Some Results on the Growth Analysis of Entire Functions Using their Maximum Terms and Relative L -orders

Transcription

1 Journal of Matematical Extension Vol. 10, No. 2, (2016), ISSN: URL: ttp:// Some Results on te Growt Analysis of Entire Functions Using teir Maximum Terms and Relative L -orders S. K. Datta University of Kalyani T. Biswas Rajbari Rabindrapalli A. Hoque University of Kalyani Abstract. In tis paper we study some comparative growt properties of composite entire functions in terms of teir maximum terms on te basis of teir relative L order ( relative L lower order ) wit respect to anoter entire function. AMS Subject Classification: 30D20; 30D30; 30D35 Keywords and Prases: Entire function, maximum term, composition, relative L order (relative L lower order ), growt. 1. Introduction Let C be te set of all finite complex numbers. Also let f be an entire function defined in te open complex plane C. Te maximum term µ f (r) and te maximum modulus M f (r) of f = a n z n on z = r are defined n=0 as µ f (r) = max ( a n r n ) and M f (r) = max z =r Received: July 2015; Accepted: February 2016 Corresponding autor f (z) respectively. We use 59

2 60 S. K. DATTA, T. BISWAS AND A. HOQUE te standard notations and definitions in te teory of entire functions wic are available in 10]. In te sequel we use te following notation: ( ) log k] x = log log k] x, k = 1, 2, 3,...and log 0] x = x. If f is non-constant ten M f (r) is strictly increasing and continuous and its inverse M f (r) : ( f (0), ) (0, ) exists and is suc tat lim (s) =. Bernal 1] introduced te definition of relative order M s f of f wit respect to g, denoted by ρ g (f) as follows: ρ g (f) = inf {µ > 0 : M f (r) < M g (r µ ) for all r > r 0 (µ) > 0} = log M g M f (r). log r Similarly, one can define te relative lower order of f wit respect to g denoted by λ g (f) as follows: λ g (f) = lim inf log M g M f (r). log r If we consider g (z) = exp z, te above definition coincides wit te classical definition 9] of order ( lower order) of an entire function f wic is as follows: Definition 1.1. Te order ρ f and te lower order λ f of an entire function f are defined as log 2] M f (r) ρ f = log r and λ f = lim inf log 2] M f (r). log r Using te inequalities µ f (r) M f (r) R R r µ f (R) for 0 r < R 8] one may give an alternative definition of entire function in te following manner: log 2] µ f (r) ρ f = log r and λ f = lim inf log 2] µ f (r). log r

3 SOME RESULTS ON THE GROWTH ANALYSIS Now let L L (r) be a positive continuous function increasing slowly L (ar) L (r) as r for every positive constant a. Sing and Barker 5] defined it in te following way: Definition ] A positive continuous function L (r) is called a slowly canging function if for ε (> 0), 1 L (kr) kε L (r) kε for r r (ε) and uniformly for k ( 1). If furter, L (r) is differentiable, te above condition is equivalent to rl (r) lim L (r) = 0. Somasundaram and Tamizarasi 6] introduced te notions of L-order for entire function were L L (r) is a positive continuous function increasing slowly L (ar) L (r) as r for every positive constant a. Te more generalised concept for L-order for entire function is L - order and its definition is as follows: Definition ] Te L -order f entire function f are defined as and te L -lower order λ L f of an f = log 2] M f (r) log rel(r)] and λl f log 2] M f (r) = lim inf. In view of te inequalities µ f (r) M f (r) one may verify tat R R r µ f (R) for 0 r < R 8] f = log 2] µ f (r) log re L(r)] and λl f log 2] µ f (r) = lim inf. In te line of Somasundaram and Tamizarasi 6] and Bernal 1], Datta and Biswas 2] gave te definition of relative L -order of an entire function in te following way:

4 62 S. K. DATTA, T. BISWAS AND A. HOQUE Definition ] Te relative L -order of an entire function f wit respect to anoter entire function g, denoted by g (f) in te following way { g (f) = inf {µ > 0 : M f (r) < M g re L(r)} µ } for all r > r0 (µ) > 0 = log Mg M f (r). Similarly, one can define te relative L -lower order of f wit respect to g denoted by λg L (f) as follows: λ L g (f) = lim inf log Mg M f (r). In te case of relative L -order (relative L -lower order), it terefore seems reasonable to define suitably an alternative definition of relative L -order (relative L -lower order) of entire function in terms of its maximum terms. Datta, Biswas and Ali 4] also introduced suc definition in te following way: Definition ] Te relative order g (f) and te relative lower order λ g (f) of an entire function f wit respect to anoter entire function g are defined as g (f) = g µ f (r) and λl g (f) = lim inf g µ f (r). In tis paper we wis to establis some results relating to te growt rates of composite entire functions in terms of teir maximum terms on te basis of relative L -order (relative L -lower order). 2. Main Results In te following we present some lemmas wic will be needed in te sequel.

5 SOME RESULTS ON THE GROWTH ANALYSIS Lemma ] Let f and g be any two entire functions. Ten for every α > 1 and 0 < r < R, µ f g (r) α ( ) αr α 1 µ f R r µ g (R). Lemma ] If f and g are any two entire functions wit g (0) = 0. Ten for all sufficiently large values of r, µ f g (r) 1 ( 1 ( r ) ) 2 µ f 8 µ g g (0). 4 Lemma ] If f be an entire and α > 1, 0 < β < α, ten for all sufficiently large r, µ f (αr) βµ f (r). Teorem 2.4. Let f, g and be any tree entire functions and g (0) = 0. If tere exist α and β, satisfying 0 < α < 1, β > 0 and α (β + 1) > 1, suc tat Ten (i) (ii) lim inf (µ g(r)) ( ) log re L(r) α = A, a real number > 0, ( log M (r)) = B, a real number > 0. β+1 (f g) =. Proof. From (i), we ave for a sequence of values of r tending to infinity (µ g(r)) (A ε) (log re L(r)) α (1) and from (ii), we obtain for all sufficiently large values of r tat (B ε) ( (r)) β+1.

6 64 S. K. DATTA, T. BISWAS AND A. HOQUE Since µ g (r) is continuous, increasing and unbounded function of r, we get from above for all sufficiently large values of r tat (µ f (µ g (r))) (B ε) ( (µ g (r)) ) β+1. (2) Also µ (r) is an increasing function of r, it follows from Lemma 2.2, Lemma 2.3, ( 1) and (2) for a sequence of values of r tending to infinity tat { ( 1 ( r µ f 24 µ g 2) )} ( ( r ))} {µ f µ g ( ( ( r (B ε) µ g ( ( r (B ε) (A ε) log ( ( (B ε) (A ε) β+1 r log µ β+1 f g(r) log (B ε) (A ε) log ( r re L(r)] ))) β+1 ) e L( r ) ) α] β+1 ) e L( r ) ) α(β+1) ) e L( r ) ] α(β+1) (B ε) (A ε) β+1 log re L(r) + O(1) ] α(β+1) lim inf. Since ε (> 0) is arbitrary and α (β + 1) > 1, it follows from above tat wic proves te teorem. (f g) =, In te line of Teorem 2.4, one may state te following two teorems witout teir proofs :

7 SOME RESULTS ON THE GROWTH ANALYSIS Teorem 2.5. Let f, g and be any tree entire functions and g (0) = 0. If tere exist α and β, satisfying 0 < α < 1, β > 0 and α (β + 1) > 1, suc tat (i) lim inf (ii) (µ g(r)) ( ) log re L(r) α = A, a real number > 0, ( (r)) = B, a real number > 0. β+1 Ten (f g) =. Teorem 2.6. Let f, g and be any tree entire functions and g (0) = 0. If tere exist α and β, satisfying 0 < α < 1, β > 0 and α (β + 1) > 1, suc tat (i) lim inf (ii) lim inf (µ g(r)) ( ) log re L(r) α = A, a real number > 0, ( (r)) = B, a real number > 0. β+1 Ten λ L (f g) =. Teorem 2.7. Let f, g and be any tree entire functions and g (0) = 0. If tere exist α and β, satisfying α > 1, 0 < β < 1 and αβ > 1, suc tat (i) (µ g(r)) ( ) α = A, a real number > 0, log 2] r log (ii) lim inf (r) ] (r)] β = B, a real number > 0.

