Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 SCITECH Volume 3, Issue 2 RESEARCH ORGANISATION ublished online: March 29, 208 Journal of rogressive Research in Mathematics www.scitecresearch.com/journals On The Quotient Function Of Entire Function Represented By Dirichlet Series Of Two Complex Variables Aseel Hameed Abed Sadaa Department of Mathematics, College of Basic Education, Mustansiriha University, Baghdad, Iraq E-mail of the corresponding author: Aseel.hh@uomustansiriyah.edu.iq Abstract. The object of this paper is study a few result involving the maximum term and the rank of the maximum term of entire function represented by Dirichlet series. - Introduction Let E be the set of all entire functions f(s, s 2 ) represented by a double entire Dirichlet series,where f(s, s 2 ) can be written as[4] :- f s, s 2 = a i,j i,j=0 e λ i s +µ j s 2 (.) where a i,j C, s, s 2 C 2, and 0 λ 0 < λ < < λ i,0 µ 0 < µ < < µ j, and lim i log i λ i Let[], = 0 = lim i log j µ j (.2) M(σ, σ 2, f) = sup <tj < f σ + it, σ 2 + it 2, (σ, σ 2 ) R 2 (.3) be the maximum modulus of f s, s 2 on the tube Re s = σ, Re s 2 = σ 2. and let[], U(σ, σ 2, f ) = max i,j 0 a i,j e λ iσ +µ j σ 2, (σ, σ 2 ) R 2 (.4) also if f(s, s 2 )be entire Dirichlet series defined by (.), then the order and lower order λof f(s, s 2 )can be defined as[4]: Volume 3, Issue 2 available at www.scitecresearch.com/journals/index.php/jprm 2263
Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 = lim σ,σ 2 sup loglogm (σ,σ 2,f) log (e σ +e σ 2) λ = lim inf loglogm σ, σ 2, f σ,σ 2 log e σ + e σ 2 (.5) (.6) In this paper the definitions of the rank and quotient function are extended to several complex variables. In the last section of this paper some remarks and theorems are given, where the characterizations of the order and lower order in terms of the rank and quotient function are studied. 2- Some important definitions In this section two definitions, which are generalized of the definition of the rank and quotient function[2], are given. Definition (2.) :- Let f(s, s 2 )be entire Dirichlet series defined by (.), and let U(σ, σ 2 ) be the maximum term of f(s, s 2 ), the rank of the maximum term can be defined as:- V σ, σ 2, f = max i,j N λ i, µ j : U σ, σ 2 = a i,j e λ iσ +µ j σ 2 (2.) Where Ndenote the set of all nature numbers. Definition (2.2):- Let f(s, s 2 ) be entire Dirichlet series defined by (.), and let Z +, ( Z + be the set of all positive integers),then for every entire functionf(s, s 2 ), there exist the quotient function of the th order A, where A can be defined as:- A σ, σ 2, f = U σ,σ 2,f ( ) U(σ,σ 2,f), (σ, σ 2 ) R 2 (2.2) 3-Theorems and Remarks In this section the following theorems and remarks are given. Theorem(3.): For every entire function f(s, s 2 ) E that, lim sup ( A σ,σ 2,f ) lim inf ( A σ,σ 2,f ) σ,σ 2 λ V(σ,σ 2,f () ) 2 µ + V(σ,σ 2,f () ) 2 σ,σ 2 λ V(σ,σ 2,f) 2 + µ (3.) V(σ,σ 2,f) 2 where V(σ, σ 2, f) 2 = λ i 2 + µ j 2 roof Now from [6] that, for any Z +, λ V(σ,σ 2,,f) 2 + µ V(σ,σ 2,,f) 2 U σ,σ 2,f () U(σ,σ 2,f) λ V(σ,σ 2,f () ) 2 + µ V(σ,σ 2,f () ) 2 (3.2) Volume 3, Issue 2 available at www.scitecresearch.com/journals/index.php/jprm 2264
Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 dividing both side of the first inequality in (3.2) by λ V(σ,σ 2,,f) 2 + µ V(σ,σ 2,f) 2, and proceeding to limits, to get lim inf ( A σ,σ 2,f ), σ,σ 2 λ V(σ,σ 2,f) 2 + µ (3.3) V(σ,σ 2,f) 2 and dividing both side of the second inequality in (3.2) by λ V(σ,σ 2,f () ) 2 + µ V(σ,σ 2,f () ) 2 to get, lim sup ( A σ,σ 2,f ) σ,σ 2 λ V (σ,σ2,f () ) 2 µ (3.4) + V (σ,σ2,f () ) 2 Combining (3.3) and (3.4) to get (3.). Remark:- If f(s, s 2 ) of order R + {0},(R + is the set of extended positive real numbers), and lower order λ R + {0}, it follows from (3.2) and the following result [3] and, that, = λ = lim sup loglogm(σ, σ 