Relative order and type of entire functions represented by Banach valued Dirichlet series in two variables

Transcript

1 Int J Nonlinear Anal Appl 7 (2016) No 1, 1-14 ISSN: (electronic) Relative der type of entire functions represented by Banach valued Dirichlet series in two variables Dibyendu Banerjee a,, Nilkanta Mondal b a Department of Mathematics, Visva-Bharati, Santiniketan , India b Ballavpur RGSVidyapith, Ballavpur , Raniganj, India (Communicated by A Ghaffari) Abstract In this paper, we introduce the idea of relative der type of entire functions represented by Banach valued Dirichlet series of two complex variables to generalize some earlier results Keywds: Banach valued Dirichlet series; relative der; relative type; entire function 2010 MSC: Primary 30B50; Secondary 30D15 1 Introduction F entire function f, let F (r) = max{ f(z) : z = r} If f is non-constant then F (r) is strictly increasing continuous function of r its inverse exists F 1 : ( f(0), ) (0, ) lim F 1 (R) = R Let f g be two entire functions Bernal [3] introduced the definition of relative der of f with respect to g, denoted by ρ g (f), as follows: ρ g (f) = inf{µ > 0 : F (r) < G(r µ ) f all r > r 0 (µ) > 0} Cresponding auth addresses: dibyendu192@rediffmailcom Mondal) Received: November 2014 Revised: June 2015 (Dibyendu Banerjee), nilkanta1986@gmailcom (Nilkanta

2 2 Banerjee, Mondal After this several papers on relative der of entire functions have appeared in the literature where growing interest of wkers on this topic has been noticed {see f example [1], [2], [4], [5], [6], [7], [8]} Let f(s) be an entire function of the complex variable s = σ + it defined by everywhere absolutely convergent Dirichlet series a n e sλn, n=1 (11) where 0 < λ n < λ n+1 (n 1), λ n as n a, ns are complex constants Let F (σ) = lub <t< f(σ + it) Then the Ritt der [10] of f(s), denoted by ρ(f) is given by ρ(f) = inf{µ > 0 : log F (σ) < exp(σµ) f all σ > R(µ)} In other wds, ρ(f) = lim sup σ log log F (σ) σ During the past decades, several auths made close investigations on the properties of entire Dirichlet series related to Ritt der In 2010, Lahiri Banerjee [9] introduced the idea of relative Ritt der as follows: Let f(s) be an entire function defined by everywhere absolutely convergent Dirichlet series (11) g(s) be an entire function Then the relative Ritt der of f(s) with respect to entire g(s) denoted by ρ g (f) is defined as ρ g (f) = inf{µ > 0 : log F (σ) < G(σµ)f all large σ}, where G(r) = max{ g(s) : s = r} Recently Srivastava [11] defined the growth parameter such as relative der, relative type, relative lower type of entire functions represented by vect valued Dirichlet series of the fm (11) as follows: Let f(s) g(s) be two entire functions defined by everywhere absolutely convergent vect valued Dirichlet series of the fm (11), where a n s belong to a Banach space (E, ) λ n s are nonnegative real numbers such that as n satisfy the conditions 0 λ 1 < λ 2 < < λ n log n lim sup n λ n log a n lim sup n λ n = D < = Also F (σ), G(σ) denote their respective maximum moduli The relative der of f(s) with respect to g(s) denoted by ρ g (f) is defined as ρ g (f) = inf{µ > 0 : F (σ) < G(σµ) f all σ > σ 0 (µ)}

