MATEMATIQKI VESNIK
originalni nauqni rad
research paper
65, 4 (2013), 547–554
December 2013
THE UNIVALENCE OF SOME INTEGRAL OPERATORS
USING THE BESSEL FUNCTIONS
Nicoleta Ularu
Abstract. In this paper we will introduce some new integral operators using the generalized
Bessel functions and analytic functions. For this operators we will prove the univalence condition.
1. Preliminaries and definitions
Let A the class of all functions of the form
∞
X
f (z) = z +
an+1 z n+1
(1)
n=1
which are analytic in the unit disk U = {z : |z| < 1} and satisfy the condition
f (0) = f 0 (0) − 1 = 0.
We denote by S the class of univalent functions.
We will consider the generalized Bessel function of the first kind and order ν as
the particular solution of the second-order linear homogenous differential equation
z 2 ω 00 (z) + bzω 0 (z) + [cz 2 − ν 2 + (1 − b)ν]ω(z) = 0.
This solution is denoted by ων,b,c (z) and has the familiar infinite sum representation
∞
³ z ´2n+ν
X
(−c)n
z ∈ C,
(2)
ω(z) = ων,b,c (z) =
2
n! Γ(ν + n + b+1
2 )
n=0
where Γ is the Euler gamma function. Considering this series we can study the
Bessel, modified Bessel and spherical Bessel functions. Geometric properties for
this were obtained by Baricz in [4].
The Bessel functions are obtained for b = c = 1 in (2) and are defined by
(see [4])
∞
³ z ´2n+ν
X
(−1)n
Jν (z) =
,
n! Γ(ν + n + 1) 2
n=0
2010 AMS Subject Classification: 30C45
Keywords and phrases: Bessel functions; integral operator; univalent; analytic functions.
547
548
N. Ularu
for z ∈ C. For b = −c = 1 in (2) are obtained the modified Bessel functions and
are defined by (see [4])
Iν (z) =
³ z ´2n+ν
1
, z ∈ C.
n! Γ(ν + n + 1) 2
n=0
∞
X
The spherical Bessel functions are obtained for b − 1 = c = 1 in (2) and are defined
by (see [4])
r
∞
³ z ´2n+ν
X
2
(−1)n
Jν+ 12 (z) =
Kν (z) =
, z ∈ C.
3
z
n! Γ(ν + n + 2 ) 2
n=0
In particular cases Bessel functions of the first kind reduces to some elementary
functions, like sine and cosine and modified Bessel functions of the first kind are
reduced to hyperbolic sine and cosine.
The generalized Bessel function of the first kind were studied by Á. Baricz
in [3]. Recently, in 2010 Baricz and Frasin proved in [6] the univalence of some
integral operators involving generalized Bessel functions and with Ponnusamy some
conditions of starlikeness and convexity of generalized Bessel functions in [7].
We consider the function defined by
√
b + 1 ν/2
)z ων,b,c ( z)
2
and using the Pochhammer symbol, defined in terms of Euler-Gamma function
½
1,
ν = 0, λ ∈ C − 0
Γ(λ + µ)
(λ)ν =
=
Γ(λ)
λ(λ + 1) . . . (λ + n − 1), µ = n, λ ∈ C
uν,b,c (z) = 2ν Γ(ν +
we obtain for the function uν,b,c (z) the following series representation
uν,b,c (z) = z +
for κ := ν +
b+1
2
∞
X
(−c)n n+1
z
4n (κ)n n!
n=1
(3)
6∈ Z0 .
More results about function uν,b,c (z) of generalized Bessel function ων,b,c (z)
we find in the papers of Baricz in [4] and [5], where he proves some geometric
properties and also some interesting inequalities that involving generalized Bessel
functions.
To prove our main results we will use Ahlfors [1] and Becker [2] univalence
criterion:
Theorem 1.1. Let d be a complex number, |d| ≤ 1, d 6= −1. If f (z) =
z + a2 z 2 + . . . is a regular function in U and
¯
¯
00
¯
¯
¯d|z|2 + (1 − |z|2 ) zf (z) ¯ ≤ 1
(4)
¯
f 0 (z) ¯
for all z ∈ U , then the function f is regular and univalent in U.
549
The univalence of some integral operators
Also we will use some lemmas that were proven by E. Deniz, H. Orhan and
H. M. Srivastava in [8]. These were obtained by the generalization of the results
obtained by Baricz and Ponnusamy in [7].
