On Relative Order and Relative Type of an Entire Function Represented by Dirichlet Series

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Available free online at http://www.geman.in

On Relative Order and Relative Type of an Entire Function Represented by Dirichlet

Series

Bibhas Chandra Mondal

Department of Mathematics, Surendranath College, 24/2, M.G.Road, Kolkata-700009, India.

E-mail: [email protected] (Received: 20-4-11/Accepted: 25-6-11)

Abstract

In this paper we have introduced relative type Tg

( )

f of an entire function f represented by Dirichlet Series relative to another entire function g represented by Dirichlet series. We obtained formulas for Tg

( )

f in terms of maximum modulus function as well as in terms of coefficients and exponents of the series.

We have also found out a property of the relative type of sum and difference of two functions. We obtained a relation between Tg

( ) ( )

f ^,T f and^{T g}

( )

^where

( )

T f and ^{T g}

( )

are classical type of f and g respectively. We have established a relation connecting Tg

( ) ( ) ( ) ( )

f ^,

ρ

g f ^,

ρ

f ^,

ρ

g where

ρ

g

( )

f is the relative order of f relative to g and

^ρ ( ) ( )

^f ^,

^ρ

^g are classical orders of f and g respectively.

Keywords: Dirichlet entire function, relative order, relative type.

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1 Introduction

Let

( )

0

,

ns n n

f s ^∞ a e^λ s σ it

=

∑

= + ^… ^… ^… ^(1.1);

0 1

0≤

λ λ

< < <...

λ

_n → ∞as n→ ∞ be an entire function represented by Dirichlet series ( called Dirichlet entire function ) where we assume that log

lim sup 0

n n

n λ

→∞ = .

Then [2] log

lim sup ⁿ

n n

a

λ

→∞ = −∞.

The order of f , denoted by

^ρ ( )

^f , is defined as

( )

^loglog

(

^,

)

lim sup M f

f σ

ρ σ

σ

= →∞ … … … (1.2)

and the lower order of f , denoted by

^λ ( )

^f , is defined as

( )

^loglog

(

^,

)

lim inf M f

f σ

λ σ

σ

= →∞ … … … (1.3)

where

(

^,

)

^sup

( )

t

M

σ

f f s

−∞< <∞

= ^.

f is said to be of regular growth if

^ρ ( ) ( )

^f ⁼

^λ

^f ^.

When ⁰^<

^ρ ( )

^f ^{< ∞}, the type of f denoted by ^{T f}

( )

, is defined as

( )

_{lim sup}^log^M

(

^, ^f

)

T f σ e^ρσ

σ

= →∞ … … … (1.4)

and lower type

^τ ( )

^f is defined as

( )

^log

(

^,

)

lim inf M f

f ^σ e^ρσ

τ σ

= →∞ … … … (1.5)

f is said to be of perfectly regular growth if

^ρ ( ) ( )

^f ⁼

^λ

^f ^and^{T f}

( ) ( )

⁼

^τ

^f ^.

From the definition of order it is evident that any Dirichlet polynomial [3] has same order '0' although a Dirichlet polynomial with higher degree grows faster than a Dirichlet polynomial with lower degree. To overcome the situation Mondal [3] introduced relative order of a Dirichlet entire function with respect to another

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On Relative Order and Relative Type of an… 25 Dirichlet entire function. We now introduce relative type of a Dirichlet entire function with respect to another Dirichlet entire function to measure growth-rate of the function more precisely.

2 Definitions and Notations

Let ,f g be two entire functions represented by Dirichlet series in the form (1.1).

Then M

( σ

^,f

)

=Mf

( ) ( σ

^,M

σ

^,g

)

=Mg

( ) σ

are strictly increasing continuous functions of

σ

and increase to ∞. The inverse function Mg⁻¹^:

(

L^,∞ → −∞ ∞

) (

^,

)

^is

strictly increasing where L ^lim Mg

( )

σ

= →−∞ and ^limMg¹

( )

σ ⁻

σ

→∞ = ∞.

Definition 2.1. [3]. Let f g be two Dirichlet entire functions in the form (1.1). , Then the relative order of f relative to g , denoted by

ρ

g

( )

f , is defined as

( )

^inf

{

^{0 :}

( ) ( )

⁰

( ) }

g f k Mf Mg k for all k

ρ = > σ < σ σ σ> .

