TAIWANESE JOURNAL OF MATHEMATICS Vol. 15, No. 2, pp. 883-917, April 2011 This paper is available online at http://www.tjm.nsysu.edu.tw/
SOME SUFFICIENT CONDITIONS FOR UNIVALENCE OF CERTAIN FAMILIES OF INTEGRAL OPERATORS INVOLVING GENERALIZED BESSEL FUNCTIONS Erhan Deniz, Halit Orhan and H. M. Srivastava* Abstract. The main object of this paper is to give sufficient conditions for certain families of integral operators, which are defined here by means of the normalized form of the generalized Bessel functions, to be univalent in the open unit disk. In particular cases, we find the corresponding simpler conditions for integral operators involving the Bessel function, the modified Bessel function and the spherical Bessel function.
1. INTRODUCTION, DEFINITIONS AND PRELIMINARIES Let A be the class of functions of the form: f (z) = z +
∞
an z n ,
n=2
which are analytic in the open unit disk U := {z : z ∈ C
and |z| < 1}
and satisfy the usual normalization condition: f (0) = f (0) − 1 = 0. We denote by S the subclass of A consisting of functions f (z) ∈ A which are univalent in U. In the past two decades, many authors have determined various sufficient conditions for the univalence of various general families of integral operators as follows. Received December 15, 2010, accepted December 30, 2010. Communicated by Jen-Chih Yao. 2010 Mathematics Subject Classification: Primary 30C45, 33C10; Secondary 30C20, 30C75. Key words and phrases: Analytic functions, Open unit disk, Univalent functions, Univalence conditions, Integral operators, Generalized Bessel functions, Bessel, Modified Bessel and spherical Bessel functions, Ahlfors-Becker and Becker univalence criteria, Schwarz lemma. *Corresponding Author.
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The first family of integral operators, studied by Seenivasagan and Breaz (see [26]), is defined as follows (see also the recent investigations on this subject by Baricz and Frasin [7] and Srivastava et al. [27]): 1/β 1/αj z n fj (t) tβ−1 dt , (1.1) Fα1,··· ,αn ,β (z) = β t 0 j=1
where each of the functions fj (j = 1, · · · , n) belongs to the class A and the parameters αj ∈ C \ {0} (j = 1, · · · , n) and β ∈ C are so constrained that the integral operators in (1.1) exist. Remark 1. We note that, if αj = α (j = 1, · · · , n), then the integral operator Fα1 ,··· ,αn ,β (z) reduces to the operator Fα,β (z) which is related closely to some known integral operators investigated earlier in Geometric Functions Theory (see, for details, [28]). The operators Fα,β (z) and Fα,α (z) were studied by Breaz and Breaz (see [12]) and Pescar (see [23]), respectively. Upon setting β = 1 and α = β = 1 in Fα,β (z), we can obtain the operators Fα,1(z) and F1,1 (z) which were studied by Breaz and Breaz (see [11]) and Alexander (see [2]), respectively. Furthermore, in their special cases when 1 (j = 1, · · · , n), α the integral operators in (1.1) would obviously reduce to the operator F1/α,1(z) which was studied by Pescar and Owa (see [24]). In particular, for α ∈ [0, 1], a special case of the operator F1/α,1(z) was studied by Miller et al. (see [18]). n=β=1
αj =
and
Remark 2. The second family of integral operators was introduced by Breaz and Breaz (see [13]) and it has the following form (see also a recent investigation on this subject by Breaz et al. [15]): 1/(nγ+1) z n [gj (t)]γ dt , (1.2) Gn,γ (z) = (nγ + 1) 0 j=1
where the functions g j ∈ A (j = 1, · · · , n) and the parameter γ ∈ C is so constrained that the integral operators in (1.2) exist. In particular, for n = 1, the integral operator G1,γ (z) was studied by Moldoveanu and Pascu (see [19]). Remark 3. The third family of integral operators was introduced by Breaz and Breaz (see [14]) and it has the following form: 1/µ z n
δ j tµ−1 hj (t) dt , (1.3) Hδ1 ,··· ,δn,µ (z) = µ 0
j=1
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where the functions hj ∈ A (j = 1, · · · , n) and the parameters µ ∈ C and δj ∈ C (j = 1, · · · , n) are so constrained that the integral operators in (1.3) exist. In particular, for µ = 1 in (1.3), the integral operator Hδ1 ,··· ,δn,µ (z) reduces to the operator Hδ1 ,··· ,δn (z) which was studied by Breaz et al. (see [16]). We observe also that, for n = µ = 1, the integral operator H(z) was introduced and studied by Pfaltzgraff (see [25]) and Kim and Merkes (see [17]). Remark 4. The fourth family of integral operators was introduced by Pescar [22] as follows: z
λ 1/λ λ−1 q(t) e t dt , (1.4) Qλ (z) = λ 0
where the function q ∈ A and the parameter λ ∈ C is so constrained that the integral operators in (1.4) exist. Two of the most important and known univalence criteria for analytic functions defined in the open unit disk U were obtained by Ahlfors [1] and Becker [10] and by Becker (see [9]). Some extensions of these two univalence criteria were given by Pescar (see [21]) involving a parameter β (which, for β = 1, yields the Ahlfors-Becker univalence criterion) and by Pascu (see [20]) involving two parameters α and β (which, for β = α = 1, yields Becker’s univalence criterion). In our present investigation, we need these two univalence criteria which we recall here as Lemmas 1 and 2 below. Lemma 1. (see [21]). Let β and c be complex numbers such that (β) > 0
|c| 1 (c = −1).
and
If the function f ∈ A satisfies the following inequality:
zf (z) 2β 2β 1 c |z| + 1 − |z| (z ∈ U), βf (z) then the function F β defined by (1.5)
Fβ (z) = β
0
z
β−1
t
1/β f (t)dt
is in the class S of normalized univalent functions in U. Lemma 2. (see [20]). If f ∈ A satisfies the following inequality: 1 − |z|2(α) zf (z) 1 z ∈ U; (α) > 0 , (α) f (z) then, for all β ∈ C such that (β) (α), the function F β defined by (1.5) is in the class S of normalized univalent functions in U.
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Lemma 3 below is a consequence of the above-mentioned Becker’s univalence criterion (see [9]) and the well-known Schwarz lemma. Lemma 3. (see [22]). Let the parameters λ ∈ C and θ ∈ R be so constrained that (λ) 1, θ > 1
and
√ 2θ |λ| 3 3.
If the function q ∈ A satisfies the following inequality: zq (z) θ (z ∈ U), then the function Q λ : U → C, defined by z
λ 1/λ λ−1 q(t) t dt , e Qλ (z) = λ 0
is in the class S of normalized univalent functions in U. We next consider the following second-order linear homogeneous differential equation (see, for details, [30]):
(b, d, ν ∈ C). (1.6) z 2 ω (z) + bzω (z) + dz 2 − ν 2 + (1 − b)ν ω(z) = 0 A particular solution of the differential equation (1.6), which is denoted by ω ν,b,d (z), is called the generalized Bessel function of the first kind of order ν. In fact, we have the following familiar series representation for the function ων,b,d (z): (1.7)
ων,b,d (z) =
∞
(−d)n n!Γ ν + n + n=0
b+1 2
z 2n+ν 2
(z ∈ C),
where Γ(z) stands for the Euler gamma function. The series in (1.7) permits us to study the Bessel, the modified Bessel and the spherical Bessel functions in a unified manner. Each of these particular cases of the function ων,b,d (z) is worthy of mention here. • For b = d = 1 in (1.7), we obtain the familiar Bessel function Jν (z) defined by (see [30]; see also [3]) (1.8)
Jν (z) =
∞ n=0
z 2n+ν (−1)n n!Γ (ν + n + 1) 2
(z ∈ C).
• For b = −d = 1 in (1.7), we obtain the modified Bessel function Iν (z) defined by (see [30]; see also [3]) (1.9)
Iν (z) =
∞ n=0
z 2n+ν 1 n!Γ (ν + n + 1) 2
(z ∈ C).
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• For b − 1 = d = 1 in (1.7), we obtain the spherical Bessel function, which we choose to denote here by Kν (z), defined by (see [3]; see also Remark 5 below) ∞
z 2n+ν 2 (−1)n Jν+ 1 (z) = (z ∈ C). (1.10) Kν (z) := 2 z 2 n!Γ ν + n + 32 n=0 Remark 5. The spherical Bessel function j n (z) of the first kind is usually defined by (see, for details, [30]) π J 1 (z) (n ∈ Z := {0, ±1, ±2, · · ·}). jn (z) := 2z n+ 2 On the other hand, in the theory of Bessel functions (see, for example, [30]), the notation Kν (z) is usually meant for the modified Bessel function of the third kind (or the Macdonald function). Here, in our present investigation, we have found it to be convenient to use the same notation Kν (z) which is defined here markedly differently by (1.10). We now introduce the function ϕν,b,d (z) defined, in terms of the generalized Bessel function ων,b,d (z), by √ ν b+1 ν z 1− 2 ων,b,d ( z). (1.11) ϕν,b,d (z) = 2 Γ ν + 2 By using the well-known Pochhammer symbol (or the shifted factorial) (λ)µ defined, for λ, µ ∈ C and in terms of the Euler Γ-function, by 1 (µ = 0; λ ∈ C \ {0}) Γ(λ + µ) = (λ)µ := Γ(λ) λ(λ + 1) · · ·(λ + n − 1) (µ = n ∈ N; λ ∈ C), it being understood conventionally that (0) 0 := 1, we obtain the following series representation for the function ϕν,b,d (z) given by (1.11): ∞ b+1 (−d)n z n+1 κ := ν + ∈ / Z0 (1.12) ϕν,b,d (z) = z + 4n (κ)n n! 2 n=1
where N := {1, 2, 3, · · ·}
and
Z0 := {0, −1, −2, · · ·}.
