A Special Type Of Differential Polynomial And Its Comparative Growth Properties

Transcription

1 Sanjib Kumar Datta 1, Ritam Biswas 2 1 Department of Mathematics, University of Kalyani, Kalyani, Dist- Nadia, PIN , West Bengal, India 2 Murshidabad College of Engineering Technology, Banjetia, Berhampore, P O-Cossimbazar Raj, PIN , West Bengal, India Abstract: Some new results depending upon the comparative growth rates of composite entire meromorphic function a special type of differential polynomial as considered by Bhooshnurmath Prasad[3] generated by one of the factors of the composition are obtained in this paper AMS Subject Classification (2010) : 30D30, 30D35 Keywords phrases: Order (lower order), hyper order (hyper lower order), growth rate, entire function, meromorphic function special type of differential polynomial I INTRODUCTION, DEFINITIONS AND NOTATIONS Let C be the set of all finite complex numbers Also let f be a meromorphic function g be an entire function defined on C In the sequel we use the following two notations: i log [k] x = log(log [k 1] x) for k = 1,2,3, ; log [0] x = x ii exp [k] x = exp exp k 1 x for k = 1,2,3, ; exp [0] x = x The following definitions are frequently used in this paper: Definition 1 The order lower order of a meromorphic function f are defined as If f is entire, one can easily verify that = = log log = log[2] M(r, f) = log[2] M(r, f) Definition 2 The hyper order hyper lower order of a meromorphic function f are defined as follows If f is entire, then = log[2] = log[2] = log[3] M(r, f) = log[3] M(r, f) Definition 3 The type ς f of a meromorphic function f is defined as follows ς f = r ρ, 0 < ρ f f < When f is entire, then ς f = log M(r, f), 0 < < r Definition 4 A function (r) is called a lower proximate order of a meromorphic function f of finite lower order if (i) (ii) A Special Type Of Differential Polynomial And Its Comparative Growth Properties (r) is non-negative continuous for r r 0, say (r) is differentiable for r r 0 except possibly at isolated points at which (r + 0) (r 0) exists, wwwijmercom 2606 Page

2 (iii) (iv) (v) r =, r r = 0 = 1 Definition 5 Let a be a complex number, finite or infinite The Nevanlinna s deficiency Valiron deficiency of a with respect to a meromorphic function f are defined as N(r, a; f) m(r, a; f) δ a; f = 1 = N(r, a; f) m(r, a; f) Δ a; f = 1 = Let f be a non-constant meromorphic function defined in the open complex plane C Also let n 0j, n 1j,, n kj (k 1) be nonnegative integers such that for each j, i=0 n ij 1 We call k M j f = A j (f) n 0j (f (1) ) n 1j (f (k) ) n kj where T r, A j = S(r, f), to be a differential monomial generated by f The numbers Γ Mj = γ Mj = k i=0 k i=0 r (r) n ij (i + 1)n ij are called the degree weight of M j [f] {cf [4]} respectively The expression P f = j =1 is called a differential polynomial generated by f The numbers γ P = max 1 j s γ M j Γ P = max 1 j s Γ M j are respectively called the degree weight of P[f] {cf [4]} Also we call the numbers γ P = min 1 j s γ M j k ( the order of the highest derivative of f) the lower degree the order of P[f] respectively If γ P = γ P, P[f] is called a homogeneous differential polynomial Bhooshnurmath Prasad [3] considered a special type of differential polynomial of the form F = f n Q[f] where Q[f] is a differential polynomial in f n = 0, 1, 2, In this paper we intend to prove some improved results depending upon the comparative growth properties of the composition of entire meromorphic functions a special type of differential polynomial as mentioned above generated by one of the factors of the composition We do not explain the stard notations definitions in the theory of entire meromorphic functions because those are available in [9] [5] II LEMMAS In this section we present some lemmas which will be needed in the sequel Lemma 1 [1] If f is meromorphic g is entire then for all sufficiently large values of r, T r, g T r, f o g 1 + o 1 T M r, g, f log M r, g Lemma 2 [2] Let f be meromorphic g be entire suppose that 0 < μ < ρ g Then for a sequence of values of r tending to infinity, T(r, f o g) T(exp( r μ ), f) Lemma 3 [3] Let F = f n Q[f] where Q[f] is a differential polynomial in f If n 1 then ρ F = λ F = Lemma 4 Let F = f n Q[f] where Q[f] is a differential polynomial in f If n 1 then T(r, F) = 1 The proof of Lemma 4 directly follows from Lemma 3 s M j [f] wwwijmercom 2607 Page

