Certain Geometric Properties of Some Bessel Functions arXiv

arXiv:1712.01687v1 [math.CV] 2 Dec 2017

Certain Geometric Properties of Some Bessel Functions Rabha M. El-Ashwah1 and Alaa H. El-Qadeem2 1

Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt 2

Department of Mathematics, Faculty of Science, Zagazig University, Zagazig 44519, Egypt r [email protected] [email protected] & [email protected]

Abstract In this paper, we determine necessary and sufficient conditions for the generalized Bessel function to be in certain subclasses of starlike and convex functions. Also, we obtain several corollaries as special cases of the main results, these corollaries gives the corresponding results of the familiar, modified and spherical Bessel functions.

Keywords: analytic function; starlike function; convex function; Bessel functions. 2010 MSC: 30C45.

1. Introduction Let A denotes the class of analytic functions which are defined on the open unit disc U = {z ∈ C : |z| < 1}, normalized by f (0) = f 0 (0) − 1 = 0, of the form f (z) = z +

∞ X

ak z k ,

(1.1)

k=2

For 0 ≤ α < 1 and 0 < β ≤ 1, let S ∗ (α, β) denotes the subclass of A consisting of functions f (z) of the form (1.1) and satisfy zf 0 (z) − 1 f (z) (1.2) < β; z ∈ U, zf 0 (z) + (1 − 2α) f (z) and f ∈ K(α, β) denotes the subclass of A consisting of functions f (z) of the form (1.1) such that zf 00 (z) f 0 (z) (1.3) zf 00 (z) < β; z ∈ U. 0 + 2 (1 − α) f (z) 1

The subclasses S ∗ (α, β) and K(α, β) are the well-known subclasses of starlike and convex functions of order α and type β, respectively, introduced by Gupta and Jain [19]. It is clear that: f (z) ∈ K(α, β) ⇐⇒ zf 0 (z) ∈ S ∗ (α, β). (1.4) We note that S ∗ (α, 1) = S ∗ (α) and K(α, 1) = K(α) are the subclasses of starlike and convex functions of order α(0 ≤ α < 1), respectively, also, S ∗ (0, 1) = S ∗ (0) = S ∗ and K(0, 1) = K are the subclasses of starlike and convex functions. These subclasses were introduced by Robertson [27], Schild [28], MacGregor [21]. Recently, Baricz [3] defined a generalized Bessel function wp,b,c (z) ≡ w(z) as follows: wp,b,c (z) =

∞ X k=0

(−c)k k!Γ k + p +

  z 2k+p  b+1
p, b, c ∈ C; p + 6= 0, −1, −2, ... , b+1 2 2 2

(1.5)

which is the particular solution of the following second-order linear homogeneous differential equation:   z 2 w00 (z) + bzw0 (z) + cz 2 − p2 + (1 − b)p w(z) = 0 (p, b, c ∈ C) , (1.6) which, in turn, is a natural generalization of the classical Bessel’s equation. Solutions of (1.6) are regarded as the generalized Bessel function of order p. The particular solution given by (1.5) is called the generalized Bessel function of the first kind of order p. The series (1.5) permits the study of Bessel function, modified Bessel functions and the spherical Bessel functions in a unified manner, as follows: (i) Taking b = c = 1 in (1.5), we have the familiar Bessel function of the first kind of order p defined by (see [31] and [4]) Jp (z) =

∞ X k=0

 z 2k+p (−1)k ; z ∈ C. k!Γ (p + k + 1) 2

(1.7)

(ii) Taking b = 1 and c = −1 in (1.5), we obtain the modified Bessel function of the first kind of order p defined by (see [31] and [4]) Ip (z) =

∞ X k=0

 z 2k+p 1 ; z ∈ C. k!Γ (p + k + 1) 2

(1.8)

√ √ (iii) Taking b = 2 and c = 1 in (1.5), the function wb,c,p (z) reduces to 2jp (z)/ π, where jp is the spherical Bessel function of the first kind of order p defined by (see [4]) r jp (z) =

∞  z 2k+p πX (−1)k
; z ∈ C. 2 k=0 k!Γ p + k + 32 2

2

(1.9)

Now, we will consider the function up,b,c (z), which is the normalized form of wb,c,p (z), as following: up,b,c (z) = Γ p +

b+1 2




√ p z 1− 2 wp,b,c (2 z).

