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1. 1. International Journal of Engineering Research and DevelopmentISSN: 2278-067X, Volume 1, Issue 10 (June 2012), PP.56-64www.ijerd.com Some Comparative Growth Properties of Entire Functions with Zero Order as Defined by Datta and Biswas Sanjib Kumar Datta1, Tanmay Biswas2, Soumen Kanti Deb3 1 Department of Mathematics, University of Kalyani, Kalyani, Dist-Nadia, PIN- 741235, West Bengal, India. 2 Rajbari, Rabindrapalli, R. N. Tagore Road, P.O. Krishnagar, Dist-Nadia, PIN- 741101, West Bengal, India. 3 Bahin High School, P.O. Bahin, Dist-Uttar Dinajpur, PIN- 733157, West Bengal, India.Abstract—In this paper we compare the maximum term of composition of two entire functions with their corresponding leftand right factors.AMS Subject Classification (2010): 30D35, 30D30.Keywords— Entire function, maximum term, composition, growth. I. INTRODUCTION, DEFINITIONS AND NOTATIONS Let f be an entire function defined in the open complex plane . The maximum term  (r , f ) of  f   a n z n on z  r is defined by  (r , f )  max ( an r n ) . To start our paper we just recall the following n 0 n 0definitions.Definition 1 The order  f and lower order  f of an entire function f is defined as follows: log [ 2] M (r , f ) log [ 2] M (r , f )  f  lim sup and  f  lim inf r  log r r  log r where log [ k ] x  log(log [ k 1] x) for k  1,2,3,... and log [ 0] x  x. If  f   then f is of finite order. Also  f  0 means that f is of order zero. In this connection Liao andYang [2] gave the following definition.Definition 2 [2]Let f be an entire function of order zero. Then the quantities  * and *f f of an entire function f aredefined as: log [ 2] M (r , f ) log [ 2] M (r , f )  *  lim sup f and *f  lim inf . r  log [ 2] r r  log [ 2] r Datta and Biswas [1] gave an alternative definition of zero order and zero lower order of an entire function in thefollowing way:Definition 3 [1] Let f be an entire function of order zero. Then the quantities  ** and *f* of f f are defined by: log M (r , f ) log M (r , f )  **  lim sup f and *f*  lim inf . r  log r r  log r Since for 0  r  R,  (r , f )  M (r , f )  R  ( R, f ) {cf .[4]} 1 Rr it is easy to see that 56
2. 2. Some Comparative Growth Properties of Entire Functions with Zero Order as.... log [ 2]  (r , f ) log [ 2]  (r , f )  f  lim sup ,  f  lim inf r  log r r  log r log [ 2]  (r , f ) log [ 2]  (r , f )  *  lim sup f , *f  lim inf r  log [ 2] r r  log [ 2] r and log  (r , f ) log  (r , f )  **  lim sup f , *f*  lim inf . r  log r r  log r In this paper we investigate some aspects of the comparative growths of maximum terms of two entire functionswith their corresponding left and right factors. We do not explain the standard notations and definitions on the theory ofentire function because those are available in [5]. II. LEMMASIn this section we present some lemmas which will be needed in the sequel.Lemma 1 [3] Let f and g be two entire functions withg (0)  0. Then for all sufficiently large values of r , 1 1 r  (r , f  g )   (  ( , g )  g (0) , f ). 4 8 4Lemma 2 [3] Let f and g be two entire functions. Then for every   0 and 0  r  R,  R  (r , f  g )  (  ( R, g ), f ).  