set of all finite complex numbers Let f be a meromorphic function and g be an entire function defined on C We use the standard notations and definitions in the theory of entire and meromorphic functions which are available ( in [11] ) and [5] In the sequel we use the following notation : log [k] x = log log [k 1] x for k =1, 2, 3, and log [0] x = x We recall the following definitions :

2 2 S K Datta and T Biswas Definition 1 The order ρ f and lower order λ f of an entire function f is defined as log [2] M (r, f) ρ f = and λ f = If f is meromorphic, one can easily verify that, log T (r, f) ρ f = and λ f = log [2] M (r, f) log T (r, f) Definition 2 The hyper order ρ f and hyper lower order λ f of an entire function f is defined as follows log [3] M (r, f) ρf = If f is meromorphic, then log [2] T (r, f) ρf = and λ f = and λ f = log [3] M (r, f) log [2] T (r, f) Definition 3 The type σ f of an entire function f is defined as σ f = When f is meromorphic, then σ f = log M (r, f), 0 <ρ r ρ f f < T (r, f), 0 <ρ r ρ f < f Definition 4 A function λ f (r) is called a lower proximate order of an entire function (meromorphic function) f of finite lower order λ f if (i) λ f (r) is non-negative and continuous for r r 0, say (ii) λ f (r) is differentiable for r r 0 except possibly at isolated points at which λ f (r +0)and λ f (r 0) exist, (iii) lim λ f (r) =λ f, (r) =0and (iv) lim r λ f (v) log M(r,f) r λ f (r) T (r,f) = 1( =1) r λ f (r) The existence of lower proximate order of an entire function (meromorphic function) can be proved in the line of Lahiri [8]

3 Results on growth properties 3 Definition 5 Let a be a complex number, finite or infinite The Nevanlinna deficiency and Valiron deficiency of a with respect to a meromorphic function f are defined as N (r, a; f) δ (a; f) =1 T (r, f) N (r, a; f) and Δ(a; f) =1 T (r, f) m (r, a; f) = T (r, f) m (r, a; f) = T (r, f) Let f be an entire function defined in the open complex plane C Kwon [6], Chen [3], Chen and Yang [4] studied the growth of entire functions satisfying the second order linear differential equation The purpose of this paper is to study on the growth of the solutions f 0 of the second order linear differential equation f + A (z) f + B (z) f =0, where A (z) and B (z) 0 are entire functions 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 sufficiently large values of r, T (r, g) T (r, f 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, f g) T (exp(r μ ),f) Lemma 3 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 ultimately an increasing function of r Proof Since d dr rλ f +δ λ f (r) = {λ f +δ λ f (r) rλ f (r) } rλ f +δ λ f (r) 1 > 0 for all sufficiently large values of r, the lemma follows Lemma 4 [7]Let f be an entire function of finite lower order If there exists entire functions a i (i =1, 2, n; n ) satisfying T (r, a i )= {T (r, f)} and n δ(a i,f)=1,then i=1 T (r, f) lim log M(r, f) = 1 π

4 4 S K Datta and T Biswas In order to state our next lemma we need the following notion of sharing of values of meromorphic functions Let S (f) be the set of all meromorphic functions a a (z) in z < which satisfy T (r, a) =o {T (r, f)} as r We consider C = C { } as a subset of S (f) We shall call any a S (f) as a small function (with respect to f ) Let E (f = a) ={z : f (z) a (z) =0} The meromorphic functions f and g are said to share a if and only if E (f = a) = E (g = a) Lemma 5 [10] If f and g are meromorphic functions that share six small functions a i S (f) S (g) for i =1, 2,, 6 ignoring multiplicities then outside a set of r of finite linear measure, 3 Theorems T (r, f) lim T (r, g) =1 In this section we present the main results of the paper Theorem 1 Let f be an entire function satisfying the second order linear differential equation f + A (z) f + B (z) f =0where A (z) and B (z) 0 are entire functions If ρ A and ρ B are both finite, ρ B <λ f and there exist entire functions a i (i =1, 2, n; n ) with (i) T (r, a i )= {T (r, B)} as r for i =1, 2, n and (ii) n δ(a i,b)=1,then i=1 {} 2 lim T (r, f) T (r, B) =0 Proof In view of the inequality T (r, B) log + M (r, B) and Lemma 1 we obtain for all sufficiently large values of r, T (r, A B) {1+o(1)} T (M (r, B),A) ie, log {1+o(1)} + log T (M (r, B),A) ie, o (1) + (ρ A + ε) log M (r, B) ie, o (1) + (ρ A + ε) r (ρ B+ε) (1)