8 66 S. K. DATTA, T. BISWAS AND A. HOQUE Ten (f g) =. Proof. From (i), we ave for a sequence of values of r tending to infinity we get tat ( α (µ g(r)) (A ε) log r) 2] (3) and from (ii), we obtain for all sufficiently large values of r tat log (µ ] f (r)) (r) (r) exp (B ε) (r)] β (B ε) (r)] β ]. Since µ g (r) is continuous, increasing and unbounded function of r, we get from above for all sufficiently large values of r tat (µ f (µ g (r))) (µ g (r)) exp (B ε) (µ g (r)) ] ] β. (4) Also µ (r) is increasing function of r, it follows from Lemma 2.2, Lemma 2.3, (3) and (4) for a sequence of values of r tending to infinity tat log re µ { ( f g(r) log 1 µ µf 24 L(r)] µ ( r g 4))}, µ { ( ( f g(r) log log r ))} µ µf µg rel(r)], { ( ( r ))} µf µg ( ( r )) µg ( µg ( r )),

9 SOME RESULTS ON THE GROWTH ANALYSIS log re L(r) exp (B ε) r β (A ε) log 2] µ g r α log re L(r), log re L(r) exp (B ε) (A ε) β log 2] r αβ (A ε) log 2] r α log re L(r), log re L(r) exp (B ε) (A ε) β log 2] r αβ log 2] r (A ε) log 2] r α exp (B ε) (A ε) β log 2] r αβ log 2] r (A ε) log 2] r α log re L(r) log, re L(r), log re L(r) r (B ε)(a ε)β (log 2] ( r log log re L(r), )) αβ r r (log 2] ( r (B ε)(a ε)β (log 2] ( r limliminfinfloglog (A ε) log 2] log re L(r) )) )) αβ αβ (A (A ε) ε) r α log 2] log 2] r α α r log log re L(r) re L(r).. Since ε (> 0) is arbitrary and α > 1, αβ > 1, te teorem follows from above. In te line of Teorem 2.7, one may also state te following two teorems witout teir proofs : Teorem 2.8. Let f, g and be any tree entire functions and g (0) = 0. If tere exist α and β, satisfying α > 1, 0 < β < 1 and αβ > 1, suc

10 68 S. K. DATTA, T. BISWAS AND A. HOQUE tat Ten (i) lim inf log (ii) (µ g(r)) ( ) α = A, a real number > 0, log 2] r (r) ] (r)] β = B, a real number > 0. (f g) =. Teorem 2.9. Let f, g and be any tree entire functions and g (0) = 0. If tere exist α and β, satisfying α > 1, 0 < β < 1 and αβ > 1, suc tat Ten (i) lim inf log (ii) lim inf (µ g(r)) ( ) α = A, a real number > 0, log 2] r (r) ] (r)] β = B, a real number > 0. λ L (f g) =. Teorem Let f, g and be any tree entire functions suc tat 0 < λ L (g) ρl (g) <, g (0) = 0 and Ten λ L (r) = A, a real number <. (f g) A λl (g) and ρl (f g) A ρl (g).