(1.10)

By using the well-known Pochhammer symbol (or the shifted factorial) defined, in terms of the familiar Gamma function, by  Γ (λ + µ) 1, (µ = 0) , = (λ)µ = λ (λ + 1) ... (λ + µ − 1) , (µ ∈ N) , Γ (λ) we can express up,b,c (z) as follows: up,b,c (z) =

∞ X k=0

(−c)k z k+1
, k! p + b+1 2 k

(1.11) (1)

(2)

(3)

− − − where p + b+1 ∈ C\Z− 0 (Z0 := Z ∪ {0}, Z = {−1, −2, ...}). Also, let up (z), up (z) and up (z) 2 be the normalized forms of Jp (z), Ip (z) and jp (z), respectively, which are defined as following:

u(1) p (z)

:= Γ (p + 1) z

1− p2

∞ X √ (−1)k z k+1 Jp (2 z) = ; p ∈ C\Z− , (p + 1)k k! k=0

∞ X √ 1− p2 u(2) I (2 (z) := Γ (p + 1) z z) = p p k=0

and u(3) p (z)

:= Γ p +

3 2




z

1− p2

1 z k+1 ; p ∈ C\Z− , (p + 1)k k!

∞ X √ (−1)k z k+1 1
jp (2 z) = ; p + ∈ C\Z− . 3 2 p + 2 k k! k=0

(1.11.a)

(1.11.b)

(1.11.c)

For further result on this transformation of the generalized Bessel function, we refer to the recent papers [2]-[6] and [12], where some interesting functional inequalities, integral representations, extensions of some known trigonometric inequalities, starlikeness, convexity and univalence, were established. Recently, Baricz and Frasin [5] and Deniz et al. [12] were interested in the univalence of some integral operators which involved the normalized form of the ordinary Bessel function of the first kind and the normalized form of the generalized Bessel functions of the first kind, respectively. Frasin [18] obtained various sufficient conditions for the convexity and strong convexity of the integral operators defined by the normalized form of the ordinary Bessel function of the first kind. Also, the problem of geometric properties (such as univalence, starlikeness and convexity) of some generalized integral operators has been discussed by many authors (see [1], [7]-[10], [17], [20], [29]). For further properties we refer to El-Ashwah and Hassan [13]-[16]. Very recently, Cho et al. [11] and Murugusundaramoorthy and Janani [22] (see also Porwal and Dixit [25]) introduced some characterization of generalized Bessel functions of first kind to be in certain subclasses of uniformly starlike and uniformly convex functions. Motivated by the new technique due to Mustafa [23], we determine some geometric properties of generalized Bessel function of the first type to be in S ∗ (α, β) and K(α, β), followed by sufficient conditions for the familiar, modified and spherical Bessel functions as special cases. 3

2. Main Result Unless otherwise mentioned, we assume in the reminder of this paper that, 0 ≤ α < 1, 0 < β ≤ 1, p, b, c ∈ C, p + (b + 1) /2 ∈ C\Z− 0 and z ∈ U . To establish our main results, we shall require the following lemmas: Lemma 1 ([19]). A sufficient condition for a function f of the form (1.1) to be in the class S ∗ (α, β) is that ∞ X [k − 1 + β (k + 1 − 2α)] |ak | ≤ 2β (1 − α) . (2.1) k=2

Lemma 2 ([19]). A sufficient condition for a function f of the form (1.1) to be in the class K(α, β) is that ∞ X k [k − 1 + β (k + 1 − 2α)] |ak | ≤ 2β (1 − α) . (2.2) k=2

Theorem 1. Let c < 0 and p +    

 2β (1 − α) 2 − e

−c  p+ b+1 2 +1

b+1 2

> 0 and assume the following condition is satisfied       

1  + 1 − e p + b+1 2

then up,b,c (z) ∈ S ∗ (α, β). Proof. Since up,b,c (z) = z +

−c  p+ b+1 2 +1

∞ X

 (1 + β) (−c)
e  − p + b+1 2



−c  p+ b+1 2 +1

(−c)k−1 zk
, (k − 1)! p + b+1 2 k−1

k=2

≥ 0, (2.3)