For further results on this relative ϕ ν,b,d (z) of the generalized Bessel function ων,b,d (z), we refer the reader to the recent papers (see, for example, [3-6, 8, 29]), where (among other things) some interesting functional inequalities, integral representations, extensions of some known trigonometric inequalities, and starlikeness, convexity and univalence of normalized analytic functions were
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established. In particular, Baricz and Frasin [7] investigated the univalence of some integral operators of the types given by (1.1), (1.2) and (1.4), which involve the normalized form of the ordinary Bessel function of the first kind. The main object of this paper is to give sufficient conditions for the families of integral operators of the types (1.1), (1.2), (1.3) and (1.4), which involve the normalized forms of the generalized Bessel functions of the first kind to be univalent in the open unit disk U. We also extend and improve the aforementioned results of Baricz and Frasin [7]. At least in some cases, our main results are stronger than the results obtained in [7]. Recently, Baricz and Ponnusamy [8] proved the following lemma. Lemma 4. (see [8]). If the parameters ν, b ∈ R and d ∈ C are so constrained that |d| −1 , κ > max 0, 4 then the function ϕν,b,d (z) :U→C z given by (1.12) satisfies the following inequality: 4κ(κ+1)−(2κ+1) |d|+ 18 |d|2 ϕν,b,d (z) 32κ2 −|d| (1.13) 8κ [4κ−|d|] (z ∈ U). κ [4(κ+1)−|d|] z Lemma 5. If the parameters ν, b ∈ R and d ∈ C are so constrained that |d| − 2 , κ > max 0, 4 then the function ϕν,b,d : U → C defined by (1.12) satisfies the following inequalities: (κ+1) |d| ϕν,b,d (z) (z ∈ U), (1.14) ϕν,b,d (z) − z κ [4(κ+1) − |d|] zϕ (z) 8(κ+1) |d| ν,b,d − 1 (z ∈ U), (1.15) 32κ(κ+1) − 8(2κ + 1) |d|+|d|2 ϕν,b,d (z) (1.16)
(1.17)
4κ(κ+1)+(κ+2) |d| 4κ(κ+1)−(3κ+2) |d| zϕν,b,d (z) κ [4(κ+1)−|d|] κ [4(κ+1)−|d|] |d| 4(κ + 1) + |d| 2 z ϕ ν,b,d (z) 2κ 4(κ + 1) − |d|
(z ∈ U)
(z ∈ U),
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Some Sufficient Conditions for Univalence
and (1.18)
zϕ (z) 4(κ + 1) |d| + |d|2 ν,b,d ϕν,b,d (z) 8κ(κ + 1) − 2(3κ + 2) |d|
(z ∈ U).
Proof. We first prove the assertion (1.14) of Lemma 5. Indeed, by using the well-known triangle inequality: |z1 + z2 | |z1 | + |z2 | and the inequaliy: (1.19)
n!(κ + 1)n−1 n(κ + 1)n−1
(n ∈ N),
we have ∞ ∞ n(−d)n n ϕ (z) n |d|n ν,b,d ϕ = (z) − z ν,b,d z n!4n (κ)n n!4n (κ)n n=1
n=1
n−1 ∞ |d| n |d|n−1 |d| 4n−1 n!(κ + 1)n−1 4κ 4(κ + 1) n=1 n=1 |d| − 2 (κ + 1) |d| κ> . = κ [4(κ + 1) − |d|] 4 |d| = 4κ
∞
Next, by combining the inequalities (1.13) with (1.14), we immediately see that the second assertion (1.15) of Lemma 5 holds true for all z ∈ U if 32κ(κ + 1) − 8(2κ + 1) |d| + |d|2 > 0. In order to prove the assertion (1.16) of Lemma 5, we make use of the well-known triangle inequality and the following inequaliy: (1.20)
2 · n!(κ + 1)n−1 (n + 1)(κ + 1)n−1
(n ∈ N).
We thus find that ∞ ∞ n (n + 1)(−d) (n + 1) |d|n n+1 = z + zϕ (z) z 1 + ν,b,d n!4n (κ)n n!4n (κ)n n=1
n=1
n−1 ∞ ∞ |d| (n + 1) |d|n−1 |d| |d| =1+ 1+ 2κ 4n−1 2n!(κ + 1)n−1 2κ 4(κ + 1) n=1 n=1 |d| − 2 4κ(κ + 1) + (κ + 2) |d| κ> , = κ [4(κ + 1) − |d|] 4
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Erhan Deniz, Halit Orhan and H. M. Srivastava
which is obviously positive if 4κ(κ + 1) + (κ + 2) |d| > 0. Similarly, by using the reverse triangle inequality: |z1 − z2 | |z1 | − |z2 | and the inequality (1.20), we have ∞ ∞ n (n + 1)(−d) (n + 1) |d|n n+1 = z + zϕ (z) z 1 − ν,b,d n!4n (κ)n n!4n (κ)n n=1 n=1 n−1 ∞ ∞ |d| (n + 1) |d|n−1 |d| |d| =1− 1− 2κ 4n−1 2n!(κ + 1)n−1 2κ 4(κ + 1) n=1 n=1 |d| − 2 4κ(κ + 1) − (3κ + 2) |d| κ> , = κ [4(κ + 1) − |d|] 4 which is positive if
4κ(κ + 1) − (3κ + 2) |d| > 0.
We now prove the assertion (1.17) of Lemma 5 by using again the triangle inequality and the following inequality: (1.21)
4 · (n − 1)!(κ + 1)n−1 (n + 1)(κ + 1)n−1
(n ∈ N \ {1}).
We thus have ∞ ∞ n 2 (n + 1)n(−d) (n + 1) |d|n n+1 = z ϕ (z) z ν,b,d n!4n (κ)n (n − 1)!4n(κ)n n=1 n=1 ∞ |d|n−1 |d| 1 n+1 + = κ 2 4 · (n − 1)! 4n−1 (κ + 1)n−1 n=2 n−1 ∞ |d| |d| 1 + κ 2 4(κ + 1) n=2 |d| − 2 |d| 4(κ + 1) + |d| κ> . = 2κ 4(κ + 1) − |d| 4 Finally, by combining the inequalities (1.16) and (1.17), we immediately deduce that (v) holds true for all z ∈ U. Thus the proof of Lemma 5 is completed. 2. UNIVALENCE OF INTEGRAL OPERATORS INVOLVING THE GENERALIZED BESSEL FUNCTIONS Our first main result provides an application of Lemma 5 and contains sufficient univalence conditions for integral operators of the type (1.1) when the functions fj (j = 1, · · · , n) are normalized forms of the generalized Bessel functions involving various parameters.
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Some Sufficient Conditions for Univalence
Theorem 1. Let the parameters ν1 , · · · , νn , b ∈ R and d ∈ C be so constrained that |d| − 2 b+1 κj = νj + (j = 1, · · · , n) . κj > 4 2 Consider the functions ϕ νj ,b,d : U → C defined by √ b+1 νj z 1−νj /2 ωνj ,b,d ( z). (2.1) ϕνj ,b,d (z) = 2 Γ νj + 2 Suppose also that κ = min{κ1 , · · · , κn}, (β) > 0, c ∈ C\{−1} and αj ∈ C\{0} (j = 1, · · · , n) and that these numbers satisfy the following inequality: n 1 8(κ + 1) |d| 1. |c| + 2 32κ(κ + 1) − 8(2κ + 1) |d| + |d| j=1 |βαj |
Then the function F ν1 ,··· ,νn ,b,d,α1,··· ,αn,β (z) : U → C, defined by (2.2)
Fν1 ,··· ,νn ,b,d,α1 ,··· ,αn ,β (z) = β
z
tβ−1
0
n ϕνj ,b,d(t) 1/αj t
j=1
1/β dt
is in the class S of normalized univalent functions in U. Proof.
We begin by setting β = 1 in (2.2) so that Fν1 ,··· ,νn ,b,d,α1,··· ,αn ,1 (z) =
n z ϕ
νj ,b,d (t)
0 j=1
1/αj
t
dt.