3 Relative der type of entire functions 7 (2016) No 1, ie, ρ g (f) = lim sup σ G 1 F (σ) σ The relative type relative lower type of f(s) with respect to g(s) denoted respectively by T g (f) τ g (f) when ρ g (f) = 1 (ie, ρ(f) = ρ(g) = ρ) defined as T g (f) = inf{µ > 0 : F (σ) < G[ 1 ρ log(µeρσ )] f all σ > σ 0 (µ)} exp[ρg 1 F (σ)] τ g (f) = lim inf σ exp(ρσ) exp[ρg 1 F (σ)] = lim sup σ exp(ρσ) If T g (f) = τ g (f) then f is said to be of regular type with respect to g Srivastava Sharma [12] introduced the idea of der type of an entire function represented by vect valued Dirichlet series of two complex variables Consider f(s 1, s 2 ) = a mn e (s 1λ m+s 2 µ n), (s j = σ j + it j, j = 1, 2) (12) m,n=1 where a mn s belong to the Banach space (E, ) ; 0 λ 1 < λ 2 < < λ m as m ; 0 µ 1 < µ 2 < < µ n as n log(m + n) lim sup m+n λ m + µ n = D < + Such a series is called a vect valued Dirichlet series in two complex variables If only a finite number of a mn s are non zero in (12), then we call it as a Banach valued Dirichlet polynomial of two complex variables Let f(s 1, s 2 ) defined above represent an entre function F (σ 1, σ 2 ) = sup{ f(σ 1 + it 1, σ 2 + it 2 ) ; < t j < ; j = 1, 2} be its maximum modulus Then the der ρ(f) of f(s 1, s 2 ) is defined as log log F (σ 1, σ 2 ) ρ(f) = lim sup σ 1,σ 2 log(e σ 1 + e σ 2) If (0 < ρ(f) < ), then the type T (f)(0 T (f) ) of f(s 1, s 2 ) is defined as log F (σ 1, σ 2 ) T (f) = lim sup (e ρ(f)σ 1 + e ρ(f)σ 2) σ 1,σ 2 At this stage it therefe seems reasonable to define suitably the relative der of entire functions defined by Banach valued Dirichlet series with respect to an entire function defined by Banach valued Dirichlet series of two complex variables to enquire its basic properties in the new context Proving some preliminary theems on the relative der, we obtain sum product theems we show that the relative der (finite) of an entire function represented by Dirichlet series (12) is the same as its partial derivative, under certain restrictions The following definitions are now introduced

4 4 Banerjee, Mondal Definition 11 Let f(s 1, s 2 ) g(s 1, s 2 ) be two entire functions defined by the Banach valued Dirichlet series (12) Then the relative der of f(s 1, s 2 ) with respect to g(s 1, s 2 ) denoted by ρ g (f) is defined by ρ g (f) = inf{µ > 0 : F (σ 1, σ 2 ) < exp[g(σ 1, σ 2 )] µ } where F (σ 1, σ 2 ) = sup{ f(σ 1 + it 1, σ 2 + it 2 ) ; < t j < ; j = 1, 2} If we put λ n = µ n = n 1 f n = 1, 2, 3, a 12 = a 21 = 1, all other a mn s are zero then g(s 1, s 2 ) = e s 1 + e s 2 consequently ρ g (f) = ρ(f) Definition 12 Let f(s 1, s 2 ) g(s 1, s 2 ) be two entire functions defined by the Banach valued Dirichlet series (12) such that ρ(f) = ρ(g) Then the relative type relative lower type of f(s 1, s 2 ) with respect to g(s 1, s 2 ) are denoted respectively by T g (f) τ g (f) defined as log F (σ 1, σ 2 ) T g (f) = lim sup σ 1,σ 2 G(ρσ 1, ρσ 2 ) log F (σ 1, σ 2 ) τ g (f) = lim inf σ 1,σ 2 G(ρσ 1, ρσ 2 ) where ρ = ρ(f) = ρ(g) Clearly T g (f) = T (f) if g(s 1, s 2 ) = e s 1 + e s 2 Definition 13 Let f 1 (s 1, s 2 ) f 2 (s 1, s 2 ) be two entire functions defined by the Banach valued Dirichlet series (12) Then f 1 (s 1, s 2 ) f 2 (s 1, s 2 ) are said to be asymptotically equivalent if there exists l, (0 < l < ) such that F 1 (σ 1, σ 2 ) F 2 (σ 1, σ 2 ) l as σ 1, σ 2 in this case we write f 1 f 2 If f 1 f 2 then clearly f 2 f 1 Throughout the paper we assume that f(s 1, s 2 ), f 1 (s 1, s 2 ), g(s 1, s 2 ), g 1 (s 1, s 2 ), etc are non-constant entire functions defined by Banach valued Dirichlet series (12) F (σ 1, σ 2 ), F 1 (σ 1, σ 2 ), G(σ 1, σ 2 ), G 1 (σ 1, σ 2 ) denote their respective maximum moduli The following lemma will be needed in the sequel Lemma 14 Let g(s 1, s 2 ) be a non-constant entire function defined by Banach valued Dirichlet series (12) γ > 1, 0 < µ < λ Then [G(σ 1, σ 2 )] γ lim σ 1,σ 2 G(σ 1, σ 2 ) = Proof of the lemma is omitted lim σ 1,σ 2 [G(σ 1, σ 2 )] λ [G(σ 1, σ 2 )] = µ

