MATEMATIQKI VESNIK
originalni nauqni rad
research paper
67, 2 (2015), 143–154
June 2015
GENERALIZED RELATIVE LOWER ORDER
OF ENTIRE FUNCTIONS
Sanjib Kumar Datta, Tanmay Biswas and Chinmay Biswas
Abstract. The basic properties of the generalized relative lower order of entire functions
are discussed in this paper. In fact, we improve here some results of Datta, Biswas and Biswas
[Casp. J. Appl. Math. Ecol. Econ., 1, 2 (2013), 3–18].
1. Introduction, definitions and notations
Let f and g be any two entire functions defined in the complex plane C and
Mf (r) = max{|f (z)| : |z| = r}, Mg (r) = max{|g(z)| : |z| = r}. Sato [9] defined the
[l]
[l]
generalized order ρf and generalized lower order λf of an entire function f for any
integer l ≥ 2, in the following way:
[l]
ρf = lim sup
r→∞
log[l] Mf (r)
log[l] Mf (r)
[l]
and λf = lim inf
,
r→∞
log r
log r
[k]
where log x = log(log[k−1] x), k = 1, 2, 3, . . . and log[0] x = x.
When l = 2, the above definition coincides with the classical definition of order
and lower order, which are as follows:
ρf = lim sup
r→∞
log[2] Mf (r)
log[2] Mf (r)
and λf = lim inf
.
r→∞
log r
log r
If f is non-constant then Mf (r) is strictly increasing and continuous, and its
inverse Mf −1 : (|f (0)|, ∞) → (0, ∞) exists and is such that lims→∞ Mf −1 (s) = ∞.
Bernal ([1], see also [2]) introduced the definition of relative order of g with
respect to f , denoted by ρf (g) as follows :
ρg (f ) = inf{µ > 0 : Mf (r) < Mg (rµ ) for all r > r0 (µ) > 0}
= lim sup
r→∞
log Mg−1 Mf (r)
.
log r
The definition coincides with the classical one [10] if g(z) = exp z.
2010 Mathematics Subject Classification: 30D20, 30D30, 30D35
Keywords and phrases: Entire function; generalized relative lower order; Property (A).
143
144
Generalized relative lower order of entire functions
Similarly, one can define the relative lower order of g with respect to f , denoted
by λf (g) as follows
log Mg−1 Mf (r)
.
λg (f ) = lim inf
r→∞
log r
Extending this notion, Lahiri and Banerjee [7] gave a more generalized concept of
relative order which may be given in the following way.
Definition 1. [7] If l ≥ 1 is a positive integer, then the l-th generalized
[l]
relative order of f with respect to g, denoted by ρf (g), is defined by
[l−1] µ
ρ[l]
r ) for all r > r0 (µ) > 0}
g (f ) = inf{µ > 0 : Mf (r) < Mg (exp
= lim sup
r→∞
log[l] Mg−1 Mf (r)
,
log r
[l]
[l]
If l = 1 then ρg (f ) = ρg (f ). If l = 1, g(z) = exp z then ρg (f ) = ρf , the classical
order of f (cf. [10]).
Analogously, one can define the l-th generalized relative lower order of g with
[l]
respect to f , denoted by λf (g) as follows
[l]
λf (g) = lim inf
r→∞
log[l] Mf−1 Mg (r)
log r
.
During the past decades, several authors (see [4–7]) made close investigations
on the properties of relative order of entire functions. In this connection the following definition is relevant.
Definition 2. [2] A non-constant entire function f is said have the Property
(A) if for any σ > 1 and for all sufficiently large r, [Mf (r)]2 ≤ Mf (rσ ) holds.
For examples of functions with or without the Property (A), one may see [2].
It is well known that the order of the products and of the sums of two entire
functions is not greater than the maximal order of the two functions. Bernal [2]
and Lahiri and Banerjee [7] extended these results for relative order and generalized
relative order. Our aim in this paper is to study some parallel basic properties of
generalized relative lower order of entire functions. We do not explain the standard
definitions and notations in the theory of entire functions as those are available
in [11].
2. Lemmas
In this section we present some lemmas which will be needed in the sequel.
Lemma 1. [2] Suppose f is a nonconstant entire function, α > 1, 0 < β < α,
s > 1, 0 < µ < λ and n is a positive integer. Then
(a) Mf (αr) > βMf (r).
