Slowly Changing Function Oriented Growth Analysis of Differential Monomials and Differential Polynomials

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1 Slowly Chanin Function Oriented Growth Analysis o Dierential Monomials Dierential Polynomials SANJIB KUMAR DATTA Department o Mathematics University o kalyani Kalyani Dist-NadiaPIN West Benal India sanjib kr datta@yahoocoin TANMAY BISWAS Rajbari Rabindrapalli R N Taore Road PO Krishnaar Dist-Nadia PIN- 70 West BenalIndia Tanmaybiswas math@redimailcom Tanmaybiswas math@yahoocom GOLOK KUMAR MONDAL Dhulauri Rabindra Vidyaniketan HS Vill PO- Dhulauri PS- Domkal Dist-Murshidabad PIN West Benal India olokmondal3@redimailcom Abstract: In this paper we establish some new results dependin on the comparative rowth properties o composite entire or meromorphic unctions usin -order -type dierential monomials dierential polynomials enerated by one o the actors Key Words: Meromorphic unction transcendental entire unction composition rowth -order -type dierential monomials dierential polynomial slowly chanin unction Introduction Deinitions Notations We denote by C the set o all inite complex numbers Let be a meromorphic unction deined on C We use the stard notations deinitions in the theory o entire meromorphic unctions which are available in [5] [7] In the sequel we use the ollowin notation : lo [k] x = lo lo [k ] x k = 2 3 lo [0] x = x Let L Lr be a positive continuous unction increasin slowly ie Lar Lr as r or every positive constant a Sinh Barker[3] deined it in the ollowin way: Somasundaram Thamizharasi [] introduced the notions o L-order L-order or entire unctions The more eneralized concept or L-order L-type or entire meromorphic unctions are -order -type respectively Their deinitions are as ollows: Deinition 2 [] The -order the - lower order o an entire unction are deined as lo [2] Mr = lo[re Lr ] lo [2] Mr = in lo [ re Lr] When is meromorphic one can easily veriy that Deinition [3] A positive continuous unction Lr is called a slowly chanin unction i or ε > 0 lo T r = lo [ re Lr] k ε Lkr Lr kε or r rε uniormly or k I urther Lr is dierentiable the above condition is equivalent to lo T r = in lo [ re Lr] Deinition 3 [] The -type σ L o an entire unction is deined as ollows: rl r Lr = 0 σ L = lo Mr [ ] re Lr 0 < < E-ISSN: Issue 6 Volume 2 June 203

2 For meromorphic σ L T r = [ ] re Lr 0 < < Let be a non-constant meromorphic unction deined in the open complex plane C Also let n 0j n j n kj k be non-neative inteers such k that or each j n ij We call i=0 M j [] = A j n 0j n j k n kj where T r A j = Sr to be a dierential monomial enerated by The numbers γ Mj = k n ij i=0 i=0 Γ Mj = k i n ij are called respectively the deree weiht o M j [] [][2] The expression P [] = s i=0 M j [] is called a dierential polynomial enerated by The numbers γ P = max γ M j js Γ P = max Γ Mj are called respectively the deree js weiht o P [][][2] Also we call the numbers γ P = min γ M j k the order o the hihest js derivative o the lower deree the order o P [] respectively I γ p = γ P P [] is called a homoeneous dierential polynomial Throuhout the paper we consider only the non-constant dierential polynomials we denote by P 0 [] a dierential polynomial not containin ie or which n 0j = 0 or j = 2 s We consider only those P [] P 0 [] sinularities o whose individual terms do not cancel each other We also denote by M[] a dierential monomial enerated by a transcendental meromorphic unction In the sequel the ollowin deinitions are also well known : Deinition Let a be a complex number inite or ininite The Nevanlinna deiciency the Valiron deiciency o a with respect to a meromorphic unction are deined as Nr a; δa; = T r mr a; = in T r Nr a; mr a; a; = in = T r T r Deinition 5 The quantity Θa; o a meromorphic unction is deined as ollows Nr a; Θa; = T r Deinition 6 [6] For a C { } we denote by nr a; = the number o simple zeros o a in z r Nr a; = is deined in terms o nr a; = in the usual way We put Nr a; = δ a; = T r the deiciency o a correspondin to the simple a- points o ie simple zeros o a Yan [5] proved that there exists at most a denumerable number o complex numbers a C { } or which δ a; > 0 δ a; Deinition 7 [9] For a C { } let n p r a; denotes the number o zeros o a in z r where a zero o multiplicity < p is counted accordin to its multiplicity a zero o multiplicity p is counted exactly p times; N p r a; is deined in terms o n p r a; in the usual way We deine δ p a; = N p r a; T r Deinition 8 [3] P [] is said to be admissible i i P [] is homoeneous or ii P [] is non homoeneous mr = Sr Lakshminarasimhan [6] introduced the idea o the unctions o L-bounded index Later Lahiri Bhattacharjee [8] worked on the entire unctions o L- bounded index o non uniorm L-bounded index In the paper we investiate the comparative rowth o composite entire meromorphic unctions dierential monomials dierential polynomials enerated by one o their actors usin -order - type 2 Lemmas In this section we present some lemmas which will be needed in the sequel E-ISSN: Issue 6 Volume 2 June 203

