Gen. Math. Notes, Vol. 4, No. 2, June 2011, pp. 23-36
ISSN 2219-7184; Copyright © ICSRS Publication, 2011
www.i-csrs.org
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: bibhas_2216@yahoo.co.in
(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.
24
Bibhas Chandra Mondal
1
Introduction
Let
f ( s ) = ∑ an eλn s , s = σ + it
∞
…
…
…
(1.1);
n=0
0 ≤ λ0 < λ1 < ... < λn → ∞ as n → ∞ be an entire function represented by Dirichlet
log n
series ( called Dirichlet entire function ) where we assume that lim sup
=0.
n →∞
Then [2] lim sup
n →∞
log an
λn
λn
= −∞ .
The order of f , denoted by ρ ( f ) , is defined as
ρ ( f ) = lim sup
loglog M (σ , f )
σ
σ →∞
…
…
…
(1.2)
…
…
(1.3)
and the lower order of f , denoted by λ ( f ) , is defined as
loglog M (σ , f )
λ ( f ) = lim inf
σ
σ →∞
…
where M (σ , f ) = sup f ( s ) .
−∞<t <∞
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
T ( f ) = lim sup
log M (σ , f )
σ →∞
e ρσ
…
…
…
(1.4)
…
…
…
(1.5)
and lower type τ ( f ) is defined as
τ ( f ) = lim inf
σ →∞
log M (σ , f )
e ρσ
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
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 ) = M f (σ ) , M (σ , g ) = M g (σ ) are strictly increasing continuous
functions of σ and increase to ∞ . The inverse function M g−1 : ( L, ∞ ) → ( −∞, ∞ ) is
strictly increasing where L = lim M g (σ ) and lim M g−1 (σ ) = ∞ .
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
ρ g ( f ) = inf {k > 0 : M f (σ ) < M g ( kσ ) for all σ > σ 0 ( k )} .
Evidently, ρ g ( f ) = lim sup
M g−1 ( M f (σ ) )
σ →∞
…
σ
…
…
(2.1)
The lower relative order of f with respect to g , denoted by λg ( f ) , is defined as
λg ( f ) = lim inf
σ →∞
M g−1 ( M f (σ ) )
…
σ
…
…
(2.2)
f is said to be of regular relative growth with respect to g if ρ g ( f ) = λg ( 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
{
Tg ( f ) = inf k > 0 : M f (σ ) <  M g ( ρ g ( f ) .σ )  for all l arg e σ
k
} ……
(2.3)
Theorem 2.3. Let f , g be twoDirichlet entire functions with relative order
ρ g ( f ) ( 0 < ρ g ( f ) < ∞ ) of f with respect to g . Then the relative type of f is
given by
Tg ( f ) = lim sup
σ →∞
log M f (σ )
log M g ( ρ g ( f ) .σ )
.
26
Bibhas Chandra Mondal
Proof: From the Definition 2.2. we have for any ε > 0, there exists σ 0 such that
M f (σ ) <  M g ( ρ g ( f ) .σ ) 
This implies,
That is,
(Tg ( f )+ε )
for all σ > σ 0 .
log M f (σ ) < (Tg ( f ) + ε ) log  M g ( ρ g ( f ) .σ )  for all σ > σ 0
log M f (σ )
log  M g ( ρ g ( f ) .σ ) 
< Tg ( f ) + ε for all σ > σ 0
…
…
(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 1 < σ 2 < ... < σ n < ... tending to infinity such that
M f (σ n ) >  M g ( ρ g ( f ) .σ n ) 
That is,
log M f (σ n )
log  M g ( ρ g ( f ) .σ n ) 
> Tg ( f ) − ε
(Tg ( f ) −ε )
…
…
…
(2.5)
for a sequence of values of σ , tending to infinity.
Hence from (2.4) and (2.5) we have,
Tg ( f ) = lim sup
σ →∞
log M f (σ )
log M g ( ρ g ( f ) .σ )
.
Theorem 2.4. Let f1 , f 2 be two Dirichlet entire functions with non-zero finite
relative orders ρ g ( f1 ) and ρ g ( f 2 ) and relative types Tg ( f1 ) and Tg ( f 2 ) with
respect to g . If ρ g ( f1 ) ≠ ρ g ( f 2 ) then the relative type of f1 ± f 2 is equal to the
type of the function with maximum relative order. That is, Tg ( f1 ± f 2 ) = Tg ( f1 ) if
ρ g ( f1 ) > ρ g ( f 2 ) .
Proof: Let ρ g ( f1 ) > ρ g ( f 2 ) .
From the Definition 2.2. of relative type, for ε > 0 ,
M f1 (σ ) <  M g ( ρ g ( f1 ) .σ ) 
and
(Tg ( f1 ) +ε )
M f2 (σ ) <  M g ( ρ g ( f 2 ) .σ ) 
(Tg ( f 2 ) +ε )
for sufficiently large σ .
On Relative Order and Relative Type of an…
27
Then, M f1 ± f2 (σ ) ≤ M f1 (σ ) + M f2 (σ )
<  M g ( ρ g ( f1 ) .σ ) 
(Tg ( f1 ) +ε )
+  M g ( ρ g ( f 2 ) .σ ) 
(Tg ( f 2 ) +ε )
for l arg e σ
Therefore,
M f1 ± f2 (σ ) <  M g (