First of all, we observe that, since ϕνj ,b,d ∈ A (j = 1, · · · , n), ϕνj ,b,d (0) = ϕνj ,b,d (0) − 1 = 0. Therefore, clearly, Fν1 ,··· ,νn ,b,d,α1 ,··· ,αn,1 ∈ A, that is, Fν1 ,··· ,νn ,b,d,α1 ,··· ,αn ,1 (0) = Fν 1 ,··· ,νn ,b,d,α1,··· ,αn,1 (0) − 1 = 0. On the other hand, it is easy to see that (2.3)
Fν 1 ,··· ,νn ,b,d,α1,··· ,αn,1 (z)
=
n ϕνj ,b,d(z) 1/αj j=1
z
,
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and Fν1 ,··· ,νn ,b,d,α1,··· ,αn ,1 (z) (2.4)
(1−αj )/αj n 1 ϕνj ,b,d (z) = αj z j=1 n zϕνj ,b,d (z) − ϕνj ,b,d (z) ϕνk ,b,d (z) 1/αk · . z2 z k=1 (k=j)
We thus find from (2.3) and (2.4) that zFν1 ,··· ,νn ,b,d,α1,··· ,αn ,1 (z)
n 1 = Fν 1 ,··· ,νn ,b,d,α1 ,··· ,αn,1 (z) αj
j=1
zϕνj ,b,d (z) ϕνj ,b,d (z)
−1 .
Now, by using the inequality (1.15) in Lemma 5 for each νj (j = 1, · · · , n), we obtain n zF ν1 ,··· ,νn ,b,d,α1 ,··· ,αn,1 (z) 1 zϕνj ,b,d (z) − 1 Fν1 ,··· ,νn ,b,d,α1,··· ,αn ,1 (z) |αj | ϕνj ,b,d (z)
j=1 n j=1
8(κj + 1) |d| 1 |αj | 32κj (κj + 1) − 8(2κj + 1) |d| + |d|2
n 8(κ + 1) |d| 1 |αj | 32κ(κ + 1) − 8(2κ + 1) |d| + |d|2 j=1 |d| − 2 b+1 > (j = 1, · · · , n) , z ∈ U; κ, κj := νj + 2 4 where we have also used the fact that the function |d| − 2 , ∞ → R, φ: 4
defined by φ(x) =
8(x + 1) |d|
32x(x + 1) − 8(2x + 1) |d| + |d|2 is decreasing and, consequently, we have
,
8(κ + 1) |d| 8(κj + 1) |d| . 2 32κj (κj + 1) − 8(2κj + 1) |d| + |d| 32κ(κ + 1) − 8(2κ + 1) |d| + |d|2 Finally, by using the triangle inequality and the assertion of Theorem 1, we get
zF (z) ,··· ,ν ,b,d,α ,··· ,α ,1 ν 2β 2β n n 1 1 c |z| + 1 − |z| βFν1 ,··· ,νn ,b,d,α1 ,··· ,αn ,1 (z) |c| +
n 1 8(κ + 1) |d| 1, 2 32κ(κ + 1) − 8(2κ + 1) |d| + |d| j=1 |βαj |
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which, in view of Lemma 1, implies that Fν1 ,··· ,νn ,b,d,α1 ,··· ,αn ,β ∈ S. This evidently completes the proof of Theorem 1. Upon setting α1 = · · · = αn = α in Theorem 1, we immediately arrive at the following application of Theorem 1. Corollary 1. Let the parameters ν 1 , · · · , νn , b, c, d, β and κj (j = 1, · · · , n) be as in Theorem 1. Also let the functions ϕ νj ,b,d : U → C be defined by (2.1) . Suppose that κ = min{κ1 , · · · , κn } and α ∈ C \ {0} and that the following inequality holds true: |c| +
8(κ + 1) |d| n 1 |βα| 32κ(κ + 1) − 8(2κ + 1) |d| + |d|2
Then the function F ν1 ,··· ,νn ,b,d,α,β (z) : U → C, defined by 1/β 1/α z n ϕνj ,b,d (t) tβ−1 dt , (2.5) Fν1 ,··· ,νn ,b,d,α,β (z) = β t 0 j=1
is in the class S of normalized univalent functions in U. Our second main result contains sufficient univalence conditions for an integral operator of the type (1.2) when the functions gj are the normalized forms of the generalized Bessel functions with various parameters. The key tools in the proof are Lemma 2 and the inequality (1.15) of Lemma 5. Theorem 2. Let the parameters ν1 , · · · , νn , b ∈ R and d ∈ C be so constrained |d| − 2 b+1 κj = νj + (j = 1, · · · , n) . κj > 4 2 Consider the functions ϕ νj ,b,d : U → C defined by (2.1) and let
that
κ = min{κ1 , · · · , κn }
and
(γ) > 0.
Moreover, suppose that the following inequality holds true: |γ|
1 32κ(κ + 1) − 8(2κ + 1) |d| + |d|2 (γ). n 8(κ + 1) |d|
Then the function G ν1 ,··· ,νn ,b,d,n,γ (z) : U → C, defined by 1/(nγ+1) z n γ , ϕνj ,b,d (t) dt (2.6) Gν1 ,··· ,νn ,b,d,n,γ (z) = (nγ + 1) 0 j=1
is in the class S of normalized univalent functions in U.
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Proof.
Let us consider the function Gν1 ,··· ,νn ,b,d,n,γ (z) : U → C defined by Gν1 ,··· ,νn ,b,d,n,γ (z) =
n z ϕ
νj ,b,d (t)
0 j=1
γ
t
dt.
Observe that Gν1 ,··· ,νn ,b,d,n,γ ∈ A, that is, that Gν1 ,··· ,νn ,b,d,n,γ (0) = Gν 1 ,··· ,νn ,b,d,n,γ (0) − 1 = 0. On the other hand, by using the assertion (1.15) of Lemma 5, the assertion of Theorem 2 and the fact that 8(κj + 1) |d| 32κj (κj + 1) − 8(2κj + 1) |d| + |d|2 we have
8(κ + 1) |d| 32κ(κ + 1) − 8(2κ + 1) |d| + |d|2
(j = 1, · · · , n),
z G ν1 ,··· ,νn ,b,d,n,γ (z) G ν1 ,··· ,νn ,b,d,n,γ (z) n |γ| zϕνj ,b,d (z) − 1 ϕνj ,b,d (z) (γ)
1 − |z|2(γ) (γ)
j=1
8(κ + 1) |d| n |γ| 1 (γ) 32κ(κ + 1) − 8(2κ + 1) |d| + |d|2
(z ∈ C).
Now, since (nγ + 1) > (γ) and the function Gν1 ,··· ,νn ,b,d,n,γ can be rewritten in the form: 1/(nγ+1) γ z n (t) ϕ νj ,b,d tnγ dt , Gν1 ,··· ,νn ,b,d,n,γ (z) = (nγ + 1) t 0 j=1
Lemma 2 would imply that Gν1 ,··· ,νn ,b,d,n,γ ∈ S. This evidently completes the proof of Theorem 2. Choosing n = 1 in Theorem 2, we have the following result. Corollary 2. Let the parameters ν, b ∈ R and d ∈ C be so constrained that κ := ν +
|d| − 2 b+1 > . 2 4
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Consider the function ϕ ν,b,d : U → C defined by (1.11). Moreover, suppose that (γ) > 0 and |γ|
32κ(κ + 1) − 8(2κ + 1) |d| + |d|2 (γ). 8(κ + 1) |d|
Then the function G ν,b,d,γ (z) : U → C, defined by (2.7)
Gν,b,d,γ (z) = (γ + 1)
0
z
1/(γ+1) (ϕν,b,d (t)) dt , γ
is in the class S of normalized univalent functions in U. The following result contains another set of sufficient univalence conditions for an integral operator of the type (1.3) when the functions hj are the normalized forms of the generalized Bessel functions involving various parameters. The key tools in the proof are Lemma 1 and the inequality (1.18) of Lemma 5. Theorem 3. Let the parameters ν1 , · · · , νn , b ∈ R and d ∈ C be so constrained that
|d| − 2 b+1 > (j = 1, · · · , n). 2 4 Consider the functions ϕ νj ,b,d : U → C defined by (2.1) . Also let κj := νj +
κ = min{κ1 , · · · , κn }, (µ) > 0, c ∈ C \ {−1} and δ1 , · · · , δn ∈ C. Moreover, suppose that the following inequality holds true: |δj | 4(κ + 1) |d| + |d|2 1. |c| + 8κ(κ + 1) − 2(3κ + 2) |d| |µ| n
j=1
Then the function H ν1 ,··· ,νn ,δ1,··· ,δn ,b,d,µ (z) : U → C, defined by (2.8)
Hν1 ,··· ,νn ,δ1,··· ,δn ,b,d,µ (z) = µ
z 0
1/µ n δj tµ−1 dt , ϕνj ,b,d (t) j=1
is in the class S of normalized univalent functions in U. Proof. Our demonstration of Theorem 3 is much akin to that of Theorem 1. Indeed, by considering the function Hν1 ,··· ,νn ,δ1 ,··· ,δn,b,d (z) : U → C defined by Hν1 ,··· ,νn ,δ1,··· ,δn ,b,d (z) =
n z
0 j=1
ϕνj ,b,d (t)
δj
dt,
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Erhan Deniz, Halit Orhan and H. M. Srivastava
and using the inequality (1.18) of Lemma 5 for each νj (j = 1, · · · , n) in conjunction with the following easily derivable inequality: 4(κ + 1) |d| + |d|2 4(κj + 1) |d| + |d|2 , 8κj (κj + 1) − 2(3κj + 2) |d| 8κ(κ + 1) − 2(3κ + 2) |d| we are led eventually to the sufficient univalence conditions asserted by Theorem 3 by means of the triangle inequality and the assertion of Theorem 3. Setting n = 1 in Theorem 3, we immediately obtain the following result. Corollary 3. Let ν, b ∈ R and d ∈ C such that b + 1 |d| − 2 > . 2 4
κ := ν +
Consider the function ϕ ν,b,d : U → C defined by (1.11). Also let (µ) > 0, c ∈ C \ {−1} and δ ∈ C. Moreover, suppose that the following inequality holds true: δ 4(κ + 1) |d| + |d|2 1. |c| + µ 8κ(κ + 1) − 2(3κ + 2) |d| Then the function H ν,δ,b,d,µ (z) : U → C, defined by (2.9)
Hν,δ,b,d,µ (z) = µ
z
µ−1
t
0
δ ϕν,b,d (t)
1/µ dt ,
is in the class S of normalized univalent functions in U. Next, by applying Lemma 3 and the inequality (1.16) asserted by Lemma 5, we easily get the following result. Theorem 4. Let the parameters ν, b ∈ R and d, λ ∈ C be so constrained that κ := ν +
|d| − 2 b+1 > . 2 4
Consider the generalized Bessel function ϕ ν,b,d defined by (1.11) . If (λ) 1 and √ 3 3κ [4(κ + 1) − |d|] , |λ| 8κ(κ + 1) + 2(κ + 2) |d| then the function Q ν,b,d,λ : U → C, defined by (2.10) is univalent in U.