S. K. Datta, T. Biswas, C. Biswas
145
(b) There exists K = K(s, f ) > 0 such that (Mf (r))s ≤ KMf (rs ) for r > 0.
Mf (rs )
Mf (rλ )
= ∞ = lim
.
r→∞ Mf (r)
r→∞ Mf (r µ )
(d) If f is transcendental then
(c) lim
Mf (rs )
Mf (rλ )
= ∞ = lim n
.
n
r→∞ r Mf (r)
r→∞ r Mf (r µ )
lim
Lemma 2. [2] Let f be an entire function satisfying the Property (A), and let
δ > 1 and n be a given positive integer. Then the inequality [Mf (r)]n ≤ Mf (rδ )
holds for r large enough.
Lemma 3. Let f, g and h are any three entire functions. If Mg (r) ≤ Mh (r)
[l]
[l]
for all sufficiently large values of r, then λh (f ) ≤ λg (f ), where l ≥ 1.
Proof. As Mg (r) ≤ Mh (r) and Mf (r) is an increasing function of r we get for
all sufficiently large values of r that
Mh−1 (r) ≤ Mg−1 (r)
i.e., Mh−1 Mf (r) ≤ Mg−1 Mf (r)
i.e., lim inf
r→∞
log[l] Mg−1 Mf (r)
log[l] Mh−1 Mf (r)
≤ lim inf
r→∞
log r
log r
[l]
i.e., λh (f ) ≤ λ[l]
g (f ).
This proves the lemma.
Lemma 4. [7] Every entire function f satisfying the Property (A) is transcendental.
Lemma 5. [8, p. 21] Let f (z) be holomorphic in the circle |z| = 2eR (R > 0)
with f (0) = 1 and η be an arbitrary positive number not exceeding 3e
2 . Then inside
the circle |z| = R, but outside of a family of excluded circles the sum of whose radii
is not greater than 4ηR, we have
log |f (z)| > −T (η) log Mf (2eR),
for T (η) = 2 + log
3e
2η .
3. Results
In this section we present the main results of the paper.
Theorem 1. If f1 , f2 , . . . , fn (n ≥ 2) and g are entire functions, then
[l]
[l]
λf (g) ≥ λfi (g),
146
Generalized relative lower order of entire functions
equality holds when
Pn
[l]
[l]
k=2 fk and λfi (g) = min{λfk (g) |
[l]
[l]
λfi (g) 6= λfk (g) (k = 1, 2, . . . , n and k 6=
where l ≥ 1, f = f1 ±
k = 1, 2, . . . , n}. The
i).
[l]
[l]
Proof. If λf (g) = ∞ then the result is obvious. So we suppose that λf (g) <
[l]
[l]
[l]
∞. We can clearly assume that λfi (g) is finite. By hypothesis, λfi (g) ≤ λfk (g)
for all k = 1, 2, . . . , i, . . . , n. We can suppose
[l]
λfi (g)
[l]
λfi (g)
> 0 (the proof for the case
= 0 is easier and left to the interested reader).
Now for any arbitrary ε > 0, we get for all sufficiently large values of r that
Mfk [exp[l−1] r
[l]
(λf (g)−ε)
k
] < Mg (r) where k = 1, 2, . . . , n
i.e., Mfk (r) < Mg [(log[l−1] r)
1
[l]
(g)−ε)
(λ
fk
] where k = 1, 2, . . . , n , so
1
Mfk (r) ≤ Mg [(log[l−1] r)
[l]
(λ
(g)−ε)
fk
] where k = 1, 2, . . . , n.
(1)
Now for all sufficiently large values of r,
Mf (r) <
n
P
k=1
i.e., Mf (r) <
n
P
Mfk (r)
Mg [(log
[l−1]
r)
(λ
1
[l]
(g)−ε)
fk
]
k=1
i.e., Mf (r) < nMg [(log[l−1] r)
(λ
1
[l]
(g)−ε)
fi
].