(Tg ( f1 ) +ε )   M g ρ g ( f 2 ) .σ
ρ g ( f1 ) .σ 
1 +
 M g ρ g ( f1 ) .σ



<  M g ( ρ g ( f1 ) .σ ) 
(
(
)
(Tg ( f1 ) +ε )
(Tg ( f2 )+ε ) 

Tg ( f1 ) + ε ) 


)
(
)
(1 + K ) , K > 0 , for sufficiently l arg e σ
[ Since ρ g ( f1 ) > ρ g ( f 2 ) and M g is a strictly increasing function,
 M g ( ρ g ( f 2 ) .σ ) 


 M g ( ρ g ( f1 ) .σ ) 


(Tg ( f2 ) +ε )
(Tg ( f1 ) +ε )
< K for large σ . ]
This implies,
 1

log 
M f1 ± f2 (σ )  < Tg ( f1 ) + ε  log  M g ( ρ g ( f1 ) .σ ) 
 K +1

That is,
Now,
(
log M f1 ± f2 (σ )
)
log M g ( ρ g ( f1 ) .σ )
< Tg ( f1 ) + ε
Tg ( f1 ± f 2 ) = lim sup
σ →∞
for sufficiently l arg e σ
(
(ρ ( f
log M f1 ± f2 (σ )
log M g
= lim sup
σ →∞
g
1
)
± f 2 ) .σ )
(
log M f1 ± f2 (σ )
)
log M g ( ρ g ( f1 ) .σ )
≤ Tg ( f1 ) + ε
[ Since ρ g ( f1 ± f 2 ) = ρ g ( f1 ) by Theorem (2.5) in Mondal, [3] ]
Since ε > 0 is arbitrary,
Tg ( f1 ± f 2 ) ≤ 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,
28
Bibhas Chandra Mondal
M f1 (σ n ) >  M g ( ρ g ( f1 ) .σ n ) 
(Tg ( f1 ) −ε )
.
Therefore, M f1 ± f2 (σ n ) ≥ M f1 (σ n ) − M f2 (σ n )
>  M g ( ρ g ( f1 ) .σ n ) 
(Tg ( f1 ) −ε )
−  M g ( ρ g ( f 2 ) .σ n ) 
(Tg ( f 2 ) +ε )
Therefore,
M f1 ± f2 (σ n ) >  M g (

(Tg ( f1 )−ε )   M g ρ g ( f 2 ) .σ n
ρ g ( f1 ) .σ n 
1 −
  M g ρ g ( f1 ) .σ n
 
(
(
)
for sufficiently l arg e n
(Tg ( f2 )+ε ) 


Tg ( f1 ) −ε )


)
(
)
for sufficiently l arg e n
Since ρ g ( f1 ) > ρ g ( f 2 ) , M g ( ρ g ( f1 ) .σ n ) > M g ( ρ g ( f 2 ) .σ n ) and we can
(Tg ( f2 )+ε ) 