Qν,b,d,λ (z) = λ
0
z
λ−1
t
λ 1/λ ϕν,b,d (t) dt , e
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Some Sufficient Conditions for Univalence
Taking into account the above results, we have the following particular cases. 2.1. Bessel Functions Choosing b = d = 1, in (1.6) or (1.7), we obtain the Bessel function Jν (z) of the first kind of order ν defined by (1.8). We observe also that √ √ √ √ √ 3 sin z −3 cos z, J1/2(z) = z sin z and J−1/2(z) = z cos z. J3/2(z) = √ z Corollary 4. Let the function J ν : U → C be defined by √ Jν (z) = 2ν Γ (ν + 1) z 1−ν/2 Jν ( z). Also let the following assertions hold true: 1. Let ν1 , · · · , νn > −1.25 (n ∈ N). Consider the functions J νj : U → C defined by √ (j = 1, · · · , n). (2.11) Jνj (z) = 2νj Γ (νj + 1) z 1−νj /2 Jνj ( z) Let ν = min{ν1 , · · · , νn } and let the parameters β, c, α 1, · · · , αn be as in Theorem 1. Moreover, suppose that these numbers satisfy the following inequality: 1 ν +2 1. |c| + 2 4ν + 10ν + 41/8 |βαj | n
j=1
Then the function F ν1 ,··· ,νn ,α1 ,··· ,αn,β (z) : U → C, defined by (2.12)
Fν1 ,··· ,νn ,α1 ,··· ,αn ,β (z) = β
z
tβ−1
0
n Jνj (t) 1/αj j=1
t
1/β dt
,
is in the class S of normalized univalent functions in U. In the particular case when |c| +
28 1 1, 233 |βα|
the function F 3/2,α,β(z) : U → C, defined by F3/2,α,β (z) = β
0
z
β−1
t
√ 1/α 1/β √ 3 sin t 3 cos t √ − dt , t t t
is in the class S of normalized univalent functions in U.
898
Erhan Deniz, Halit Orhan and H. M. Srivastava
2. Let ν1 , · · · , νn > −1.25 (n ∈ N) and consider the normalized Bessel functions Jνj : U → C defined by (2.11). Also let ν = min{ν 1 , · · · , νn } and (γ) > 0 and suppose that these numbers satisfy the following inequality: 1 4ν 2 + 10ν + 41/8 (γ). n ν +2
|γ|
Then the function G ν1 ,··· ,νn ,n,γ (z) : U → C, defined by 1/(nγ+1) z n γ , Jνj (t) dt (2.13) Gν1 ,··· ,νn ,n,γ (z) = (nγ + 1) 0 j=1
is in the class S of normalized univalent functions in U. In the particular case when 89 (γ), 20 the function G 1/2,γ (z) : U → C, defined by |γ|
G1/2,γ (z) = (γ + 1)
0
z
1/(γ+1)
√ √ γ t sin t dt ,
is in the class S of normalized univalent functions in U. 3. Let ν1 , · · · , νn > −1.25 (n ∈ N) and consider the normalized Bessel functions Jνj : U → C defined by (2.11). Let ν = min{ν1 , · · · , νn } and let the parameters µ, c, δ1 , · · · , δn be as in Theorem 3. Moreover, suppose that these numbers satisfy the following inequality: |δj | 4ν + 9 1 8ν 2 + 18ν + 6 |µ| n
|c| +
j=1
Then the function H ν1 ,··· ,νn ,δ1,··· ,δn ,µ (z) : U → C, defined by 1/µ z n δj tµ−1 dt , Jνj (t) (2.14) Hν1 ,··· ,νn ,δ1 ,··· ,δn ,µ (z) = µ 0
j=1
is in the class S of normalized univalent functions in U. In particular, the function H−1/2,δ,µ (z) : U → C, defined by H−1/2,δ,µ (z) = µ
z 0
√ √ δ 1/µ √ t sin t tµ−1 cos t − dt , 2
is in the class S of normalized univalent functions in U.
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Some Sufficient Conditions for Univalence
4. Let λ ∈ C and ν > −1.25 and consider the normalized Bessel function J ν (z) given by (1.8). If (λ) 1 and √ 3 3(ν + 1)(4ν + 7) , |λ| 8ν 2 + 26ν + 22 then the function Q ν,λ : U → C, defined by z
λ 1/λ λ−1 Jν (t) t dt , e (2.15) Qν,λ (z) = λ 0
is in the class S of normalized univalent functions in U. In the particular case when |λ| 1.8959 · · · , the function Q 1/2,λ (z) : U → C, defined by z 1/λ
√ √ λ λ−1 t sin t t dt , e Q1/2,λ(z) = λ 0
is in the class S of normalized univalent functions in U. Remark 6. Baricz and Frasin [7] proved that the following general integral operators: Fν1 ,··· ,νn ,α1 ,··· ,αn ,β (z),
Gν1 ,··· ,νn ,n,γ (z)
and
Qν,λ (z)
defined by (2.12), (2.13) and (2.15), respectively, are actually univalent for all ν, ν1 , · · · , νn > −0.690 98 · · · . From Corollary 4, we see that our results (with ν, ν1 , · · · , νn > −1.25) are stronger than the Baricz-Frasin results for the same integral operators (see, for details, [7]). 2.2. Modified Bessel Functions Taking b = 1 and d = −1 in (1.6) or (1.7), we obtain the modified Bessel function Iν (z) of the first kind of order ν defined by (1.9). We observe also that √ √ 3 sinh z √ , I3/2(z) = 3 cosh z − z √ √ √ and I−1/2 (z) = z cosh z. I1/2(z) = z sinh z Corollary 5. Let the function I ν : U → C be defined by √ Iν (z) = 2ν Γ (ν + 1) z 1−ν/2 Iν ( z). Also let the following assertions hold true: 1. Let ν1 , · · · , νn > −1.25 (n ∈ N). Consider the functions I νj : U → C defined by
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(2.16)
Erhan Deniz, Halit Orhan and H. M. Srivastava
√ Iνj (z) = 2νj Γ (νj + 1) z 1−νj /2 Iνj ( z)
(j = 1, · · · , n).
