Applied Mathematics Letters Univalence of integral operators ... - Core

Applied Mathematics Letters 23 (2010) 371–376

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Applied Mathematics Letters journal homepage: www.elsevier.com/locate/aml

Univalence of integral operators involving Bessel functions Árpád Baricz a , Basem A. Frasin b,∗ a

Department of Economics, Babeş-Bolyai University, Cluj-Napoca 400591, Romania

b

Department of Mathematics, Al al-Bayt University, Mafraq 130095, Jordan

article

abstract

info

Article history: Received 8 March 2009 Accepted 28 October 2009

In this note our aim is to deduce some sufficient conditions for integral operators involving Bessel functions of the first kind to be univalent in the open unit disk. The key tools in our proofs are the generalized versions of the well-known Ahlfors’ and Becker’s univalence criteria and some inequalities for the normalized Bessel functions of the first kind. © 2009 Elsevier Ltd. All rights reserved.

Keywords: Analytic functions Integral operators Bessel functions

1. Introduction and preliminaries Let A denote the class of the normalized functions of the form f (z ) = z +

X

an z n ,

n ≥2

which are analytic in the open unit disk D = {z ∈ C : |z | < 1}. Further, by S we denote the class of all functions in A which are univalent (or schlicht) in D. In the last two decades many authors (see for example [1] and the references therein) have obtained various sufficient conditions for the univalence of the following integral operators:

( Z 1/αi )1/β n  z Y fi ( t ) β−1 dt , Fα1 ,α1 ,...,αn ,β (z ) = β t 0

i=1

t

(1)

and

 Z z 1/γ
γ −1 f (t ) γ Fγ (z ) = γ t e dt ,

(2)

0

where the functions f1 , f2 , . . . , fn and f belong to the class A and the parameters α1 , α2 , . . . , αn , α, β and γ are complex numbers such that the integrals in (1) and (2) exist. Here and throughout the paper every many-valued function is taken with the principal branch. In this paper we are mainly interested in some integral operators of type (1) and (2) which involve the normalized Bessel function of the first kind. More precisely, we would like to show that by using some inequalities for the normalized Bessel functions of the first kind the univalence of some integral operators which involve Bessel functions can be derived easily via some well-known univalence criteria. In particular, we obtain simple sufficient conditions for some integral operators which involve the sine and cosine functions. In the proofs of our main results we need the following univalence criteria. The

∗

Corresponding author. E-mail addresses: [email protected] (Á. Baricz), [email protected] (B.A. Frasin).

0893-9659/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2009.10.013

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first result, i.e. Lemma 1 is a generalization of Ahlfors’ and Becker’s univalence criterion [2,3] (which corresponds to the case β = 1), while the second, i.e. Lemma 2 is a generalization of the well-known univalence criterion of Becker [4] (which in fact corresponds to the case β = α = 1). Finally, Lemma 3 is a consequence of the above mentioned criterion of Becker’s and of the well-known Schwarz lemma. Lemma 1 ([5]). Let β and c be complex numbers such that Re β > 0 and |c | ≤ 1, c 6= −1. If the function h ∈ A satisfies the inequality

00 c |z |2β + (1 − |z |2β ) zh (z ) ≤ 1 0 β h (z ) for all z ∈ D, then the function Fβ ∈ A, defined by

1/β  Z z β−1 0 t h (t )dt , Fβ (z ) = β

(3)

0

is in the class S , i.e. is univalent in D. Lemma 2 ([6]). Let α ∈ C such that Re α > 0. If h ∈ A satisfies 1 − |z |2Re α zh00 (z )

h0 (z ) ≤ 1

Re α

for all z ∈ D, then for all β ∈ C such that Re β ≥ Re α , the function, defined by (3), is in the class S .

√ Lemma 0 3 ([7]). Let γ ∈ C and α ∈ R such that Re γ ≥ 1, α > 1 and 2α|γ | ≤ 3 3. If h ∈ A satisfies the inequality zh (z ) ≤ α for all z ∈ D, then the function Fγ : D → C, defined by Fγ (z ) =

 Z z 1/γ γ t γ −1 (eh(t ) )γ dt , 0

is in the class S . The Bessel function of the first kind of order ν is defined by the infinite series Jν ( z ) =

X (−1)n (z /2)2n+ν , n!Γ (n + ν + 1) n ≥0

where Γ stands for the Euler gamma function, z ∈ C and ν ∈ R. Recently, Szász and Kupán [8] investigated the univalence of the normalized Bessel function of the first kind gν : D → C, defined by gν (z ) = 2ν Γ (ν + 1)z 1−ν/2 Jν (z 1/2 ) = z +

X n≥1

4n n

(−1)n z n+1 . !(ν + 1) . . . (ν + n)

The following result is mainly based on [8] and is one of the crucial facts in the proofs of the our main results.

