1 Introduction, Definitions and Notations

Transcription

1 Acta Universitatis Apulensis ISSN: No 34/2013 pp THE GROWTH ESTIMATE OF ITERATED ENTIRE FUNCTIONS Ratan Kumar Dutta Abstract In this paper we study growth properties of iterated entire functions which improve some earlier results 2010 Mathematics Subject Classification: 30D35 Keywords and phrases: Entire functions, growth, iteration, order, lower order, p-th order, lower p-th order 1 Introduction, Definitions and Notations Let fz and gz be two transcendental entire functions defined in the open complex log T r,fog log T r,fog plane C, it is well known [1] that lim r T r,f = and lim r T r,g = 0 Later on Singh [11] investigated some comparative growth of logt r, f og and T r, f Further in [11] he raised the problem of investing the comparative growth of logt r, fog and T r, g However some results on the comparative growth of logt r, fog and T r, g are proved in [6] Also in [7] Lahiri and Datta made close investigation on comparative growth properties of logt r, f og and T r, g together with that of log log T r, fog and T r, f k Recently Banerjee and Dutta [2] made close investigation on comparative growth properties of iterated entire functions In this paper, we study growth of iterated entire functions to generalist some results of Banerjee and Dutta [2] in terms of p-th order and lower p-th order The following definitions are well known Definition 11 The order ρ f and the lower order λ f of a meromorphic function is defined as log T r, f ρ f = If f is entire then log log Mr, f ρ f = and λ f = r and λ f = r log T r, f log log Mr, f 81

2 Definition 12 The hyper order ρ f and the hyper lower order λ f of a meromorphic function is defined as log log T r, f ρ f = and If f is entire then and λ f = r log log T r, f log [3] Mr, f ρ f = λ f = r log [3] Mr, f Notation 13 [10] log [0] x = x, exp [0] x = x and for positive integer m, log [m] x = loglog [m 1] x, exp [m] x = expexp [m 1] x Definition 14 The p-th order ρ p f and the lower p-th order λp f function f is defined as ρ p log [p] T r, f f = and If f is an entire function then = r log [p] T r, f of a meromorphic and ρ p f log [p+1] Mr, f = = r log [p+1] Mr, f Clearly ρ p f ρp 1 f and λp 1 f for all p and when p = 1 then p-th order and lower p-th order coincides with classical order and lower order respectively Definition 15 The hyper p-th order ρ p f and the hyper lower p-th order λp f of a meromorphic function f is defined as ρ p f log [p+1] T r, f = 82

3 and If f is an entire function then = r log [p+1] T r, f and ρ p f log [p+2] Mr, f = = r log [p+2] Mr, f Clearly ρ p f ρp 1 f and λp 1 f for all p and when p = 1 then hyper p-th order and hyper lower p-th order coincides with hyper order and hyper lower order respectively Definition 16 A function λ f r is called a lower proximate order of a meromorphic function f if i λ f r is nonnegative 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 r λ f r = λ f < ; iv lim r rλ f r = 0; and T r,f v r = 1 r λ f r According to Lahiri and Banerjee [4], fz and gz be two entire functions then the iteration of f with respect to g is defined as follows: f 1 z = fz f 2 z = fgz = fg 1 z f 3 z = fgfz = fg 2 z = fgf 1 z f n z = fgffz or gz, according as n is odd or even, = fg n 1 z = fgf n 2 z, 83

4 and so are g 1 z = gz g 2 z = gfz = gf 1 z g n z = gf n 1 z = gfg n 2 z Clearly all f n z and g n z are entire functions Throughout the paper we assume f, g etc are non constant entire functions having respective p-th orders ρ p f, ρp g and respective lower p-th orders, λp g Also we do not explain the standard notations and definitions of the theory of entire and meromorphic functions because those are available in [3], [12] and [13] 2 Lemmas The following lemmas will be needed in the sequel Lemma 21 [3] Let fz be an entire function For 0 r < R <, we have T r, f log + Mr, f R + r T R, f R r Lemma 22 [1] If f and g are any two entire functions, for all sufficiently large values of r, 1 M 8 M 2, g g0, f Mr, fog MMr, g, f Lemma 23 [9] Let fz and gz be two entire functions Then we have T r, fg log M 8 M 4, g + O1, f Lemma 24 [5] Let f be an entire function Then for k > 2, log [k 1] Mr, f r log [k 2] T r, f = 1 Lemma 25 [7] Let f be a meromorphic function Then for δ> 0 the function r λ f +δ λ f r is an increasing function of r 84