(2)
Now in view of the first part of Lemma 1, we obtain from (2) for all sufficiently
large values of r that
1
[l]
Mf (r) < Mg [(n + 1)(log[l−1] r) fi
]
[l]
r (λf (g)−ε)
) i
i.e., Mf [exp[l−1] (
] < Mg (r)
n+1
r (λf[l] (g)−ε)
) i
i.e., exp[l−1] (
< Mf−1 Mg (r)
n+1
r
[l]
i.e., (λfi (g) − ε) log(
) < log[l] Mf−1 Mg (r)
n+1
log[l] Mf−1 Mg (r)
[l]
i.e., λfi (g) − ε <
log r + O(1)
(λ
i.e.,
log[l] Mf−1 Mg (r)
log r + O(1)
So
[l]
λf (g) = lim inf
r→∞
(g)−ε)
[l]
> λfi (g) − ε.
log[l] Mf−1 Mg (r)
log r + O(1)
[l]
≥ λfi (g) − ε.
147
S. K. Datta, T. Biswas, C. Biswas
Since ε > 0 is arbitrary,
[l]
[l]
λf (g) ≥ λfi (g).
[l]
(3)
[l]
Next let λfi (g) < λfk (g) where k = 1, 2, . . . , n and k 6= i. As ε(> 0) is
arbitrary, from the definition of generalized lower order it follows for a sequence of
values of r tending to infinity that
Mg (r) < Mfi [exp[l−1] r
[l]
(λf (g)+ε)
i
]
1
i.e., Mg [(log[l−1] r)
[l]
(λ
[l]
(g)+ε)
fi
] < Mfi (r).
(4)
[l]
Since λfi (g) < λfk (g) where k = 1, 2, . . . , n and k 6= i, then in view of the third
part of Lemma 1 we obtain that
lim
Mg [(log
r→∞
Mg [(log
[l−1]
[l−1]
r)
1
[l]
(λ (g)+ε)
fi
]
r)
1
[l]
(λ
(g)−ε)
fk
]
= ∞ where k = 1, 2, . . . , n and k 6= i.
(5)
Therefore from (5) we obtain for all sufficiently large values of r that
Mg [(log
[l−1]
r)
1
[l]
(λ (g)+ε)
fi
] > nMg [(log
[l−1]
r)
1
[l]
(λ
(g)−ε)
fk
],
(6)
for all k ∈ {1, 2, . . . , n}\{i}.
Thus from (1), (4) and (6) we get for a sequence of values of r tending to
infinity that
Mfi (r) > Mg [(log
[l−1]
r)
1
[l]
(λ (g)+ε)
fi
]
1
i.e., Mfi (r) > nMg [(log[l−1] r)
[l]
(λ
(g)−ε)
fk
]
i.e., Mfi (r) > nMfk (r) for all k = 1, 2, . . . , n with k 6= i.
(7)
So from (4) and (7) and in view of the first part of Lemma 1 it follows for a sequence
of values of r tending to infinity that
Mf (r) ≥ Mfi (r) −
i.e., Mf (r) ≥ Mfi (r) −
n
P
k=1
k6=i
Mfk (r)
n
1 P
Mfi (r)
n k=1
k6=i
n−1
i.e., Mf (r) > Mfi (r) − (
)Mfi (r)
n
1
i.e., Mf (r) > ( )Mfi (r)
n
1
[l]
1
(λ (g)+ε)
[l−1]
fi
i.e., Mf (r) > ( )Mg [(log
r)
]
n
148
Generalized relative lower order of entire functions
(λ
1
[l]
(g)+ε)
fi
r)
].
n+1
This gives for a sequence of values of r tending to infinity that
i.e., Mf (r) > Mg [
(log
[l−1]
[l]
(λf (g)+ε)
Mf [exp[l−1] {(n + 1)r}
[l]
(λf (g)+ε)
i
i.e., {(n + 1)r}
i
] > Mg (r)
> log[l−1] Mf−1 Mg (r)
log[l] Mf−1 Mg (r)
[l]
i.e., λfi (g) + ε >
log((n + 1)r)
log[l] Mf−1 Mg (r)
[l]
i.e., λfi (g) + ε >
log r + O(1)
[l]
i.e., λfi (g) ≥ lim inf
log[l] Mf−1 Mg (r)
r→∞
log r + O(1)
[l]
[l]
i.e., λf (g) = lim inf
log Mf−1 Mg (r)
log r
r→∞
≤ λfi (g).