(2.4)

by virtue of Lemma 1, it is suffices to show that ∞ X

[k − 1 + β (k + 1 − 2α)]

k=2

Let L(α, β; p, b, c) =

∞ X

(−c)k−1
≤ 2β(1 − α). p + b+1 (k − 1)! 2 k−1

[k (1 + β) + β (1 − 2α) − 1]

k=2

(2.5)

(−c)k−1
, p + b+1 (k − 1)! 2 k−1

by simple computation, we get L(α, β; p, b, c) =

∞ X

[(k − 1) (1 + β) + 2β (1 − α)]

k=2

= (1+β)

∞ X k=2

(−c)k−1
p + b+1 (k − 1)! 2 k−1

∞ X (−c)k−1 (−c)k−1

+ 2β (1−α) .(2.6) b+1 p + b+1 (k − 2)! p + (k − 1)! 2 k−1 2 k−1 k=2

4

But p+

b+1 2 k−1




=

p+

b+1 2




p+

b+1 2

≥

p+

b+1 2




p+

b+1 2



+ 1 ... p + b+1 + k − 2 2
k−2 +1 ; k ≥ 2,

which is equivalent to 1 p+


b+1 2

≤ p+

b+1 2




p+

b+1 2

+1


k−2

,

(2.7)

k−1

then ∞

(1 + β) X (−c)k−1

k−2 b+1 p + b+1 (k − 2)! 2 k=2 p + 2 + 1 ∞ (−c)k−1 2β (1 − α) X
+
k−2 b+1 p + b+1 (k − 1)! 2 k=2 p + 2 + 1 ∞ (1 + β) (−c) X (−c)k−2
=
k−2 b+1 p + b+1 p + (k − 2)! + 1 2 k=2 2
∞ b+1 2β (1 − α) p + 2 + 1 X (−c)k−1
+
k−1 b+1 p + b+1 (k − 1)! 2 k=2 p + 2 + 1      
−c −c     b+1 2β (1 − α) p + 2 + 1  p+ b+1 (1 + β) (−c) p+ b+1  2 +1 2 +1

e = e + − 1 .  b+1 b+1 p+ 2 p+ 2

L(α, β; p, b, c) ≤

It can be varified that the last expression L(α, β; p, b, c) is bounded above by 2β(1 − α) if (2.3) is satisfied. This completes the proof of Theorem 1. Taking b = c = 1 in Theorem 1, we have the following result: Corollary 1. If p > −1, then the condition   1 p+2 (2p + 3) − (p + 2) + β + 1 ≥ 0, 2β (1 − α) e (1)

suffices that up (z) ∈ S ∗ (α, β). Putting b = 1 and c = −1 in Theorem 1, we have the following result: Corollary 2. If p > −1, then the condition   1 1 p+2 p+2 2β (1 − α) (2p + 3) − (p + 2) e + (β + 1) e ≥ 0,

5

Figure 1 (2)

suffices that up (z) ∈ S ∗ (α, β). Also, taking b = 2 and c = 1 in Theorem 1, we have the following result: Corollary 3. If p > − 23 , then the condition 

 2 ( 2p+5 ) β (1 − α) (2p + 4) 2e − 1 + αβ + 1 ≥ 0, (3)

suffices that up (z) ∈ S ∗ (α, β). Taking β = 1 in Corollary 1, we have the following result: (1) Corollary 4. Let p > −1, then up (z) ∈ S ∗ (α) if the following condition is satisfied   1 p+2 (2p + 3) − (p + 2) + 1 ≥ 0, (1 − α) e

(2.8)

Taking α = 0 in Corollary 4, we have the following result: (1) Corollary 5. The function up (z) is a starlike function for all p > −1. Proof. Taking α = 0 in (2.8) and using p > −1, then the sufficient condition for starlikeness of (1) up (z) is that (2p + 3) e

1 p+2

− (p + 1) ≥ 0,

1 x+2

but the function (2x + 3) e − (x + 1) it is non-negative for all x > x0 ' −1.5314 (see Figure 1). This proves Corollary 5. Remark 1. The result introduced in Corollary 1 gives a more improved result of this obtained by Mustafa [23, Corollary 3.2] and the result obtained by Prajapat [26, Corollary 2.8 (a)]. Taking β = 1 in Corollary 2, we have the following result:

6

Figure 2 (2)

Corollary 6. Let p > −1, then up (z) ∈ S ∗ (α) if the following condition is satisfied   1 1 p+2 p+2 ≥ 0, +e (1 − α) (2p + 3) − (p + 2) e