5 Relative der type of entire functions 7 (2016) No 1, Preliminary Theems log log F (σ Theem 21 (a) ρ g (f) = lim sup 1,σ 2 ) σ1,σ 2 log G(σ 1,σ 2 ) (b) If f(s 1, s 2 ) be a Dirichlet polynomial g(s 1, s 2 ) is not a Dirichlet polynomial, then ρ g (f) = 0 (c) If F 1 (σ 1, σ 2 ) F 2 (σ 1, σ 2 ) f all large σ 1, σ 2 then ρ g (f 1 ) ρ g (f 2 ) (d) If G 1 (σ 1, σ 2 ) G 2 (σ 1, σ 2 ) f all large σ 1, σ 2 then ρ g1 (f) ρ g2 (f) Proof (a) This follows from the definition (b) Let f be of the fm f(s 1, s 2 ) = m { n a kl e s 1λ k +s 2 µ l} Then k=1 m n F (σ 1, σ 2 ) = sup { a kl e (σ 1+it 1 )λ k +(σ 2 +it 2 )µ l } <t 1,t 2 < k=1 l=1 m n { a kl e σ 1λ k +σ 2 µ l } k=1 mn l=1 l=1 max a kl e σ 1λ m+σ 2 µ n = Me σ 1λ m+σ 2 µ n, k=1,2,3,,m;l=1,2,3,,n since we may clearly assume that σ 1, σ 2 are positive, where (21) M = mn max a kl k=1,2,3,,m;l=1,2,3,,n is a constant On the other h, since g(s 1, s 2 ) is not a Dirichlet polynomial, f all large σ 1, σ 2, p f every δ > 0 k a constant large at pleasure, [G(σ 1, σ 2 )] δ > k δ σ δp 1 σ δp 2 > log M + (σ 1 λ m + σ 2 µ n ) log F (σ 1, σ 2 ) using (21) So, f all large σ 1, σ 2 arbitrary δ > 0 log log F (σ 1, σ 2 ) < δ this gives that ρ g (f) = 0 (c) Since F 1 (σ 1, σ 2 ) F 2 (σ 1, σ 2 ) f all large σ 1, σ 2, so log lim sup σ 1,σ 2 log log F 2 (σ 1, σ 2 ) lim sup σ 1,σ 2 ie, (d) Proof is similar as that of (c) ρ g (f 1 ) ρ g (f 2 ) Remark 22 Let f 1 (s 1, s 2 ) = g(s 1, s 2 ) = e s 1+s 2 f 2 (s 1, s 2 ) = e 2(s 1+s 2 ) Then clearly F 1 (σ 1, σ 2 ) < F 2 (σ 1, σ 2 ) But ρ g (f 1 ) = ρ g (f 2 ) = 0 Let f(s 1, s 2 ) = g 1 (s 1, s 2 ) = e s 1+s 2 G 1 (σ 1, σ 2 ) < G 2 (σ 1, σ 2 ) But ρ g1 (f) = ρ g2 (f) = 0 g 2 (s 1, s 2 ) = e 2(s 1+s 2 ) Then clearly