  M g ( ρ g ( f 2 ) .σ n ) 

make 
T
f
−
ε
( g ( 1) ) 
  M g ( ρ g ( f1 ) .σ n ) 




large n.
So,
 1
arbitrarily small  <  for sufficiently
 2
M f1 ± f2 (σ n ) >  M g ( ρ g ( f1 ) .σ n ) 
(Tg ( f1 )−ε )  1 
1 − 
 2
That is, 2 M f1 ± f 2 (σ n ) >  M g ( ρ g ( f1 ) .σ n ) 
Hence,
lim sup
σ n →∞
So,
(
log 2 M f1 ± f2 (σ n )
(Tg ( f1 ) −ε )
)
log M g ( ρ g ( f1 ) .σ n )
Tg ( f1 ± f 2 ) = lim sup
σ →∞
(
(ρ ( f
≥ Tg ( f1 ) − ε
log M f1 ± f2 (σ )
log M g
g
)
1 ± f 2 ) .σ )
≥ Tg ( f1 ) − ε
[ Since ρ g ( f1 ± f 2 ) = ρ g ( f1 ) by Theorem (2.5) in Mondal [3] ]
Tg ( f1 ± f 2 ) ≥ Tg ( f1 ) …
Since ε > 0 is arbitrary,
Hence from (2.6) and (2.7) we have,
…
…
(2.7)
On Relative Order and Relative Type of an…
29
Tg ( f1 ± f 2 ) = Tg ( f1 ) , when ρ g ( f1 ) > ρ g ( f 2 )
.
Lemma 2.5. If f ( s ) is a Dirichlet entire function of order ρ ( f ) , then for any
k > 0, f ( ks ) has order k .ρ ( f ) .
Proof: Let g ( s ) = f ( ks ) , k > 0, s = σ + it .
Then
M g (σ ) = sup g ( s ) = sup f ( ks ) = sup f ( ks ) = M f ( kσ ) .
−∞<t <∞
−∞<t <∞
Therefore, ρ ( g ) = lim sup
log log M g (σ )
σ
σ →∞
= lim sup
σ →∞
log log M f ( kσ )
σ
−∞< kt <∞
= k lim sup
log log M f ( kσ )
kσ →∞
= kρ ( f ) .
kσ
Lemma 2.6. Let f ( s ) be a Dirichlet entire function and k > 1 . Then for all
α >0,
α M f (σ ) < M f ( kσ ) 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 α f (σ ) < M g (σ ) .
That is, α M f (σ ) < M g (σ ) = M f ( kσ ) for l arg e σ and for all α > 0 and k > 1 .
Definition 2.7. Two Dirichlet entire functions g1 and g 2 are said to be
asymptotically equivalent, denoted by g1 ∼ g 2 , if lim
σ →∞
M g1 (σ )
M g 2 (σ )
=1.
Theorem 2.8. Let f1 , f 2 and g be three Dirichlet entire functions with relative
orders ρ g ( f1 ) , ρ g ( f 2 )
and relative types Tg ( f1 ) , Tg ( f 2 ) of f1 and f 2 . If
f1 ∼ f 2 , then ρ g ( f1 ) = ρ g ( f 2 ) and Tg ( f1 ) = Tg ( f 2 ) .
Proof: Since f1 ∼ f 2 , lim
σ →∞
M f1 (σ )
M f 2 (σ )
= 1.
Therefore, for any ε > 0 , there exists σ 0 ( ε ) such that
(1 − ε ) M f (σ ) < M f (σ ) < (1 + ε ) M f (σ )
2
1
2
for all σ > σ 0 ( ε ) … (2.8)
30
Bibhas Chandra Mondal
Then,
ρ g ( f1 ) = lim sup
(
M g−1 M f1 (σ )
σ →∞
≤ lim sup
σ →∞
≤ lim sup
M
−1
g
σ
)
( (1 + ε ) M (σ ) )
[ by (2.8) ]
(
[ by Lemma 2.6. ]
f2
σ
M g−1 M f2 ( (1 + ε ) σ )
σ →∞
= (1 + ε ) lim sup
σ
(
)
M g−1 M f2 ( (1 + ε ) σ )
σ →∞
= (1 + ε ) ρ g ( f 2 )
(1 + ε ) σ
Since ε > 0 is arbitrary, ρ g ( f1 ) ≤ ρ g ( f 2 )
)
…
…
…
(2.9)
ρ g ( f 2 ) ≤ ρ g ( f1 )
…
…
…
(2.10)
ρ g ( f1 ) = ρ g ( f 2 )
…
…
…
(2.11)
Reversing the roles of f1 and f 2 we get,
From (2.9) and (2.10),
Now,
log M f1 (σ )
log M g ( ρ g ( f1 ) .σ )
<
=
(
log M f1 (σ )
[ by (2.11) ]
log M g ( ρ g ( f 2 ) .σ )
)
( ρ ( f ) .σ )
log (1 + ε ) M f2 (σ )
log M g
g
for all σ > σ 0 ( ε ) [ by (2.8) ]
2
…
Therefore,
Tg ( f1 ) = lim sup
σ →∞
≤ lim sup
σ →∞
= lim sup
σ →∞
…
…
(2.12)
log M f1 (σ )
log M g ( ρ g ( f1 ) .σ )
(
)
( ρ ( f ) .σ )
log (1 + ε ) M f2 (σ )
log M g
g
[ by (2.12)]
2
log M f2 (σ )
log M g ( ρ g ( f 2 ) .σ )
Since f 2 ∼ f1 , Tg ( f 2 ) ≤ Tg ( f1 )
From (2.13) and (2.14),
Tg ( f1 ) = Tg ( f 2 ) .
= Tg ( f 2 ) …
…
…
…
…
…
(2.13)
( 2.14)
On Relative Order and Relative Type of an…
31
Theorem 2.9. Let g1 , g 2 and f be three Dirichlet entire functions with ρ g1 ( f ) ,
( f ) be relative orders of f with respect to g1 and g 2 respectively and
Tg ( f ) , Tg ( f ) be the relative types of f . If g1 ∼ g 2 , then ρ g ( f ) = ρ g ( f )
and Tg ( f ) = Tg ( f ) .
ρg
2
1
2
1
1
2
2
Proof: Since g1 ∼ g 2
,
lim
σ →∞
M g1 (σ )
M g 2 (σ )
=1
Then for any ε > 0 , there exists σ 0 ( ε ) such that
(1 − ε ) M g (σ ) < M g (σ ) < (1 + ε ) M g (σ )
This implies, M g (σ ) < M g (ασ ) where α > 1
2
1
1
2
Lemma 2.6. ).
for all σ > σ 0 ( ε )
2
(
This implies, σ < M g−11 M g2 (ασ )
)
for sufficiently large σ ( by
for sufficiently large σ …
……
(2.15)
…
(2.16)
Similarly, reversing the role of g1 and g 2 , ρ g1 ( f ) ≤ ρ g2 ( f ) …
…
(2.17)
Hence, ρ g1 ( f ) = ρ g2 ( f )
…
(2.18)
Let t = M g2 (ασ ) . Then σ =
Therefore from (2.15),
This implies, lim sup
(
1
M −21 ( t ) .
α g
M g−21 ( t ) < α M g−11 ( t ) for all large σ
M g−21 M f (σ )
σ →∞
σ
) ≤ lim sup α M ( M (σ ) )
−1
g1
f
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 ) …
Again,
log M f (σ )
(
log M g1 ρ g1 ( f ) .σ
[ by (2.16) and (2.17)] …
)
=
>
Therefore,
lim sup
σ →∞
…
log M f (σ )
(
log M f (σ )
(
log M g1 ρ g2 ( f ) .σ
[ by (2.18) ]
log M f (σ )
(
)
log (1 + ε ) M g2 ρ g2 ( f ) .σ 