(2.9)

Taking α = 0 in Corollary 6, we have the following result: (2) Corollary 7. The function up (z) is a starlike function for all p > −1. Proof. Taking α = 0 in (2.9) and using p > −1, then the sufficient condition for starlikeness of (2) up (z) is that (2p + 3) − e

1 p+2

(p + 1) ≥ 0,

1 x+2

but the function (2x + 3) − e (x + 1) it is non-negative for all x > x0 = −2 (see Figure 2). This proves Corollary 7. Taking β = 1 in Corollary 3, we have the following result: (3) Corollary 8. Let p > − 32 , then up (z) ∈ S ∗ (α) if the following condition is satisfied 

 2 ( 2p+5 ) (1 − α) (2p + 4) 2e − 1 + α + 1 ≥ 0,

(2.10)

Taking α = 0 in Corollary 8, we have the following result: (3) Corollary 9. The function up (z) is a starlike function for all p > − 32 . Proof. Taking α = 0 in (2.10) and using p > − 32 , then the sufficient condition for starlikeness of (3) up (z) is that 2 ( 2p+5 ) 4 (p + 2) e − (2p + 3) ≥ 0, 2 ( 2x+5 ) but the function 4 (x + 2) e − (2x + 3) it is non-negative for all x > x0 ' −2.0314 (see Figure 3). This proves Corollary 9.

7

Figure 3 Theorem 2. Let c < 0 and p +

b+1 2

> 0 and assume that the condition

   c b+1 p+ b+1 +1 p+ +1 2 2 2β(1−α) 1+ e − b+1 p+

2

b+1 2(1+β(2−α))c 2β(1−α)(p+ 2 +1) (1+β)c2 − + b+1 b+1 b+1 b+1 (p+ )(p+ +1) p+ p+ 2 2 2 2

 ≥0, (2.11)

is satisfied, then up,b,c (z) ∈ K(α, β). Proof. By virtue of Lemma 2, it is suffices to show that ∞ X

k [k − 1 + β (k + 1 − 2α)]

k=2

(−c)k−1
≤ 2β(1 − α). p + b+1 (k − 1)! 2 k−1

Let F (α, β; p, b, c) = =

∞ X k=2 ∞ X

k [k (1 + β) + β (1 − 2α) − 1] 

(−c)k−1
(k − 1)! p + b+1 2 k−1

 k 2 (1 + β) + k (β (1 − 2α) − 1)

k=2

8

(−c)k−1
(k − 1)! p + b+1 2 k−1

(2.12)

substituting k 2 = (k − 1)(k − 2) + 3(k − 1) + 1, k = (k − 1) + 1 and using (2.7), then we have F (α, β; p, b, c) = (1 + β)

∞ X k=2





(−c)k−1 b+1 (p+ 2 ) (k−3)!

+ 2β (1 − α)

+ 2β (2 − α) + 2

∞ X

k−1

∞ X k=2

k=2

(−c)k−1

(

p+ b+1 (k−2)! 2 k−1

)

(−c)k−1 b+1 p+ ( 2 ) (k−1)! k−1

∞

≤

X (1 + β) (−c)2 (−c)k−3

k−3  p + b+1 p + b+1 + 1 k=2 p+ b+1 +1 (k−3)! 2 2 2

+

(2β (2 − α) + 2) (−c)
p + b+1 2

∞ X

(−c)k−2  k−2 b+1 p+ +1 (k−2)! k=2 2


∞ +1 X (2β (1 − α)) p + b+1 (−c)k−1 2
 k−1 + b+1 b+1 p+ 2 +1 (k−1)! k=2 p+ 2



(1 + β) (−c)2

e p + b+1 p + b+1 +1 2 2



=

−c  p+ b+1 2 +1

 (2β (1 − α)) + p+

p + b+1
2 b+1 2








+1  e

+

 

" =

2 (1+β)(−c)     + (2β(2−α)+2)(−c) b+1 b+1 b+1 p+ p+ p+ +1 2 2 2   b+1 (2β(1−α)) p+ +1 2   . − b+1 p+ 2