6 6 Banerjee, Mondal Theem 23 (a) If ρ(f 1 ) = ρ(f 2 ) = ρ(g) F 1 (σ 1, σ 2 ) F 2 (σ 1, σ 2 ) f all large values of σ 1, σ 2 then T g (f 1 ) T g (f 2 ) (b) If ρ(f) = ρ(g 1 ) = ρ(g 2 ) G 1 (σ 1, σ 2 ) G 2 (σ 1, σ 2 ) f all large values of σ 1, σ 2 then T g1 (f) T g2 (f) Proof This follows from definition Theem 24 If ρ(f 1 ) = ρ(f 2 ) = ρ(g) then τ g (f 1 ) T g (f 2 ) lim inf σ 1,σ 2 log F 2 (σ 1, σ 2 ) τ g(f 1 ) τ g (f 2 ) lim sup log F 2 (σ 1, σ 2 ) T g(f 1 ) τ g (f 2 ) σ 1,σ 2 Proof Suppose ρ(f 1 ) = ρ(f 2 ) = ρ(g) = ρ Then by definition f any ɛ > 0 f all large values of σ 1, σ 2 > (τ g (f 1 ) ɛ)g(ρσ 1, ρσ 2 ) (22) log F 2 (σ 1, σ 2 ) < (T g (f 2 ) + ɛ)g(ρσ 1, ρσ 2 ) (23) Therefe from Equation (22) (23) we get f all large σ 1, σ 2, log F 2 (σ 1, σ 2 ) > τ g(f 1 ) ɛ T g (f 2 ) + ɛ lim inf log F 2 (σ 1, σ 2 ) τ g(f 1 ) T g (f 2 ) σ 1,σ 2 (24) Again by definition f any ɛ > 0 there exist sequences {σ 1n }, σ 1n {σ 2n }, σ 2n as n such that log F 1 (σ 1n, σ 2n ) < (τ g (f 1 ) + ɛ)g(ρσ 1n, ρσ 2n ) (25) Again f all large values of σ 1, σ 2 log F 2 (σ 1, σ 2 ) > (τ g (f 2 ) ɛ)g(ρσ 1, ρσ 2 ) (26) Hence from (25) (26) we get lim inf log F 2 (σ 1, σ 2 ) τ g(f 1 ) τ g (f 2 ) σ 1,σ 2 log F 1 (σ 1n, σ 2n ) log F 2 (σ 1n, σ 2n ) < τ g(f 1 ) + ɛ τ g (f 2 ) ɛ (27) Also there exist sequences {σ 1m }, σ 1m {σ 2m }, σ 2m as m such that log F 2 (σ 1m, σ 2m ) < (τ g (f 2 ) + ɛ)g(ρσ 1m, ρσ 2m ) (28)

7 Relative der type of entire functions 7 (2016) No 1, Therefe from (22) (28) we get, lim sup log F 2 (σ 1, σ 2 ) τ g(f 1 ) τ g (f 2 ) σ 1,σ 2 log F 1 (σ 1m, σ 2m ) log F 2 (σ 1m, σ 2m ) > τ g(f 1 ) ɛ τ g (f 2 ) + ɛ (29) Again f any ɛ > 0 f all large values of σ 1, σ 2, < (T g (f 1 ) + ɛ)g(ρσ 1, ρσ 2 ) (210) log F 2 (σ 1, σ 2 ) > (τ g (f 2 ) ɛ)g(ρσ 1, ρσ 2 ) (211) Therefe from (210) (211) we get f all large σ 1, σ 2 log F 2 (σ 1, σ 2 ) < T g(f 1 ) + ɛ τ g (f 2 ) ɛ lim sup log F 2 (σ 1, σ 2 ) T g(f 1 ) τ g (f 2 ) (212) σ 1,σ 2 The theem now follows from (24), (27), (29) (212) 3 Sum Product Theems Theem 31 Let f 1 (s 1, s 2 ), f 2 (s 1, s 2 ) g(s 1, s 2 ) be three entire functions defined by the Banach valued Dirichlet series (12) Then sign of equality holds if ρ g (f 1 ) ρ g (f 2 ) ρ g (f 1 ± f 2 ) max{ρ g (f 1 ), ρ g (f 2 )}, Proof We may suppose that ρ g (f 1 ) ρ g (f 2 ) both are finite because in the contrary case the inequality follows immediately We prove the theem f addition only, because the proof f subtraction is analogous Let f = f 1 + f 2, ρ = ρ g (f), ρ i = ρ g (f i ), i = 1, 2 ρ 1 ρ 2 F arbitrary ɛ > 0 f all large σ 1, σ 2 we have from Theem 21 (a), F 1 (σ 1, σ 2 ) < exp[g(σ 1, σ 2 )] ρ 1+ɛ exp[g(σ 1, σ 2 )] ρ 2+ɛ F 2 (σ 1, σ 2 ) < exp[g(σ 1, σ 2 )] ρ 2+ɛ So, f all large σ 1, σ 2 F (σ 1, σ 2 ) F 1 (σ 1, σ 2 ) + F 2 (σ 1, σ 2 ) < 2 exp[g(σ 1, σ 2 )] ρ 2+ɛ < exp[g(σ 1, σ 2 )] ρ 2+2ɛ