2, f) σ,σ 2 log (e σ = lim + e σ sup log (λ V σ,σ 2,f 2 + µ V σ,σ 2,f 2) 2) σ,σ 2 log (e σ + e σ 2) lim inf loglogm(σ, σ 2, f) σ,σ 2 log (e σ = lim + e σ inf log (λ V σ,σ 2,f 2 + µ V σ,σ 2,f 2) 2) σ,σ 2 log (e σ + e σ 2) log U σ,σ2,f () U (σ,σ2,f) = lim σ,σ 2 sup log (e σ +e σ 2) (3.5) and log U σ,σ2,f () U (σ,σ2,f) λ = lim σ,σ 2 inf log (e σ +e σ 2 ) (3.6) Remark 2: If f(s, s 2 ) of order R + {0},(R + is the set of extended positive real numbers), and lower order λ R + {0}, it follows from the result of [5], that Volume 3, Issue 2 available at www.scitecresearch.com/journals/index.php/jprm 2265
lim inf logu(σ, σ 2, f) σ,σ 2 λ V σ,σ 2,f 2 + µ V σ,σ 2,f 2 Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 λ lim sup logu σ, σ 2, f σ,σ 2 λ V σ,σ 2,f 2 + µ (3.7) V σ,σ 2,f 2 Remark 3: For every entire function f s, s 2 E,it follows from the results of( [2],[6]) that U σ, σ 2, f U σ, σ 2, f U σ, σ 2, f (3.8) Next the following theorem is improve of Remark2. Theorem(3.2): For every entire function f(s, s 2 ) E of order R + {0},(R + is the set of extended positive real numbers), and lower order λ R + {0} and for any N, that log U σ,σ 2,f () lim σ,σ 2 inf λ V σ,σ2,f 2 + µ V σ,σ2,f 2 log U σ,σ 2,f lim λ σ,σ 2 sup λ V σ,σ2,f 2 + µ (3.9) V σ,σ2,f 2 roof Now from (3.2), that log {λ V σ,σ 2,f 2 + µ V σ,σ 2,,f 2 } {logu σ, σ 2, f logu σ, σ 2, f } from the first inequality in (3.0),it follows log λ V(σ,σ 2,f () ) 2 + µ V(σ,σ 2,f () ) 2 (3.0) log λ V σ lim,σ 2,f 2 + µ V σ,σ 2,,f 2 σ,σ 2 λ V σ,σ 2,f 2 + µ V σ,σ 2,f 2 lim σ,σ 2 logu σ, σ 2, f λ V σ,σ 2,f 2 + µ V σ,σ 2,f 2 lim σ,σ 2 logu σ, σ 2, f λ V σ,σ 2,f 2 + µ V σ,σ 2,f 2 lim log U σ,σ 2,f σ,σ 2 λ V σ,σ2,f 2 + µ V σ,σ2,f 2 λ (3.) in view of (3.7). Since λ V σ,σ 2,f 2 + µ V σ,σ 2,f 2 tends to infinity with (σ, σ 2 ),it follows from (3.), that Volume 3, Issue 2 available at www.scitecresearch.com/journals/index.php/jprm 2266
Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 lim sup σ,σ 2 log U σ,σ 2,f λ V σ,σ2,f 2 + µ V σ,σ2,f 2 λ (3.2) Also, from the second inequality in (3.0), to get lim sup logu σ, σ 2, f σ,σ 2 λ V σ,σ 2,f 2 + µ V σ,σ 2,f 2 lim inf logu σ, σ 2, f σ,σ 2 λ V σ,σ 2,f 2 + µ lim inf logu σ, σ 2, f V σ,σ 2,f 2 σ,σ 2 λ V σ,σ 2,f 2 + µ V σ,σ 2,f 2 log U σ,σ 2,f lim inf σ,σ 2 λ V σ,σ2,f 2 + µ (3.3) V σ,σ2,f 2 in view of (3.7). Since, from remark, it follows, from (3.3), that Log {λ lim sup V (σ,σ2,f +µ V σ,σ2,f } = σ,σ 2 log (e σ +e σ 2) lim inf σ,σ 2 log U σ,σ 2,f λ V σ,σ2,f 2 + µ V σ,σ2,f 2 (3.4) Combining (3.2) and (2.4), to get (3.9) Now the corollaries are immediate from (3.9):- Corollary :- Iff is of infinite order, then log U σ,σ 2,f lim inf = 0. σ,σ 2 λ V σ,σ2,f 2 + µ (3.5) V σ,σ2,f 2 Corollary 2 :- If f is of lower order zero, then References log U σ,σ 2,f lim sup = +. σ,σ 2 λ V σ,σ2,f 2 + µ (3.6) V σ,σ2,f 2 [] S.Daoud,"Entire Functions Represented By Dirichlet Series Of Two Complex Variables", ortugaliae Mathematica 43(4),985,pp:407-45. [2] J.S.Gupta, Bala and Shakti,"Growth numbers of entire functions defined by Dirichlet Series",Bull.Soc.Math.Grece, 4(973),pp:9-24. [3] Q.I.Rahman,"On the maximum modulus and the coefficients of an entire Dirichlet series", Tohku Math.J,8(956),pp:08-3. Volume 3, Issue 2 available at www.scitecresearch.com/journals/index.php/jprm 2267
Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 [4] F.Q.Salimov,"On (R) Growth Of Entire Functions Represented By Dirichlet series", Caspian Journal of Applied Mathematics, Ecology and Economics,V.2,No.2,(204), pp:3-4. [5] R..Srivastava,"On the entire functions and their derivatives represented by Dirichletseries",Ganita,(958),pp:83-93. [6] R.S.L.Srivastavaand J.S.Gupta,"On the maximum term of integral function defined by Dirichlet series",math.ann.,74(967),pp:240-246. Volume 3, Issue 2 available at www.scitecresearch.com/journals/index.php/jprm 2268