−c  p+ b+1 2 +1





(2β (2 − α) + 2) (−c)
e p + b+1 2  

−c  p+ b+1 2 +1

+





 − 1

 # −c   b+1 (2β(1−α)) p+ +1 p+ b+1 2 2 +1   e b+1 p+ 2

The last expression F (α, β; p, b, c) is bounded above by 2β(1 − α) if (2.11) is satisfied. This completes the proof of Theorem 2. Taking b = c = 1 in Theorem 2, we have the following result: Corollary 10. If p > −1, then the condition    1  p+2 2(1+β(2−α)) p+2 1+β 2β(1−α) 1+ p+1 e − (p+1)(p+2) + 2β(1−α)(p+2) − ≥0, p+1 p+1 (1)

suffices that up (z) ∈ K(α, β). Putting b = 1 and c = −1 in Theorem 2, we have the following result: 9

Figure 4 Corollary 11. If p > −1, then the condition    1  p+2 (1+β) 2(1+β(2−α)) 2β(1−α)(p+2) − e ≥0, + + 2β(1−α) 1+ p+2 p+1 (p+1)(p+2) p+1 p+1 (2)

suffices that up (z) ∈ K(α, β). Also, taking b = 2 and c = 1 in Theorem 2, we have the following result: Corollary 12. If p > − 32 , then the condition β(1−α)



1+ 2p+5 2p+3



e

2 2p+5

−



2(1+β) + β(1−α)(2p+5) − 2(1+β(2−α)) (2p+3)(2p+5) 2p+3 2p+3



≥0,

(3)

suffices that up (z) ∈ K(α, β). Taking β = 1 in Corollary 10, we have the following result: (1) Corollary 13. Let p > −1, then up (z) ∈ K(α) if the following condition is satisfied   1   p+2 (1−α)(p+2) 3−α 1 (1−α) 1+ p+2 e − + − ≥0, p+1 (p+1)(p+2) p+1 p+1

(2.13)

Taking α = 0 in Corollary 13, we have the following result: (1) Corollary 14. The function up (z) is a convex function for all p > −1. Proof. Taking α = 0 in (2.13) and using p > −1, then the sufficient condition for starlikeness of (1) up (z) is that 1
p+2
2p2 + 7p + 6 e − p2 + p − 1 ≥0, (2.13) 1 x+2

but the function (2x2 + 7x + 6) e − (x2 + x − 1) it is non-negative for all x > x0 ' −1.5254 (see Figure 4). This proves Corollary 14. 10

Figure 5 Remark 2. The result introduced in Corollary 14 gives a more improved result of this obtained by Mustafa [23, Corollary 3.4]. Taking β = 1 in Corollary 11, we have the following result: (2) Corollary 15. Let p > −1, then up (z) ∈ K(α) if the following condition is satisfied    1  p+2 p+2 3−α (1−α)(p+2) 1 (2.14) e ≥0, (1−α) 1+ p+1 − (p+1)(p+2) + p+1 + p+1 Taking α = 0 in Corollary 15, we have the following result: (2) Corollary 16. The function up (z) is a convex function for all p > x0 ' 3.8523. Proof. Taking α = 0 in (2.14) and using p > −1, then the sufficient condition for starlikeness of (2) up (z) is that 1

p+2 2p2 + 7p + 6 − p2 + 7p + 11 e ≥0, 1 x+2

it is non-negative for all x > x0 ' 3.8523 but the function (2x2 + 7x + 6) − (x2 + 7x + 11) e (see Figure 5). This proves Corollary 16. Taking β = 1 in Corollary 12, we have the following result: (3) Corollary 17. Let p > − 32 , then up (z) ∈ K(α) if the following condition is satisfied   2   2p+5 (1−α)(2p+5) 2(3−α) 4 (1−α) 1+ 2p+5 e − + − ≥0, 2p+3 (2p+3)(2p+5) 2p+3 2p+3

(2.15)

Taking α = 0 in Corollary 17, we have the following result: (3) Corollary 18. The function up (z) is a convex function for all p > − 32 . Proof. Taking α = 0 in (2.15) and using p > − 32 , then the sufficient condition for starlikeness of (3) up (z) is that 2
2p+5
8p2 + 36p + 40 e − 4p2 +8p − 1 ≥0, 2 2x+5

but the function (8x2 + 36x + 40) e − (4x2 + 8x − 1) it is non-negative for all x > x0 ' −2.0254 (see Figure 6). This proves Corollary 18. 11

Figure 6

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