8 8 Banerjee, Mondal Therefe, log log F (σ 1, σ 2 ) f all large σ 1, σ 2 Since ɛ > 0 is arbitrary, we obtain (ρ 2 + 2ɛ), ρ ρ 2 (31) This proves the first part of the theem F the second part, let ρ 1 < ρ 2 suppose that ρ 1 < µ 1 < µ < λ < ρ 2 Then f all large σ 1, σ 2 F 1 (σ 1, σ 2 ) < exp[g(σ 1, σ 2 )] µ (32) there exist non decreasing sequences {σ 1n }, σ 1n such that {σ 2n }, σ 2n as n F 2 (σ 1n, σ 2n ) > exp[g(σ 1n, σ 2n )] λ (33) Using Lemma 14 we see that [G(σ 1, σ 2 )] λ > 2[G(σ 1, σ 2 )] µ f all large σ 1, σ 2 (34) So, from (32), (33) (34), F 2 (σ 1n, σ 2n ) > 2F 1 (σ 1n, σ 2n ) f n = 1, 2, 3, Therefe, F (σ 1n, σ 2n ) F 2 (σ 1n, σ 2n ) F 1 (σ 1n, σ 2n ) > 1 2 F 2(σ 1n, σ 2n ) > 1 2 exp[g(σ 1n, σ 2n )] λ > exp[g(σ 1n, σ 2n )] µ 1 by (33) Therefe, ρ ρ 2 (35) So, from (31) (35) we get ρ = ρ 2 this proves the theem Remark 32 F Banach valued Dirichlet series to hold the equality, the condition ρ g (f 1 ) ρ g (f 2 ) is not necessary Because if we take f 1 (s 1, s 2 ) = 2e s 1+s 2, f 2 (s 1, s 2 ) = e s 1+s 2 g(s 1, s 2 ) = e s 1+s 2 then clearly F 1 (σ 1, σ 2 ) = 2e σ 1+σ 2, F 2 (σ 1, σ 2 ) = e σ 1+σ 2 G(σ 1, σ 2 ) = e σ 1+σ 2 Therefe ρ g (f 1 ) = ρ g (f 2 ) = 0

9 Relative der type of entire functions 7 (2016) No 1, On the other h Therefe Thus, f 1 + f 2 = e s 1+s 2 ρ g (f 1 + f 2 ) = 0 ρ g (f 1 + f 2 ) = max{ρ g (f 1 ), ρ g (f 2 )} Also if we take f 1 (s 1, s 2 ) = 2e s 1+s 2, f 2 (s 1, s 2 ) = g(s 1, s 2 ) = e s 1+s 2 Then clearly ρ g (f 1 f 2 ) = max{ρ g (f 1 ), ρ g (f 2 )} Theem 33 Let f 1 (s 1, s 2 ) g(s 1, s 2 ) be two entire functions defined by the Banach valued Dirichlet series (12) f 2 (s 1, s 2 ) be an entire function defined by (12) where the coefficients a mn C Then ρ g (f 1 f 2 ) max{ρ g (f 1 ), ρ g (f 2 )} Proof Let f = f 1 f 2 the notations ρ, ρ 1, ρ 2 have the analogous meanings as in Theem 31 Without loss of generality let ρ 1 ρ 2 both are finite ρ 1 ρ 2 Then f arbitrary ɛ > 0 f all large σ 1, σ 2 Therefe, F (σ 1, σ 2 ) F 1 (σ 1, σ 2 )F 2 (σ 1, σ 2 ) < exp[g(σ 1, σ 2 )] ρ 1+ɛ exp[g(σ 1, σ 2 )] ρ 2+ɛ exp{2[g(σ 1, σ 2 )] ρ 2+ɛ } exp[g(σ 1, σ 2 )] ρ 2+2ɛ log log F (σ 1, σ 2 ) < (ρ 2 + 2ɛ) f all large σ 1, σ 2 Since ɛ > 0 is arbitrary so ρ ρ 2, which proves the theem Remark 34 F Banach valued Dirichlet series the equality may hold F example, suppose f 1 (s 1, s 2 ) = f 2 (s 1, s 2 ) = g(s 1, s 2 ) = e s 1+s 2 Then clearly ρ g (f 1 ) = ρ g (f 2 ) = 0 On the other h f 1 f 2 = e 2(s 1+s 2 ) Therefe, ρ g (f 1 f 2 ) = 0 Thus ρ g (f 1 f 2 ) = max{ρ g (f 1 ), ρ g (f 2 )} Theem 35 Let f 1 (s 1, s 2 ), f 2 (s 1, s 2 ) g(s 1, s 2 ) be three entire functions defined by the Banach valued Dirichlet series (12) such that T g (f 1 ), T g (f 2 ) T g (f 1 ± f 2 ) are defined Then T g (f 1 ± f 2 ) max{t g (f 1 ), T g (f 2 )}, the equality holds if T g (f 1 ) T g (f 2 ) Proof We may suppose that T g (f 1 ) T g (f 2 ) both are finite because in the contrary case the inequality follows immediately We prove the theem f addition only, because the proof f subtraction is analogous Let f = f 1 + f 2, T = T g (f), T i = T g (f i ) ρ = ρ(f i ) = ρ(f) = ρ(g), i = 1, 2 suppose that T 1 T 2 F arbitrary ɛ > 0 f all large σ 1, σ 2 we have by definition, < (T 1 + ɛ)g(ρσ 1, ρσ 2 ) F 1 (σ 1, σ 2 ) < exp[(t 1 + ɛ)g(ρσ 1, ρσ 2 )] exp[(t 2 + ɛ)g(ρσ 1, ρσ 2 )]