3 Lemma 9 [] I be meromorphic be entire then or all suiciently lare values o r T r { o } T r T M r lo M r Lemma 0 [] Let be any two entire unctions or all r > 0 T r { r } 3 lo M 8 M o Lemma [7] Let be an entire unction with λ < a i i = 2 3 n; n are entire unctions satisyin T r a i = o {T r } I n T r δ a i = then lo Mr = π i= Lemma 2 [3] Let P 0 [] be admissiblei is o inite order or o non zero lower order Θ a; = 2 then T r P 0 [] = Γ T r P0 [] Lemma 3 [3] Let be either o inite order or o non-zero lower order such that Θ ; = δ p a; = or δ ; = δ a; = or homoeneous P 0 [] T r P 0 [] = γ T r P0 [] Lemma Let be a meromorphic unction o inite order or o non zero lower order I Θ a; = 2 then the -order -lower order o admissible P 0 [] is same as that o Also the -type o P 0 [] is Γ P0 [] times that o when is o inite positive -order Proo By Lemma 2 equal to P 0 [] = lo T rp 0 [] lo T r lo T r P 0 [] lo [ re Lr] exists is lo T r = lo [ lo T r P 0 [] relr] lo T r = = In a similar manner P 0 [] = λl σ L Aain by Lemma 2 T r P 0 [] P 0 [] = [ ] re Lr P 0 [] T r P 0 [] = T r = Γ P0 []σ L This proves the lemma T r [ re Lr ] Lemma 5 Let be a meromorphic unction o inite order or o non zero lower order such that Θ ; = δ p a; = or δ ; = δ a; = the -order -lower order o homoeneous P 0 [] are same Also the -type o P 0 [] is γ P0 [] times that o when is o inite positive - order We omit the proo o the lemma because it can be carried out in the line o Lemma with the help o Lemma 3 Lemma 6 [0] Let be a transcendental meromorphic unction o inite order or o non-zero lower order δ a; = T r M[] = Γ M Γ M γ M Θ ; T r where Nr Θ ; = T r Lemma 7 I be a transcendental meromorphic unction o inite order or o non-zero lower order δ a; = then -order -lower order o M[] are same as those o Also the -type o M[] is Γ M Γ M γ M Θ ; times that o when is o inite positive -order We omit the proo o the lemma because it can be carried out in the line o Lemma with the help o Lemma 6 3 Theorems In this section we present the main results o the paper It is needless to mention that in the paper the admissibility homoenity o P 0 [] will be needed as per the requirements o the theorems E-ISSN: Issue 6 Volume 2 June 203