Evidently,

( )

_{lim sup} ^g¹

(

^f

( ) )

g

M M

f σ

ρ σ

σ

−

= →∞ … … … (2.1)

The lower relative order of f with respect to g , denoted by

λ

g

( )

f , is defined as

( )

¹

( ( ) )

lim inf ^g ^f

g

M M

f σ

λ σ

σ

−

= →∞ … … … (2.2)

f is said to be of regular relative growth with respect to g if

ρ

g

( )

f =

λ

g

( )

f . We define the relative type as:

Definition 2.2. Let f g be twoDirichlet entire functions with , ⁰<

ρ

g

( )

f < ∞^. Then the relative type of f with respect to g , denoted by Tg

( )

f , is defined as

( )

^inf

{

^{0 :}

( ) ( ( )

^.

)

^k ^arg

}

g f g g

T f = k> M

σ

<^M

ρ

f

σ

^ for all l e

σ

… … (2.3)

Theorem 2.3. Let f g be twoDirichlet entire functions with relative order ,

( )

g f

ρ

⁽⁰<

ρ

g

( )

f < ∞) of f with respect to g . Then the relative type of f is

given by

( ) ( )

( ( ) )

lim sup log

log .

f g

g g

T f M

M f

σ

ρ σ

= →∞ .

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Proof: From the Definition 2.2. we have for any ε >0, there exists

σ

₀ such that

( ) ( ( )

^.

)

⁽^T^g^{( )}^f ⁾ ⁰

f g g

M σ <^M ρ f σ ^ ⁺^ε for allσ σ> .

This implies, ^log^M^f

( ) ^σ

^<

(

^T^g

( )

^f ⁺

^ε )

^log^^M^g

( ^ρ

^g

^{( )}

^f ^.

^σ )

^ ^{for all}

^{σ σ}

^> ⁰

That is,

( )

( ( ) ) ^{( )}

⁰

log

log .

f

g

g g

M T f for all

M f

σ ε σ σ

ρ σ

^< ⁺ ^>

 

 

… … (2.4 )

Also from the Definition 2.2., corresponding to

ε

>0, there exists a sequence

{ } σ

n n of values of

σ

where

σ σ

₁< ₂< <...

σ

_n <... tending to infinity such that ^M^f

( )

^σⁿ ^>^^M^g

(

^ρ^g

( )

^f ^.^σⁿ

)

^⁽^T^g^{( )}^f ⁻^ε⁾

That is,

( )

( ( ) ) ^{( )}

log

log .

f n

g

g g n

M T f

M f

σ ε

ρ σ

^> ⁻

 

 

… … … (2.5)

for a sequence of values of

σ

, tending to infinity.

Hence from (2.4) and (2.5) we have,

( ) ( )

( ( ) )

lim sup log

log .

f g

g g

T f M

M f

σ

ρ σ

= →∞ .

Theorem 2.4. Let f f be two Dirichlet entire functions with non-zero finite ₁, ₂ relative orders

ρ

g

( )

f1 and

ρ

g

( )

f2 and relative types Tg

( )

f1 and Tg

( )

f2 with respect to g . If

ρ

g

( )

f1 ≠

ρ

g

( )

f2 then the relative type of f₁± f₂ is equal to the type of the function with maximum relative order. That is, Tg

(

f1± f2

)

=Tg

( )

f1 if

( )

1

( )

2

g f g f

ρ

>

ρ

^.

Proof: Let

ρ

g

( )

f1 >

ρ

g

( )

f2 ^.

From the Definition 2.2. of relative type, for

ε

>0^,

( ) ( ( ) )

⁽ ^{( )}¹ ⁾

1 1 . ^T^g ^f

f g g

M σ <^M ρ f σ ^ ⁺^ε

and M_f2

( )

σ <^M_g

(

ρ_g

( )

f2 .σ

)

^⁽^T^g^{( )}^f²⁺^ε⁾ for sufficiently large

σ

.

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On Relative Order and Relative Type of an… 27 Then, Mf1_±f2

( ) σ

≤Mf1

( ) σ

+Mf2

( ) σ

( ( )

¹ ^.