10 10 Banerjee, Mondal So, f all large σ 1, σ 2, F 2 (σ 1, σ 2 ) < exp[(t 2 + ɛ)g(ρσ 1, ρσ 2 )] Therefe, Since ɛ > 0 is arbitrary so T T 2 F (σ 1, σ 2 ) F 1 (σ 1, σ 2 ) + F 2 (σ 1, σ 2 ) < 2 exp[(t 2 + ɛ)g(ρσ 1, ρσ 2 )] log F (σ 1, σ 2 ) < log 2 + (T 2 + ɛ)g(ρσ 1, ρσ 2 ) log F (σ 1, σ 2 ) G(ρσ 1, ρσ 2 ) < (T 2 + ɛ) + o(1) This proves the first part of the theem F the second part, let T 1 < T 2 suppose that T 1 < µ < λ < T 2 Then f all large σ 1, σ 2 (36) F 1 (σ 1, σ 2 ) < exp[µg(ρσ 1, ρσ 2 )] (37) there exist non decreasing sequences {σ 1n }, σ 1n, {σ 2n }, σ 2n as n such that F 2 (σ 1n, σ 2n ) > exp[λg(ρσ 1n, ρσ 2n )] (38) Now, by using (37) (38) F (σ 1n, σ 2n ) F 2 (σ 1n, σ 2n ) F 1 (σ 1n, σ 2n ) > exp[λg(ρσ 1n, ρσ 2n )] exp[µg(ρσ 1n, ρσ 2n )] > 2 exp[µg(ρσ 1n, ρσ 2n )] exp[µg(ρσ 1n, ρσ 2n )] > exp[µg(ρσ 1n, ρσ 2n )] Therefe, T T 2 log F (σ 1n, σ 2n ) > µg(ρσ 1n, ρσ 2n ) log F (σ 1n, σ 2n ) G(ρσ 1n, ρσ 2n ) > µ From (36) (39) we get T = T 2 this proves the theem (39) Theem 36 Let f 1 (s 1, s 2 ), g(s 1, s 2 ) be two entire functions defined by the Banach valued Dirichlet series (12) f 2 (s 1, s 2 ) be an entire function defined by (12) where the coefficients a mn C such that T g (f 1 ), T g (f 2 ) T g (f 1 f 2 ) are defined Then T g (f 1 f 2 ) T g (f 1 ) + T g (f 2 ) Proof Let f = f 1 f 2 the notations ρ, T, T 1 T 2 have the analogous meanings as in Theem 35 Suppose T 1 T 2 both are finite because in the contrary case the theem is obvious Then f arbitrary ɛ > 0 f all large σ 1, σ 2 F (σ 1, σ 2 ) F 1 (σ 1, σ 2 )F 2 (σ 1, σ 2 ) < exp[(t 1 + ɛ)g(ρσ 1, ρσ 2 )] exp[(t 2 + ɛ)g(ρσ 1, ρσ 2 )] = exp[(t 1 + T 2 + 2ɛ)G(ρσ 1, ρσ 2 )] Since ɛ > 0 is arbitrary we obtain T T 1 + T 2 this proves the theem