Lemma 1.1. If the parameters ν, b ∈ R and c ∈ C are so constrained that
½
¾
|c| − 2
κ > max 0,
,
4
then the function ϕν,b,c : U → C,
∞
X
(−c)n n+1
ϕν,b,c (z) = z +
z
n
4 (κ)n n!
n=1
satisfies the following inequalities:
¯
¯
¯
¯ 0
(κ + 1) |c|
¯ϕν,b,c (z) − ϕν,b,c (z) ¯ ≤
¯ κ[4(κ + 1) − |c|] , z ∈ U ,
¯
z
¯ 0
¯
¯ zϕν,b,c (z)
¯
8(κ + 1) |c|
¯
¯≤
−
1
¯ ϕν,b,c (z)
¯ 32κ(κ + 1) − 8(2κ + 1) |c| + |c|2 , z ∈ U ,
¯ 4κ(κ + 1) + (κ + 2) |c|
4κ(κ + 1) − (3κ + 2) |c| ¯¯ 0
≤ zϕν,b,c (z)¯ ≤
, z ∈ U,
κ[4(κ + 1) − |c|]
κ[4(κ + 1) − |c|]
¯ 2 00 ¯ |c| 4(κ + 1) + |c|
¯z ϕν,b,c ¯ ≤
, z ∈ U,
2κ4(κ + 1) − |c|
and
¯
¯
2
¯ zϕ00 (z) ¯
4(κ + 1) |s| + |c|
¯ ν,b,c ¯
, z ∈ U.
¯≤
¯ 0
¯ ϕν,b,c (z) ¯ 8κ(κ + 1) − 2(3κ + 2) |c|
(5)
(6)
Lemma 1.2. If the parameters ν, b ∈ R and c ∈ C are so constrained that
½
¾
|c|
k > max 0,
−1
8
then the function ϕν,b,c : U → C,
∞
X
(−c)n n+1
z
ϕν,b,c (z) = z +
4n (κ)n n!
n=1
satisfies the following inequalities:
¯
¯
µ
¶
¯ 0
¯
¯ϕν,b,c (z) − ϕν,b,c (z) ¯ ≤ |c| 8(κ + 1) + |c| , z ∈ U ,
¯
¯ 4κ 8(κ + 1) − |c|
z
¯ 0
¯
2
¯ zϕν,b,c (z)
¯
8(κ + 1) |c| + |c|
¯
¯≤
−
1
¯ ϕν,b,c (z)
¯ 32κ(κ + 1) − 4(2κ + 3) |c| ,
(7)
2
¯
¯
32κ(κ + 1) − 8(3κ + 2) |c| − |c|
≤ ¯zϕ0ν,b,c (z)¯
4κ[8(κ + 1) − |c|]
2
≤
32κ(κ + 1) + 4(3κ + 4) |c| + |c|
, z ∈ U,
4κ[8(κ + 1) − |c|]
550
N. Ularu
¯ 2 00
¯
¯z ϕν,b,c (z)¯ ≤ |c|
2κ
Ã
2
|c|
8(κ + 2) + |c| 8(κ + 1) + |c|
·
+
8(κ + 1) 8(κ + 2) − |c| 8(κ − 1) − |c|
!
, z∈U
and¯
¯
¯ zϕ00 (z) ¯ |c|
¯ ν,b,c ¯
×
¯ 0
¯≤
¯ ϕν,b,c (z) ¯
2
2
×
64(k + 1)2 [8(κ + 2) − |c|] + 128(κ + 1)(κ + 2) − [8(κ + 2) + |c|] |c|
. (8)
2(κ + 1)[8(κ + 1) − |c|]16(κ + 1)(2κ − |c|) − |c| (4κ + |c|)
In this paper, using the generalized Bessel functions, we will introduce the
operators:
¶γ
Z zY
n µ
uνi ,b,c (t) i
Iνi (u, g)(z) =
dt,
(9)
gi (t)
0 i=1
Z zY
n
(uνi ,b,c (t))γi
Jνi (u, g)(z) =
dt
(10)
(gi (t))σi
0 i=1
Z z
Tν (u, g)(z) =
(u0ν,b,c (t)eg(t) )α dt,
(11)
0
where uνi ,b,c (z) are generalized Bessel functions and gi (z) ∈ A.
These operators are derived from the operators defined in [9] and [10].
2. Main results
Theorem 2.1. Let νi , b, c ∈ R, κi > |c|−2
for κi = νi + b+1
4
2 and the function
1,
n.
If
κ
=
min{κ
,
.
.
.