3 In the line of Lemma 3 we may prove the following lemma: Lemma 5 Let F = f n Q[f] where Q[f] is a differential polynomial in f If n 1 then ρ F = λ F = Lemma 6 For a meromorphic function f of finite lower order, lower proximate order exists The lemma can be proved in the line of Theorem 1 [7] so the proof is omitted Lemma 7 Let f be a meromorphic function of finite lower order Then for δ > 0 the function r +δ (r) is an increasing function of r Proof Since d dr r +δ (r) = + δ r r r +δ r 1 > 0 for sufficiently large values of r, the lemma follows Lemma 8 [6] Let f be an entire function of finite lower order If there exists entire functions a i (i= 1, 2, 3,, n; n ) satisfying T r, a i = o{} then n i=1 δ(a i, f) = 1, log M(r, f) = 1 π III In this section we present the main results of the paper THEOREMS Theorem 1 Let f be a meromorphic function g be an entire function satisfying i, λ g are both finite ii for n 1, G = g n Q[g] Then 3 ρ f 2 λ g Proof If =, the result is obvious So we suppose that < Since T r, g log + M r, g, in view of Lemma 1 we get for all sufficiently large values of r that T r, f o g 1 + o 1 T(M r, g, f) ie, log{1 + o(1)} + log T(M r, g, f) ie, o ε log M(r, g) ie, log M(r, g) f + ε Since ε(> 0) is arbitrary, it follows that log M(r, g) f 1 As by condition (v) of Definition 4 = 1, r λ g (r) so for given ε(0 < ε < 1) we get for a sequence of values of r tending to infinity that T r, g 1 + ε r λ g r (2) for all sufficiently large values of r, T r, g > 1 ε r λ g r (3) Since log M(r, g) 3T(2r, g) {cf [5]}, we have by (2), for a sequence of values of r tending to infinity, log M(r, g) 3T 2r, g ε 2r λ g 2r (4) Combining (3) (4), we obtain for a sequence of values of r tending to infinity that log M(r, g) 3(1 + ε) (1 ε) (2r)λg (2r) r λ g (r) Now for any δ(> 0), for a sequence of values of r tending to infinity we obtain that log M(r, g) 3(1 + ε) (1 ε) (2r) λg +δ (2r) λ g +δ λ g (2r) 1 r λ g (r) ie, wwwijmercom 2608 Page

4 log M(r, g) 3(1 + ε) (1 ε) 2λ g +δ (5) because r λ g +δ λ g (r) is an increasing function of r by Lemma 7 Since ε(> 0) δ(> 0) are arbitrary, it follows from (5) that log M(r, g) 3 2 λ g (6) Thus from (1) (6) we obtain that 3 ρ f 2 λ g (7) Now in view of (7) Lemma 3, we get that = log T(r, f og) 3 2 λ g This proves the theorem Theorem 2 Let f be meromorphic g be entire such that <, λ g < for n 1, G = g n Q g Then 1 log Proof Since T r, g log + M r, g, in view of Lemma 1, we get for all sufficiently large values of r that log T(M r, g, f) + log{1 + o(1)} ie, + ε log M(r, g) + o(1) ie, log [2] T r, f o g log [2] M r, g + O 1 (8) It is well known that for any entire function g, log M(r, g) 3T 2r, g cf 5 Then for 0 < ε < 1 δ > 0, for a sequence of values of r tending to infinity it follows from (5) that log 2 M r, g log T r, g + O 1 9 Now combining (8) (9), we obtain for a sequence of values of r tending to infinity that log 2 T(r, f o g) log T r, g + O(1) ie, log 2 T(r, f o g) 1 (10) log T r, g As by Lemma 4, log log exists is equal to 1, then from (10) we get that = log[2] T(r, f o g) log log log log 11 = 1 Thus the theorem is established Remark 1 The condition < is essential in Theorem 2 which is evident from the following example Example 1 Let f = exp [2] z g = exp z Then f o g = exp [3] z for n 1, G = g n Q g Taking n = 1, A j = 1, n 0j = 1 n 1j = = n kj = 0; we see that G = exp 2z Now we have Again from the inequality {cf p18, [5]} we obtain that = log[2] M(r, f) = log[2] (exp [2] r) = λ g = log[2] M(r, g) = log 2 (exp r) = 1 log + M(r, f) 3T(2r, f) wwwijmercom 2609 Page