)

⁽^T^g^{( )}^f¹ ⁾

( ^{( )}

² ^.

)

⁽^T^g^{( )}^f² ⁾ ^arg

g g g g

M ρ f σ ⁺^ε M ρ f σ ⁺^ε for l e σ

   

<  + 

Therefore,

( ) ( ( ) )

⁽ ^{( )} ⁾

( ^{( )} )

⁽ ^{( )} ⁾

( ( ) )

⁽ ^{( )} ⁾

2 1

1 2 1

2 1

1

.

. 1

.

g g

g

T f

T f g g

f f g g T f

g g

M f

M M f

M f

ε ε

ε

ρ σ

σ ρ σ

ρ σ

+ +

± +

   

   

 

<   + 

 

   

 

( ( )

¹ ^.

)

⁽^T^g^{( )}^f¹ ⁾

⁽

¹

⁾

^, ^{0 ,} ^arg

g g

M ρ f σ ⁺^ε K K for sufficiently l eσ

 

<  + >

[ Since

ρ

g

( )

f1 >

ρ

g

( )

f2 ^andM is a strictly increasing function, g

( ( ) )

⁽ ^{( )} ⁾

( ( ) )

⁽ ^{( )} ⁾

2

1

2

1

. .

g

T f

g g

T f

g g

M f

K

M f

ε

ρ σ

+

 

  <

 

 

for large

σ

^{. ]}

This implies,

( ) ( ) ( ( ) )

1 2 1 1

log 1 log .

1M^f ^f T^g f M^g ^g f

K _± σ ε ρ σ

 < +   

 +     

 

That is,

( ( ) )

(

¹

( )

² ¹

) ^{( )}

¹

log

log . arg

f f

g

g g

M

T f for sufficiently l e

M f

σ ε σ

ρ σ

± < +

Now,

( ) ( ( ) )

( )

(

¹ ²

)

1 2

log lim sup

log .

f f g

g g

M

T f f

M f f

σ

ρ σ

±

± = →∞

±

( ( ) )

(

¹

( )

² ¹

) ^{( )}

¹

log lim sup

log .

f f

g

g g

M

T f

M f

σ

σ ε

ρ σ

±

= →∞ ≤ +

[ Since

ρ

g

(

f1± f2

)

=

ρ

g

( )

f1 by Theorem (2.5) in Mondal, [3] ]

Since

ε

>0 is arbitrary, Tg

(

f1± f2

)

≤Tg

( )

f1 … … … (2.6) Again, from Definition 2.2., for

ε

>0 there exists a sequence

{ } σ

n n of values of

σ

, tending to infinity, such that,

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( ) ( ( ) )

⁽ ^{( )}¹ ⁾

1 1 . ^g

T f

f n g g n

M σ >^M ρ f σ ^ ⁻^ε .

Therefore, Mf1_±f2

( ) σ

n ≥Mf1

( ) σ

n −Mf2

( ) σ

n

( ( )

¹ ^.

)

⁽^T^g^{( )}^f¹ ⁾

( ^{( )}

² ^.

)

⁽^T^g^{( )}^f² ⁾ ^arg

g g n g g n

M ρ f σ ⁻^ε M ρ f σ ⁺^ε for sufficiently l e n

   

>  − 

Therefore,

( ) ( ( ) )

⁽ ^{( )} ⁾

( ^{( )} )

⁽ ^{( )} ⁾

( ( ) )

⁽ ^{( )} ⁾

2 1

1 2 1

2 1

1

.

. 1 arg

.

g g

g

T f

T f g g n

f f n g g n T f

g g n

M f

M M f for sufficiently l e n

M f

ε ε

ε

ρ σ

σ ρ σ

ρ σ

+

−

± −

   

   

 

>   − 

 

   

 

Since^ρ^g

( )

^f¹ ^>^ρ^g

( )

^f² ^,^M^g

(

^ρ^g

( )

^f¹ ^.^σⁿ

)

^>^M^g

(

^ρ^g

^{( )}

^f² ^.^σⁿ

)

and we can

make

( ( ) )

⁽ ^{( )} ⁾

( ( ) )

⁽ ^{( )} ⁾

2

1

2

1

. .

g

T f

g g n

T f

g g n

M f

ε

ρ σ

+

−

  

  

 

 

   

 

arbitrarily small 1 2

 

< 

  for sufficiently large n.