5 Results on growth properties 5 Again we get for all sufficiently large values of r log T (r, f) > (λ f ε) ie, T (r, f) > r (λ f ε) (2) Now combining (1) and (2) it follows for all sufficiently large values of r T (r, f) o (1) + (ρ A + ε) r (ρ B+ε) (3) r (λ f ε) Since ρ B <λ f, we can choose ε (> 0) in such a way that So in view of (3) and (4) it follows that ρ B + ε<λ f ε (4) lim T (r, f) =0 (5) Again from Lemma 4 we get for all sufficiently large values of r, T (r, B) o (1) + (ρ A + ε) log M (r, B) T (r, B) ie, T (r, B) log M (r, B) (ρ A + ε) T (r, B) ie, T (r, B) Since ε (> 0) is arbitrary it follows from above that T (r, B) Combining (5) and (6) we obtain that This proves the theorem {} 2 T (r, f) T (r, B) = lim T (r, f) 0πρ A =0 (ρ A + ε) π πρ A (6) T (r, B)

6 6 S K Datta and T Biswas Theorem 2 If f be an entire function satisfying the second order linear differential equation f + A (z) f + B (z) f =0with A (z) and B (z) 0 as entire functions and (i) λ A > 0, (ii) ρ f <, (iii) 0<λ B ρ B, then λ B max, ρ B log [2] T (r, f) λ ρf f r, Proof We know that for r>0 ([9]) and for all sufficiently large values of T (r, A B) 1 { 1 ( r ) } 3 log M 8 M 2,B + o (1),A (7) Since λ A and λ B are the lower orders of A and B respectively then for given ε (> 0) and for all large values of r we obtain that log M (r, A) >r λa ε and log M (r, B) >r λb ε where 0 <ε<min {λ A,λ B } So from (7) we have for all sufficiently large values of r, T (r, A B) 1 { 1 ( r ) } λa ε 3 8 M 4,B + o (1) ie, T (r, A B) 1 3 { 1 9 M ( r 4,B )} λa ε ( r ) ie, O (1) + (λ A ε) log M 4,B ( r λb ε ie, O (1) + (λ A ε) 4) ie, O (1) + (λ B ε) (8) Again we get for a sequence of values of r tending to infinity, ( ) λf log [2] T (r, f) + ε (9) Combining (8) and (9), it follows for a sequence of values of r tending to infinity, log [2] T (r, f) O (1) + (λ B ε) ( λf + ε)

7 Results on growth properties 7 Since ε (> 0) is arbitrary, we obtain that log [2] T (r, f) λ B λ f (10) Again from (7) we get for a sequence of values of r tending to infinity, ( r ρb ε O (1) + (λ A ε) 4) ie, O (1) + (ρ B ε) (11) Also for all sufficiently large values of r, we have ( log [2] ρf ) T (r, f) + ε (12) Now from (11) and (12) it follows for a sequence of values of r tending to infinity that log [2] T (r, f) O (1) + (ρ B ε) ( ρf + ε) As ε (0 <ε<ρ B ) is arbitrary, we obtain from above that ρ B (13) log [2] T (r, f) ρf Therefore from (10) and (13) we get that λ B max, ρ B log [2] T (r, f) λ ρf f Thus the theorem is established Theorem 3 Let f be an entire function satisfying the second order linear differential equation f + A (z) f + B (z) f =0where A (z) and B (z) 0 are entire functions If (i) 0< λ f < ρ f, (ii) ρ B < and (iii) ρ A < then λ B min, ρ B log [2] T (r, f) λ ρf f