4 Theorem 8 Let be a meromorphic unction be an entire unction o inite order or o non zero lower order such that i 0 < < ii σ L < iii 0 < σ L < iv Θ ; = δ p a; = or δ ; = δ a; = a i L M r = o {T r P 0 []} then in lo T r T r P 0 [] L M r L γ P0 [] b i T r P 0 [] = o {L M r } then in lo T r T r P 0 [] L M r L Proo Since T r lo M r in view o Lemma 9 we obtain or all suiciently lare values o r that ie ie T r { o } T M r lo T r lo { o } lo T M r lo T r o {lo M r L M r } Usin the deinition o -type we obtain rom or all suiciently lare values o r that lo T r o {re σ L Lr} L L M r 2 Aain rom the deinition o -type in view o Lemma 3 Lemma 5 we et or a sequence o values o r tendin to ininity that T r P 0 [] ie T r P 0 [] ie {re Lr} { σp L 0 [] ε { γ P0 []σ L ε re Lr} P 0 [] } {re Lr} L T r P 0 [] γp0 []σ L ε 3 Now rom 2 3 it ollows or a sequence o values o r tendin to ininity that lo T r o σ L L M r ie lo T r T r P 0 [] L M r o T r P 0 [] L M r T r P 0 [] γp0 []σ L ε ε σ L ε γ P0 []σ L ε LMr T rp 0 [] T rp 0[] LMr I L M r = o {T r P 0 []} then rom we obtain that in lo T r T r P 0 [] L M r σ γ P0 [] σ L ε Since ε > 0 is arbitrary it ollows rom above that in lo T r T r P 0 [] L M r L γ P0 [] Thus the irst part o Theorem 8 ollows Aain i T r P 0 [] = o {L M r } then rom it ollows that in lo T r T r P 0 [] L M r As ε > 0 is arbitrary we obtain rom above that in lo T r T r P 0 [] L M r L Thus the second part o Theorem 8 ollows Remark 9 With the help o Lemma 5 the conclusion o Theorem 8 can also be drawn under the hypothesis Θ ; = δ p a; = E-ISSN: Issue 6 Volume 2 June 203

5 or δ ; = δ a; = Proo In view o condition ii we obtain rom 2 or all suiciently lare values o r that instead o Θ a; = 2 In the line o Theorem 8 with the help o Lemma 7 we may state the ollowin theorem without proo : Theorem 20 Let be a meromorphic unction be a transcendental entire unction o inite order or o non zero lower order such that i 0 < < ii σ L < iii 0 < σ L < iv δ a; = Thus a i L M r = o {T r M []} then in lo T r T r M [] L M r Γ M Γ M γ M Θ ; b i T r M [] = o {L M r } then in lo T r T r M [] L M r L Theorem 2 Let be meromorphic with inite order or non zero lower order be entire with i 0 < < ii = iii σ L < iv 0 < σ L < v Θ a; = 2 a i L M r = o { r α e αlr} as r or some positive α < then in lo T r T r P 0 [] L M r L σ L Γ P0 [] σ L b i T r P 0 [] = o {L M r } then in lo T r T r P 0 [] L M r L lo T r o {re σ L Lr} L L M r 5 Aain rom the deinition o -type in view o Lemma 2 Lemma we et or a sequence o values o r tendin to ininity that T r P 0 [] ie T r P 0 [] ie {re Lr} { σl L ε { Γ P0 []σ L ε re Lr} P 0 [] } {re Lr} L T r P 0 [] 6 Γ P0 []σ L ε Now rom 5 6 it ollows or a sequence o values o r tendin to ininity that lo T r o ie σ L L M r lo T r T r P 0 [] L M r o T r P 0 [] L M r T r P 0 [] Γ P0 []σ L ε ε σ L ε Γ P0 []σ L ε LMr T rp 0 [] T rp 0[] LMr 7 I L M r = o {T r L} then rom 7 we et that in lo T r T r P 0 [] L M r Γ P0 [] σ σ L ε E-ISSN: Issue 6 Volume 2 June 203