Let ν = min{ν1 , · · · , νn } and let the parameters β, c, α 1, · · · , αn be as in Theorem 1. Moreover, suppose that these numbers satisfy the following inequality: 1 ν +2 1. |c| + 2 4ν + 10ν + 41/8 |βαj | n
j=1
Then the function F ν1 ,··· ,νn ,α1 ,··· ,αn,β (z) : U → C, defined by (2.17)
Fν1 ,··· ,νn ,α1 ,··· ,αn ,β (z) = β
z
tβ−1
0
n Iνj (t) 1/αj j=1
t
1/β dt
,
is in the class S of normalized univalent functions in U. In the particular case when |c| +
28 1 1, 233 |βα|
the function F 3/2,α,β(z) : U → C, defined by F3/2,α,β(z) = β
z
tβ−1
0
√ 1/α 1/β √ 3 cosh t 3 sinh t √ − dt , t t t
is in the class S of normalized univalent functions in U. 2. Let ν1 , · · · , νn > −1.25 (n ∈ N) and consider the normalized modified Bessel functions I νj : U → C defined by (2.16) . Let ν = min{ν1 , · · · , νn } and (γ) > 0 and suppose that these numbers satisfy the following inequality: 1 4ν 2 + 10ν + 41/8 (γ). |γ| n ν +2 Then the function G ν1 ,··· ,νn ,n,γ (z) : U → C, defined by (2.18)
Gν1 ,··· ,νn ,n,γ (z) = (nγ + 1)
n z
0 j=1
Iνj (t)
γ
1/(nγ+1) dt
,
is in the class S of normalized univalent functions in U. In the particular case when |γ|
89 (γ), 20
Some Sufficient Conditions for Univalence
901
the function G 1/2,γ (z) : U → C, defined by 1/(γ+1) z √ γ √ t sinh t dt , G1/2,γ (z) = (γ + 1) 0
is in the class S of normalized univalent functions in U. 3. Let ν1 , · · · , νn > −1.25 (n ∈ N) and consider the normalized modified Bessel functions I νj : U → C defined by (2.16) . Also let ν = min{ν 1 , · · · , νn } and let the parameters µ, c, δ 1 , · · · , δn be as in Theorem 3. Suppose that these numbers satisfy the following inequality: |δj | 4ν + 9 1. 8ν 2 + 18ν + 6 |µ| n
|c| +
j=1
Then the function H ν1 ,··· ,νn ,δ1,··· ,δn ,µ (z) : U → C, defined by 1/µ z n δj tµ−1 dt , Iν j (t) Hν1 ,··· ,νn ,δ1 ,··· ,δn ,µ (z) = µ 0
j=1
is in the class S of normalized univalent functions in U. In particular, the function H−1/2,δ,µ (z) : U → C, defined by H−1/2,δ,µ (z) = µ
z 0
√ √ δ 1/(γ+1) √ t sinh t tµ−1 cosh t + dt , 2
is in the class S of normalized univalent functions in U. 4. Let λ ∈ C and ν > −1.25 and consider the normalized modified Bessel function Iν (t) given by (1.9). If (λ) 1 and √ 3 3(ν + 1)(4ν + 7) , |λ| 8ν 2 + 26ν + 22 then the function Q ν,λ : U → C, defined by z
λ 1/λ λ−1 Iν (t) t dt , e Qν,λ (z) = λ 0
is in the class S of normalized univalent functions in U. In the particular case when |λ| 1.1809 · · · , the function Q −1/2,λ (z) : U → C, defined by z 1/λ
√ λ λ−1 t cosh t t dt , e Q−1/2,λ (z) = λ 0
is in the class S of normalized univalent functions in U.
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Erhan Deniz, Halit Orhan and H. M. Srivastava
2.3. Spherical Bessel Functions Upon setting b = 2 and d = 1 in (1.6) or (1.7), we obtain the spherical Bessel function Kν (z) of the first kind of order ν defined here by (1.10). Corollary 6. Let K ν : U → C be defined by √ Kν (z) = 2ν Γ (ν + 1) z 1−ν/2 Kν ( z). Also let the following assertions hold true: 1. Let ν1 , · · · , νn > −2.25 (n ∈ N). Consider the functions K νj : U → C defined by √ (j = 1, · · · , n). (2.19) Kνj (z) = 2νj Γ (νj + 1) z 1−νj /2 Kνj ( z) Let ν = min{ν1 , · · · , νn } and let the parameters β, c, α 1, · · · , αn be as in Theorem 1. Moreover, suppose that these numbers satisfy the following inequality: 1 2ν + 5 1. 2 8ν + 28ν + 89/4 |βαj | n
|c| +
j=1
Then the function F ν1 ,··· ,νn ,α1 ,··· ,αn,β (z) : U → C, defined by (2.20)
Fν1 ,··· ,νn ,α1 ,··· ,αn ,β (z) = β
0
z
tβ−1
n Kνj (t) 1/αj j=1
t
1/β dt
,
is in the class S of normalized univalent functions in U. 2. Let ν1 , · · · , νn > −2.25 (n ∈ N) and consider the normalized spherical Bessel functions K νj : U → C defined by (2.19). Let ν = min{ν1 , · · · , νn } and (γ) > 0 and suppose that these numbers satisfy the following inequality: 1 8ν 2 + 28ν + 89/48 (γ). |γ| n ν +5 Then the function G ν1 ,··· ,νn ,n,γ (z) : U → C, defined by 1/(nγ+1) z n γ , Kνj (t) dt (2.21) Gν1 ,··· ,νn ,n,γ (z) = (nγ + 1) 0 j=1
is in the class S of normalized univalent functions in U. 3. Let ν1 , · · · , νn > −2.25 (n ∈ N) and consider the normalized spherical Bessel functions K νj : U → C defined by (2.19). Also let ν = min{ν 1 , · · · , νn } and let
Some Sufficient Conditions for Univalence
903
the parameters µ, c, δ 1 , · · · , δn be as in Theorem 3. Suppose that these numbers satisfy the following inequality: |δj | 4ν + 11 1 |c| + 2 8ν + 26ν + 17 |µ| n
j=1
Then the function H ν1 ,··· ,νn ,δ1,··· ,δn ,µ (z) : U → C, defined by 1/µ z n δj tµ−1 dt , Kν j (t) (2.22) Hν1 ,··· ,νn ,δ1 ,··· ,δn ,µ (z) = µ 0
j=1
is in the class S of normalized univalent functions in U. 4. Let λ ∈ C and ν > −2.25 and consider the normalized spherical Bessel function Kν (t) given by (1.10). If (λ) 1 and √ 3 3(2ν + 3)(4ν + 9) , |λ| 16ν 2 + 68ν + 74 then the function Q ν,λ : U → C, defined by z
λ 1/λ tλ−1 eKν (t) dt , (2.23) Qν,λ (z) = λ 0
is in the class S of normalized univalent functions in U. 3. UNIVALENCE OF INTEGRAL OPERATORS FOR SHARP BOUNDS OF THE GENERALIZED BESSEL FUNCTIONS In our proof of Lemma 5, the key tools were the following inequalities [see also (1.19), (1.20) and (1.21)]: n!(κ + 1)n−1 n(κ + 1)n−1 2 · n!(κ + 1)n−1 (n + 1)(κ + 1)n−1
(κ > −1; n ∈ N), (κ > −1; n ∈ N)
and 4 · (n − 1)!(κ + 1)n−1 (n + 1)(κ + 1)n−1
(κ > −1; n ∈ N \ {1}).
Clearly, the above inequalities can be improved easily by using the following well-known inequality: n! 2n−1
(κ > −1; n ∈ N).
More precisely, we have (3.1)
n!(κ + 1)n−1 [2(κ + 1)]n−1
(κ > −1; n ∈ N)
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Erhan Deniz, Halit Orhan and H. M. Srivastava
and (3.2)
2 · (n − 1)!(κ + 1)n−1 [2(κ + 1)]n−1
(κ > −1; n ∈ N \ {1}).
The following lemma was proven by Baricz and Ponnusamy [8]. that
Lemma 6. (see [8]). If the parameters ν, b ∈ R and d ∈ C are so constrained |d| −1 , κ > max 0, 8
then the function inequalities: (3.3) and (3.4)
ϕν,b,d (z) : U → C defined by (1.12) satisfies the following z
8κ(κ + 1) − (2κ + 3) |d| ϕν,b,d (z) 8κ + |d| 8κ − |d| κ [8(κ + 1) − |d|] z ϕν,b,d (z) |d| 8(κ + 1) + |d| 4κ 8(κ + 1) − |d| z
(z ∈ U)
(z ∈ U).
Lemma 7. If the parameters ν, b ∈ R and d ∈ C are so constrained that |d| −1 , κ > max 0, 8 then the function ϕ ν,b,d : U → C defined by (1.12) satisfies the following inequalities: ϕν,b,d (z) |d| 8(κ + 1) + |d| (z ∈ U), (3.5) 4κ 8(κ + 1) − |d| ϕν,b,d (z) − z zϕ (z) 8(κ + 1) |d| + |d|2 ν,b,d − 1 (z ∈ U), (3.6) 32κ(κ + 1) − 4(2κ + 3) |d| ϕν,b,d (z)
(3.7)
32κ(κ + 1) − 8(3κ + 2) |d| − |d|2 4κ [8(κ + 1) − |d|] zϕ (z) ν,b,d
32κ(κ + 1) + 4(3κ + 4) |d| + |d|2 (z ∈ U), 4κ [8(κ + 1) − |d|] 2 |d| 2 |d| 8(κ+2)+|d| 8(κ+1)+|d| z ϕν,b,d (z) + 2κ 8(κ+1) 8(κ+2)−|d| 8(κ+1)−|d|
(3.8)
(z ∈ U)
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Some Sufficient Conditions for Univalence
and
(3.9)
zϕ (z) |d| ν,b,d ϕν,b,d (z) 2 2
64(κ+1)2 [8(κ+2)−|d|]+128(κ+1)(κ+2)−[8(κ+2) + |d|] |d| · 2(κ+1) [8(κ+1)−|d|] {16(κ+1)(2κ−|d|)−|d| (4κ+|d|)}
(z ∈ U).