5 ie, ie, T r, G log M r, G = log(exp 2r) Combining the above two inequalities, we have Therefore which is contrary to Theorem 2 log + O(1) T r, f o g 1 3 log M r 2, f og = 1 3 exp[2] ( r 2 ) log [2] T r, f o g r 2 + O 1 log r 2 + O(1) + O(1) =, log Theorem 3 Let f g be any two entire functions such that ρ g < < for n 1, F = f n Q[f] G = g n Q g Also there exist entire functions a i (i = 1, 2,, n; n ) with i T r, a i = o T r, g as r for i = 1, 2,, n Then n ii δ a i ; g = 1 i=1 {} 2 T r, F = 0 Proof In view of the inequality log + M(r, g) Lemma 1, we obtain for all sufficiently large values of r that T r, f o g {1 + o 1 }T(M r, g, f) ie, log{1 + o(1)} + log T(M r, g, f) ie, o ε log M(r, g) ie, o ε r (ρ g +ε) (11) Again in view of Lemma 3, we get for all sufficiently large values of r that log T r, F > λ F ε ie, log T r, F > ε ie, T r, F > r λf ε (12) Now combining (11) (12), it follows for all sufficiently large values of r that o 1 + ( + ε)r (ρ g +ε) r ε (13) T(r, F) Since ρ g <, we can choose ε(> 0) in such a way that ρ g + ε < ε (14) So in view of (13) (14), it follows that T(r, F) Again from Lemma 4 Lemma 8, we get for all sufficiently large values of r that o ε log M(r, g) ie, ie, ( + ε) = 0 (15) log M(r, g) wwwijmercom 2610 Page

6 sup log M r, g f + ε T r, g ie, f + ε π Since ε(> 0) is arbitrary, it follows from above that f π (16) Combining (15) (16), we obtain that {} 2 T r, F = log T(r, f og) T(r, F) ie, This proves the theorem 0 π = 0, {} 2 T r, F = 0 Theorem 4 Let f g be any two entire functions satisfying the following conditions: i > 0 ii < iii 0 < λ g g also let for n 1, F = f n Q f Then log [2] max{ λ g } T(r, F) Proof We know that for r > 0 {cf [8]} for all sufficiently large values of r, T r, f o g 1 3 log M 1 8 M r, g + o 1, f (17) 4 Since λ g are the lower orders of f g respectively then for given ε(> 0) for all sufficiently large values of r we obtain that log M r, f > r ε log M r, g > r λ g ε where 0 < ε < min, λ g So from (17) we have for all sufficiently large values of r, T(r, f o g) M r λf ε 4, g + o 1 ie, ie, ie, T(r, f o g) 1 3 {1 9 M(r 4, g)} ε O 1 + ε log M( r 4, g) O 1 + ε ( r 4 )λ g ε ie, log 2 T r, f o g O 1 + λ g ε 18 Again in view of Lemma 1, we get for a sequence of values r tending to infinity that log 2 T r, F λ F + ε ie, log 2 T r, F + ε (19) Combining (18) (19), it follows for a sequence of values of r tending to infinity that O 1 + λ g ε + ε Since ε(> 0) is arbitrary, we obtain that Again from (17), we get for a sequence of values of r tending to infinity that λ g (20) wwwijmercom 2611 Page

7 O 1 + ( ε)( r 4 )ρ g ε ie, log 2 T r, f o g O 1 + ρ g ε (21) Also in view of Lemma 5, for all sufficiently large values of r we have log [2] T r, F F + ε ie, log [2] T r, F + ε (22) Now from (21) (22), it follows for a sequence of values of r tending to infinity that As ε(0 < ε < ρ g ) is arbitrary, we obtain from above that Therefore from (20) (23), we get that Thus the theorem is established O 1 + ρ g ε + ε log [2] T(r, F) ρ g (23) max{ λ g } Theorem 5 Let f be meromorphic g be entire such that i 0 < <, ii ρ g <, iii < iv for n 1, F = f n Q f Then log [2] min{ λ g } T(r, F) Proof In view of Lemma 1 the inequality T r, g log + M r, g, we obtain for all sufficiently large values of r that log T r, f o g o ε log M(r, g) (24) Also for a sequence of values of r tending to infinity, log M(r, g) r λ g +ε (25) Combining (24) (25), it follows for a sequence of values of r tending to infinity that log T r, f o g o 1 + ( + ε)r λ g +ε ie, log T r, f o g r λ g +ε {o 1 + ( + ε)} ie, log [2] T r, f o g O 1 + λ g + ε (26) Again in view of Lemma 5, we obtain for all sufficiently large values of r that log 2 T r, F > λ F ε ie, log 2 T r, F > ε (27) Now from (26) (27), we get for a sequence of values of r tending to infinity that log [2] T r, f o g log 2 T r, F O 1 + λ g + ε ε As ε(> 0) is arbitrary, it follows that log [2] T r, f o g log 2 T r, F λ g (28) In view of Lemma 1, we obtain for all sufficiently large values of r that log 2 T r, f o g O 1 + ρ g + ε (29) Also by Remark 1, it follows for a sequence of values of r tending to infinity that log 2 T r, F > ρ F ε ie, log 2 T r, F > ε 30 Combining (29) (30), we get for a sequence of values of r tending to infinity that wwwijmercom 2612 Page