6 Since ε > 0 is arbitrary it ollows rom above that in lo T r T r P 0 [] L M r L σ L Γ P0 [] σ L Thus the irst part o Theorem 2 ollows Aain i T r L = o {L M r } then rom 7 it ollows that in lo T r T r P 0 [] L M r As ε > 0 is arbitrary we obtain rom above that in lo T r T r P 0 [] L M r L Thus the second part o Theorem 2 ollows Now in the line o Theorem 2 one may state the ollowin two theorems without proo : Theorem 22 Let be a meromorphic unction with inite order or non zero lower order be entire such that i 0 < < ii = iii σ L < iv 0 < σ L < v Θ ; = δ p a; = or δ ; = δ a; = a i L M r = o { r α e αlr} as r or some positive α < then in lo T r T r P 0 [] L M r L σ L γ P0 [] σ L b i T r P 0 [] = o {L M r } then in lo T r T r P 0 [] L M r L Theorem 23 Let be a transcendental meromorphic unction with inite order or non zero lower order be entire such that i 0 < < ii = iii σ L < iv 0 < σ L < v δ a; = a i L M r = o { r α e αlr} as r or some positive α < then in lo T r T r M[] L M r σ L Γ M Γ M γ M Θ ; σ L b i T r M[] = o {L M r } then in lo T r T r M[] L M r L Theorem 2 Let be an entire unction o inite order or o non zero lower order be an entire unction with 0 < < 0 < < Θ ; = δ p a; = or δ ; = δ a; = lo [2] T r lo T r P 0 [] L 8 M r L Proo In view o Lemma 0 we have or all suiciently lare values o r T r 3 lo M { r } 8 M o ie lo T r o { r } lo [2] M 8 M o 8 ie lo T r o ε [ { r } lo 8 M o r ] L 8 M ie lo T r o ε [ lo { r 8 M o 8 M r } r ] L 8 M E-ISSN: Issue 6 Volume 2 June 203

7 ie lo T r lo M r lo o M r 8 lo M r ie lo [2] T r r λ lo [2] M ε [ { λ ε lo exp L lo ε lo M r L 8 M r r L 8 M 8 M r }] ε [lo M r L 8 M r lo M r ie lo [2] T r r λ lo [2] M ε ] o r L 8 M [lo ε M r L 8 M r ] o lo { λ exp L ε L 8 M r } lo M r ε ie r lo [2] T r lo [2] M λ ε r L 8 M 9 Now rom 9 it ollows or a sequence o values o r tendin to ininity that { r lo [2] T r ε lo el r } ε r L 8 M 0 In view o Lemma 5 we et or all suiciently lare values o r that lo T r P 0 [] P 0 [] lo {re Lr} ie lo T r P 0 [] lo {re Lr} ie lo T r P 0 [] { r lo el r } lo Hence rom 0 it ollows or all suiciently lare values o r that ie lo [2] T r ε lo T r P 0 [] lo ε r L 8 M ie lo [2] T r [ ε lo T r P 0 [] L ε lo 8 M r ] ε lo [2] T r ie lo T r P 0 [] L 8 M r ε lo ε lo T r P 0 [] L 8 M r 2 Since ε > 0 is arbitrary it ollows rom 2 that lo [2] T r lo T r P 0 [] L 8 M r L This proves the theorem In the line o Theorem 2 the ollowin theorem may be proved : Theorem 25 Let be an entire unction o inite order or o non zero lower order be an entire unction with 0 < < 0 < < Θ ; = δ p a; = or δ ; = δ a; = in lo [2] T r lo T r P 0 [] L 8 M r λl Remark 26 By Lemma the conclusion o Theorem 2 Theorem 25 can also be drawn under the hypothesis Θ a; = 2 instead o Θ ; = δ p a; = or δ ; = δ a; = E-ISSN: Issue 6 Volume 2 June 203