log M g1 ρ g1 ( f ) .σ
This implies, Tg1 ( f ) ≥ Tg2 ( f )
)
)
≥ lim sup
σ →∞
log M f (σ )
(
for all σ > σ 0 .
log M g2 ρ g2 ( f ) .σ
…
…
)
….
(2.19)
32
Bibhas Chandra Mondal
Since g 2 ∼ g1 , Tg2 ( f ) ≥ Tg1 ( f )
Hence,
Tg1 ( f ) = Tg2 ( f )
p
Theorem 2.10. Let f ( s ) = ∑ an e
λn s
n=0
…
…
…
(2.20)
…
…
…
(2.21)
q
and g ( s ) = ∑ bn e µn s be two non-constant
n =0
Dirichlet polynomials of degrees λ p , µ q respectively. Then the relative type
Tg ( f ) of f with respect to g is 1 .
f ( s ) = a0 eλ0 s + a1eλ1s + . . . + a p e
Proof: Let
g ( s ) = b0e µ0 s + b1e µ1s + . . . + bq e
Then M f (σ ) ∼ a p e
µq s
and
.
and M g (σ ) ∼ bq e
λ pσ
λps
µ qσ
.
For any ε > 0 , there exists σ 0 ( ε ) such that
(1 − ε ) < M f (σ ) < a p eλ σ (1 + ε )
µσ
µσ
e (1 − ε ) < M g (σ ) < bq e (1 + ε )
ap e
and
bq
λ pσ
p
q
q
Since ρ g ( f ) =
Now ,
for σ > σ 0 ( ε ) … … (2.22)
for σ > σ 0 ( ε )
… … (2.23)
λp
(Mondal, [3]), 0 < ρ g ( f ) < ∞ .
µq
Tg ( f ) = lim sup
σ →∞
log M f (σ )
log M g ( ρ g ( f ) .σ )
λ σ
log (1 + ε ) a p e p 