(8)
So from (3) and (8), we finally obtain that
[l]
[l]
λf (g) = λfi (g),
[l]
[l]
whenever λfi (g) 6= λfk (g) for all k ∈ {1, 2, . . . , n}\{i}.
Theorem 2. Let n, l be two positive integers with n, l ≥ 2. Then
1 [l]
[l]
[l]
λ (g) ≤ λf n (g) ≤ λf (g).
n f
Proof. From the first and second parts of Lemma 1, we obtain that
{Mf (r)}n ≤ KMf (rn ) < Mf ((K + 1)rn ), n > 1 and r > 0
(9)
where K = K(n, f ) > 0. Therefore from (9) we obtain that
Mf−1 (rn ) < (K + 1){Mf−1 (r)}n
So
[l]
λf n (g) ≥
log[l]
−1
1
n
(K+1) Mf Mg (r )
log rn
1 [l]
λ (g).
(10)
n f
On the other hand since {Mf (r)}n > Mf (r) for all sufficiently large values of r, we
have by Lemma 3
[l]
[l]
λf n (g) ≤ λf (g).
(11)
[l]
i.e., λf n (g) ≥
Thus the theorem follows from (10) and (11).
149
S. K. Datta, T. Biswas, C. Biswas
Proposition 1. Let n, l be two positive integers with n, l ≥ 2. Then
1 [l]
[l]
[l]
ρ (g) ≤ ρf n (g) ≤ ρf (g).
n f
The proof is omitted as it can be carried out under the lines of Theorem 2.
[l]
Theorem 3. Let P be a polynomial. If f is transcendental then λP f (g) =
[l]
λf (g),
and if
[l]
g is transcendental, then λf (P g)
[l]
[l]
[l]
then λP f (g) = λf (P g) = λf (g)
[l]
λf (g). If
[l]
λP f (P g).
=
f and g are both
transcendental
=
Here P f and P g
denote the ordinary product of P with f and g respectively and l ≥ 1.
Proof. Let m be the degree of P (z). Then there exists α such that 0 < α < 1
and a positive integer n(> m) for which
2α ≤ |P (z)| ≤ rn
holds on |z| = r for all sufficiently large values of r. Now by the first part of Lemma
1
Mg (αr), i.e.,
1 we obtain that Mg ( α1 · αr) > 2α
Mg (αr) < 2αMg (r).
(12)
Now let us consider h(z) = P (z) · f (z). Then from (2) and in view of the fourth
part of Lemma 1 we get for any s(> 1) and for all sufficiently large values of r that
Mg (αr) < 2αMg (r) ≤ Mh (r) ≤ rn Mg (r) < Mg (rs ).
So
lim inf
r→∞
log[l] Mf−1 Mg (αr)
log r
≤ lim inf
log[l] Mf−1 Mh (r)
log r
r→∞
≤ lim inf
r→∞
log[l] Mf−1 Mg (rs )
log r
i.e.
lim inf
r→∞
log[l] Mf−1 Mg (αr)
log(αr) + O(1)
≤ lim inf
log[l] Mf−1 Mh (r)
r→∞
[l]
log r
[l]
≤ lim inf
r→∞
log[l] Mf−1 Mg (rs )
log rs
·s
[l]
i.e., λf (g) ≤ λh (g) ≤ s · λf (g),
and letting s → 1+ we get
[l]
[l]
λP f (g) = λf (g).
(13)
Similarly, when g is transcendental one can easily prove that
[l]
[l]
λf (P g) = λf (g).
(14)
If f and g are both transcendental then the conclusion of the theorem can easily
be obtained by combining (13) and (14), and the theorem follows.
150
Generalized relative lower order of entire functions
Theorem 4. If f1 , f2 , . . . , fn (n ≥ 2), g are entire functions and g has the
Property (A), then
[l]
[l]
λf (g) ≥ λfi (g)
Qn
[l]
[l]
where f = k=1 fk and λfi (g) = min{λfk (g) | k = 1, 2, . . . , n}. The equality
[l]
[l]
6 λfk (g) (k = 1, 2, . . . , n and k 6= i). Finally, assume that F1
holds when λfi (g) =
1
and F2 are entire functions such that f := F
F2 is also an entire function. Then
[l]
[l]
[l]
λf (g) = min{λF1 (g), λF2 (g)} .