11 Relative der type of entire functions 7 (2016) No 1, Relative der type of the partial derivatives Theem 41 Let f(s 1, s 2 ) g(s 1, s 2 ) be two entire functions defined by the Banach valued Dirichlet series (12) Then ρ g (f) = ρ g ( f ) Proof We write, F s1 (σ 1, σ 2 ) = sup{ f(σ 1 + it 1, σ 2 + it 2 ) ; < t j < ; j = 1, 2} (σ 1 + it 1 ) From [[11], p68], we may write f fixed σ 2 f all large values of σ 1 F (σ 1, σ 2 ) < σ 1 Fs1 (σ 1, σ 2 ) + O(1) log F (σ 1, σ 2 ) < log F s1 (σ 1, σ 2 ) + log σ 1 + O(1) (41) Therefe, log log F (σ 1, σ 2 ) f fixed σ 2 f all large σ 1, so, ( ) f ρ g (f) ρ g < log log F s1 (σ 1, σ 2 ) + o(1) (42) To obtain the reverse inequality we have from [[11], p68] f fixed σ 2 f large σ 1 F s1 (σ 1, σ 2 ) ɛ 1 δ F (σ 1 + δ, σ 2 ), where ɛ > 0 is arbitrary δ > 0 is fixed So, log F s1 (σ 1, σ 2 ) log F (σ 1 + δ, σ 2 ) + O(1) (43) log log F s1 (σ 1, σ 2 ) log log F (σ 1 + δ, σ 2 ) + o(1) Since σ 2 is any fixed real number, σ 1 is large δ is any fixed number so, ρ g ( f ) ρ g (f) From (42) (44) we get ( ) f ρ g (f) = ρ g (44) Remark 42 In Theem 41 putting g(s 1, s 2 ) = e s 1 + e s 2, we get ρ(f) = ρ( f ) Theem 43 Let f(s 1, s 2 ) g(s 1, s 2 ) be two entire functions defined by the Banach valued Dirichlet series (12) such that ρ(f) = ρ(g) Then T g (f) = T g ( f )

12 12 Banerjee, Mondal Proof From Remark 42, we have, ρ(f) = ρ( f ) Therefe, ρ( f ) = ρ(g) so T g ( f ) exists Suppose ρ(f) = ρ(g) = ρ( f ) = ρ As in Theem 41 we write { } f(σ 1 + it 1, σ 2 + it 2 ) F s1 (σ 1, σ 2 ) = sup (σ 1 + it 1 ) ; < t j < ; j = 1, 2 Then from (41) we get log F (σ 1, σ 2 ) < log F s1 (σ 1, σ 2 ) + log σ 1 + O(1) So taking σ 1, σ 2 we get T g (f) T g ( f ) log F (σ 1, σ 2 ) G(ρσ 1, ρσ 2 ) < log F s1 (σ 1, σ 2 ) + o(1) G(ρσ 1, ρσ 2 ) Again f a fixed σ 2 large σ 1 we get from (43) f a fixed δ > 0 log F s1 (σ 1, σ 2 ) G(ρσ 1, ρσ 2 ) log F (σ 1 + δ, σ 2 ) G(ρσ 1, ρσ 2 ) + o(1) Since σ 2 is any fixed real number, σ 1 is large δ is any fixed number so, T g ( f ) T g (f) From (45) (46) we get T g (f) = T g ( f ) (45) (46) 5 Asymptotic behavi Theem 51 Let f(s 1, s 2 ), g 1 (s 1, s 2 ) g 2 (s 1, s 2 ) be three entire functions defined by the Banach valued Dirichlet series (12) suppose g 1 g 2 Then ρ g1 (f) = ρ g2 (f) Proof Let ɛ > 0 Then by definition, f all large σ 1, σ 2, there exists l (0 < l < ) such that G 1 (σ 1, σ 2 ) < (l + ɛ)g 2 (σ 1, σ 2 ) (51) Now f all large σ 1, σ 2 log log F (σ 1, σ 2 ) < (ρ g1(f) + ɛ) log G 1 (σ 1, σ 2 ) using (51), F (σ 1, σ 2 ) < exp[g 1 (σ 1, σ 2 )] ρg 1 (f)+ɛ < exp[(l + ɛ)g 2 (σ 1, σ 2 )] ρg 1 (f)+ɛ < exp[g 2 (σ 1, σ 2 )] ρg 1 (f)+2ɛ Therefe, log log F (σ 1, σ 2 ) log G 2 (σ 1, σ 2 ) < ρ g1 (f) + 2ɛ Since ɛ > 0 is arbitrary small, so ρ g2 (f) ρ g1 (f) The reverse inequality is clear because g 2 g 1 so ρ g1 (f) = ρ g2 (f)