≤ lim sup
µ ρ ( f )σ
σ →∞ log (1 − ε ) b e q g

q


= lim sup
σ →∞
Also,
Tg ( f ) = lim sup
σ →∞
λ pσ
= 1
µ q ρ g ( f ) .σ
log M f (σ )
log M g ( ρ g ( f ) .σ )
λ σ
log (1 − ε ) a p e p 


≥ lim sup
µq ρ g ( f )σ
σ →∞ log (1 + ε ) b e

q


…
… …
(2.24)
On Relative Order and Relative Type of an…
λ pσ
= lim sup
= 1
µ q ρ g ( f ) .σ
σ →∞
By (2.24) and (2.25),
33
…
…
…
(2.25)
Tg ( f ) = 1 .
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 Tg ( f ) ≥
g is of perfectly regular growth, then Tg ( f ) =
T(f)
T (g)
T(f)
T (g)
. Moreover, if
.
Proof: From the definition of type we have for any ε > 0 , there exists σ 0 ( ε )
such that
log M f (σ ) < (T ( f ) + ε ) eσ . ρ ( f )
and
log M g (σ ) < (T ( g ) + ε ) eσ . ρ ( g )
for σ > σ 0 ( ε )
…
……
(2.26)
for σ > σ 0 ( ε )
…… …
(2.27)
Also there exists a sequence {σ n }n of values of σ tending to infinity, such that
log M f (σ n ) > (T ( f ) − ε ) eσ n . ρ ( f )
By Theorem 2.3.,
Tg ( f ) = lim sup
σ →∞
…… …
log M f (σ )
log M g ( ρ g ( f ) .σ )
T ( f ) − ε  eσ n ρ ( f )
≥ lim sup
σ n →∞ log M g ( ρ g ( f ) .σ n )
[ by (2.28)]
T ( f ) − ε  e
σ . ρ g .ρ f
T ( g ) + ε  e n ( ) g ( )
σ nρ ( f )
≥ lim sup
σ n →∞
(2.28)
T ( f ) − ε  e
≥ lim sup 
σ .ρ ( f )
σ n →∞ T ( g ) + ε  e n


T ( f )−ε
=
T (g)+ε
σnρ( f )
Since ε > 0 is arbitrary, Tg ( f ) ≥
T(f)
T (g)
[ by (2.27)]
[Since ρ g ( f ) =
…
…
ρ( f )
,[3]]
ρ (g)
…
(2.29)
34
Bibhas Chandra Mondal
Moreover, if g be of perfectly regular growth, we have, for any ε > 0 , there
exists σ 0 ( ε ) such that
log M g (σ ) > T ( g ) − ε  eσρ ( g )
Hence,
Tg ( f ) = lim sup
σ →∞
for σ > σ 0 ( ε )
… … (2.30)
log M f (σ )
log M g ( ρ g ( f ) .σ )
T ( f ) + ε  eσ . ρ ( f )
≤ lim sup
σ . ρ g .ρ f
σ →∞
T ( g ) − ε  e ( ) g ( )
=
…
T ( f )−ε
[ by (2.30)]
[Since ρ g ( f ) =
T (g)+ε
Since ε > 0 is arbitrary, Tg ( f ) ≤
By (2.29) and (2.31), Tg ( f ) =
T(f)
…
T (g)
T(f)
T (g)
ρ( f )
,[3]]
ρ (g)
…
…
(2.31)
.
∞
∞
n=0
n =0
Theorem 2.12. Let f ( s ) = ∑ an eλn s and g ( s ) = ∑ bn e µn s 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 λn +1 ∼ λn and
χn =
log
bn
bn +1
λn − λn
is monotonic non decreasing. Then the relative type Tg ( f ) of f
is given by
a
1
Tg ( f ) =
lim sup  n
ρ g ( f ) n→∞  bn