,
κ
},
γi ∈ C − 0, d ∈ C,
uνi ,b,c (z) be defined by (3) ¯for i =
1
n
¯
¯ zgi0 (z) ¯
d 6= −1, gi (z) ∈ A with ¯ gi (z) ¯ ≤ Mi for Mi ≥ 1, i = 1, n and if we have the
inequality
Ã
!
n
X
8(κ + 1) |c|
≤ 1,
(12)
|d| +
|γi | 1 +
2 + Mi
32κ(κ + 1) − 8(2κ + 1) |c| + |c|
i=1
then the operator Iνi (u, g)(z) defined by (9) is in the univalent function class S.
Proof. From the definition of Iνi (u, g)(z) we obtain that
µ 0
¶
n
zuνi ,b,c (z) zgi0 (z)
zIν00i (u, g)(z) X
=
γ
−
.
i
Iν0 i (u, g)(z)
uνi ,b,c (z)
gi (z)
i=1
It follows that
¯ 00
¯
¯ ¯
¯¶
µ¯ 0
n
¯ zIνi (u, g)(z) ¯ X
¯ zuνi ,b,c (z) ¯ ¯ zgi0 (z) ¯
¯
¯≤
¯
¯
¯
¯
|γi | ¯
¯ I 0 (u, g)(z) ¯
¯ + ¯ gi (z) ¯ .
u
(z)
ν
,b,c
νi
i
i=1
We will use relation (5) from Lemma 1.1 and we obtain
¯
¯ 0
¯ zuνi ,b,c (z) ¯
8(κi + 1) |c|
¯
¯
¯ uν ,b,c (z) ¯ ≤ 1 + 32κ (κ + 1) − 8(2κ + 1) |c| + |c|2 .
i
i i
i
(13)
(14)
551
¯ 0 ¯
¯ zg (z) ¯
From (14) and from the hypothesis that ¯ gii z ¯ ≤ Mi the relation (13) is equivalent
with
!
Ã
¯ 00
¯
n
¯ zIνi (u, g)(z) ¯ X
8(κ
+
1)
|c|
i
¯
¯
|γi | 1 +
2 + Mi .
¯ I 0 (u, g)(z) ¯ ≤
32κ (κ + 1) − 8(2κ + 1) |c| + |c|
The univalence of some integral operators
νi
i=1
³
´
i
i
i
We define the function G : |c|−2
4 , ∞ → R, G(x) =
is a decreasing function, so it follows that
8(κi + 1) |c|
32κi (κi + 1) − 8(2κi + 1) |c| + |c|
2
≤
8(x+1)|c|
.
32x(x+1)−8(2x+1)|c|+|c|2
8(κ + 1) |c|
This
2.
32κ(κ + 1) − 8(2κ + 1) |c| + |c|
In order to prove the univalence we will use Theorem 1.1. So we have that
¯
¯
00
¯
¯
¯d|z|2 + (1 − |z|2 ) zIνi (u, g)(z) ¯
¯
0
Iνi (u, g)(z) ¯
Ã
!
n
X
8(κ + 1) |c|
≤ |d| +
|γi | 1 +
2 + Mi . (15)
32κ(κ + 1) − 8(2κ + 1) |c| + |c|
i=1
Using (5) in (15) we obtain that
¯
¯
00
¯
¯
¯d|z|2 + (1 − |z|2 ) zIνi (u, g)(z) ¯ ≤ 1,
¯
0
Iνi (u, g)(z) ¯
so the operator Iνi (u, g)(z) is in the univalent function class S.
For γ1 = · · · = γn = γ and M1 = · · · = Mn = M in Theorem 2.1 we obtain
Corollary 2.1. Let νi , b, c ∈ R, κi > |c|−2
for κi = νi + b+1
4
2 and the function
uνi ,b,c (z) be defined by (3) ¯for i =¯ 1, n. If κ = min{κ1 , . . . , κn }, γ ∈ C − 0, d ∈ C,
¯ zg0 (z) ¯
d 6= −1, gi (z) ∈ A with ¯ gii(z) ¯ ≤ M for M ≥ 1, i = 1, n and if we have the
inequality
Ã
!
8(κ + 1) |c|
|d| + n |γ| 1 +
≤ 1,
2 +M
32κ(κ + 1) − 8(2κ + 1) |c| + |c|
´
R z Qn ³ u
(t) γ
dt is in the univalent function
then the operator Iνi (u, g)(z) = 0 i=1 νgi ,b,c
i (t)
class S.