8 8 S K Datta and T Biswas Proof In view of Lemma 1 and the inequality T (r, B) log + M (r, B), we obtain for all sufficiently large values of r, o (1) + (ρ A + ε) log M (r, B) (14) Also for a sequence of values of r tending to infinity, log M (r, B) r λ B+ε (15) Combining (14) and (15) it follows for a sequence of values of r tending to infinity, o (1) + (ρ A + ε) r λ B+ε ie, r λ B+ε {(ρ A + ε)+o (1)} ie, O (1) + (λ B + ε) (16) Again we have for all sufficiently large values of r, ( ) λf log [2] T (r, f) > ε (17) Now from (16) and (17) we get for a sequence of values of r tending to infinity, log [2] T (r, f) O (1) + (λ B + ε) ( λf ε) As ε (> 0) is arbitrary, it follows that log [2] T (r, f) In view of Lemma 1 we get for all sufficiently large values of r, λ B λ f (18) O (1) + (ρ B + ε) (19) Also for a sequence of values of r tending to infinity, ( log [2] ρf ) T (r, f) > ε (20)

9 Results on growth properties 9 Combining (19) and (20) we have for a sequence of values of r tending to infinity, log [2] T (r, f) O (1) + (ρ B + ε) ( ρf ε) Since ε (> 0) is arbitrary, it follows from above that log [2] T (r, f) Now from (18) and (21) we get that log [2] T (r, f) ρ B ρf (21) λ B min, ρ B λ ρf f This proves the theorem The following theorem is a natural consequence of Theorem 2 and Theorem 3 Theorem 4 Let f be an entire function satisfying the second order linear differential equation f + A (z) f + B (z) f =0where A (z) and B (z) 0 are entire functions If (i) 0< λ f ρ f <, (ii) 0<λ A ρ A < and (iii) 0 <λ B ρ B < then λ B min, ρ B log [2] T (r, f) λ ρf f λ B max, ρ B λ ρf log [2] T (r, f) f Theorem 5 If f be an entire function satisfying the second order linear differential equation f + A (z) f + B (z) f =0with A (z) and B (z) 0 as entire functions and (i) λ A > 0,(ii) ρ f < then log [2] T (exp (r ρ B ),A B) log [2] T (exp (r μ ),f) =, where 0 <μ<ρ B Proof Let 0 <μ <ρ B Then in view of Lemma 2 we get for a sequence of values of r tending to infinity, ( ( log T exp r μ ) ),A

10 10 S K Datta and T Biswas { )} μ ie, (λ A ε) log exp (r ie, (λ A ε) r μ ie, O (1) + μ So for a sequence of values of r tending to infinity, log [2] T (exp (r ρ B ),A B) O (1) + μ log {exp (r ρ B )} ie, log [2] T (exp (r ρ B ),A B) O (1) + μ r ρ B (22) Again we have for all sufficiently large values of r, log T (exp (r μ ),f) (ρ f + ε) log {exp (r μ )} ie, log T (exp (r μ ),f) (ρ f + ε) r μ ie, log [2] T (exp (r μ ),f) O (1) + μ (23) Now combining (22) and (23) we obtain for a sequence of values of r tending to infinity, log [2] T (exp (r ρ B ),A B) log [2] T (exp (r μ ),f) O (1) + μ r ρ B, μ from which the theorem follows Theorem 6 Let f be an entire function satisfying the second order linear differential equation f + A (z) f + B (z) f =0where A (z) and B (z) 0 are entire functions If λ A and λ B are both finite and also if f and B share six small functions a i S (f) S (B) for i =1, 2,, 6 ignoring multiplicities then outside a set of r of finite linear measure, T (r, f) 3ρ A 2 λ B Proof If ρ A =, the result is obvious So we suppose that ρ A < Since T (r, B) log + M (r, B), in view of Lemma 1, we get for all sufficiently large values of r, T (r, A B) {1+o (1)} T (M (r, B),A)