13 Relative der type of entire functions 7 (2016) No 1, Remark 52 F Banach valued Dirichlet series the condition g 1 g 2 is not necessary F example, let f(s 1, s 2 ) = g 1 (s 1, s 2 ) = e s 1+s 2 g 2 (s 1, s 2 ) = e 2(s 1+s 2 ) Then clearly F (σ 1, σ 2 ) = G 1 (σ 1, σ 2 ) = e σ 1+σ 2 G 2 (σ 1, σ 2 ) = e 2(σ 1+σ 2 ) Therefe, g 1 g 2 does not hold but ρ g1 (f) = ρ g2 (f) = 0 Theem 53 Let f 1 (s 1, s 2 ), f 2 (s 1, s 2 ) g(s 1, s 2 ) be three entire functions defined by the Banach valued Dirichlet series (12) suppose f 1 f 2 Then ρ g (f 1 ) = ρ g (f 2 ) Proof Let ɛ > 0 Then by definition, f all large σ 1, σ 2, there exists l, (0 < l < ) such that F 1 (σ 1, σ 2 ) < (l + ɛ)f 2 (σ 1, σ 2 ) Now f all large σ 1, σ 2 Therefe, ie < log F 2 (σ 1, σ 2 ) + log(l + ɛ) log < log log F 2(σ 1, σ 2 ) ρ g (f 1 ) ρ g (f 2 ) + o(1) The reverse inequality is clear because f 2 f 1 so ρ g (f 1 ) = ρ g (f 2 ) Remark 54 F Banach valued Dirichlet series the condition f 1 f 2 is not necessary which follows from the following example Let f 1 (s 1, s 2 ) = g(s 1, s 2 ) = e s 1+s 2 f 2 (s 1, s 2 ) = e 2(s 1+s 2 ) Then clearly f 1 f 2 does not hold but ρ g (f 1 ) = ρ g (f 2 ) = 0 Theem 55 Let f 1 (s 1, s 2 ), f 2 (s 1, s 2 ) g(s 1, s 2 ) be three entire functions defined by the Banach valued Dirichlet series (12) such that T g (f 1 ) T g (f 2 ) are defined suppose f 1 f 2 Then T g (f 1 ) = T g (f 2 ) Proof Let ɛ > 0 ρ(f 1 ) = ρ(f 2 ) = ρ(g) = ρ Then by definition, f all large σ 1, σ 2, there exists l (0 < l < ) such that F 1 (σ 1, σ 2 ) < (l + ɛ)f 2 (σ 1, σ 2 ) < log F 2 (σ 1, σ 2 ) + O(1) G(ρσ 1, ρσ 2 ) < log F 2(σ 1, σ 2 ) G(ρσ 1, ρσ 2 ) + o(1) Since ɛ > 0 is arbitrary small, so T g (f 1 ) T g (f 2 ) The reverse inequality is clear because f 2 f 1 so T g (f 1 ) = T g (f 2 )

14 14 Banerjee, Mondal References [1] D Banerjee RK Dutta, Relative der of entire functions of two complex variables, International J Math Sci Engg Appls 1 (2007) [2] D Banerjee S Jana, Meromphic functions of relative der (p, q), Soochow J Math 33 (2007) [3] L Bernal, Orden relative de crecimiento de functones enteras, Collect Math 39 (1988) [4] BK Lahiri D Banerjee, Relative der of entire meromphic functions, Proc Nat Acad Sci, India 69(A)(III)(1999) [5] BK Lahiri D Banerjee, Generalised relative der of entire functions,proc Nat Acad Sci, India 72(A)(IV)(2002) [6] BK Lahiri D Banerjee, Entire functions of relative der (p, q), Soochow J Math 31 (2005) [7] BK Lahiri D Banerjee, Relative der of meromphic functions, Proc Nat Acad Sci, India 75(A)(II) (2005) [8] BK Lahiri D Banerjee, A note on relative der of entire functions, Bull Cal Math Soc 97 (2005) [9] BK Lahiri D Banerjee, Relative Ritt der of entire Dirichlet series, Int J contemp Math Sci 5 (2010) [10] QI Rahaman, The Ritt der of the derivative of an entire function, Annales Polonici Math 17 (1965) [11] GS Srivastava, A note on relative type of entire functions represented by vect valued Dirichlet series, J Classical Anal 2 (2013) [12] GS Srivastava A Sharma, Some growth properties of entire functions represented by vect valued Dirichlet series in two variables, Gen Math Notes 2 (2011)