Theorem 2.2. Let νi , b, c ∈ R, κi > |c|−2
for κi = νi + b+1
4
2 and the function
uνi ,b,c (z) be defined by (3) ¯for i =
1,
n.
If
κ
=
min{κ
,
.
.
.
,
κ
},
γi ∈ C − 0, d ∈ C,
1
n
¯
¯ zgi0 (z) ¯
d 6= −1, gi (z) ∈ A with ¯ gi (z) ¯ ≤ Mi for Mi ≥ 1, i = 1, n and if we have the
inequality
Ã
!
n
n
X
X
8(κ + 1) |c|
|d| +
|γi | 1 +
+
σi Mi ≤ 1,
2
32κ(κ + 1) − 8(2κ + 1) |c| + |c|
i=1
i=1
then the operator Jνi (u, g)(z) defined by (10) is in the univalent function class S.
552
N. Ularu
Proof. Similar with the proof of Theorem 2.1.
b+1
Theorem 2.3. Let νi , b, c ∈ R, κi > |c|
8 − 1 for κi = νi + 2 and the function
uνi ,b,c (z) be defined by (3) ¯for i =
¯ 1, n. If κ = min{κ1 , . . . , κn }, γi ∈ C − 0, d ∈ C,
¯ zgi0 (z) ¯
d 6= −1, gi (z) ∈ A with ¯ gi (z) ¯ ≤ Mi for Mi ≥ 1, i = 1, n and if we have the
inequality
!
Ã
n
2
X
8(κ + 1) |c| + |c|
|d| +
|γi | 1 +
+ Mi ≤ 1,
(16)
32κ(κ + 1) − 4(2κ + 3) |c|
i=1
then the operator Iνi (u, g)(z) defined by (9) is in the univalent function class S.
Proof. From (9) we obtain that
n
zIν00i (u, g)(z) X
=
γi
Iν0 i (u, g)(z)
i=1
and
µ
zu0νi ,b,c (z) zgi0 (z)
−
uνi ,b,c (z)
gi (z)
¶
¯ 00
¯
¯ ¯
¯¶
µ¯ 0
n
¯ zIνi (u, g)(z) ¯ X
¯ zuνi ,b,c (z) ¯ ¯ zgi0 (z) ¯
¯
¯≤
¯
¯
¯
¯
|γi | ¯
¯ I 0 (u, g)(z) ¯
¯ + ¯ gi (z) ¯ .
u
(z)
ν
,b,c
νi
i
i=1
Now we will use the relation (7) from Lemma 1.2 and we obtain that
¯ 0
¯
2
¯ zuνi ,b,c (z) ¯
8(κi + 1) |c| + |c|
¯
¯≤1+
.
¯ uν ,b,c (z) ¯
32κi (κi + 1) − 4(2κi + 3) |c|
i
¯ 0 ¯
¯ zg (z) ¯
Using this and the hypothesis that ¯ gii(z) ¯ ≤ Mi it follows that
!
Ã
¯
¯ 00
n
2
¯ zIνi (u, g)(z) ¯ X
8(κi + 1) |c| + |c|
¯≤
¯
.
|γi | 1 +
¯ I 0 (u, g)(z) ¯
32κi (κi + 1) − 4(2κi + 3) |c|
νi
i=1
´
³
8(x+1)|c|+|c|
We consider the function H : |c|
8 − 1, ∞ , H(x) = 32x(x+1)−4(2x+3)|c| that is a
decreasing function, so
2
2
8(κi + 1) |c| + |c|
8(κ + 1) |c| + |c|
≤
.
32κi (κi + 1) − 4(2κi + 3) |c|
32κ(κ + 1) − 4(2κ + 3) |c|
Using Theorem 1.1 it follows that
¯
¯
00
¯
¯
¯d|z|2 + (1 − |z|2 ) zIνi (u, g)(z) ¯
¯
0
Iνi (u, g)(z) ¯
n
X
Ã
2
8(κ + 1) |c| + |c|
+ Mi
≤ |d| +
|γi | 1 +
32κ(κ + 1) − 4(2κ + 3) |c|
i=1
!
≤ 1.
From the above relation it follows that Iνi (u, g)(z) ∈ S.
b+1
Theorem 2.4. Let νi , b, c ∈ R, κi > |c|
8 − 1 for κi = νi + 2 and the function
uνi ,b,c (z) be defined by (3) for i = 1, n. If κ = min{κ1 , . . . , κn }, γi ∈ C − 0, d ∈ C,
553
¯ 0 ¯
¯ zg (z) ¯
d 6= −1, gi (z) ∈ A with ¯ gii(z) ¯ ≤ Mi for Mi ≥ 1, i = 1, n and if we have the
inequality
Ã
!
n
n
2
X
X
8(κ + 1) |c| + |c|
|d| +
|γi | 1 +
+
σi Mi ≤ 1,
(17)
32κ(κ + 1) − 4(2κ + 3) |c|
i=1
i=1
The univalence of some integral operators
then the operator Iνi (u, g)(z) defined by (9) is in the univalent function class S.