So, 1 2

( ) ( ( )

1

)

⁽ ^{( )}¹ ⁾

. 1 1

2

Tg f

f f n g g n

M _±

σ

>^M

ρ

f

σ

^ ⁻^ε ^ − ^ That is, 2 1 2

( ) ( ( )

1 .

)

⁽ ^g^{( )}¹ ⁾

T f

f f n g g n

M _± σ >^M ρ f σ ^ ⁻^ε Hence,

( ( ) )

(

¹

( )

² ¹

) ^{( )}

¹

log 2 lim sup

log .

n

f f n

g

g g n

M

T f

M f

σ

σ ε

ρ σ

±

→∞ ≥ −

So,

( ) ( ( ) )

( )

(

¹ ²

) ^{( )}

1 2 1

1 2

lim sup log

log .

f f

g g

T f f M T f

M f f

σ

σ ε

ρ σ

±

± = →∞ ≥ −

±

[ Since

ρ

g

(

f1± f2

)

=

ρ

g

( )

f1 by Theorem (2.5) in Mondal [3] ]

Since

ε

>0 is arbitrary, Tg

(

f1± f2

)

≥Tg

( )

f1 ^… ^… ^… ^(2.7)

Hence from (2.6) and (2.7) we have,

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On Relative Order and Relative Type of an… 29

(

1 2

) ( )

1

g g

T f ± f =T f ^{, when}

ρ

g

( )

f1 >

ρ

g

( )

f2 _.

Lemma 2.5. If ^{f s}

( )

is a Dirichlet entire function of order

^ρ ( )

^f , then for any

( )

0,

k> f ks has order ^k^.

^ρ ( )

^f ^.

Proof: Let ^{g s}

( )

⁼ ^{f ks}

( )

^, ^k^>^0,^s^{= +}

^σ

^it^.

Then g

( )

^sup

( )

^sup

( )

^sup

( )

f

( )

t t kt

M

σ

g s f ks f ks M k

σ

−∞< <∞ −∞< <∞ −∞< <∞

= = = = ^.

Therefore,

( )

^{log log}

( )

lim sup M^g

g σ

ρ σ

σ

= →∞

( ) ( ) ( )

log log log log

lim sup ^f lim sup ^f

k

M k M k

k k f

k

σ σ

σ σ ρ

→∞ →∞

= = = .

Lemma 2.6. Let ^{f s}

( )

be a Dirichlet entire function and k>1. Then for all

α

>0,

( ) ( )

f f

M M k

α σ

<

σ

for sufficiently large

σ

^.

Proof: From the definition of order,

^ρ ( )

^f ⁼

^{ρ α} ( )

^f ^,

^α

^>⁰^.

Let k >1 and ^{g s}

( )

⁼ ^{f ks}

( )

. Then by Lemma (2.1),

^ρ ( )

^g ⁼^k

^ρ ( )

^f ^.

Therefore,

^{ρ α} ( )

^f ⁼

^ρ ( )

^f ^<^k

^ρ ( )

^f ⁼

^ρ ( )

^g ^.

Hence for sufficiently large

σ

^,M_αf

( ) σ

<Mg

( ) σ

^.

That is,

α

Mf

( ) σ

<Mg

( ) σ

=Mf

( )

k

σ

for l^arge

σ

and for all

α

>⁰and k >¹. Definition 2.7. Two Dirichlet entire functions g and ₁ g are said to be ₂ asymptotically equivalent, denoted by g₁∼g₂, if

( )

1

( )

2

lim ^g 1

g

M

σ M

σ σ

→∞ = .

Theorem 2.8. Let f , ₁ f and g be three Dirichlet entire functions with relative ₂ orders

ρ

g

( )

f1 ,

ρ

g

( )

f2 and relative types Tg

( )

f1 , Tg

( )

f2 of f and ₁ f . If ₂

1 2

f ∼ f , then

ρ

g

( )

f1 =

ρ

g

( )

f2 and Tg

( )

f1 =Tg

( )

f2 . Proof: Since f₁∼ f₂,

( )

1

( )

2

lim ^f 1

f

M

σ M

σ σ

→∞ = .