√

Lemma 4. Let ν > (−5 + 5)/4 and consider the normalized Bessel function of the first kind gν : D → C, defined by gν (z ) = 2ν Γ (ν + 1)z 1−ν/2 Jν (z 1/2 ), where Jν stands for the Bessel function of the first kind. Then the following inequalities hold for all z ∈ D

0 zgν (z ) ν+2 g (z ) − 1 ≤ 4ν 2 + 10ν + 5 , ν

(4)

0 4ν 2 + 13ν + 11 zg (z ) ≤ . ν (ν + 1)(4ν + 7)

(5)

Proof. It is known [8] that for all ν > (−5 +

√

5)/4 and z ∈ D we have 2 0 gν (z ) ν+2 g (z ) − gν (z ) ≤ ≥ 4ν + 10ν + 5 . and ν z (ν + 1)(4ν + 7) z (ν + 1)(4ν + 7)

Á. Baricz, B.A. Frasin / Applied Mathematics Letters 23 (2010) 371–376

373

Combining these inequalities we immediately √ get that (4) holds. Finally, by using the definition of the normalized Bessel function, we obtain that for all ν > (−5 + 5)/4 and z ∈ D

X X (−1)n (n + 1)z n+1 0 n+1 zg (z ) = z + ≤ 1 + ν n n!(ν + 1) . . . (ν + n) n n!(ν + 1) . . . (ν + n) 4 4 n ≥1 n ≥1 ≤ 1+

1

1

X

2(ν + 1) n≥1 [4(ν + 2)]

n −1

=

4ν 2 + 13ν + 11

(ν + 1)(4ν + 7)

. 

2. Univalence of some integral operators involving Bessel functions Our first main result is an application of Lemma 1 and contains sufficient conditions for an integral operator of the type (1) when the functions fi are the normalized Bessel functions with various parameters. Theorem 1. Let n be a natural number and let ν1 , ν2 , . . . , νn > (−5 + νi

gνi (z ) = 2 Γ (νi + 1)z

1−νi /2

J νi ( z

1/2

√

5)/4. Consider the functions gνi : D → C, defined by

).

Let ν = min{ν1 , ν2 , . . . , νn }, β ∈ C with Re β > 0, c ∈ C with c 6= −1 and let α1 , α2 , . . . , αn be nonzero complex numbers. Moreover, suppose that these numbers satisfy the following inequality

|c | +

n X ν+2 n ≤ 1. 4ν + 10ν + 5 i=1 |βαi | 2

Then the function Fν1 ,...,νn ,α1 ,...,αn ,β : D → C, defined by

( Z 1/αi )1/β n  z Y gνi (t ) β−1 Fν1 ,...,νn ,α1 ,...,αn ,β (z ) = β t dt , 0

i=1

t

is in S , i.e. is univalent in D. Proof. Let us consider the function Fν1 ,...,νn ,α1 ,...,αn : D → C, defined by Fν1 ,...,νn ,α1 ,...,αn (z ) =

1/αi Z zY n  gνi (t ) 0

t

i =1

dt .

First observe that, since for all i ∈ {1, 2, . . . , n} we have gνi ∈ A, i.e. gνi (0) = gν0 i (0) − 1 = 0, clearly Fν1 ,...,νn ,α1 ,...,αn ∈ A, i.e. Fν1 ,...,νn ,α1 ,...,αn (0) = Fν0 1 ,...,νn ,α1 ,...,αn (0) − 1 = 0. On the other hand, it is easy to see that Fν0 1 ,...,νn ,α1 ,...,αn (z ) =

1/αi n  Y gν (z ) i

i=1

z

and zFν001 ,...,νn ,α1 ,...,αn (z ) Fν0 1 ,...,νn ,α1 ,...,αn (z )

=

 0  n X 1 zgνi (z ) −1 . αi gνi (z ) i=1

Now, by using the inequality (4) for each νi , where i ∈ {1, 2, . . . , n}, we obtain