[l]
Proof. By Lemma 4, g is transcendental. Suppose that λf (g) < ∞. Otherwise
[l]
[l]
if λf (g) = ∞ then the result is obvious. We can clearly assume that λfi (g) is finite.
[l]
[l]
[l]
Also suppose that λfi (g) ≤ λfk (g) where k = 1, 2, . . . , n. We can suppose λfi (g) > 0
[l]
(the proof for the case λfi (g) = 0 is easier and left to the interested reader).
[l]
Now for any arbitrary ε > 0, with ε < λfi (g), we have for all sufficiently large
values of r that
Mfk [exp[l−1] r
[l]
(λf (g)− 2ε )
k
i.e., Mfk (r) < Mg [(log
[l−1]
] < Mg (r) where k = 1, 2, . . . , n
r)
1
[l]
(λ
(g)− ε )
2
fk
] where k = 1, 2, . . . , n, so
1
Mfk (r) ≤ Mg [(log[l−1] r)
(λ
[l]
(g)− ε )
2
fi
] for k = 1, 2, . . . , n.
(15)
From (15) we have for all sufficiently large values of r that
n
Q
Mf (r) <
k=1
i.e., Mf (r) <
n
Q
Mfk (r),
Mg [(log[l−1] r)
(λ
1
[l]
(g)− ε )
2
fk
]
k=1
i.e., Mf (r) < [Mg [(log[l−1] r)
Observe that
[l]
δ :=
λfi (g) −
[l]
ε
2
λfi (g) − ε
(λ
1
[l]
(g)− ε )
2
fi
]]n .
(16)
> 1.
(17)
Since g has the Property (A), in view of Lemma 2 and (17) we obtain from (16)
for all sufficiently large values of r that
Mf (r) < Mg (log[l−1] r)
(λ
δ
[l]
(g)− ε )
2
fi
i.e., Mf [exp[l−1] r
i.e., r
[l]
(λf (g)−ε)
i
= Mg [(log[l−1] r)
[l]
(λf (g)−ε)
i
] < Mg (r)
< log[l−1] Mf−1 Mg (r)
(λ
1
[l]
(g)−ε)
fi
]
151
S. K. Datta, T. Biswas, C. Biswas
[l]
i.e., (λfi (g) − ε) log r < log[l] Mf−1 Mg (r)
[l]
i.e., λfi (g) − ε <
So
[l]
λf (g) = lim inf
log[l] Mf−1 Mg (r)
log r
log[l] Mf−1 Mg (r)
log r
r→∞
[l]
≥ λfi (g) − ε.
Since ε > 0 is arbitrary,
[l]
[l]
λf (g) ≥ λfi (g).
[l]
[l]
Next, let λfi (g) < λfk (g)
[l]
[l]
ε < 14 min{λfk (g) − λfi (g) :
(18)
where k = 1, 2, . . . , n and k 6= i. Fix ε > 0 with
k ∈ {1, . . . , n}\{i}}. Without loss of any generality, we
may assume that fk (0) = 1 where k = 1, 2, . . . , n and k 6= i.
Now from the definition of relative lower order we obtain for a sequence of
values of R tending to infinity that
Mg (R) < Mfi [exp[l−1] R
[l]
(λf (g)+ε)
i
]
1
i.e., Mfi (R) > Mg [(log[l−1] R)
[l]
(λ (g)+ε)
fi
].
(19)
Also for all sufficiently large values of r we get that
Mfk [exp[l−1] r
[l]
(λf (g)−ε)
k
i.e., Mfk (r) < Mg [(log
[l]
] < Mg (r) where k = 1, 2, . . . , n and k 6= i
[l−1]
r)
1
[l]
(λ
(g)−ε)
fk
] where k = 1, 2, . . . , n and k 6= i.
[l]
Since λfi (g) < λfk (g), we get from above that
Mfk (r) < Mg [(log
[l−1]
r)
1
[l]
(λ (g)−ε)
fi
where k = 1, 2, . . . , n and k 6= i.
Now in view of Lemma 5, taking fk (z) for f (z), η =
for the values of z specified in the lemma that
]
(20)
1
16
and 2R for R, it follows
log |fk (z)| > −T (η) log Mfk (2e · 2R),
where
T (η) = 2 + log(
3e
1 ) = 2 + log(24e).