Therefore, for any

ε

>0, there exists

σ ε

0

( )

such that

(

¹−

ε )

Mf₂

( ) σ

<Mf₁

( ) ( σ

< +¹

ε )

Mf₂

( ) σ

for all

σ σ ε

> 0

( )

… (2.8)

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Then,

( )

¹

(

¹

( ) )

1

lim sup

^g ^f

g

M M f

σ

ρ σ

σ

−

=

→∞

( ) ( )

(

²

)

1 1

lim supM^g M^f

σ

ε σ

σ

−

→∞

≤ + [ by (2.8) ]

( )

(

²

)

1 1

lim supM^g M^f

σ

ε σ σ

−

→∞

≤ + [ by Lemma 2.6. ]

( ) ( ( ( ) ) )

(

²

)

1 1

1 lim sup

1

g f

M M

σ

ε ε σ

ε σ

−

→∞

= + +

+

(

1

ε ρ ) ( )

_g f2

= +

Since

ε

>0 is arbitrary,

ρ

g

( )

f1 ≤

ρ

g

( )

f2 ^… ^… ^… ^(2.9)

Reversing the roles of f and ₁ f we get, ₂

ρ

g

( )

f2 ≤

ρ

g

( )

f1 … … … (2.10) From (2.9) and (2.10),

ρ

g

( )

f1 =

ρ

g

( )

f2 … … … (2.11) Now,

( ) ( )

(

¹ ¹

) (

¹

^{( )} ^{( )}

²

)

log log

log . log .

f f

g g g g

M M

M f M f

σ σ

ρ σ ⁼ ρ σ [ by (2.11) ]

( ) ( )

( )

( ( )

²²

)

log 1

log .

f

g g

M

M f

ε σ

ρ σ

< +

for all

^{σ σ ε}

^> ⁰

( )

[ by (2.8) ]

… … … (2.12)

Therefore,

( ) ( )

(

¹

( ) )

1

lim sup log

log .

f g

g g

T f M

M f

σ

ρ σ

= →∞

( ) ( )

( )

( ( )

²²

)

log 1 lim sup

log .

f

g g

M

M f

σ

ε σ

ρ σ

→∞

≤ + [ by (2.12)]

( ) ( )

(

² ²

) ^{( )}

²

lim sup log

log .

f

g

g g

M T f

M f

σ

ρ σ

= →∞ = … … … (2.13)

Since f₂ ∼ f₁ , Tg

( )

f2 ≤Tg

( )

f1 ^… ^… ^… ^{( 2.14)}

From (2.13) and (2.14),

( )

1

( )

2

g g

T f =T f .

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On Relative Order and Relative Type of an… 31 Theorem 2.9. Let g , ₁ g and f be three Dirichlet entire functions with ₂

ρ

g1

( )

f ^,

2

( )

g f

ρ

be relative orders of f with respect to g¹ and g respectively and ₂

1

( )

Tg f , Tg2

( )

f be the relative types of f . If g₁∼ g₂, then

ρ

g1

( )

f =

ρ

g2

( )

f and Tg1

( )

f =Tg2

( )

f .

Proof: Since g₁∼ g₂ ,

( ) ( )

1

2

lim ^g 1

g

M

σ M

σ σ

→∞ =

Then for any

ε

>0, there exists

σ ε

0

( )

such that

(

1−

ε )

M_g2

( ) σ

<M_g1

( ) ( σ

< +1

ε )

M_g2

( ) σ

for all

σ σ ε

> 0

( )

This implies, Mg1

( ) σ

<Mg2

( ) ασ

^where

α

>1 for sufficiently large

σ

( by Lemma 2.6. ).

This implies,

^σ

^<^M^g⁻¹¹

(

^M^g²

( ) ^ασ )

for sufficiently large

σ

… … … (2.15) Let t=Mg₂

( ) ασ

^{. Then} 2

( )

1 1

Mg t σ ⁼α ⁻ ^.