0 X n n n zF 00 X 1 νi + 2 n ν+2 ν1 ,...,νn ,α1 ,...,αn (z ) X 1 zgνi (z ) ≤ ≤ − 1 0 = 2 2 Fν1 ,...,νn ,α1 ,...,αn (z ) |αi | gνi (z ) |αi | 4νi + 10νi + 5 |αi | 4ν + 10ν + 5 i =1 i=1 i=1 √ √ for all z ∈ D and ν, ν1 , ν2 , . . . , νn > (−5 + 5)/4. Here we used that the function ϕ : ((−5 + 5)/4, ∞) → R, defined by ϕ(x) =

x+2 4x2

+ 10x + 5

,

is decreasing and consequently for all i ∈ {1, 2, . . . , n} we have

νi + 2 ν+2 ≤ 2 . 4ν + 10ν + 5 4ν + 10νi + 5 2 i

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Now, by using the triangle inequality and the hypothesis, we obtain

n X zFν001 ,...,νn ,α1 ,...,αn (z ) n ν+2 2β 2β ≤ 1, c |z | + (1 − |z | ) 0 ≤ |c | + 2 β Fν1 ,...,νn ,α1 ,...,αn (z ) 4ν + 10ν + 5 i=1 |βαi | which in view of Lemma 1 implies that Fν1 ,...,νn ,α1 ,...,αn ,β ∈ S . With this the proof is complete.



Choosing α1 = α2 = · · · = αn = α in Theorem 1, we have the following result. Corollary 1. Let the numbers β, c , ν, ν1 , . . . , νn be as in Theorem 1 and let α be a nonzero complex number. Moreover, suppose that the functions gνi ∈ A are as in Theorem 1 and the following inequality

|c | +

(ν + 2)n2 ≤1 |αβ| 4ν 2 + 10ν + 5 1

is valid. Then, the function Fν1 ,...,νn ,α,β : D → C, defined by

( Z 1/α )1/β n  z Y g νi ( t ) β−1 t dt , Fν1 ,...,νn ,α,β (z ) = β 0

is in S , i.e. is univalent in D.

t

i=1

√

√

Observe that g1/2 (z ) = z sin z and g−1/2 (z ) = z cos immediately obtain the following result.

√

z. Thus, taking n = 1 in Theorem 1 or in Corollary 1, we

√

Corollary 2. Let ν > (−5 + 5)/4, β ∈ C with Re β > 0, c ∈ C with c 6= −1 and let α 6= 0 be a complex number. Moreover, suppose that these numbers satisfy the following inequality

|c | +

1 ν+2 ≤ 1. |αβ| 4ν 2 + 10ν + 5

Then the function Fν,α,β : D → C, defined by

( Z  1/α )1/β z β−1 gν (t ) Fν,α,β (z ) = β t dt , t

0

is in S . In particular, if |c | + (5/22)/|αβ| ≤ 1, then the function F1/2,α,β : D → C, defined by

  Z z F1/2,α,β (z ) = β t β−1  0

 √ !1/α 1/β t dt , √  t

sin

is in S . Moreover, if |c | + (3/2)/|αβ| ≤ 1, then the function F−1/2,α,β : D → C, defined by F−1/2,α,β (z ) =

 Z z  √ 1/α 1/β dt , β t β−1 cos t 0

is in S . The following result contains another sufficient conditions for integrals of the type (1) to be univalent in the unit disk D. The key tools in the proof are Lemma 2 and the inequality (4). Theorem 2. Let n be a natural number, ν1 , ν2 , . . . , νn > (−5 + C, defined by

√

5)/4 and consider the normalized Bessel functions gνi : D →

gνi (z ) = 2νi Γ (νi + 1)z 1−νi /2 Jνi (z 1/2 ). Let ν = min{ν1 , ν2 , . . . , νn }, α ∈ C with Re α > 0, and suppose that these numbers satisfy the following inequality

|α| ≤

1 4ν 2 + 10ν + 5 n

ν+2

Re α.

Then the function Fν1 ,...,νn ,α,n : D → C, defined by

( Fν1 ,...,νn ,α,n (z ) =

(nα + 1)

Z zY n 0

is in S , i.e. is univalent in D.

i=1

gνi (t )


α

)1/(nα+1) dt

,

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375

Proof. Let us consider the auxiliary function Fν1 ,...,νn ,α : D → C, defined by Fν1 ,...,νn ,α (z ) =

α Z zY n  g νi ( t ) 0

t

i=1

dt .

