Sufficient conditions for integral operator defined by ... - Ele-Math

J ournal of Mathematical I nequalities Volume 4, Number 2 (2010), 301–306

SUFFICIENT CONDITIONS FOR INTEGRAL OPERATOR DEFINED BY BESSEL FUNCTIONS

B. A. F RASIN (Communicated by N. Elezovi´c) Abstract. The main object of the present paper is to derive several sufficient conditions for integral operator defined by Bessel functions of the first kind to be convex and strongly convex of given order in the open unit disk.

1. Introduction and definitions Let A denote the class of functions of the form: ∞

f (z) = z + ∑ an zn ,

(1.1)

n=2

which are analytic in the open unit disk D = {z : |z| < 1}. A function f (z) belonging to A is said to be convex of order γ if it satisfies   z f  (z) Re 1 +  (z ∈ D) >γ f (z) for some γ (0  γ < 1). We denote by C (γ ) the subclass of A consisting of functions which are convex of order γ in D. If f (z) ∈ A satisfies       arg 1 + z f (z) − γ  < π β (z ∈ D)   2  f (z) for some γ (0  γ < 1) and β (0 < β  1), then f (z) is said to be strongly convex of order β and type γ in D , and we denote by C γ (β ) the class of such functions. It is clear that C 0 (β ) ≡ C (β ) the class of strongly convex of order β in D and C 0 (1) ≡ C the class of all convex functions in D. The Bessel function of the first kind of order ν is defined by the infinite series Jν (z) =

∞

(−1)n (z/2)2n+ν , n=0 n!Γ(n + ν + 1)

∑

Mathematics subject classification (2010): 30C45, 30A20, 33C10. Keywords and phrases: Analytic functions, Convex functions, Bessel functions, Integral operator. c   , Zagreb Paper JMI-04-26

301

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B. A. F RASIN

where Γ stands for the Euler gamma function, z ∈ C and ν ∈ R. Recently, Sz´asz and Kup´an [8] investigated the univalence of the normalized Bessel function of the first kind gν : D → C, defined by ∞

(−1)n zn+1 . n n=1 4 n!(ν + 1) . . . (ν + n)

gν (z) = 2ν Γ(ν + 1)z1−ν /2 Jν (z1/2 ) = z + ∑

(1.2)

Baricz and Frasin [1] have obtained various sufficient conditions for the univalence of the following integral operators defined by Bessel functions of the first kind: ⎧ ⎫ 1/αi ⎬1/β ⎨ z n  (t) g ν i dt , Fν1 ,...,νn ,α1 ,...,αn ,β (z) = β t β −1 ∏ ⎩ ⎭ t i=1 0

Fν1 ,...,νn ,α ,n (z) = and

⎧ ⎨ ⎩

(nα + 1)

⎫1/(nα +1) ⎬

z n 0

∏ (gνi (t))α dt ⎭ i=1

⎫1/γ ⎧ ⎨ z

γ ⎬ Fν ,γ (z) = γ t γ −1 egν (t) dt . ⎭ ⎩ 0

In this paper we are mainly interested on some integral operators of the following types which involve the normalized Bessel function of the first kind: Fν1 ,...,νn ,α1 ,...,αn (z) =

 z n  gνi (t) αi

∏

0

i=1

t

dt

(1.3)

More precisely, we would like to obtain some sufficient conditions for Fν1 ,...,νn ,α1 ,...,αn (z) to be in the classes C (γ ). Also, we prove new results, which involves strongly convexity of the integral operator of the type (1.3) when n = 1 . In particular, we obtain simple sufficient conditions for some integral operators which involve the sine and cosine functions. For the integral operators of the form (1.3) which involve the normalized analytic function of the form (1.1) see the references [2, 3, 4, 5, 6, 7]. In the proofs of our results we need the following result based on [8]. √ L EMMA 1.1. 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)z1−ν /2Jν (z1/2 ), where Jν stands for the Bessel function of the first kind. Then the following inequality hold for all z ∈ D      zgν (z) ν +2   (1.4)  gν (z) − 1  4ν 2 + 10ν + 5