1
ρ( f )
 λn
 a ρg ( f ) 
1
 =

lim sup  n
ρ
f
n →∞
(
)

 bn 
g
ρ(g)
Proof: By Theorem 2.11.
T(f)
Tg ( f ) =
T (g)
ρ( f )
λn
λ
an
eρ ( f )
n →∞
=
ρ(g )
λn
λ
lim sup
bn
eρ ( f )
n →∞
lim sup
n
n
( Ritt,[4])
ρ(g)
λn
.
On Relative Order and Relative Type of an…
= lim sup
n →∞
λn
an
eρ ( f )

a
≤ lim sup  n

n →∞
 bn
ρ( f )
ρ(g)
λn
λn
ρ( f )
λn
.lim inf
n →∞
35
eρ ( g )
λn bn
ρ(g)
λn

ρ (g)
.
ρ ( f )

a
lim sup  n
≤
ρ g ( f ) n→∞  bn

1
1
ρ( f )


ρ( g )

λn
 a ρ g ( f ). ρ ( g ) 
lim sup  n ρ ( g ) 
=
ρ g ( f ) n→∞  bn


[Since ρ g ( f ) =
ρ( f )
,[3]]
ρ (g)
…
…
…
(2.32)
…
…
…
(2.33)
1
1
 a ρg ( f ) 
1

lim sup  n
=
ρ g ( f ) n→∞  bn 


= µ (say)
ρ( g )
λn
λn
Then for any ε > 0 , there exists a sequence {nk }k of values of n tending to
infinity such that

1  ank
ρg ( f )  b
 nk
ρ( f )
This implies,
So,
ank
λnk
e.ρ ( f )
ρ( f )


ρ( g ) 

ρ( g)
>
bnk
λnk
e.ρ ( g )
 λn
T ( f ) = lim sup 
an
n →∞
 e.ρ ( f )
 λnk
≥ lim sup ( µ − ε ) 
bnk
nk →∞
 e.ρ ( g )
1
λnk
> µ −ε
.( µ − ε )
ρ( f )
ρ( g )
λn
λnk
[Since, ρ g ( f ) =
ρ( f )
,[3]]
ρ (g)

 λnk
ank
 ≥ lim sup 
nk →∞  e.ρ ( f )


ρ( f )
λnk






ρ(g)
 λnk

λnk
≥ ( µ − ε ) lim inf 
bnk

nk →∞
 e.ρ ( g )

ρ(g) 
 λn
λn
≥ ( µ − ε ) lim inf 
bn

n →∞
e
.
g
ρ
 ( )

= ( µ − ε )T ( g )
36
Bibhas Chandra Mondal
[Since λn +1 ∼ λn and χ n =
log
bn
bn +1
λn − λn
is monotonic non decreasing for n ≥ n0 and
g is of perfectly regular growth, [1]]
T(f)
Therefore,
≥ µ −ε
T (g)
Since ε > 0 is arbitrary, Tg ( f ) ≥ µ
By (2.33) and (2.34) we have,
a
1
lim sup  n
Tg ( f ) =
ρ g ( f ) n→∞  bn

…
ρ( f )
ρ(g)
…
1
…
(2.34)
 λn
 a ρg ( f ) 
1
 =

lim sup  n
ρ
f
n →∞

 bn 
g ( )
ρ(g)
λn
.
Acknowledgement
I would like to thank Prof. B.C. Chakraborty, my supervisor, for constantly
encouraging me and supervising this paper.
References
[1]
[2]
[3]
[4]
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.
A.I. Markushevich, Theory of Functions of a Complex Variable, PrenticeHall, INC., Englewood Cliffs, N.J. II, (1965).
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.
J.F. Ritt, On certain points in the theory of Dirichlet series, Amer. J. Math,
50(1928), 73-86.