Observe that Fν1 ,...,νn ,α ∈ A, i.e. Fν1 ,...,νn ,α (0) = Fν0 1 ,...,νn ,α (0) − 1 = 0. On the other hand, by using (4) and the fact that for all i ∈ {1, 2, . . . , n}

νi + 2 4νi2 + 10νi + 5

≤

ν+2 4ν 2 + 10ν + 5

,

we obtain that for all z ∈ D





n 00 zgν0 i (z ) 1 − |z |2Re α zFν1 ,...,νn ,α (z ) ν+2 |nα| |α| X − 1 ≤ · 2 ≤ 1. 0 ≤ Re α Fν1 ,...,νn ,α (z ) Re α i=1 gνi (z ) Re α 4ν + 10ν + 5





Now, since Re (nα + 1) > Re α and the function Fν1 ,...,νn ,α,n can be rewritten in the form

( Fν1 ,...,νn ,α,n (z ) =

(nα + 1)

z

Z

t

nα

α n  Y gν (t ) i

0

t

i =1

applying Lemma 2, the required results follows.

)1/(nα+1) dt

,



Now, by choosing n = 1 in Theorem 2 we obtain the following particular result.

√

Corollary 3. Let ν > (−5 + 5)/4 and consider the normalized Bessel function gν : D → C, defined by gν (z ) = 2ν Γ (ν + 1)z 1−ν/2 Jν (z 1/2 ). Moreover, let α ∈ C such that Re α > 0 and

|α| ≤

4ν 2 + 10ν + 5

ν+2

Re α.

Then the function Fν,α : D → C, defined by Fν,α (z ) =

 1/(α+1) Z z (α + 1) , (gν (t ))α dt 0

is univalent in D. In particular, if |α| ≤ (22/5)Re α , then F1/2,α : D → C, defined by F1/2,α (z ) =



(α + 1)

Z z √

t sin

1/(α+1)

√ α t

dt

,

0

is in S . Moreover, if |α| ≤ (2/3)Re α, then the function F−1/2,α : D → C, defined by F−1/2,α (z ) =



(α + 1)

Z z

t cos

√ α t

1/(α+1) dt

,

0

is univalent in D. Finally, by applying Lemma 3 and the inequality (5), we easily get the following result. Theorem 3. Let γ ∈ C, ν > (−5 +

√

5)/4 and let gν be the normalized Bessel function of the first kind. If Re γ ≥ 1 and

√

|γ | ≤

3 3(ν + 1)(4ν + 7) 8ν 2 + 26ν + 22

,

then the function Fν,γ : D → C, defined by Fν,γ (z ) =

 Z z 1/γ
γ γ t γ −1 egν (t ) dt , 0

is univalent in D. Choosing ν = 1/2, ν = −1/2 in the above theorem, we obtain the following particular cases.

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Á. Baricz, B.A. Frasin / Applied Mathematics Letters 23 (2010) 371–376

√

Corollary 4. If γ ∈ C such that Re γ ≥ 1 and |γ | ≤ 81 3/74, then F1/2,γ : D → C, defined by F1/2,γ (z ) =

 Z z  √ √ γ 1/γ t γ −1 e t sin t dt , γ 0

is univalent in D.

√

Corollary 5. If γ ∈ C such that Re γ ≥ 1 and |γ | ≤ 15 3/22, then F−1/2,γ : D → C, defined by

 Z z 1/γ  √ γ t cos t γ −1 e t F−1/2,γ (z ) = γ dt , 0

is univalent in D. Acknowledgement The research of Árpád Baricz was supported by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences. References [1] [2] [3] [4] [5] [6] [7] [8]

D. Breaz, H.Ö. Güney, On the univalence criterion of a general integral operator, J. Inequal. Appl. (2008) 8. Art. ID 702715. L.V. Ahlfors, Sufficient conditions for quasiconformal extension, Ann. of Math. Stud. 79 (1974) 23–29. J. Becker, Löwnersche Differentialgleichung und Schlichtheitskriterien, Math. Ann. 202 (4) (1973) 321–335. J. Becker, Löwnersche Differentialgleichung und quasikonform fortsetzbare schlichte Funktionen, J. Reine Angew. Math. 255 (1972) 23–43. V. Pescar, A new generalization of Ahlfor’s and Becker’s criterion of univalence, Bull. Malays. Math. Soc. 19 (2) (1996) 53–54. N. Pascu, An improvment of Becker’s univalence criterion, in: Proceedings of the Commemorative Session Simion Stoilow, Brasov, 1987 pp. 43–98. V. Pescar, Univalence of certain integral operators, Acta Univ. Apulensis Math. Inform. 12 (2006) 43–48. R. Szász, P. Kupán, About the univalence of the Bessel functions, Stud. Univ. Babeş-Bolyai Math. 54 (1) (2009) 127–132.