Proof. By using the inequality (3.4) in Lemma 6, we obtain ϕν,b,d (z) ϕν,b,d (z) ϕν,b,d (z) ϕ = z ν,b,d (z) − z z z |d| |d| 8(κ + 1) + |d| z ∈ U; κ > −1 , 4κ 8(κ + 1) − |d| 8 which proves the inequality (3.5). Combining the inequality (3.5) of Lemma 7 with the inequality (3.3) of Lemma 6, we immediately arrive at the assertion (3.6) of Lemma 7. By using the triangle inequality and the inequalies (3.1) and (3.2) above, we obtain zϕ ν,b,d (z) ∞ ∞ (n + 1)(−d)n n+1 (n + 1) |d|n z 1 + = z + n!4n (κ)n n!4n (κ)n n=1 n=1 ∞ ∞ |d| |d|n−1 |d|n−1 1+2 =1+ + 4κ 4n−1 2(n − 1)!(κ + 1)n−1 4n−1 n!(κ + 1)n−1 n=2 n=1 n−1 n−1 ∞ ∞ |d| |d| |d| 1+2 + 1+ 4κ 8(κ + 1) 8(κ + 1) n=2 n=1 |d| 32κ(κ + 1) + 4(3κ + 4) |d| + |d|2 z ∈ U; κ > −1 , = 4κ [8(κ + 1) − |d|] 8 which yields the inequality (3.6). Similarly, by using the reverse triangle inequality in conjunction with the inequalities (3.1) and (3.2) above, we find that zϕ ν,b,d (z) ∞ ∞ (n + 1)(−d)n n+1 (n + 1) |d|n z 1 − = z + n!4n (κ)n n!4n (κ)n n=1 n=1 ∞ ∞ |d| |d|n−1 |d|n−1 1+2 =1− + 4κ 4n−1 (n − 1)!(κ + 1)n−1 n=1 4n−1 n!(κ + 1)n−1 n=2
906
Erhan Deniz, Halit Orhan and H. M. Srivastava
n−1 n−1 ∞ ∞ |d| |d| |d| 1− + 1+2 4κ 8(κ + 1) 8(κ + 1) n=2 n=1 |d| 32κ(κ + 1) − 8(3κ + 2) |d| − |d|2 >0 z ∈ U; κ > −1 , = 4κ [8(κ + 1) − |d|] 8 provided that
32κ(κ + 1) − 8(3κ + 2) |d| − |d|2 > 0.
This proves the inequality (3.7). From (1.12) , we obtain the following derivative formulas (see [3, p. 161]): ϕν,b,d (z) d ϕν+1,b,d (z) =− z 4κ z ϕν+2,b,d (z) ϕν,b,d (z) d2 . = z 16κ(κ + 1) z Moreover, we can write ϕν,b,d (z) ϕν,b,d (z) 2 ϕν,b,d (z) − = z z z z
and
or, equivalently, z 2 ϕν,b,d (z)
=z
3
ϕν,b,d (z) z
+ 2z
2
ϕν,b,d (z) z
.
Now, from the last relation, (3.3) and (3.4), we find that 2 3 ϕν,b,d (z) 2 ϕν,b,d (z) z ϕ + 2 |z| ν,b,d (z) = |z| z z ϕν+2,b,d (z) ϕν,b,d (z) |d|2 +2 16κ(κ + 1) z z 8(κ + 2) + |d| |d| 8(κ + 1) + |d| |d|2 + 16κ(κ + 1) 8(κ + 2) − |d| 2κ 8(κ + 1) − |d| |d|2 8(κ + 2) + |d| 8(κ + 1) + |d| |d| + , = 2κ 8(κ + 1) 8(κ + 2) − |d| 8(κ + 1) − |d| which proves the inequality (3.8). Finally, by combining the inequalities (3.7) and (3.8) of Lemma 7, we immediately obtain the assertion (3.9) of Lemma 7. Our proof of Lemma 7 is thus completed. Our first main result in this section is an application of Lemma 7 and contains sufficient conditions for integral operators of the type (2.2).
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Some Sufficient Conditions for Univalence
Theorem 5. Let the parameters ν1 , · · · , νn , b ∈ R and d ∈ C be so constrained that
|d| b+1 > −1 (j = 1, · · · , n). 2 8 Consider the functions ϕ νj ,b,d : U → C defined by (2.1) . Also let κj := νj +
κ = min{κ1 , · · · , κn}, (β) > 0, c ∈ C\{−1} and αj ∈ C\{0} (j = 1, · · · , n). Moreover, suppose that these numbers satisfy the following inequality: 1 8(κ + 1) |d| + |d|2 1. 32κ(κ + 1) − 4(2κ + 3) |d| |βαj | n
|c| +
j=1
Then the function F ν1 ,··· ,νn ,b,d,α1 ,··· ,αn ,β (z) : U → C, defined by (2.2), is in the class S of normalized univalent functions in U. Proof. We begin by expressing the function Fν1 ,··· ,νn ,b,d,α1,··· ,αn (z) : U → C as follows: z n ϕνj ,b,d (t) 1/αj dt. Fν1 ,··· ,νn ,b,d,α1 ,··· ,αn (z) = t 0 j=1
First of all, we observe that, since ϕνj ,b,d ∈ A, that is, ϕνj ,b,d (0) = ϕνj ,b,d (0) − 1 = 0, we have Fν1 ,··· ,νn ,b,d,α1 ,··· ,αn ∈ A, that is,
Fν1 ,··· ,νn ,b,d,α1 ,··· ,αn (0) = Fν 1 ,··· ,νn ,b,d,α1,··· ,αn (0) − 1 = 0.
On the other hand, it is easy to see that n 1 = Fν1 ,··· ,νn ,b,d,α1,··· ,αn (z) αj
zFν1 ,··· ,νn ,b,d,α1 ,··· ,αn (z)
j=1
zϕνj ,b,d (z) ϕνj ,b,d (z)
−1 .
Thus, by using the inequality (3.6) of Lemma 7 for each νj (j = 1, · · · , n), we obtain n zF ν1 ,··· ,νn ,b,d,α1 ,··· ,αn (z) 1 zϕνj ,b,d (z) − 1 Fν1 ,··· ,νn ,b,d,α1,··· ,αn (z) |αj | ϕνj ,b,d (z) j=1 n 8(κj + 1) |d| + |d|2 1 |αj | 32κj (κj + 1) − 4(2κj + 3) |d| j=1 n 8(κ + 1) |d| + |d|2 1 |αj | 32κ(κ + 1) − 4(2κ + 3) |d| j=1
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Erhan Deniz, Halit Orhan and H. M. Srivastava
|d| b+1 > − 1 (j = 1, · · · , n) . z ∈ U; κ = min{κ1 , · · · , κn }; κj := νj + 2 8
− 1, ∞ → R, defined by Here we have used the fact that the function φ : |d| 8
φ(x) =
8(x + 1) |d| + |d|2 , 32x(x + 1) − 4(2x + 3) |d|
is decreasing and, consequently, that 8(κ + 1) |d| + |d|2 8(κj + 1) |d| + |d|2 32κj (κj + 1) − 4(2κj + 3) |d| 32κ(κ + 1) − 4(2κ + 3) |d|
(j = 1, · · · , n).
Finally, by using the triangle inequality and the assertion of Theorem 5, we obtain
zF (z) ,··· ,ν ,b,d,α ,··· ,α ν 2β 2β n n 1 1 c |z| + 1 − |z| βFν1 ,··· ,νn ,b,d,α1,··· ,αn (z) 1 8(κ + 1) |d| + |d|2 1, |c| + 32κ(κ + 1) − 4(2κ + 3) |d| |βαj | n
j=1
which, in view of Lemma 1, implies that Fν1 ,··· ,νn ,b,d,α1 ,··· ,αn ,β ∈ S. This evidently completes the proof of Theorem 5. Upon setting α1 = · · · = αn = α in Theorem 5, we immediately arrive at the following application of Theorem 5. Corollary 7. Let the parameters ν 1 , · · · , νn , b, c, d, β and κj (j = 1, · · · , n) be prescribed as in Theorem 5. Also let κ = min{κ1 , · · · , κn }
and
α ∈ C \ {0}.
Moreover, suppose that the functions ϕ νj ,b,d ∈ A are defined by (2.1) and the following inequality: 8(κ + 1) |d| + |d|2 n 1 |c| + |αβ| 32κ(κ + 1) − 4(2κ + 3) |d| holds true. Then the function F ν1 ,··· ,νn ,b,d,α,β (z) : U → C, defined by (2.5), is in the class S of normalized univalent functions in U. Our second result in this section provides sufficient conditions for the integral operator in (2.6) . The key tools in the proof are Lemma 2 and the inequality (3.6) of Lemma 7.