8 Theorem 27 Let be a transcendental entire unction o inite order or o non zero lower order be an entire unction with 0 < < 0 < < δ a; = lo [2] T r lo T r P 0 [] L 8 M r L Theorem 28 Let be a transcendental entire unction o inite order or o non zero lower order be an entire unction with 0 < < 0 < < δ a; = in lo [2] T r lo T r M[] L 8 M r λl The proo o the above two theorems can be established in the line o Theorem 2 Theorem 25 respectively with the help o Lemma 7 thereore is omitted Theorem 29 Let be meromorphic be entire o inite order or o non zero lower order with L < 0 < < δ ; = δ a; = a i L M r = o {lo T r P 0 []} then lo [2] T r lo T r P 0 [] L M r L b i T r P 0 [] = o {L M r } then lo [2] T r lo T r P 0 [] L M r = 0 Proo For all suiciently lare values o r we obtain in view o T r lo M r by Lemma 9 that T r { o } T M r ie lo T r lo { o } ie lo T r o lo T M r lo T M r ie lo T r o {lo M r L M r } ie lo T r o { } L M r lo M r lo M r ie lo [2] T r o lo lo { L M r lo M r ie lo [2] T r o lo lo { L M r lo M r } lo [2] M r } lo {re Lr} ie lo [2] T r o {lo r Lr} L M r lo M r 3 Aain in view o Lemma 5 we et rom the deinition o -lower order or all suiciently lare values o r that lo T r P 0 [] P 0 [] ε lo [re Lr] ie lo T r P 0 [] ie lo T r P 0 [] ε [re Lr] ε lo [lo r Lr] ie lo r Lr lo T r P 0 [] λ ε Hence rom 3 it ollows or all suiciently lare values o r that lo [2] T r o lo T r P 0 [] ε L M r lo M r E-ISSN: Issue 6 Volume 2 June 203

9 lo [2] T r ie lo T r P 0 [] L M r lo T r P 0 [] o ε lo T r P 0 [] L M r ie L M r [lo T r P 0 [] L M r ] lo M r lo [2] T r lo T r P 0 [] L M r o [ ε ε LMr lo T rp 0 [] lo T rp 0[] LMr ] lo M r 5 Since L M r = o {lo T r P 0 []} as r ε > 0 is arbitrary we obtain rom 5 that lo [2] T r lo T r P 0 [] L M r L 6 Aain i lo T r = o {L M r } then rom 5 we et that lo [2] T r lo T r P 0 [] L M r = 0 7 Thus rom 6 7 the theorem is established Corollary 30 Let be meromorphic be entire o inite order or o non zero lower order with L < 0 < < δ ; = δ a; = a i L M r = o {lo T r P 0 []} then in lo [2] T r T r P 0 [] L M r b i T r P 0 [] = o {L M r } then in lo [2] T r T r P 0 [] L M r = 0 We omit the proo o Corollary 30 because it can be carried out in the line o Theorem 29 Remark 3 The equality sin in Theorem 29 Corollary 30 cannot be removed as we see in the ollowin example Example 32 Let = = exp z Lr = p exp r where p is any positive real number = = = δ ; = δ a; = Also let s = A = { or i = n i = 0 or i Now So Hence T r P 0 [] = exp z exp r 2π 3 r 2 r T r = r M r = exp r π L M r = L exp r = p exp exp r in lo [2] T r lo T r P 0 [] L M r lo [2] T r = lo T r P 0 [] L M r = = lo [ r 2 lo r O ] lo r O p exp exp r Remark 33 The conclusion o Theorem 29 Corollary 30 can also be drawn under the hypothesis Θ ; = δ p a; = or Θ a; = 2 instead o δ ; = δ a; = In the line o Theorem 29 with the help o Lemma 7 we may state the ollowin theorem without proo : Theorem 3 Let be meromorphic be transcendental entire o inite order or o non zer lower order with < 0 < < δ a; = E-ISSN: Issue 6 Volume 2 June 203