5 Lemma 26 [8] Let f be an entire function of finite lower order If there exist entire functions a i i = 1, 2, 3n; n satisfying T r, a i = o{t r, f} and n T r, f δa i, f = 1 then lim r log Mr, f = 1 π i=1 Lemma 27 Let fz and gz be two non constant entire functions such that 0 < ρ p f < and 0 < ρp g < Then for all sufficiently large r and ε > 0, { ρ p log [n 1p] T r, f n f + ε log Mr, g + O1 when n is even ρ p g + ε log Mr, f + O1 when n is odd where p 1 Proof First suppose that n is even Then from Lemma 21 and second part of Lemma 22 also from definition of p-th order, it follows that for all sufficiently large values of r, T r, f n log Mr, f n log MMr, g n 1, f ie, log [p] T r, f n log [p+1] MMr, g n 1, f log[mr, g n 1 ] ρp f +ε So, log [p+1] T r, f n log [2] Mr, gf n 2 + O1 Taking repeated logarithms p-1 times, we get log [2p] T r, f n log [p+1] MMr, f n 2, g + O1 log[mr, f n 2 ] ρp g+ε + O1 ie, log [2p+1] T r, f n log [2] Mr, f n 2 + O1 Again taking repeated logarithms p-1 times, we get log [3p] T r, f n log[mr, g n 3 ] ρp f +ε + O1 Finally, after taking repeated logarithms n-4p times more, we have for all sufficiently large values of r, log [n 1p] T r, f n log[mr, g] ρp f +ε + O1 ie, log [n 1p] T r, f n ρ p f + ε log Mr, g + O1 Similarly if n is odd then for all sufficiently large values of r This proves the lemma log [n 1p] T r, f n ρ p g + ε log Mr, f + O1 85

6 Lemma 28 Let fz and gz be two non constant entire functions such that 0 < < and 0 < λp g < Then for any ε 0 < ε <min{, λp g} and p 1, { λ p log [n 1p] r T r, f n f ε log M, g +O1 when n is even λ p g ε log M, f +O1 when n is odd for all sufficiently large values of r Proof To prove this lemma we first consider n is even Then from Lemma 21 and Lemma 23 we get for ε 0 < ε <min{, λp g} and for all large values of r T r, f n = T r, fg n log M 8 M 4, g n 1 log [p] T r, f n log [p+1] M [ 1 log 8 M 4, g n 1 [ 1 log 9 M 4, g n 1 + O1, f 1 8 M 4, g n 1 + O1, f ] λ p + O1 ] ε + O1 λ p r f ε log M 4, g n 1 + O1 λ p r f εt 4, g n 1 + O1 λ p 1 f ε1 3 log M 8 M 4 2, f n 2 that is, log [2p] T r, f n log [p+1] M [ 1 log 8 M 4 2, f n 2 [ 1 log 9 M 4 2, f n 2 f ε + O1 + O1 + O1, g + O1, 1 8 M 4 2, f n 2 + O1, g + O1 ] λ p g ε + O1 + O1 ] λ p g ε + O1 ie, log [2p] T r, f n λ p g ε log M 4 2, f n 2 + O1 r Therefore, log [n 2p] T r, f n λ p g ε log M 4 n 2, fg + O1 So, log [n 1p] T r, f n λ p r f ε log M, g + O1 when n is even 86

7 Similarly log [n 1p] T r, f n λ p g ε log M This proves the lemma, f + O1 when n is odd 3 Theorems Theorem 31 Let f and g be two non-constant entire functions having finite lower orders Then i ii when n is even and iii iv when n is odd log [n 1p] T r, f n r T r, g log [n 1p] T r, f n r T r, g log [n 1p] T r, f n r T r, f log [n 1p] T r, f n r T r, f 3ρ p f 2λg, λg 3ρ p g2 λ f, λ p g λ f Proof We may clearly assume 0 < ρp f < and 0 < λp g ρ p g < Now from Lemma 27 for arbitrary ε > 0 log [n 1p] T r, f n ρ p f + ε log Mr, g + O1 31 when n is even Let 0 < ε <min{1,, λp g} Since T r, g r r λgr = 1, there is a sequence of values of r tending to infinity for which and for all large values of r T r, g < 1 + εr λgr 32 T r, g > 1 εr λgr 33 87