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Some Sufficient Conditions for Univalence
Theorem 6. Let the parameters ν1 , · · · , νn , b ∈ R and d ∈ C be so constrained that
|d| b+1 > −1 (j = 1, · · · , n). 2 8 Consider the functions ϕ νj ,b,d : U → C defined by (2.1) . Also let κj := νj +
κ = min{κ1 , · · · , κn }
(γ) > 0.
and
Moreover, suppose that these numbers satisfy the following inequality: 1 32κ(κ + 1) − 4(2κ + 3) |d| (γ). |γ| n 8(κ + 1) |d| + |d|2 Then the function G ν1 ,··· ,νn ,b,d,n,γ (z) : U → C, defined by (2.6) , is in the class S of normalized univalent functions in U. Proof.
Let us consider the function Gν1 ,··· ,νn ,b,d,n,γ (z) : U → C defined by Gν1 ,··· ,νn ,b,d,n,γ (z) =
n z ϕ 0 j=1
νj ,b,d (t)
γ
t
dt.
Observe that Gν1 ,··· ,νn ,b,d,n,γ ∈ A, that is, that Gν1 ,··· ,νn ,b,d,n,γ (0) = Gν 1 ,··· ,νn ,b,d,n,γ (0) − 1 = 0. On the other hand, by using the inequality (3.6) of Lemma 7, the assertion of Theorem 6 and the fact that 8(κj + 1) |d| + |d|2 8(κ + 1) |d| + |d|2 32κj (κj + 1) − 4(2κj + 3) |d| 8κ(κ + 1) − 4(2κ + 3) |d| we obtain
z G (z) ν1 ,··· ,νn ,b,d,n,γ G ν1 ,··· ,νn ,b,d,n,γ (z) n |γ| zϕνj ,b,d (z) − 1 (γ) ϕνj ,b,d (z) j=1 8(κ + 1) |d| + |d|2 n |γ| 1 (γ) 32κ(κ + 1) − 4(2κ + 3) |d|
(j = 1, · · · , n),
1 − |z|2(γ) (γ)
Now, since (nγ + 1) > (γ)
(n ∈ N),
(z ∈ U).
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Erhan Deniz, Halit Orhan and H. M. Srivastava
the function Gν1 ,··· ,νn ,b,d,n,γ can be rewritten in the form: 1/(nγ+1) γ z n (t) ϕ νj ,b,d tnγ dt Gν1 ,··· ,νn ,b,d,n,γ (z) = (nγ + 1) t 0 j=1
which, in view of Lemma 2, implies that Gν1 ,··· ,νn ,b,d,n,γ ∈ S. This evidently completes the proof of Theorem 6. Choosing n = 1 in Theorem 6, we have the following result. Corollary 8. Let the parameters ν, b ∈ R and d ∈ C be so constrained that |d| b+1 > − 1. 2 8 : U → C defined by (1.11). Moreover, suppose that
κ := ν +
Consider the function ϕ ν,b,d (γ) > 0 and 32κ(κ + 1) − 4(2κ + 3) |d| (γ). |γ| 8(κ + 1) |d| + |d|2
Then the function G ν,b,d,γ (z) : U → C, defined by 1/(γ+1) z γ (ϕν,b,d (t)) dt , Gν,b,d,n,γ (z) = (γ + 1) 0
is in the class S of normalized univalent functions in U. The following result contains another set of sufficient conditions for integral operators of the type (2.8). The key tools in the proof are Lemma 1 and the inequality (3.9) of Lemma 7. Theorem 7. Let the parameters ν1 , · · · , νn , b ∈ R and d ∈ C be so constrained that
|d| b+1 > −1 (j = 1, · · · , n). 2 8 Consider the functions ϕ νj ,b,d : U → C defined by (2.1) . Also let κj := νj +
κ = min{κ1 , · · · , κn }, (µ) > 0, c ∈ C\{−1} and δj ∈ C (j = 1, · · · , n). Moreover, suppose that these numbers satisfy the following inequality: |d| 2
2
64(κ + 1)2 [8(κ + 2) − |d|] + 128(κ + 1)(κ + 2) − [8(κ + 2) + |d|] |d| 2(κ + 1) [8(κ + 1) − |d|] {16(κ + 1)(2κ − |d|) − |d| (4κ + |d|)}
n |δj | j=1
µ
1 − |c| .
Then the function H ν1 ,··· ,νn ,δ1 ,··· ,δn,b,d,µ (z) : U → C, defined by (2.8), is in the class S of normalized univalent functions in U.
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Some Sufficient Conditions for Univalence
Proof. The proof of Theorem 7 is much akin to that of Theorem 3, so we omit the details involved in this case. By setting n = 1 in Theorem 7, we immediately obtain the following result. Corollary 9. Let the parameters ν, b ∈ R and d ∈ C be so constrained that κ := ν +
|d| b+1 > − 1. 2 8
Consider the function ϕ ν,b,d : U → C defined by (1.11). Also let (µ) > 0, c ∈ C \ {−1}
and
δ ∈ C.
Moreover, suppose that these numbers satisfy the following inequality: |δd| 64(κ + 1)2 [8(κ + 2) − |d|] + 128(κ + 1)(κ + 2) − [8(κ + 2) + |d|] |d|2 2 |µ| 2(κ + 1) [8(κ + 1) − |d|] {16(κ + 1)(2κ − |d|) − |d| (4κ + |d|)} 1 − |c| . Then the function H ν,δ,b,d,µ (z) : U → C, defined by (2.9) , is in the class S of normalized univalent functions in U. By applying Lemma 3 and the inequality (3.7) of Lemma 7, we easily get the following result. Theorem 8. Let the parameters ν, b ∈ R and d, λ ∈ C be so constrained that κ := ν +
|d| b+1 > − 1. 2 8
Consider the generalized Bessel function ϕ ν,b,d defined by (1.11) . If (λ) 1 and √ 6 3κ [8(κ + 1) − |d|] , |λ| 32κ(κ + 1) + 4(3κ + 4) |d| + |d|2 then the function Q ν,b,d,λ : U → C, defined by (2.10) , is in the class S of normalized univalent functions in U. Remark 7. In their special cases, Theorems 5 to 8 would readily yield the the corresonding results for the Bessel function (b = d = 1) , the modified Bessel function (b = −d = 1) and the spherical Bessel function (b−1 = d = 1) as asserted by Corollary 4, Corollary 5 and Corollary 6, respectively. We now consider another set of inequalities (see [8, p. 12]): (3.10) (κ)n > κ(κ + α0 )n−1
and n! > (1 + α0 )n−1
(κ > 0; n ∈ N \ {1, 2}),
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Erhan Deniz, Halit Orhan and H. M. Srivastava
where α0 ∼ = 1.302775637 · · ·
(3.11)
is the greatest root of the following quadratic equation: α2 + α − 3 = 0.
(3.12)
Thus, by using the inequalities in (3.10) and the same steps as in the proofs of Lemma 5 and Lemma 7, we obtain some improved versions of Lemma 5 and Theorems 1 to 4. Lemma 8. (see [8]). If the parameters ν, b ∈ R and d ∈ C are so constrained that |d| − α0 , κ > max 0, 4(1 + α0 ) where α0 is given by (3.11) and (3.12), then the function ϕν,b,d (z) :U→C z defined by (1.12) satisfies the following inequalities: ϕν,b,d (z) (z ∈ U) (3.13) 1 − Φ(κ) < z and ϕν,b,d (z) |d| < (3.14) 4κ [1 + Φ(κ + 1)] z ∈ U, z where Φ(κ) is defined by (3.15)
Φ(κ) = −
2
2
|d| |d| |d| + + 16(1 + α0 )κ(κ+α0 ) 32κ(κ+1) κ
(1+α0 )(κ+α0 ) . 4(1+α0 )(κ+α0 )−|d|
Lemma 9. If the parameters ν, b ∈ R and d ∈ C are so constrained that |d| − α0 , κ > max 0, 4(1 + α0 ) then the function ϕ ν,b,d : U → C defined by (1.12) satisfies the following inequalities: ϕν,b,d (z) |d| (3.16) < 4κ [1 + Φ(κ + 1)] (z ∈ U), ϕν,b,d (z) − z |d| 1 + Φ(κ + 1) zϕ (z) ν,b,d − 1 < (z ∈ U), (3.17) 4κ ϕν,b,d (z) 1 − Φ(κ)
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Some Sufficient Conditions for Univalence
|d| [1 − Φ(κ + 1)] + 1 − Φ(κ) zϕν,b,d (z) 4κ |d| (3.19) [1+Φ(κ+1)]+1+Φ(κ) (z ∈ U), 4κ
(3.18)
(3.20)
|d| 2 z ϕ ν,b,d (z) 2κ
and (3.21)
zϕ (z) ν,b,d ϕν,b,d (z)
|d| 2κ
|d| [1+Φ(κ+2)]+1+Φ(κ+1) 8(κ+1)
|d| 8(κ+1) |d| 4κ
(z ∈ U)
[1 + Φ(κ + 2)] + 1 + Φ(κ + 1)
[1 − Φ(κ + 1)] + 1 − Φ(κ)
(z ∈ U),
where Φ(κ) defined by (3.15) . Proof. Our proof of Lemma 9 is very similar to that of Lemma 7. It is based upon the known inequalities in (3.10) and Lemma 8. We choose to omit the details involved. By the aid of Lemmas 1 to 3 and Lemma 9, we prove the following resuts (Theorem 9 to 12). The proofs of Theorems 9 to 12 are very similar to the proofs of Theorems 1 to 4, respectively, so we omit the details involved in these cases. Theorem 9. Let the parameters ν, b ∈ R and d ∈ C be so constrained that κ := ν +
|d| b+1 > − α0 , 2 4(1 + α0 )
where α0 is given by (3.11) and (3.12). Consider the function ϕ ν,b,d : U → C defined by (1.11). Also let (β) > 0, c ∈ C \ {−1}
and
α ∈ C \ {0}.