Proof. The proof is similar with the proof of Theorem 2.3.
Theorem 2.5. Let ν, b, c ∈ R, κ > |c|−2
for κ = ν + b+1
4
2 and the function
uν,b,c (z) be defined by (3). If α ∈ C − 0, d ∈ C − −1, g(z) ∈ A with |zgi0 (z)| ≤ M
and if we have the inequality
Ã
!
2
4(κ + 1) |c| + |c|
|d| + |α|
+ M ≤ 1,
8κ(κ + 1) − 2(3κ + 2) |c|
then the operator Tν (u, g)(z) defined by (11) is in the class of univalent function S.
Proof. From the definition of Tν (u, g)(z) we obtain that
µ 00
¶
zuν,b,c (z)
zTν00 (u, g)(z)
0
=α
+ zg (z)
Tν0 (u, g)(z)
uν,b,c (z)
and
¯ 00
¯
¯
µ¯ 00
¶
¯ zTν (u, g)(z) ¯
¯ zu
(z) ¯
¯
¯ ≤ |α| ¯ ν,b,c ¯ + |zg 0 (z)| .
(18)
¯ T 0 (u, g)(z) ¯
¯ uν,b,c (z) ¯
ν
For the function uν,b,c (z) we will use the relation (6) from Lemma 1.1 and we obtain
that
¯
¯ 00
2
¯ zuν,b,c (z) ¯
4(κ + 1) |c| + |c|
¯≤
¯
¯ uν,b,c (z) ¯ 8κ(κ + 1) − 2(3κ + 2) |c| .
Now using the hypothesis that |zg(z)| ≤ M and the above relation from (18) we
obtain
!
Ã
¯ 00
¯
2
¯ zTν (u, g)(z) ¯
4(κ
+
1)
|c|
+
|c|
¯
¯
¯ T 0 (u, g)(z) ¯ ≤ |α| 8κ(κ + 1) − 2(3κ + 2) |c| + M .
ν
To prove the univalence we will use Theorem 1.1. So from Theorem 1.1 and from
the theorem hypothesis it follows that
Ã
!
¯
¯
2
00
¯
¯
zT
(u,
g)(z)
4(κ
+
1)
|c|
+
|c|
ν
2
2
¯d|z| + (1 − |z|)
¯ ≤ |d| + |α|
+ M ≤ 1,
¯
T 0 (u, g)(z) ¯
8κ(κ + 1) − 2(3κ + 2) |c|
ν
which implies that the operator is in the univalent function class S.
b+1
Theorem 2.6. Let ν, b, c ∈ R, κ > |c|
8 − 1 for κ = ν + 2 and the function
uν,b,c (z) be defined by (3). If γ ∈ C − 0, d ∈ C, d 6= −1, g(z) ∈ A with |zg 0 (z)| ≤ M
for M ≥ 1 and if we have the inequality
|d| + |α| ×
Ã
!
2
|c| 64(κ + 1)2 [8(κ + 2) − |c|] + 128(κ + 1)(κ + 2) − [8(κ + 2) + |c|] |c|
×
·
+M
2
2(κ + 1)[8(κ + 1) − |c|]16(κ + 1)(2κ − |c|) − |c| (4κ + |c|)
≤ 1,
554
N. Ularu
then the operator Tν (u, g)(z) defined by (11) is in the univalent function class S.
Proof. The proof is similar with the proof of Theorem 2.5, or the results from
Lemma 1.2 can be used.
Acknowledgements. This work was partially supported by the strategic project POSDRU 107/1.5/S/77265, inside POSDRU Romania 2007-2013 cofinanced by the European Social Fund-Investing in People. The author would like
to thank the referees for the useful comments and suggestions.
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[9] N. Ularu, D. Breaz, Univalence conditions and some properties for some new integral operators, submitted.
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Letters 25 (2012), 658–661.
(received 20.01.2012; in revised form 19.05.2012; available online 01.10.2012)
University of Pitesti, str Târgul din Vale, no. 1, Piteşti, România
E-mail: nicoletaularu@yahoo.com