8 Thus for a sequence of values of r tending to infinity we get for any δ> 0 log Mr, g T r, g 3T 2r, g T r, g 31 + ε 1 ε 2λg+δ 31 + ε 1 ε because r λg+δ λgr is an increasing function of r Since ε, δ > 0 be arbitrary, we have Therefore from 31 and 34 we get 2r λg+δ 2r λg+δ λg2r 1 r λgr log Mr, g 32 λg 34 r T r, g log [n 1p] T r, f n 3ρ p r T r, g f 2λg when n is even Again for even n we have from Lemma 28 log [n 1p] T r, f n λ p r f ε log M, g + O1 λ p r f εt, g + O1 λg+δ ε1 ε1 + O1 λg+δ λ r, by 33 g Since r λg+δ λgr is an increasing function of r, we have λgr log [n 1p] T r, f n λ p r f ε1 ε1 + O1 λg+δ for all large values of r So by 32 for a sequence of values of r tending to infinity log [n 1p] T r, f n ε1 ε 1 + ε 1 + O1 T r, g λg+δ Since ε and δ are arbitrary, it follows from the above that log [n 1p] T r, f n r T r, g λg Similarly for odd n we get the second part of the theorem This proves the theorem 88

9 Theorem 32 Let f and g be two non-constant entire functions such that and λ p g> 0 are finite Also there exist entire functions a i i = 1, 2, 3n; n satisfying T r, a i = o{t r, g} as r and n δa i, g = 1 Then when n is even i=1 π log [n 1p] T r, f n πρ p λg r T r, g f Proof If = 0 then the first inequality is obvious Now we suppose that λp f > 0 For 0 < ε <min{1,, λp g} we have from Lemma 28 for all large values of r log [n 1p] T r, f n T r, g εlog M, g T r, g εlog M, g T, g T + O1 when n is even, g + O1 35 T r, g Also from 32 and 33 we get for a sequence of values of r and for δ > 0 T, g > 1 ε λg+δ 1 T r, g 1 + ε λg+δ λ r g r λgr 1 ε ε λg+δ because r λg+δ λgr is an increasing function of r Since ε, δ > 0 be arbitrary, so using Lemma 26, we have from 35 log [n 1p] T r, f n r T r, g πλp f λg If ρ p f =, the second inequality is obvious So we may assume ρp f < Then the second inequality follows from Lemma 26 and Lemma 27 This proves the theorem Theorem 33 Let f and g be two non-constant entire functions such that > 0 and λ p g are finite Also there exist entire functions a i i = 1, 2, 3n; n satisfying T r, a i = o{t r, f} as r and n δa i, f = 1 i=1 89

10 Then when n is odd πλ p g λ f log [n 1p] T r, f n πρ p g r T r, f Theorem 34 Let f and g be two non-constant entire functions such that 0 < ρ p f < and 0 < λp g ρ p g < Then for k = 0, 1, 2, 3, λ p g ρ p g when n is even and log [np+1] T r, f n log [np+1] T r, f n r log [p] T r, g k r log [p] ρp g T r, g k λ p g ρ p f log [np+1] T r, f n log [np+1] T r, f n r log [p] T r, f k r log [p] ρp f T r, f k when n is odd, where f k denote the k-th derivative of f Proof First suppose that n is even Then for given ε0 < ε < min{, λp g} we get from Lemma 28 for all large values of r log [n 1p] T r, f n λ p r f ε log M, g + O1 λ p r f εt, g + O1 ie, log [np] T r, f n log [p] T, g + O1 So, log [np+1] T r, f n log [p+1] T, g + O1 So for all large values of r Since log [np+1] T r, f n log [p] log[p+1] T, g log r T r, g k log r log [p] + o1 36 T r, g k log [p] T r, g k = ρ p g, so for all large values of r and arbitrary ε > 0 we have log [p] T r, g k < ρ p g + ε 37 90