8 O 1 + ρ g + ε ε Since ε(> 0) is arbitrary, it follows from above that Now from (28) (31), we get that g (31) min{ λ g } This proves the theorem The following theorem is a natural consequence of Theorem 4 Theorem 5: Theorem 6 Let f g be any two entire functions such that i 0 < < <, ii 0 < <, iii 0 < λ g g < iv for n 1, F = f n Q f Then min{ λ g } max{ λ g } log[2] T(r, f o g) log [2] T(r, F) Theorem 7 Let f be meromorphic g be entire satisfying 0 < <,ρ g > 0 also let for n 1, F = f n Q[f] Then log [2] T(exp r ρ g, f o g) log [2] T(exp r μ, F) =, where 0 < μ < ρ g Proof Let 0 < μ < ρ g Then in view of Lemma 2, we get for a sequence of values of r tending to infinity that log T(exp(r μ ), f) ie, ε log{exp(r μ )} ie, ε r μ ie, log [2] T r, f o g O 1 + μ (32) Again in view of Lemma 3, we have for all sufficiently large values of r, log T(exp r μ, F) F + ε log{exp(r μ )} ie, log T(exp r μ, F) + ε r μ ie, log T(exp r μ, F) O 1 + μ (33) Now combining (32) (33), we obtain for a sequence of values of r tending to infinity that from which the theorem follows log [2] T(exp r ρ g, f o g) log [2] T(exp r μ, F) O 1 + μ r ρg O 1 + μ Remark 2 The condition ρ g > 0 is necessary in Theorem 7 as we see in the following example Example 2 Let f = exp z, g = z μ = 1 > 0 Then f o g = exp z for n 1, F = f n Q f Taking n = 1, A j = 1, n 0j = 1 n 1j = = n kj = 0; we see that F = exp 2z Then Also we get that = log[2] M(r, f) = 1, = log[2] M(r, f) = 1 ρ g = log[2] M(r, g) = 0 T r, f = r π wwwijmercom 2613 Page

9 Therefore T exp r ρ g, f o g = e π T exp r μ, F = So from the above two expressions we obtain that log [2] T(exp r ρ g, f o g) log [2] T(exp r μ, F) ie, which contradicts Theorem 7 2 exp r π = log [2] T(exp r ρ g, f o g) log [2] T(exp r μ, F) O(1) + O(1) REFERENCES [1] Bergweiler, W: On the Nevanlinna Characteristic of a composite function, Complex Variables, Vol 10 (1988), pp [2] Bergweiler, W: On the growth rate of composite meromorphic functions, Complex Variables, Vol 14 (1990), pp [3] Bhooshnurmath, S S Prasad, K S L N: The value distribution of some differential polynomials, Bull Cal Math Soc, Vol 101, No 1 (2009), pp [4] Doeringer, W: Exceptional values of differential polynomials, Pacific J Math, Vol 98, No 1 (1982), pp [5] Hayman, W K: Meromorphic functions, The Clarendon Press, Oxford, 1964 [6] Lin, Q Dai, C: On a conjecture of Shah concerning small functions, Kexue Tongbao (English Ed), Vol 31, No 4 (1986), pp [7] Lahiri, I: Generalised proximate order of meromorphic functions, Mat Vensik, Vol 41 (1989), pp 9-16 [8] Niino, K Yang, C C: Some growth relationships on factors of two composite entire functions, Factorization Theory of Meromorphic Functions Related Topics, Marcel Dekker, Inc (New York Basel), (1982), pp [9] Valiron, G: Lectures on the general theory of integral functions, Chelsea Publishing Company, 1949 = 0, wwwijmercom 2614 Page