Therefore from (2.15), Mg⁻₂¹

( )

t <

α

Mg⁻₁¹

( )

t for all large

σ

This implies, 2

( ( ) )

1

( ^{( )} )

1 1

lim supM^g M^f lim sup M^g M^f

σ σ

σ α σ

σ σ

− −

→∞ ≤ →∞

This implies,

ρ

g2

( )

f ≤

αρ

g1

( )

f ^{for any}

α

>1^.

Taking limit as α →1⁺, we have,

ρ

g2

( )

f ≤

ρ

g1

( )

f ^… ^… ^… ^(2.16) Similarly, reversing the role of g and ₁ g , ₂

ρ

g1

( )

f ≤

ρ

g2

( )

f ^{… …} ^(2.17) Hence,

ρ

g₁

( )

f =

ρ

g₂

( )

f [ by (2.16) and (2.17)] … … (2.18)

Again,

( )

( ( ) ) ( ^{( )} ^{( )} )

1 1 1 2

log log

log . log .

f f

g g g g

M M

M f M f

σ σ

ρ σ

⁼

ρ σ

[ by (2.18) ]

(

^log

)

²

( ( )

²

( ) )

log 1 .

f

g g

M

M f

σ

ε ρ σ

>  + 

for all

σ σ

> ₀. Therefore,

( ) ( )

( ) ( ^{( )} ^{( )} )

1 1 2 2

log log

lim sup lim sup

log . log .

f f

g g g g

M M

M f M f

σ σ

ρ σ ρ σ

→∞ ≥ →∞

This implies, Tg₁

( )

f ≥Tg₂

( )

f

… … …. (2.19)

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Since g₂ ∼g₁, Tg2

( )

f ≥Tg1

( )

f _… _… _… _(2.20) Hence,

( ) ( )

1 2

g g

T f =T f ^… ^… ^… ^(2.21)

Theorem 2.10. Let

( )

0

n

p s n n

f s a e^λ

=

∑

^and

( )

0

n

q s n n

g s b e^µ

=

∑

be two non-constant Dirichlet polynomials of degrees λ_p, µ_qrespectively. Then the relative type

( )

Tg f of f with respect to g is 1.

Proof: Let f s

( )

=a e0 ^λ⁰^s+a e1 ^λ¹^s + +. . . a e_p ^λ^p^s and

( )

0 ⁰^s 1 ¹^s . . . _q ^q^s g s =b e^µ +b e^µ + +b e^µ .

Then ^M^f

( )

^σ ^∼ ^{a e}^p ^{λ σ}^p ^and^M^g

( )

^σ ^∼ ^{b e}^q ^{µ σ}^q ^.

For any

ε

>0, there exists

σ ε

0

( )

such that

(

¹

) ( ) (

¹

)

⁰

( )

p p

p f p

a e^{λ σ} −ε <M σ < a e^{λ σ} +ε forσ σ ε> … … (2.22) and ^{b e}^q ^{µ σ}^q

(

¹⁻^ε

)

^<^M^g

( )

^σ ^< ^{b e}^q ^{µ σ}^q

(

¹⁺^ε

)

^for^{σ σ ε}^> ⁰

( )

… … (2.23) Since ^g

( )

^p

q

f λ

ρ ⁼ µ (Mondal, [3]), ⁰<

ρ

g

( )

f < ∞^.

Now ,

( ) ( )

( ( ) )

lim sup log

log .

f g

g g

T f M

M f

σ

ρ σ

= →∞

( )

^{( )}

log 1 lim sup

log 1

p

q g

p f q

a e b e

λ σ µ ρ σ σ

ε ε

→∞

 + 

 

≤  − 

lim sup

( )

1

.

p

q g f

σ

λ σ

µ ρ σ

= →∞ =

… … … (2.24)

Also,

( ) ( )

( ( ) )

lim sup log

log .

f g

g g

T f M

M f

σ

ρ σ

= →∞

( )

^{( )}

log 1 lim sup

log 1

p

q g

p f q

a e b e

λ σ µ ρ σ σ

ε ε

→∞

 − 

 

≥  + 

(11)

lim sup

( )

1

.

p

q g f

σ

λ σ

µ ρ σ

= →∞ =

… … … (2.25)

By (2.24) and (2.25),

( )

¹

Tg f = ^.