2 · 16
Therefore
log |fk (z)| > −(2 + log(24e)) log Mfk (4eR)
holds within and on |z| = 2R but outside a family of excluded circles the sum of
1
whose radii is not greater than 4 · 16
· 2R = R2 . If r ∈ (R, 2R) then on |z| = r
log |fk (z)| > −7 log Mfk (4e · R).
(21)
152
Generalized relative lower order of entire functions
Since r > R, we have from above and (19) for a sequence of values of r tending to
infinity that
1
[l]
(λ (g)+ε)
fi
1
r (λ[l]
(g)+ε)
Mfi (r) > Mfi (R) > Mg [(log
R)
] > Mg [(log
) fi
]. (22)
2
Let zr be a point on |z| = r such that Mfi (r) = |fi (zr )|. Therefore as r > R,
from (20), (21) and (22) it follows for a sequence of values of r tending to infinity
that
n
Q
|fk (z)| : |z| = r}, so
Mf (r) = max{|f (z)| : |z| = r} = max{
[l−1]
[l−1]
k=1
n
Q
Mf (r) ≥
|fk (zr )||fi (zr )|
k=1
k6=i
n
Q
i.e., Mf (r) ≥
|fk (zr )|Mfi (r)
k=1
k6=i
i.e., Mf (r) ≥
n
Q
k=1
k6=i
≥
n
Q
[Mg [(log
[l−1]
[Mfk (4eR)]−7 Mg [(log[l−1] ( 2r ))
(4eR))
(λ
1
[l]
(g)−ε)
fk
k=1
k6=i
=
n
Q
[Mg [(log
[l−1]
(4eR))
(λ
1
[l]
(g)−ε)
fk
(λ
1
[l]
(g)+ε)
fi
]]−7 Mg [(log[l−1] ( 2r ))
]
1
[l]
(λ (g)+ε)
fi
]
1
]]
k=1
k6=i
−7
[l]
(λ (g)+ε)
fi
Mg [(log[l−1] ( 4er
],
8e ))
hence
Mf (r) ≥
n
Q
[Mg [(log[l−1] 4er)
(λ
1
[l]
(g)−ε)
fk
1
]]−7 Mg [(log[l−1]
k=1
k6=i
[l]
4er (λfi (g)+ε)
].
8e )
1
[l]
(λ (g)+ε)
fi
(log[l−1] ( 4er
8e ))
(23)
1
[l]
(λ (g)+2ε)
fi
[l−1]
On the other hand, we have
≥ (log
(4er))
asymptotically. By using this fact together with Lemma 2 (with n = 2 and
[l]
δ :=
λf (g)+3ε
i
[l]
λf (g)+2ε
> 1) we get for r large enough that
i
1
1
[l]
4er (λ[l]
(λ (g)+3ε) 2
(g)+ε)
)) fi
] ≥ Mg [{(log[l−1] (4er)) fi
} ].
(24)
8e
[l]
Let L := min{λfk (g) : k 6= i}. Now, by choosing this time δ := [l]L−ε (which is
Mg [(log[l−1] (
λf (g)+3ε
i
> 1 due to our selection of ε), a further application of Lemma 2 yields
Mg [(log
[l−1]
(4er))
1
[l]
λ (g)+3ε
fi
1
] ≥ Mg [(log[l−1] (4er)) L−ε ]7n
≥
n
Q
k=1
k6=i
Mg [[(log[l−1] (4er))
λ
1
[l]
(g)−ε
fk
]]7
(25)
153
S. K. Datta, T. Biswas, C. Biswas
for r large enough. Now from (23), (24) and (25), it follows for a sequence of values
of r tending to infinity that
Mf (r) ≥ Mg [(log[l−1] (4er))
i.e., Mf [exp[l−1] r
i.e., r
[l]
(λf (g)+3ε)
i
[l]
(λf (g)+3ε)
i
1
[l]
(λ (g)+3ε)
fi
]
] ≥ Mg (4er)
≥ log[l−1] Mf−1 Mg (4er)
[l]
i.e., (λfi (g) + 3ε) log r ≥ log[l] Mf−1 Mg (4er)
log[l] Mf−1 Mg (4er)
[l]
i.e., λfi (g) + 3ε ≥
log(4er) + O(1)
+
If we let ε → 0 then we get
log[l] Mf−1 Mg (4er)
[l]
λfi (g) ≥ lim inf
log(4er) + O(1)
r→∞
Therefore
[l]
λf (g) = lim inf
log[l] Mf−1 Mg (r)
log r
So from (18) and (26) we finally obtain that
r→∞
[l]
.