1 R  rLemma 3 [1] Let f and g be two entire functions such that  f   and  g  0. then  fog  . III. THEOREMSIn this section we present the main results of the paper.Theorem 1 Let f and g be two entire functions with 0  *f*   **   and  g  0 . Then for any A  0, f log  (r , f  g ) lim  . log  (r A , f ) r Proof. In view of Lemma 1 we obtain for all sufficiently large values of r , 1 1 r log  (r , f  g )  log{  (  ( , g ), f )} 4 8 4 1 r i.e., log  (r , f  g )  (*f*   ) log{  ( , g )}  O(1) 8 4 r i.e., log  (r , f  g )  (*f*   ) log  ( , g )  O(1) 4 g  r i.e., log  (r , f  g )  (   ) ** f  O(1) . (2) 4 Again from the definition of  ** we have for arbitrary positive  and for all sufficiently large values of r , f log  (r A , f )  A(  **   ) log r . f (3) Therefore it follows from (2) and (3) for all sufficiently large values of r that 57
3. 3. Some Comparative Growth Properties of Entire Functions with Zero Order as....  g  r (   ) **  O(1) log  (r , f  g ) f  4 log  (r A , f ) A(  **   ) log r f log  (r , f  g ) i.e., lim  . r  log  (r A , f )This completes the proof.Remark 1 If we take  g  0 instead of  g  0 in Theorem 1 and the other conditions remain the same, then in the lineof Theorem 1 one can easily verify that log  (r , f  g ) lim sup  , r  log  (r A , f ) where A  0.Remark 2 Also if we consider 0  *f*   or 0   **   instead f of 0  *f*   **   in Theorem 1 and the fother conditions remain the same, then in the line of Theorem 1 one can easily verify that log  (r , f  g ) lim sup  , r  log  (r A , f ) where A  0.Theorem 2 Let f and g be two an entire functions such that 0  *f*   and 0   g   g  . Then log  (r , f  g ) lim  , r  log [ 2 ]  ( r A , g ) where A  1,2,3... .Proof. Since g is the order of g , for given  and for all sufficiently large values of r we get from the definition oforder that log [ 2]  (r A , g )  A(  g   ) log r . (4) Now we obtain from (2) and (4) for all sufficiently large values of r that  g  r (*f*   )   O(1) log  (r , f  g ) 4  . (5) log [ 2]  (r A , g ) A(  g   ) log r As  g  0 , the theorem follows from (5) .Remark 3 If we take 0   **   instead of 0  *f*   in Theorem 2 and the other conditions remain the same, then fin the line of Theorem 2 one can easily verify that log  (r , f  g ) lim sup  , r  log [ 2]  (r A , g ) where A  1,2,3... . 58
4. 4. Some Comparative Growth Properties of Entire Functions with Zero Order as....Remark 4 Also if we consider 0   g  . or 0   g  . instead of 0   g   g  . in Theorem 2 and theother conditions remain the same, then in the line of Theorem 2 one can easily verify that log  (r , f  g ) lim sup  , r  log [ 2]  (r A , g ) where A  1,2,3... .Theorem 3 Let f and g be two an entire functions such that 0   f   f  . and  g*  . Then * log [ 2]  (r , f  g ) lim sup  0, r  log  (r A , f ) where A  1,2,3... .Proof. By Lemma 2 and Lemma 3 we get for all sufficiently large values of r that R log [ 2]  (r , f  g )  log [ 2]  (  ( R, g ), f )  O(1) Rr  R  i.e., log [ 2]  (r , f  g )  (  f   ) log  ( R, g )   O(1) Rr  i.e., log  (r, f  g )  (  f   ) log  ( R, g )  O(1) [ 2] i.e., log [ 2]  (r, f  g )  (  f   )(  g*   ) log R  O(1). * (6)Again from the definition of  f we have for arbitrary positive  and for all sufficiently large values of r, log [ 2]  (r A , f )  A( f   ) log r i.e., log  (r A , f )  r A(  f  ) . 7 Therefore it follows from (6) and (7) for all sufficiently large values of r that log [ 2]  (r , f  g ) (  f   )(  g   ) log R  O(1) **  A(   ) . (8) log  (r A , f ) r f As  f  0 , the theorem follows from (8) .Remark 5 If we take 0   f   or 0   f   instead of 0   f   f   in Theorem 3 and the otherconditions remain the same, then in the line of Theorem 3 one can easily verify that log [ 2]  (r , f  g ) lim inf  0, r  log  (r A , f ) where A  1,2,3... .Remark 6 Also if we take **  . instead of  g*  . in Theorem 3 and the other conditions remain the same, then in g *the line of Theorem 3 one can easily verify that log [ 2]  (r , f  g ) lim inf  0, r  log  (r A , f ) where A  1,2,3... . 59