11 SOME RESULTS ON THE GROWTH ANALYSIS Proof. Since µ (r) is an increasing function of r, it follows from Lemma 2.2 for all sufficiently large values of r tat log rel(r)] {µ f (µ g (26r))}, {µ f (µ g (26r))} (µ g (26r)) (µ g (26r)) log rel(r)], (5) lim inf lim inf {µ f (µ g (26r))} (µ g (26r)) lim inf {µ f (µ g (26r))} (µ g (26r)) λ L (µ ] g (26r)), lim inf (µ g (26r)), (f g) A λl (g). (6) Also from (5), we obtain for all sufficiently large values of r tat {µ f (µ g (26r))} (µ g (26r)) (µ ] g (26r)) {µ f (µ g (26r))} (µ g (26r)) (µ g (26r))

12 70 S. K. DATTA, T. BISWAS AND A. HOQUE Terefore te teorem follows from (6) and (7). (fog) A ρl (g). (7) Teorem Let f, g and be any tree entire functions suc tat 0 < λ L (g) <, g (0) = 0 and Ten (r) = A, a real number <. (f g) B λl (g). Proof. Since µ (r) is an increasing function of r, it follows from Lemma 2.2 for all sufficiently large values of r tat µ { ( ( f g(r) log log r ))} µ µf µg rel(r)], { ( ( r ))} µf µg ( ( r )) µmg ( ( log r )) µ µmg, { ( ( r ))} µf µg ( ( r )) µmg { ( ( r ))} µf µg ( ( r )) lim inf µmg ( ( log r µ µmg ))] ( ( r )) µmg,, (f g) B λl (g).

13 SOME RESULTS ON THE GROWTH ANALYSIS Tus te proof is complete. Teorem Let f, g and be any tree entire functions suc tat 0 < λ L (g) ρl (g) <, g (0) = 0 and Ten lim inf (r) = B, a real number <. λ L (f g) B ρl (g). Teorem Let f, g and be any tree entire functions suc tat 0 < (g) <, g (0) = 0 and for a particular value of δ > 0. T en (r) = A, a real number < (f g) A ρl (g). Te proof of Teorem 2.12 and Teorem 2.13 are omitted because tose can be carried out in te line of Teorem 2.10 and Teorem 2.11 respectively. Acknowledgements Te autors are tankful to te referees for teir valuable suggestions towards te improvement of te paper. References 1] L. Bernal, Orden relative de crecimiento de funciones enteras,collect. Mat., 39 (1988), ] S. K. Datta and T. Biswas, Growt of entire functions based on relative order, International Journal of Pure and Applied Matematics, 51 (1) (2009),

14 72 S. K. DATTA, T. BISWAS AND A. HOQUE 3] S. K. Datta and A. R. Maji, Relative order of entire functions in terms of teir maximum terms, Int. Journal of Mat. Analysis, 43 (5) (2011), ] S. K. Datta, T. Biswas, and S. Ali, Growt estinmates of composite entire functions based on maximum terms using teir relative L- order, Advances in Applied Matematical Analysis, 7 (2) (2012), ] S. K. Sing and G. P. Barker, Slowly canging functions and teir applications, Indian J. Mat., 19 (1) (1977), ] D. Somasundaram and R. Tamizarasi, A note on te entire functions of L-bounded index and L type, Indian J. Pure Appl. Mat., 19 (3) (1988), ] A. P. Sing, On maximum term of composition of entire functions, Proc. Nat. Acad. Sci. India, 59 (A)(part I) (1989), ] A. P. Sing and M. S. Baloria, On maximum modulus and maximum term of composition of entire functions, Indian J. Pure Appl. Mat., 22 (12) (1991), ] E. C. Titcmars, Te Teory of Functions, 2nd ed., Oxford University Press, Oxford, ] G. Valiron, Lectures on te General Teory of Integral Functions, Celsea Publising Company, Sanjib Kumar Datta Department of Matematics Assistant Professor of Matematics University of Kalyani P.O.-Kalyani, Dist-Nadia, PIN , West Bengal, India sanjib kr datta@yaoo.co.in

15 SOME RESULTS ON THE GROWTH ANALYSIS Tanmay Biswas Researc Scolar of Matematics Rajbari, Rabindrapalli P.O.-Krisnagar, Dist-Nadia, PIN , West Bengal, India tanmaybiswas Asanul Hoque Department of Matematics Researc Scolar of Matematics University Of Kalyani P.O.-Kalyani, Dist-Nadia, PIN , West Bengal, India