[l]
≤ λfi (g).
(26)
[l]
λf (g) = λfi (g),
[l]
[l]
if one assumes that λfi (g) 6= λfk (g) for all k ∈ {1, 2, . . . , n}\{i}.
Let now f =
F1
F2
[l]
[l]
with F1 , F2 , f entire, and suppose λF1 (g) ≥ λF2 (g). We
[l]
[l]
[l]
[l]
have F1 = f.F2 . Thus λF1 (g) = λf (g) if λf (g) < λF2 (g). So it follows that
[l]
[l]
[l]
[l]
λF1 (g) < λF2 (g), which contradicts the hypothesis “λF1 (g) ≥ λF2 (g)”. Hence
[l]
[l]
[l]
[l]
[l]
[l]
λf (g) = λ F1 (g) ≥ λF2 (g) = min{λF1 (g), λF2 (g)}. Also suppose that λF1 (g) >
[l]
F2
[l]
[l]
[l]
[l]
[l]
[l]
λF2 (g). Then λF1 (g) = min{λf (g), λF2 (g)} = λF2 (g), if λf (g) > λF2 (g), which
[l]
[l]
[l]
[l]
also a contradiction. Thus λf (g) = λ F1 (g) = min{λF1 (g), λF2 (g)}. This proves
the theorem.
F2
Acknowledgement. The authors are thankful to the referee for his/her
valuable suggestions towards the improvement of the paper.
REFERENCES
[1] L. Bernal, Crecimiento relativo de funciones enteras. Contribución al estudio de lasfunciones
enteras con ı́ndice exponencial finito, Doctoral Dissertation, University of Seville, Spain, 1984
[2] L. Bernal, Orden relativo de crecimiento de funciones enteras, Collect. Math., 39 (1988),
209–229.
[3] S.K. Datta, T. Biswas and R. Biswas, Some results on relative lower order of entire functions,
Casp. J. Appl. Math. Ecol. Econ., 1, 2 (2013), 3–18.
154
Generalized relative lower order of entire functions
[4] S. Halvarsson, Growth properties of entire functions depending on a parameter, Ann. Polon.
Math., 14, 1 (1996), 71–96.
[5] C.O. Kiselman, Order and type as measure of growth for convex or entire functions, Proc.
Lond. Math. Soc., 66, (1993), 152–186.
[6] C.O. Kiselman, Plurisubharmonic functions and potential theory in several complex variable,
a contribution to the book project, Development of Mathematics, 1950–2000, edited by HeanPaul Pier.
[7] B.K. Lahiri and D. Banerjee, Generalised relative order of entire functions, Proc. Nat. Acad.
Sci. India, 72(A), 4 (2002), 351–371.
[8] B.J. Levin, Distribution of Zeros of Entire Functions, American Mathematical Society, Providence (1980).
[9] D. Sato, On the rate of growth of entire functions of fast growth, Bull. Amer. Math. Soc.,
69 (1963), 411–414.
[10] E.C. Titchmarsh, The Theory of Functions, 2nd ed. Oxford University Press, Oxford (1968).
[11] G. Valiron, Lectures on the General Theory of Integral Functions, Chelsea Publishing Company (1949).
(received 19.11.2013; in revised form 01.03.2014; available online 15.04.2014)
S. K. Datta, Department of Mathematics, University of Kalyani, P.O. Kalyani, Dist-Nadia, PIN741235, West Bengal, India
E-mail: sanjib kr datta@yahoo.co.in
T. Biswas, Rajbari, Rabindrapalli, R. N. Tagore Road, P.O. Krishnagar, Dist-Nadia, PIN- 741101,
West Bengal, India
E-mail: tanmaybiswas math@rediffmail.com
C. Biswas, Taraknagar Jamuna Sundari High School, Vill+P.O. Taraknagar, P.S. Hanskhali, Dist.Nadia, PIN-741502, West Bengal, India
E-mail: chinmay.shib@gmail.com