Theorem 2.11. Let f g be two Dirichlet entire functions with non-zero finite , orders

^ρ ( ) ( )

^f ^,

^ρ

^g ^{and types}^{T f}

( ) ( ) ( )

^,^{T g} ^≠⁰ , where g is of regular growth.

Then the relative type Tg

( )

f satisfies the inequality

( ) ( ) ( )

g

T f T f

≥T g . Moreover, if g is of perfectly regular growth, then

( ) ( )

( )

g

T f T f

=T g .

Proof: From the definition of type we have for any

ε

>0, there exists

σ ε

0

( )

such that

( ) ( ( ) )

^. ^{( )} ⁰

( )

logM_f σ < T f +ε e^{σ ρ} ^f forσ σ ε> … … … (2.26) and ^log^M^g

( )

^σ ^<

(

^{T g}

( )

⁺^ε

)

^e^{σ ρ}^. ^{( )}^g ^for^{σ σ ε}^> ⁰

^{( )}

… … … (2.27) Also there exists a sequence

{ } σ

n n of values of

σ

tending to infinity, such that

( ) ( ( ) )

^. ^{( )}

logM_f σ_n > T f −ε e^{σ ρ}ⁿ ^f … … … (2.28) By Theorem 2.3.,

( ) ( )

( ( ) )

lim sup log

log .

f g

g g

T f M

M f

σ

ρ σ

= →∞

( )

^{( )}

( ( ) )

lim sup

log .

n

f

g g n

T f e

M f

σ ρ

σ

ε

ρ σ

→∞

 − 

 

≥

[ by (2.28)]

( )

^{( )}

( )

^. ^{( ) ( )}^.

lim sup

n

n g

n

f

g f

T f e

T g e

σ ρ σ ρ ρ σ

ε ε

→∞

 − 

 

≥  +  [ by (2.27)]

( )

^{( )}

( )

^. ^{( )}

lim sup

n

n n

f f

T f e

T g e

σ ρ σ σ ρ

ε ε

→∞

 − 

 

≥  + 

[Since

( ) ( )

( )

g

f f

g ρ ρ

= ρ ,[3]]

( ) ( )

T f T g

ε ε

= −

+

Since

ε

>0 is arbitrary,

( ) ( ) ( )

g

T f T f

≥T g … … … (2.29)

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Moreover, if g be of perfectly regular growth, we have, for any

ε

>0^{, there} exists

σ ε

0

( )

such that

( ) ( )

^{( )} ⁰

( )

logM_g σ >^T g −ε^e^σρ ^g forσ σ ε> … … … (2.30) Hence,

( ) ( )

( ( ) )

lim sup log

log .

f g

g g

T f M

M f

σ

ρ σ

= →∞

( )

^{( )}

( )

^{( ) ( )}

.

. .

lim sup

g

f

g f

T f e

T g e

σ ρ σ ρ ρ σ

ε ε

→∞

 + 

 

≤  −  [ by (2.30)]

( ) ( )

T f T g

ε ε

= −

+

[Since

( ) ( )

( )

g

f f

g ρ ρ

= ρ ,[3]]

Since

ε

>0 is arbitrary,

( ) ( ) ( )

g

T f T f

≤T g … … … (2.31)

By (2.29) and (2.31),

( ) ( ) ( )

g

T f T f

= T g . Theorem 2.12. Let

( )

0

ns n n

f s a e^λ

∞

=

∑

^and

_{( )}

0

ns n n

g s b e^µ

∞

=

∑

be two non-constant Dirichlet entire functions with non-zero finite orders

^ρ ( ) ( )

^f ^,

^ρ

^g and types

( ) ( )

^,

T f T g where g is of perfectly regular growth and

λ

_n₊₁∼

λ

_n and

1

log ⁿ

n n

b χ b

λ λ⁺

= − is monotonic non decreasing. Then the relative type Tg

( )

f of f is given by

( ) ( )

( )

( ) ( )

1

1 1

lim sup lim sup

n g n

g

f f

n n

g g

n n

g n g n

a a

T f

f b f b

ρ λ ρ ρλ

ρ ^→∞ ρ ρ ^→∞

   

   

= =

   

   

.