11 Since ε > 0 is arbitrary, so from 36 and 37 we have log [np+1] T r, f n log [p+1] T, g 4 log 4 n 1 r log [p] n 1 T r, g k r log ρ p g r λp g ρ p g 38 Again from Lemma 27 we get for all large values of r Since ie log [n 1p] T r, f n ρ p f + ε log Mr, g + O1 log [np+1] T r, f n log [p] T r, g k log[p+2] Mr, g log [p] T r, g k log [p] T r, g k = λ p g, so for all large values of r and arbitrary ε0 < ε < λ p g we have + o1 39 log [p] T r, g k > λ p g ε 310 Since ε > 0 is arbitrary, so from 39 and 310 we have log [np+1] T r, f n r log [p] T r, f k ρp g λ p 311 g Combining 38 and 311 we obtain the first part of the theorem Similarly when n is odd then we have the second part of the theorem This proves the theorem Theorem 35 Let f and g be two non-constant entire functions such that 0 < ρ p f < and 0 < λp g ρ p g < Then i λ p g ρ p g log [np] T r, f n log [np] T r, f n r log [p] 1 T r, g r log [p] ρp g T r, g λ p g when n is even and ii ρ p f log [np] T r, f n log [np] T r, f n r log [p] 1 T r, f r log [p] ρp f T r, f when n is odd 91

12 Proof First suppose that n is even Then for given ε0 < ε < min{, λp g} we get from Lemma 27 and Lemma 28 for all large values of r Also, So ie log [n 1p] T r, f n ρ p f + ε log Mr, g + O1 ie log [np] T r, f n log [p+1] Mr, g + O1 ie log [np] T r, f n log [p] T r, g log [np] T r, f n ie r log [p] T r, g log[p+1] Mr, g log [p] T r, g + o [by Lemma 24] 313 log [n 1p] T r, f n λ p r f ε log M, g + O1 ie log [np] T r, f n log [p+1] M, g + O1 log [np] T r, f n log [p] T r, g log [np] T r, f n r log [p] T r, g log[p] T, g 4 log 4 n 1 n 1 log ρ p g λp g ρ p g Also from 312, we get for all large values of r, log [np] T r, f n log [p] T r, g log [np] T r, f n r log [p] T r, g Again from Lemma 28, r + o1 314 log[p+1] Mr, g ρp g λ p g log [p] T r, g + o1 315 log [n 1p] T r, f n λ p r f ε log M, g + O1 ie log [np] T r, f n log [p+1] M, g + O1 316 From 33 we obtain for all large values of r and for δ > 0 and ε0 < ε < 1 log M, g > 1 ε λg+δ λg+δ λ g 1 ε rλgr λg+δ 92

13 because r λg+δ λgr is an increasing function of r So by 32 we get for a sequence of values of r tending to infinity log M, g ie log [2] M, g 1 ε ε T r, g λg+δ log T r, g + O1 Taking repeated logarithms p-1 times, we get log [p+1] M, g log [p] T r, g + O1 317 Now from 316 and 317 log [np] T r, f n r log [p] T r, f So the theorem follows from 313, 314, 315 and 318 when n is even Similarly when n is odd we get ii Corollary 36 Using the hypothesis of Theorem 35 if f and g are of regular growth then log [np] T r, f n log [np] T r, f n lim r log [p] = lim T r, g r log [p] = 1 T r, f Remark 37 The conditions, λp g > 0 and ρ p f, ρp g < are necessary for Theorem 35 and Corollary 36, which are shown by the following examples Example 38 Let f = z, g = exp [p] z Then = ρp f = 0 and 0 < λp g = ρ p g < Now when n is even then f n = exp [ np 2 ] z Therefore, So, T r, f n log Mr, f n = exp [ np 2 1] r log [np] T r, f n log [np] exp [ np np [np = log 2 +1] r = log [ np 2 +1] r 2 1] r Also when n is odd n 1 [ f n = exp 2 p] z 93