10 a i L M r = o {lo T r M []} then lo [2] T r lo T r M [] L M r L b i T r M [] = o {L M r } then lo [2] T r lo T r M [] L M r = 0 The proo o the ollowin corollary may also be deduced in view o Theorem 3 : Corollary 35 Let be meromorphic be transcendental entire o inite order or o non zero lower order with < 0 < < δ a; = a i L M r = o {lo T r M []} then in lo [2] T r T r M [] L M r b i T r M [] = o {L M r } then in lo [2] T r T r M [] L M r = 0 Theorem 36 Let be meromorphic with < be entire with inite order or non zero inite lower order Θ a; = 2 Also let there exists entire unctions a i i = 2 3 n; n such that T r a i = o {T r } n δ a i = i= I L M r = o { r α e αlr} as r or some α with 0 < α < then otherwise lo T r in T r P 0 [] in πλl Γ P0 [] lo T r T r P 0 [] L M r = 0 Proo In view o the inequality T r lo Mr by Lemma 9 we et or a sequence o values o r tendin to ininity that T r { o } T M r ie ie ie ie lo T r lo { o } lo T M r lo T r ε lo M r L M r O lo T r T r P 0 [] ε lo M r L M r O lo T r T r P 0 [] T r P 0 [] lo M r L M r ε T r P 0 [] O 8 Case I Let L M r = o { r α e αlr} as r or some α with 0 < α < Since α < we can choose ε > 0 in such a way that α < ε 9 As L M r = o { r α e αlr} as r we et in view o 9 that L M r [ ] = 0 20 re Lr ε Aain in view o Lemma 3 we obtain or all suiciently lare values o r lo T r P 0 [] P 0 [] ε lo {re Lr} ie lo T r P 0 [] ε lo {re Lr} ie T r P 0 [] [re Lr] λ ε 2 Now rom 8 2 we obtain or a sequence o values o r tendin to ininity that lo T r T r P 0 [] E-ISSN: Issue 6 Volume 2 June 203

11 ε lo M r L M r T r P 0 [] [ ] O re Lr ε ie lo M r T r lo T r T r P 0 [] T r T r P 0 [] ε L M r [ ] re Lr ε O 22 Now combinin in view o Lemma Lemma 2 it ollows that lo T r in T r P 0 [] πλl 23 Γ P0 [] Case II I L M r o { r α e αlr} as r or some α with 0 < α < then rom 8 we et or a sequence o values o r tendin to ininity that lo T r T r P 0 [] L M r lo M r ε T r P 0 [] L M r O T r P 0 [] ie in lo T r T r P 0 [] L M r = 0 Thus combinin Case I Case II the theorem ollows Remark 37 In view o Lemma 5 one can easily veriy that the conclusion o Theorem 35 can also be deduced i we replace Θ a; = 2 by Θ ; = δ p a; = or δ ; = δ a; = In the line o Theorem 36 the ollowin theorem can be proved : Theorem 38 Let be meromorphic with < be entire with inite order or non zero inite lower order Θ ; = δ p a; = or δ ; = δ a; = Also let there exists entire unctions a i i = 2 3 n; n such that T r a i = o {T r } n δ a i = i= I L M r = o { r α e αlr} as r or some α with 0 < α < then lo T r T r P 0 [] πl γ P0 [] otherwise lo T r T r P 0 [] L M r = 0 Remark 39 In view o Lemma one can easily veriy that the conclusion o Theorem 38 can also be deduced i we replace Θ ; = δ p a; = or δ ; = δ a; = by Θ a; = 2 In the line o Theorem 36 Theorem 38 with the help o Lemma 7 we may state the ollowin two theorems without proo : Theorem 0 Let be meromorphic with < be transcendental entire with inite order or non zero inite lower order δ a; = Also let there exists entire unctions a i i = 2 3 n; n such that T r a i = o {T r } n δ a i = i= I L M r = o { r α e αlr} as r or some α with 0 < α < then lo T r in T r M [] otherwise in π Γ M Γ M γ M Θ ; lo T r T r M [] L M r = 0 Theorem Let be meromorphic with < be transcendental entire with inite order or non zero inite lower order δ a; = Also let there exist entire unctions a i i = 2 3 n; n such that T r a i = o {T r } n δ a i = i= I L M r = o { r α e αlr} as r or some α with 0 < α < then lo T r T r P 0 [] π Γ M Γ M γ M Θ ; otherwise lo T r T r P 0 [] L M r = 0 E-ISSN: Issue 6 Volume 2 June 203