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2. Convexity of integral operators involving Bessel functions Our first result provides sufficient conditions for integral operator of the type (1.3) to be convex of given order δ . √ T HEOREM 2.1. Let n be a natural number and let ν1 , ν2 , . . . , νn > (−5 + 5)/4. Consider the functions gνi : D → C, defined by gνi (z) = 2νi Γ(νi + 1)z1−νi /2 Jνi (z1/2 ).

(2.1)

Let ν = min{ν1 , ν2 , . . . , νn } and α1 , α2 , . . . , αn be positive real numbers. Moreover, suppose that these numbers satisfy the following inequality 0  1−

n 2+ν αi < 1. ∑ 4ν 2 + 10ν + 5 i=1

Then the function Fν1 ,...,νn ,α1 ,...,αn : D → C, defined by Fν1 ,...,νn ,α1 ,...,αn (z) =

 z n  gνi (t) αi

∏

0

is in C (δ ) , where

δ =1 −

t

i=1

dt,

(2.2)

n

2+ν

4ν 2 + 10ν + 5

∑ αi .

i=1

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

n zFν1 ,...,νn ,α1 ,...,αn (z) = αi ∑ Fν1 ,...,νn ,α1 ,...,αn (z) i=1



 zgνi (z) −1 . gνi (z)

or, equivalently, 1+

n zFν1 ,...,νn ,α1 ,...,αn (z) = ∑ αi  Fν1 ,...,νn ,α1 ,...,αn (z) i=1



 n zgνi (z) + 1 − ∑ αi . gνi (z) i=1

(2.3)

Taking the real part of both terms of (2.3), we have        n n zFν1 ,...,νn ,α1 ,...,αn (z) zgνi (z) Re 1 +  = ∑ αi Re + 1 − ∑ αi . Fν1 ,...,νn ,α1 ,...,αn (z) gνi (z) i=1 i=1

(2.4)

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B. A. F RASIN

Now, by using the inequality (1.4) for each νi , where i ∈ {1, 2, . . . , n}, we obtain        n n zgνi (z) zFν1 ,...,νn ,α1 ,...,αn (z) Re 1 +  = ∑ αi Re + 1 − ∑ αi Fν1 ,...,νn ,α1 ,...,αn (z) gνi (z) i=1 i=1     n n νi + 2 > ∑ αi 1 − 2 + 1 − ∑ αi 4νi + 10νi + 5 i=1 i=1   n νi + 2 = 1 − ∑ αi 2 4νi + 10νi + 5 i=1 > 1−

n

2+ν

4ν 2 + 10ν + 5

∑ αi

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 2 4νi + 10νi + 5



ν +2 4ν 2 + 10ν + 5

.

(2.5)

n

2+ν Because 0  1 − 4ν 2 +10 α < 1,we get Fν1 ,...,νn ,α1 ,...,αn (z) ∈ C (δ ) , where ν +5 ∑ i

δ =1 −

2+ν 4ν 2 +10ν +5

n

i=1

∑ αi . This completes the proof.

i=1



Choosing α1 = α2 = . . . = αn = α in Theorem 2.1, we have the following result. C OROLLARY 2.2. Let the numbers ν , ν1 , . . . , νn be as in Theorem 2.1 and let α1 , α2 , . . . , αn be positive real numbers. Moreover, suppose that the functions gνi ∈ A defined by (2.1) and the following inequality 0  1−

(2 + ν )nα < 1. 4ν 2 + 10ν + 5

is valid. Then the function Fν1 ,...,νn ,α1 ,...,αn (z) defined by (2.2) is in C (ζ ) , where