Proof: By Theorem 2.11.

( ) ( ) ( )

g

T f T f

=T g

( )

lim sup lim sup

n

f n

n n

g n

n n

e f a e f b

ρ λ

λ ρ

→∞

= ( Ritt,[4])

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( )

lim sup ⁿ.lim inf ( )

n

f n

n n g

n

n n

e g

e f a

b

ρ λ

λ ρ

ρ ^→∞ λ

= →∞

( )

lim sup .

( )

n

f n n g

n

g a

b f

ρ λ

ρ ρ

→∞

 

 

≤  

 

 

( )

( ) ( )

1

1 lim sup

f n

n n g

g n

a

f b

ρ λ

ρ

^→∞ ρ

 

 

≤  

[Since

( ) ( ) ( )

g

f f

g ρ ρ

= ρ ,[3]]

… … … (2.32)

( )

( ) ( ) ( )

1

1 .

lim sup

n

g f g

n n g

g n

a

f b

ρ ρ λ

ρ

^→∞ ρ

 

 

=  

( )

( ) ^{( )}

1 lim sup

g g f n

n

g n n

a

f b

ρ

λ ρ

ρ

^→∞

 

 

=  

µ

= (say) … … … (2.33)

Then for any

ε

>0, there exists a sequence

{ }

nk k of values of n tending to infinity such that

( )

( ) ( )

1

1 k ^nk

k

f n

g

g n

a

f b

λ ρ

ρ

µ ε

ρ

 

  > −

 

 

This implies,

( )

( ) (

^.

)

. .

nk nk

k k

f g

n n

a b

e f e g

ρ ρ

λ λ

ρ

^>

ρ µ ε

⁻ ^[Since,

( ) ( ) ( )

g

f f

g ρ ρ

= ρ ,[3]]

So,

( ) ( )

( )

lim sup lim sup

. .

k nk

n

k k

f f n n

n n

T f a a

e f e f

ρ ρ

λ λ λ

λ

ρ ρ

→∞ →∞

   

=  ≥  

   

   

( ) ( )

( )

lim sup

.

k nk

k k

g n

n n

e g b

ρ λ

µ ε λ ρ

→∞

 

≥ −  

 

 

( ) ( )

( )

lim inf .

k nk

k k

g n

n bn

e g

ρ λ

µ ε λ

ρ

→∞

 

≥ −  

 

 

( ) ( )

lim inf ( )

.

n

g n

n bn

e g

ρ λ

µ ε λ

ρ

→∞

 

≥ −  

 

 

=

( ^{µ ε}

⁻

) ( )

^{T g}

(14)

[Since

λ

_n₊₁∼

λ

_n and ¹ log ⁿ

n n

b χ b

λ λ⁺

= − is monotonic non decreasing for n≥n₀ and g is of perfectly regular growth, [1]]

Therefore,

( ) ( )

T f

T g ≥ −µ ε

Since

ε

>0 is arbitrary, Tg

( )

f ≥

µ

^… ^… ^… ^(2.34) By (2.33) and (2.34) we have,

( ) ( )

( )

( ) ( )

1

1 1

lim sup lim sup

n g n

g

f f

n n

g g

n n

g n g n

a a

T f

f b f b

ρ

ρ λ ρ λ

ρ ^→∞ ρ ρ ^→∞

   

   

= =

   

    .

Acknowledgement

I would like to thank Prof. B.C. Chakraborty, my supervisor, for constantly encouraging me and supervising this paper.

References

[1] P.K. Kamthan, A note on the maximum term and the rank of an entire function represented by Dirichlet series, Math. Student, 31(1963), 17-33.

[2] A.I. Markushevich, Theory of Functions of a Complex Variable, Prentice- Hall, INC., Englewood Cliffs, N.J. II, (1965).

[3] B.C. Mondal, Relative order and lower relative order of an entire function represented by Dirichlet series, International J. of Math. Sci. & Engg.

Appls. (IJMSEA), 5(1) (2011), 365-378.

[4] J.F. Ritt, On certain points in the theory of Dirichlet series, Amer. J. Math, 50(1928), 73-86.