14 Therefore, So, n 1 [ T r, f n log Mr, f n = exp 2 p 1] r log [np] T r, f n log [np] n 1 [ exp 2 p 1] r n 1 [np = log 2 p+1] r n+1 [ = log 2 p+1] r Now and log [p] T r, f = log [p+1] r Therefore when n is even 3T 2r, g log Mr, g = exp [p 1] r ie log [p] T r, g + O1 log [np] T r, f n log [p] T r, g np log[ 2 +1] r 0 as r, + O1 and when n is odd log [np] T r, f n log [p] T r, f log[ n+1 2 p+1] r log [p+1] r 0 as r Example 39 Let f = exp [2p] z, g = exp [p] z Then = ρp f =, λp g = ρ p g = 1 Now when n is even f n = exp [ 3np 2 ] z Therefore 3T 2r, f n log Mr, f n = exp [ 3np 2 1] r ie T r, f n 1 3np exp[ 2 1] r 3 2 log [np] T r, f n log [np] exp [ 3np 2 1] r 2 + o1 = exp [ np 2 1] r 2 + o1 Also when n is odd 3n+1 [ f n = exp 2 p] z 94

15 Therefore Also Therefore when n is even and when n is odd 3n+1 [ 3T 2r, f n log Mr, f n = exp 2 p 1] r ie T r, f n 1 3n+1 p 1] r exp[ log [np] T r, f n log [np] 3n+1 [ p 1] exp r o1 n+1 [ = exp 2 p 1] r 2 + o1 log [p] T r, f exp [p 1] r and log [p] T r, g log [np] T r, f n log [p] T r, g log [np] T r, f n log [p] T r, f np exp[ 2 1] r 2 + o1 as r, exp[ n+1 2 p 1] r 2 + o1 exp [p 1] r as r Theorem 310 Let f and g be two entire functions such that 0 < ρp f < and 0 < λ p g ρ p g < Then for k = 0, 1, 2, 3, i λ p g ρ p f when n is even log [np] T r, f n log [np] T r, f n r log [p] T r, f k r log [p] T r, f k ρ p g ii when n is odd ρ p g log [np] T r, f n log [np] T r, f n r log [p] T r, g k r log [p] T r, g k ρ p f λ p g Proof First suppose that n is even Then for given ε0 < ε < min{, λp g} we have from Lemma 27 for all large values of r, log [n 1p] T r, f n ρ p f + ε log Mr, g + O1 ie log [np] T r, f n log [p+1] Mr, g + O1 Also we know that log [p] T r, g k = λ p g 95

16 Now log [np] T r, f n r log [p] T r, f k log [p+1] Mr, g r log [p] T r, f k [ ] log [p+1] Mr, g log [p] T r, f k = ρp g 319 Again from Lemma 28 we have for all large values of r, Also Therefore, log [n 1p] T r, f n λ p r f ε log M, g + O1 ie, log [np] T r, f n log [p+1] M, g + O1 λ p g ε + O1 Since ε > 0 is arbitrary we get log [p] T r, f k < ρ p f + ε log [np] T r, f n log [p] T r, f k λ p g ε + O1 ρ p f + ε log [np] T r, f n r log [p] T r, f k λ Therefore from 319 and 320 we have the result for even n Similarly for odd n we have ii This proves the theorem References p g ρ p f 320 [1] J Clunie, The composition of entire and meromorphic functions, Mathematical essays dedicated to A J Macintyre, Ohio Univ Press, 1970, [2] D Banerjee and R K Dutta, The growth of iterated entire functions, Bulliten of Mathematical Analysis and Applications, Volume , [3] W K Hayman, Meromorphic Functions, The Clarendon Press, Oxford,

17 [4] B K Lahiri and D Banerjee, Relative fix points of entire functions, J Indian Acad Math, , [5] I Lahiri, Generalised proximate order of meromorphic functions, Matematnykn Bechnk, , 9-16 [6] I Lahiri, Growth of composite integral functions, Indian J Pure and Appl Math, , [7] I Lahiri and S K Datta, On the growth of composite entire and meromorphic functions, Indian J Pure and Appl Math, , [8] Q Lin and C Dai, On a conjecture of Shah concerning small functions, Kexue Tong English Ed, , [9] K Niino and C C 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, [10] D Sato, On the rate of growth of entire functions of fast growth, Bull Amer Math Soc, , [11] A P Singh, Growth of composite entire functions, Kodai Math J, , [12] G Valiron, Lectures on the general theory of Integral functions, Chelsea Publishing Company, 1949 [13] C C Yang and H X Yi, Uniqueness Theory of Meromorphic Functions, Kluwer Academic Publishers and Science Press, Beijing, 2003 Ratan Kumar Dutta Department of Mathematics Patulia High School Patulia, Kolkata-119 West bengal India ratan 3128@yahoocom 97