On the Growth of Entire Functions of Several Complex Variables

Transcription

1 Int Journal of Math Analysis Vol no On the Growth of Entire Functions of Several Complex Variables Huzo H Khan and Rifaqat Ali Department of Mathematics Aligarh Muslim University Aligarh India huzokhan@yahoocom rifaqatali1@gmailcom Abstract In this paper we generalize and improve the results of R K Srivastava Vinod Kumar [4] and S S Dalal [1] Here we considered the Tayl series expansion of an entire function in terms of homogeneous polynomials of degree m + n) in two complex variables Mathematics Subject Classification: 30E10 Keywds Homogeneous polynomials der and type lower der and lower type 1 Introduction If ν : C 2 R + =[0 [ be a real-valued function such that the following conditions hold: i) νz + z ) νz)+νz ) z z C 2 ii) νλz) λ νz) λ C iii) νz) =0 z =0 then ν is a nm Let fz 1 z 2 )= mn=0 P mn z 1 z 2 ) the Tayl series expansion of fz 1 z 2 ) in terms of homogeneous polynomials P mn z 1 z 2 ):C 2 C of degree m+n) We have Mr 1 r 2 ) = sup fz 1 z 2 ) t =1 2 r = max r 1 r 2 ) νz t) r

2 2142 H H Khan and R Ali is the maximum modulus of fz 1 z 2 ) r 1 r 2 R + with respect to the nm ν Define C mn = sup P mn z 1 z 2 ) νz t) 1 The der lower der and type of the function are defined respectively by ρ λ = lim r 1 r 2 sup inf log log Mr 1 r 2 ) logr 1 r 2 ) T t = lim r 1 r 2 sup inf log Mr 1 r 2 ) r ρ 1 + r ρ 2) In [2] we have proved the following results ρ = lim sup log[m + n)α m+n ] 11) logc mn ) 1/m+n where eρt = lim sup 12) C mn ) ρ/m+n [m m n n ] 1/m+n if m n 1 m+n) α mn = 0 if m n =0 Analogously the lower der and lower type are defined by λ = linf log[ ] 13) logc mn ) 1/m+n eρt = linf 14) C mn ) ρ/m+n In this paper we have generalized and improved the results of R K Srivastava Vinod Kumar [4] and S S Dalal [1] Surprisingly they have not mention the fact α mn in their results Here I have defined the ders and types different from those of above auth To reduce the mechanical labour we have considered only two variables though the results can easily be extended to several complex variables

3 On the growth of entire functions Main Results Theem 1 f i z 1 z 2 )= mn=0 P i) mn z 1z 2 ) where i =1 k be k entire functions of finite ders ρ 1 ρ 2 ρ k and nonzero lower ders λ 1 λ 2 λ k respectively Then the function f z 1 z 2 )= mn=0 P mn z 1 z 2 ) where C mn Π k C i) mi mn) are constant is an entire function such that { k 1 1 ρ m ) k 1 ρ k λ m ) } k 1 k λ k i 1 λ i 21) Proof In can be easily seen [3 p 9] that necessary and sufficient condition f fz 1 z 2 ) to represent an entire function of two complex variables z 1 and z 2 is lim sup C mn ) 1/m+n =0 since f i z 1 z 2 ) are entire functions so it leads Also which gives lim sup C mn) i) 1/m+n =0 f i =1 k C mn Π k C i) mn) mi lim sup Cmn α mn Hence fz 1 z 2 ) is an entire function ) 1/m+n [ ) C i) mi ] 1/m+n Π k mn lim sup 22) α mn i) Now applying 11)and13) f functions f i z 1 z 2 ) we get lim sup inf log[ i) ] logc mn) = i) 1/m+n λ i logc i) mn ) m + n)log ) + ε) f m+ n>m 0 + n 0 23)

4 2144 H H Khan and R Ali and logc mn) i) m + n)log ) 24) λ i + ε) f an infinite sequence of values of mn Taking i =1 k in 23) and adding we get log Π k Ci) mn ) m + n)log ) using 22) we have k + ε) log C mn m + n)log ) log[m + n)α m+n ] lim sup logc mn ) 1/m+n k ρ ) 1 k ) k 1 mi + ε) k 1 i=0 1 ρ m k ρ k which proves a part of left hand side of 21) In the same way by taking i =1 k 1 in 23) and i = k in 24) and adding we obtain k 1 1 λ m k λ k which proves the second part of left hand side of 21) i=0 Similarly the right hand side can be prove Theem 2 f i z 1 z 2 )= mn=0 P i) mn z 1z 2 ) where i =1 k be k entire functions of finite nonzero ders ρ 1 ρ 2 ρ k and types T 1 T 2 T k and lower type t 1 t 2 t k respectively 0 t i T i )i =1 k) Then the function fz 1 z 2 )= mn=0 P mn z 1 z 2 )

5 On the growth of entire functions 2145 is an entire function such that Π k 1 t i ) / { ρt) 1/ρ ρ k t k ) m k/ρ k ρt ) 1/ρ } ρ k T k ) m k/ρ k Π k 1 T i ) / 25) where ρ T and t are der type and lower type of fz 1 z 2 ) respectively Proof Using 12) and 14) f functions f i z 1 z 2 ) we have C i) mn) [ [e T i + ε)] m+n ] mi / f m+ n>m m + n)α mn )) m+n 0 + n 0 26) and C i) mn) [ [e t i + ε)] m+n ] mi / 27) m + n)α mn )) m+n f an infinite sequence of values of m n Taking i =1 k in 26) and multiplying we have [ k k C mn) i) [e T i + ε)] m+n ] mi / m + n)α mn )) m+n Using 22) we get [ ] m+n eρk T k + ε) C mn m k /ρ k [ ] k 1 m+n m eρi T i + ε) i / C mn ) ρ/m+n {[ eρk T k + ε) ] ρ } mk /ρ k k 1 {[ eρi T i + ε) ] ρ } mi / Using the fact that if any kl s out of L L 1 L k are equal to one then all the k +1) L s are equal to one and We get k 1 1 ρ = L= ρ λ L i = λ i C mn ) ρ/m+n e{[ρ kt k + ε)] ρ } m k/ρ k k 1 {[ T i + ε)] ρ } /

6 2146 H H Khan and R Ali In view of 12) we get ρt ) 1/ρ k 1 ρ ρ k T k ) T i ) / 28) k/ρ k Now taking i =1 k 1 in 26) and i = k in 27) and multiplying we get ρt) 1/ρ k 1 ρ ρ k t k ) T i ) / 29) k/ρ k 28) and 29) together proves that left hand side of 25) Similarly the right hand side can be proved Hence the proof of Theem 2 is complete References [1] S S Dalal On the der and type of integral functions of several complex variables J Indian Math Soc ) [2] D Kumar and K N Ara On the pq) der and pq) type of homogeneous polynomials of two complex variables Math Sci Res J 9 no7 2005) [3] P Lelong and L Gruman Entire Functions of several complex variables A series of comprehensive studies in Mathematics 282 Springer Verlag Berlin 1986 [4] R K Srivastava and V Kumar On the der and type of integral functions of several complex variables Comp Math 17 Fasc 21966) Received: April 2011