ζ =1 −

(2 + ν )nα . 4ν 2 + 10ν + 5

√ √ √ Observe that g1/2(z) = z sin z and g−1/2(z) = z cos z. Thus, taking n = 1 in Theorem 2.1 or in Corollary 2.2, we immediately obtain the following result. √ C OROLLARY 2.3. Let ν > (−5 + 5)/4 and α > 0 be a real number. Moreover, suppose that these numbers satisfy the following inequality 0  1−

(2 + ν )α < 1. 4ν 2 + 10ν + 5

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

z  0

gν (t) t

α dt,

is in C (η ) , where

(2 + ν )α . 4ν 2 + 10ν + 5 In particular, if 0 < α  22/5, then the function F1/2,α : D → C, defined by

η =1 −

F1/2,α (z) =

z  0

√ α sin t √ dt, t

is in C (ξ ), where ξ =1 − (5/22)α . Moreover, if 0 < α  2/3, then the function F−1/2,α : D → C, defined by F−1/2,α (z) =

z 

√ α cos t dt,

0

is in C (λ ), where λ =1 − (3/2)α . 3. Strongly convexity In this section, we prove the following results, which involves strongly convexity of the integral operator of the type (1.3) when n = 1 . by

√ T HEOREM 3.1. Let ν > (−9 + 33)/8 . Then the function Fν ,α : D → C, defined Fν ,α (z) =

z  0

gν (t) t

α dt,

is in C ρ (1) = C (ρ ) , where ρ = 1 − α ; 0 < α  1. Proof. It follows from (2.3), (1.4) and (2.5) that             zF  (z) zgν (z)   zgν (z)  arg 1 + ν ,α   − (1 − α )  = arg α = arg  Fν ,α (z) gν (z)   gν (z)    2+ν  arcsin 2 4ν + 10ν + 5 π  arcsin 1 = , 2 so that Fν ,α (z) ∈ C (ρ ), where ρ = 1 − α ; 0 < α  1, which proves Theorem 3.1.



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B. A. F RASIN

C OROLLARY 3.2. The function F1/2,α : D → C, defined by F1/2,α (z) =

z  0

√ α sin t √ dt, t

is in C (ρ ), where ρ = 1 − α ; 0 < α  1 . In particular, when α = 1, the function

z sin √t √ 0

t

dt is convex in D.

Acknowledgement. The author would like to thank the referee for his helpful comments and suggestions. REFERENCES [1] A. BARICZ AND B. A. F RASIN, Univalence of integral operators involving Bessel functions, Applied Mathematics Letters (in press) [2] D. B REAZ, A convexity property for an integral operator on the class Sα (β ) , Gen. Math., 15, 2–3 (2007), 177–183. [3] D. B REAZ AND N. B REAZ, Two integral operator, Studia Universitatis Babes-Bolyai, Mathematica, Cluj-Napoca, 3 (2002), 13–21. ¨ [4] D. B REAZ AND H. G UNEY , The integral operator on the classes Sα∗ (b) and Cα (b) , J. Math. Ineq., 2, 1 (2008), 97–100. [5] D. B REAZ , S.O WA AND N. B REAZ, A new integral univalent operator, Acta Univ. Apulensis Math., 16 (2008), 11–16. [6] S. B ULUT, A note on the paper of Breaz and G¨uney, J. Math. Ineq., 2, 4 (2008), 549–553. [7] B.A. F RASIN, Some sunfficient conditions for certain integral operators, J. Math. Ineq., 2, 4 (2008), 527–535. ´ , P. K UP AN ´ , About the univalence of the Bessel functions, Stud. Univ. Babes¸-Bolyai Math., [8] R. S Z ASZ 54, 1 (2009), 127–132.

(Received March 2, 2009)

Journal of Mathematical Inequalities

www.ele-math.com [email protected]

B. A. Frasin Department of Mathematics Al al-Bayt University P.O. Box: 130095 Mafraq Jordan e-mail: [email protected]