12 Theorem 2 Let be a meromorphic unction be entire o inite order or o non zero lower order such that < = δ ; = δ a; = lo T r lo T r P 0 [] = Proo Let us suppose that the conclusion o the theorem does not hold we can ind a constant β > 0 such that or a sequence o values o r tendin to ininity lo T r β lo T r P 0 [] 2 Aain rom the deinition o P 0 [] it ollows that or all suiciently lare values o r in view o Lemma 6 lo T r P 0 [] P 0 [] lo {re Lr} ie lo T r P 0 [] { lo re Lr} 25 Thus rom 2 25 we have or a sequence o values o r tendin to ininity that lo T r β lo {re Lr} Proo By Theorem 2 or Remark 3 we obtain or all suiciently lare values o r or K > lo T r > K lo T r P 0 [] ie T r > {T r P 0 []} K rom which the corollary ollows Remark 5 The condition = is necessary in Theorem 2 Corollary which is evident rom the ollowin example : Example 6 Let = z = exp z Lr = p exp r where p is any positive real number Also let s = A = { or i = n i = 0 or i Also Now P 0 [] = exp z δ ; = δ a; = = < = < T r = T r exp z = r π T r P 0 [] = T r exp z = r π ie lo T r lo { re Lr} β lo { re Lr} lo { re Lr} lo T r ie in lo { re Lr} = λl < This is a contradiction This proves the theorem Thereore lo T r lo T r P 0 [] T r T r P 0 [] lo r O = lo r O = r π = r π = Remark 3 Theorem 2 is also valid with it superior instead o it i = is replaced by = the other conditions remainin the same Corollary Under the assumptions o Theorem 2 or Remark 3 T r T r P 0 [] = Remark 7 Considerin = z = exp z A = Lr = p exp r where p is any positive real number s = A = { or i = n i = 0 or i one may also veriy that the condition = in Remark 3 Corollary is essential E-ISSN: Issue 6 Volume 2 June 203

13 Remark 8 The conclusion o Theorem 2 Remark 3 Corollary can also be drawn under the hypothesis Θ ; = δ p a; = or Θ a; = 2 instead o δ ; = δ a; = In the line o Theorem 2 the ollowin theorem may be deduced: Theorem 9 Let be meromorphic be transcendental entire o inite order or non zero lower order such that < = δ a; = lo T r lo T r M [] = Remark 50 Theorem 9 is also valid with it superior instead o it i = is replaced by = the other conditions remainin the same Corollary 5 Under the assumptions o Theorem 9 or Remark 50 T r T r M [] = The proo is omitted because it can be carried out in the line o Corollary Reerences [] W Berweiler On the Nevanlinna Characteristic o a composite unction Complex Variables pp [2] W Berweiler On the rowth rate o composite meromorphic unctions Complex Variables 990 pp87-96 [3] N Bhattacharjee I Lahiri Growth value distribution o dierential polynomials Bull Math Soc Sc Math Roumanie Tome pp85-0 [] W Doeriner Exceptional values o dierential polynomials Paciic J Math pp55-62 [5] W K Hayman Meromorphic Functions The Clarendon Press Oxord 96 [6] TV Lakshminarasimhan A note on entire unctions o bounded index J Indian Math Soc pp 3-9 [7] Q Lin C Dai On a conjecture o Shah concernin small unctions Kexue Ton bao Enlish Ed pp [8] I Lahiri NR Bhattacharjee Functions o L- bounded index o non-uniorm L-bounded index Indian J Math pp 3-57 [9] I Lahiri Deiciencies o dierential polynomials Indian J Pure Appl Math pp35-7 [0] I Lahiri SK Datta Growth value distribution o dierential monomials Indian J Pure Appl Math pp 83-8 [] K Niino CC Yan Some rowth relationships on actors o two composite entire unctions Factorization theory o meromorphic unctions related topics Marcel Dekker IncNew York Basel 982 pp [2] LR Sons Deiciencies o monomials Math Z 969 pp53-68 [3] SK Sinh GP Barker Slowly chanin unctions their applications Indian J Math pp -6 [] D Somasundaram R Thamizharasi A note on the entire unctions o L-bounded index L-type Indian J Pure ApplMath pp [5] L Yan Value distribution theory new research on it Science Press Beijin 982 [6] H X Yi On a result o Sinh Bull Austral Math Soc 990 pp7-20 [7] G Valiron Lectures on the eneral theory o interal unctions Chelsea Publishin Company 99 E-ISSN: Issue 6 Volume 2 June 203