11 Results on growth properties 11 ie, log {1+o (1)} + log T (M (r, B),A) ie, o (1) + (ρ A + ε) log M (r, B) ie, T (r, B) Since ε (> 0) is arbitrary, it follows that T (r, B) (ρ A + ε) ρ A log M (r, B) T (r, B) log M (r, B) (24) T (r, B) T (r,b) As =1, so for given ε (0 <ε<1) we get for a sequence of values r λ B (r) of r tending to infinity, T (r, B) (1 + ε) r λ B(r) (25) and for all sufficiently large values of r, T (r, B) > (1 ε) r λ B(r) (26) Since log M (r, B) 3T (2r, B), we have by (25) for a sequence of values of r tending to infinity, log M (r, B) 3T (2r, B) 3(1+ε)(2r) λ B(2r) (27) Combining (26) and (27) we obtain for a sequence of values of r tending to infinity, log M (r, B) 3(1+ε) (2r)λB(2r) T (r, B) (1 ε) r λ B(r) Now for any δ>0, for a sequence of values of r tending to infinity, log M (r, B) T (r, B) 3(1+ε) (1 ε) (2r) λb+δ (2r) λ B+δ λ B (2r) 1 r λ B(r) ie, log M (r, B) T (r, B) 3(1+ε) (1 ε) 2λ B+δ, (28) because r λ B+δ λ B (r) is ultimately an increasing function of r by Lemma 3 Since ε (> 0) and δ (> 0) are arbitrary, it follows from (28) that log M (r, B) T (r, B) 32 λ B (29)

12 12 S K Datta and T Biswas Thus from (24) and (29) we obtain that T (r, B) 3ρ A 2 λ B (30) Since f and B share six small functions a i S (f) S (B) for i =1, 2,, 6 ignoring multiplicities then in view of Lemma 5 we may write outside a set of r of finite linear measure, T (r, f) lim T (r, B) =1 (31) Thus the theorem follows from (30) and (31) Theorem 7 Let f be an entire function satisfying the second order linear differential equation f + A (z) f + B (z) f =0where A (z) and B (z) 0 are entire functions If ρ A < and λ B < and also if f and B share six small functions a i S (f) S (B) for i =1, 2,, 6 ignoring multiplicities then outside a set of r of finite linear measure, log [3] T (r, A B) log [2] T (r, f) 1 Proof Since T (r, B) log + M (r, B), in view of Lemma 1, we get for all sufficiently large values of r, log T (M (r, B),A) + log {1+o(1)} ie, (ρ A + ε) log M (r, B)+o (1) ie, log [3] T (r, A B) log [3] M (r, B)+O (1) (32) It is well known that for any entire function B, log M (r, B) 3T (2r, B) {cf [5]} Then for 0 <ε<1 and δ (> 0), for a sequence of values of r tending to infinity it follows from (28) that log [3] M (r, B) log [2] T (r, B)+O (1) (33) Now combining (32) and (33) we obtain for a sequence of values of r tending to infinity, log [3] T (r, A B) log [2] T (r, B)+O (1) log [3] T (r, A B) ie, log [2] T (r, B) 1 (34)

13 Results on growth properties 13 As f and B share six small functions a i S (f) S (B) for i =1, 2,, 6 ignoring multiplicities then in view of Lemma 5 we may write outside a set of r of finite linear measure, log [2] T (r, f) lim =1 (35) log [2] T (r, B) Thus the theorem is established from (34) and (35) References [1] W Bergweiler : On the Nevanlinna Characteristic of a composite function,complex Variables,10 (1988),pp [2] W Bergweiler : On the growth rate of composite meromorphic functions, Complex Variables,14 (1990),pp [3] Zong-Xuan Chen : The growth of solutions of second order linear differential equations with meromorphic coefficients, Kodai Math J, 22(1999), pp [4] Zong-Xuan Chen and Chung-Chun Yang : Some further results on the zeros and growths of entire solutions of second order linear differential equations, Kodai Math J, 22 (1999),pp [5] WK Hayman : Meromorphic Functions, The Clarendon Press, Oxford 1964 [6] Ki-Ho Kwon, : On the growth of entire functions satisfying second order linear differential equations, Bull Korean Math Soc, 33(1996), pp [7] Q Lin, and C Dai : On a conjecture of Shah concerning Small functions, Kexue Tong bao (English Ed) 31(1986), No4,pp [8] I Lahiri : Generalised proximate order of meromorphic functions, MatVesnik, 41 (1989),pp9 16 [9] K Niino, and CC Yang : 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 [10] AP, Singh, KS Charak, and JL Sharma : Meromorphic functions that share small functions, J Ramanujan Math Soc, Vol 11, No 1 (1996),pp73 84

14 14 S K Datta and T Biswas [11] G Valiron : Lectures on the General Theory of Integral Functions, Chelsea Publishing Company,1949 Received: September 8, 2008