5. 5. Some Comparative Growth Properties of Entire Functions with Zero Order as....Theorem 4 Let f and g be two an entire functions such that 0  *f*   **   and  g*  0. Then f * log  (r , f  g )  f  g ** ** lim sup  , r  log  (r A , f ) A ** f where A  1,2,3... .Proof. Suppose 0    min{*f* ,  g* }. *Now from Lemma 1 we have for a sequence of values of r tending to infinity that 1 1 r log  (r , f  g )  log{  (  ( , g ), f )} 4 8 4 1 r i.e., log  (r , f  g )  (*f*   ) log{  ( , g )}  O(1) 8 4 1 r i.e., log  (r , f  g )  (*f*   ) log  (*f*   ) log  ( , g )  O(1) 8 4 r i.e., log  (r , f  g )  (*f*   )(  g*   ) log  O(1) * 4 i.e., log  (r, f  g )  ( f   )(  g   ) log r  O(1) . ** ** (9)So combining (3) and (9) we get for a sequence of values of r tending to infinity that log  (r , f  g ) ( f   )(  g   ) log r  O(1) ** **  log  (r A , f ) A(  **   ) log r f log  (r , f  g )  f  g ** ** i.e., lim sup  . r  log  (r A , f ) A ** f This completes the proof.Remark 7 Under the same conditions of Theorem 4 if *f*   ** ,then f log  (r , f  g )  g ** lim sup  . r  log  (r A , f ) ARemark 8 In Theorem 4 if we take **  0 instead of  g*  0 and the other conditions remain the same then it can be g *shown that log  (r , f  g )  f  g ** ** lim inf  . r  log  (r A , f ) A ** f In addition if *f*   ** , then f log  (r , f  g )  g ** lim inf  . r  log  (r A , f ) ARemark 9 Also if we consider 0  *f*   or 0   **   instead f of 0  *f*   **   in Theorem 4 and the fother conditions remain the same, then one can easily verify that 60
6. 6. Some Comparative Growth Properties of Entire Functions with Zero Order as.... log  (r , f  g )  g ** lim sup  . r  log  (r A , f ) ATheorem 5 Let f and g be two an entire functions such that 0  **   g*   and  **  0. Then g * f log  (r , f  g )  f  g ** ** lim sup  , r  log  (r A , g ) A g* * where A0.Proof. Let us choose  in such a way that 0    min{  ** , ** }. f gNow from Lemma 1 we have for a sequence of values of r tending to infinity that 1 1 r log  (r , f  g )  log{  (  ( , g ), f )} 4 8 4 1 r i.e., log  (r , f  g )  (  **   ) log{  ( , g )}  O(1) f 8 4 1 r i.e., log  (r , f  g )  (  **   ) log  (  **   ) log  ( , g )  O(1) f f 8 4 r i.e., log  (r , f  g )  (  **   )(**   ) log  O(1) f g 4 i.e., log  (r, f  g )  (  f   )( g   ) log r  O(1) . ** ** (10) Also we have for all sufficiently large values of r, log  (r A , g )  A(  g*   ) log r . * (11)Therefore from (10) and (11) we get for a sequence of values of r tending to infinity that log  (r , f  g ) (  f   )( g   ) log r  O(1) ** **  log  (r A , g ) A(  g*   ) log r * log  (r , f  g )  f  g ** ** i.e., lim sup  . r  log  (r A , g ) A g* * This proves the theorem.Remark 10 Under the same conditions of Theorem 5 if **   g* , g * then log  (r , f  g )  f ** lim sup  . r  log  (r A , g ) ARemark 11 In Theorem 5 if we take *f*  0. instead of  **  0. and the other conditions remain the same then it can fbe shown that log  (r , f  g )  f  g ** ** lim inf  , r  log  (r A , g ) A g* * In addition if **   g* , then g * 61