Moreover, suppose that these numbers satisfy the following inequality: 1 + Φ(κ + 1) |d| 1. |c| + 4κ |βα| 1 − Φ(κ) where Φ(κ) is defined by (3.15). Then the function F ν,b,d,α,β (z) : U → C, defined by 1/α 1/β z β−1 ϕν,b,d (t) t dt , Fν,b,d,α,β (z) = β t 0 is in the class S of normalized univalent functions in U.
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Erhan Deniz, Halit Orhan and H. M. Srivastava
Theorem 10. Let the parameters ν, b ∈ R and d ∈ C be so constrained that κ := ν +
|d| b+1 > − α0 , 2 4(1 + α0 )
where α0 is given by (3.11) and (3.12). Consider the function ϕ ν,b,d : U → C defined by (1.11). Also let (γ) > 0. Moreover, suppose that these numbers satisfy the following inequality: 4κ [1 − Φ(κ)] (γ). |γd| 1 + Φ(κ + 1) where Φ(κ) is defined by (3.15). Then the function G ν,b,d,γ (z) : U → C, defined by (2.7), is in the class S of normalized univalent functions in U. Theorem 11. Let the parameters ν, b ∈ R and d ∈ C are so constrained that κ := ν +
|d| b+1 > − α0 , 2 4(1 + α0 )
where α0 is given by (3.11) and (3.12). Consider the function ϕ ν,b,d : U → C defined by (1.11). Also let (µ) > 0, c ∈ C \ {−1}
and
δ ∈ C.
Moreover, suppose that these numbers are satisfy the following inequality:
|d| |d| δ 2κ 8(κ+1) [1 + Φ(κ + 2)] + 1 + Φ(κ + 1) 1, |c| + |d| µ [1 − Φ(κ + 1)] + 1 − Φ(κ) 4κ
where Φ(κ) is defined by (3.15). Then the function H ν,δ,b,d,µ (z) : U → C, defined by (2.9), is in the class S of normalized univalent functions in U. By applying Lemma 3 and the inequality (3.18) of Lemma 9, we arrive at the following result. Theorem 12. Let the parameters ν, b ∈ R and d, λ ∈ C be so constrained that κ := ν +
|d| b+1 > − α0 , 2 4(1 + α0 )
where α0 is given by (3.11) and (3.12). Consider the generalized Bessel function ϕν,b,d defined by (1.11) . If (λ) 1 and √ 6 3κ , |λ| |d| [1 + Φ(κ + 1)] + 4κ [1 + Φ(κ)] then the function Q ν,b,d,λ : U → C, defined by (2.10) , is in the class S of normalized univalent functions in U.
Some Sufficient Conditions for Univalence
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Remark 8. By suitably specializing Theorems 9 to 12, we can obtain the corresponding results for the Bessel function (b = d = 1) , for the modified Bessel function (b = −d = 1) and for the spherical Bessel function (b − 1 = d = 1) as asserted by Corollary 4, Corollary 5 and Corollary 6, respectively. ACKNOWLEDGMENTS The present investigation was completed during the third-named author’s visit to Atatu¨ rk University in Erzurum in November 2010. The research of the firstand the second-named authors (Erhan Deniz and Halit Orhan) was supported by the Atatu¨ rk University Rectorship under the BAP Project (The Scientific and Research Project of Atatu¨ rk University) Number 2010/28. REFERENCES 1. L. V. Ahlfors, Sufficient conditions for quasiconformal extension, Ann. of Math. Stud., 79 (1974), 23-29. 2. J. W. Alexander, Functions which map the interior of the unit circle upon simple regions, Ann. of Math. (Ser. 2), 17 (1915), 12-22. ´ Baricz, Geometric properties of generalized Bessel functions, Publ. 3. A. Debrecen, 73 (2008), 155-178.
Math.
´ Baricz, Functional inequalities involving special functions, J. Math. Anal. Appl., 4. A. 319 (2006), 450-459. ´ Baricz, Functional inequalities involving special functions. II, J. Math. Anal. 5. A. Appl., 327 (2007), 1202-1213. ´ Baricz, Some inequalities involving generalized Bessel functions, Math. Inequal. 6. A. Appl., 10 (2007), 827-842. ´ Baricz and B. A. Frasin, Univalence of integral operators involving Bessel 7. A. functions, Appl. Math. Lett., 23 (2010), 371-376. ´ Baricz and S. Ponnusamy, Starlikeness and convexity of generalized Bessel 8. A. functions, Integral Transforms Spec. Funct., 21 (2010), 641-653. 9. J. Becker, Lo¨ wnersche Differentialgleichung und Schlichtheitskriterien, Math. Ann., 202 (1973), 321-335. 10. J. Becker, Lo¨ wnersche Differentialgleichung und quasikonform fortsetzbare schlichte funktionen, J. Reine Angew. Math., 255 (1972), 23-43. 11. D. Breaz and N. Breaz, Two integral operators, Stud. Univ. Babes¸-Bolyai. Math., 47(3) (2002), 13-19. 12. D. Breaz and N. Breaz, The univalent conditions for an integral operator on the classes Sp and T2 , J. Approx. Theory Appl., 1 (2005), 93-98.
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13. D. Breaz and N. Breaz, Univalence of an integral operator, Mathematica (Cluj), 47(70) (2005), 35-38. 14. D. Breaz and N. Breaz, Univalence conditions for certain integral operators, Stud. Univ. Babes¸-Bolyai Math., 47(2) (2002), 9-15. 15. D. Breaz, N. Breaz and H. M. Srivastava, An extension of the univalent condition for a family of integral operators, Appl. Math. Lett., 22 (2009), 41-44. 16. D. Breaz, S. Owa and N. Breaz, A new integral univalent operator, Acta Univ. Apulensis Math. Inform., 16 (2008), 11-16. 17. Y. J. Kim and E. P. Merkes, On an integral of powers of a spirallike function, Kyungpook Math. J., 12 (1972), 249-252. 18. S. S. Miller, P. T. Mocanu and M. O. Reade, Starlike integral operators, Pacific J. Math., 79 (1978), 157-168. 19. S. Moldoveanu and N. N. Pascu, Integral operators which preserve the univalence, Mathematica (Cluj), 32(55) (1990), 159-166. 20. N. N. Pascu, An improvement of Becker’s univalence criterion, in: Proceedings of the Commemorative Session: Simion Stoilow (Brasov, 1987), pp. 43-48. 21. V. Pescar, A new generalization of Ahlfor’s and Becker’s criterion of univalence, Bull. Malaysian Math. Soc., 19 (1996), 53-54. 22. V. Pescar, Univalence of certain integral operators, Acta Univ. Apulensis Math. Inform., 12 (2006), 43-48. 23. V. Pescar, New criteria for univalence of certain integral operators, Demonstratio Math., 33 (2000), 51-54. 24. V. Pescar and S. Owa, Sufficient conditions for univalence of certain integral operators, Indian J. Math., 42 (2000), 347-351 λ 25. J. A. Pfaltzgraff, Univalence of the integral of f (z) , Bull. London Math. Soc., 7 (1975), 254-256. 26. N. Seenivasagan and D. Breaz, Certain sufficient conditions for univalence, Gen. Math., 15(4) (2007), 7-15. 27. H. M. Srivastava, E. Deniz and H. Orhan, Some general univalence criteria for a family of integral operators, Appl. Math. Comput., 215 (2010), 3696-3701. 28. H. M. Srivastava and S. Owa (Editors), Current Topics in Analytic Function Theory, World Scientifc Publishing Company, Singapore, New Jersey, London and Hong Kong, 1992. 29. R. Sza´ sz and P. Kupa´ n, About the univalence of the Bessel functions, Stud. Univ. Babes¸-Bolyai Math., 54(1) (2009), 127-132. 30. G. N. Watson, A Treatise on the Theory of Bessel Functions, Second edition, Cambridge University Press, Cambridge, London and New York, 1944.
Some Sufficient Conditions for Univalence
Erhan Deniz and Halit Orhan Department of Mathematics Faculty of Science Atatu¨ rk University TR-25240 Erzurum Turkey E-mail:
[email protected] [email protected] H. M. Srivastava Department of Mathematics and Statistics University of Victoria Victoria, British Columbia V8W 3R4 Canada E-mail:
[email protected]
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