7. 7. Some Comparative Growth Properties of Entire Functions with Zero Order as.... log  (r , f  g )  f ** lim inf  . r  log  (r A , g ) ARemark 12 Also if we consider 0  **   or 0   g*   instead of 0  **   g*   in Theorem 5 and the g * g *other conditions remain the same, then one can easily verify that log  (r , f  g )  f ** lim sup  . r  log  (r A , g ) ATheorem 6 Let f and g be two an entire functions such that 0  *f*   **   and  g*  0. Then f * log  (r , f  g )  f  g ** ** lim sup  , r  log  (r A , f ) A*f* where A  1,2,3... .Proof. From the definition of *f* we have for arbitrary positive  and for all sufficiently large values of r, log  (r A , f )  A(  **   ) log r . f (12) Again by Lemma 2 it follows for all sufficiently large values of r that  R  log  (r , f  g )  log    ( R, g ), f   O(1) Rr   R  i.e., log  (r , f  g )  (  **   ) log  ( R, g )   O(1) Rr f  i.e., log  (r, f  g )  (  f   ) log  ( R, g )  O(1) ** i.e., log  (r, f  g )  (  **   )(  g*   ) log R  O(1). f * (13) Now combining (12) and (13) we get for all sufficiently large values of r, log  (r , f  g ) (  f   )(  g   ) log R  O(1) ** **  log  (r A , f ) A(  **   ) log r f log  (r , f  g )  f  g ** ** i.e., lim sup  . r  log  (r A , f ) A*f*This completes the proof. Similarly we may state the following theorem without proof for the right factor g of the composite function f g:Theorem 7 Let f and g be two an entire functions such that  **   and 0  **   g*   . Then f g * log  (r , f  g )  f  g ** ** lim sup  , r  log  (r A , g ) A**g where A  1,2,3... .Remark 13 Under the same conditions of Theorem 6 if *f*   ** , f then 62
8. 8. Some Comparative Growth Properties of Entire Functions with Zero Order as.... log  (r , f  g )  g ** lim sup  . r  log  (r A , f ) ARemark 14 In Theorem 6 if we take **  0 instead of  g*  0 and the other conditions remain the same then it can be g *shown that log  (r , f  g )  f  g ** ** lim inf  , r  log  (r A , f ) A*f* In addition if *f*   ** , then f log  (r , f  g )  g ** lim inf  . r  log  (r A , f ) ARemark 15 If we take 0  *f*   or 0   **   instead f of 0  *f*   **   in f Theorem 6 and the otherconditions remain the same, then one can easily verify that log  (r , f  g )  g ** lim inf  . r  log  (r A , f ) ARemark 16 Under the same conditions of Theorem 7 if **   g* , then g * log  (r , f  g )  f ** lim sup  . r  log  (r A , g ) ARemark 17 In Theorem 7 if we take *f*   instead of  **   and the other conditions remain the same then it can be fshown that log  (r , f  g )  f  g ** ** lim inf  , r  log  (r A , g ) A**g In addition if **   g* , then g * log  (r , f  g )  f ** lim inf  . r  log  (r A , g ) ARemark 18 If we take 0  **   or 0   g*   instead g * of 0  **   g*   in g * Theorem 7 and the otherconditions remain the same, then one can easily verify that log  (r , f  g )  f ** lim inf  . r  log  (r A , g ) A 63
9. 9. Some Comparative Growth Properties of Entire Functions with Zero Order as.... REFERENCES[1]. S.K. Datta and T. Biswas, On the definition of a meromorphic function of order zero, Int. Math. Forum, Vol.4, No. 37(2009) pp.1851-1861.[2]. L. Liao and C. C. Yang, On the growth of composite entire functions, Yokohama Math. J., Vol. 46(1999), pp. 97- 107.[3]. A. P. Singh, On maximum term of composition of entire functions, Proc. Nat. Acad. Sci. India, Vol. 59(A), Part I (1989), pp. 103-115.[4]. A. P. Singh and M. S. Baloria, On maximum modulus and maximum term of composition of entire functions, Indian J. Pure Appl. Math., Vol. 22(1991), No 12, pp. 1019-1026.[5]. G. Valiron, Lectures on the general theory of integral functions, Chelsea Publishing Company (1949). 64