An Application of Generalized Bessel Functions on

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Turkish Journal of Analysis and Number Theory, 2015, Vol. 3, No. 1, 1-6 Available online at http://pubs.sciepub.com/tjant/3/1/1 © Science and Education Publishing DOI:10.12691/tjant-3-1-1

An Application of Generalized Bessel Functions on Certain Subclasses of Analytic Functions G. Murugusundaramoorthy*, T. Janani School of Advanced Sciences, VIT University Vellore - 632014, Tamilnadu, India *Corresponding author: [email protected]

Abstract The purpose of the present paper is to investigate some characterization for generalized Bessel functions of first kind to be in the new subclasses of β uniformly starlike and β uniformly convex functions of order α. Further we point out consequences of our main results. Keywords: univalent, starlike, convex, uniformly starlike functions, uniformly convex functions, Bessel functions Cite This Article: G. Murugusundaramoorthy, and T. Janani, “An Application of Generalized Bessel Functions on Certain Subclasses of Analytic Functions.” Turkish Journal of Analysis and Number Theory, vol. 3, no. 1 (2015): 1-6. doi: 10.12691/tjant-3-1-1.  zf''( z )  R 1 +  >α f'( z )  

1. Introduction Let  be the class of analytic functions of the form ∞

f ( z) = z + ∑ n = 2 an z n

z ∈ .

(1)

As usual, we denote by  the subclass of  consisting of functions which are normalized by f (0)= 0= f'(0) − 1 and also univalent in  Denote by  the subclass of  consisting of functions whose nonzero coefficients from second on, is given by ∞

f ( z) = z − ∑ n = 2 an z n , an ≥ 0

(2)

For functions f ∈  given by (1) and g ∈  given by ∞

g ( z )= z + ∑ n = 2 bn z n ‍ , we define the Hadamard product (or convolution) of f and g by ∞

( f *g )( z ) = z + ∑ n = 2 an bn z n ‍ , z ∈ .

(3)

A function f ∈  is said to be starlike of order α (0 ≤ α < 1) , if and only if

 zf'( z )  R  > α ( z ∈ ).  f ( z)  This function class is denoted by  * (α ) . We also write

 * (0) =:  * where  * denotes the class of functions f ∈  such that f () is starlike with respect to the origin. A function f ∈  is said to be convex of order

α (0 ≤ α < 1) if and only if

( z ∈ ).

This class is denoted by (α ) Further,  = (0) , the well-known standard class of convex functions. It is an established fact that

f ∈ (α ) ⇔ zf' ∈  * (α ). Let  * (α ) and  (α ) are the class of starlike and convex functions of order α (0 ≤ α < 1) , introduced and studied by Silverman [21]. The class β −  was introduced by Kanas and Wi´sniowska [12], where its geometric definition and connections with the conic domains were considered. The class β −  was defined pure geometrically as a subclass of univalent functions, that map each circular arc contained in the unit disk  with a center ξ ,| ξ |≤ β (0 ≤ β < 1) , onto a convex arc. The notion of β − uniformly convex function is a natural extension of the classical convexity. Observe that, if β = 0 then the center ξ is the origin and the class β −  reduces to the class of convex univalent functions  . Moreover for β = 1 corresponds to the class of uniformly convex functions  introduced by Goodman [10,11], and studied extensively by Rønning [19,20]. The class β − P is related to the class β −  by means of the wellknown Alexander equivalence between the usual classes of convex  and starlike  * functions. Further the analytic criterion for functions in these classes are given as below(also see [6,19,20,24]). For −1 < α ≤ 1 and β ≥ 0 a function f ∈  is said to be in the class

2

Turkish Journal of Analysis and Number Theory

(i) β − uniformly starlike functions of order α is denoted by P (α , β ) if it satisfies the condition

 zf'( z )  zf'( z ) −α  > β −1 , z ∈  R f ( z ) f ( z)  

(4)

and (ii) β − uniformly convex functions of order α denoted

Theorem 1. A function f ( z ) of the form (1) is in P (λ , α , β ) if ∞

∑ n=2 [‍ n(1 + β ) − (α + β )(1 + nλ − λ )] | an |≤ 1 − α .

Theorem 2. A function f ( z ) of the form (2) is in P (λ , α , β ) if and only if ∞

∑ n=2 ‍[n(1 + β ) − (α + β )(1 + nλ − λ )] | an |

by  (α , β ), if it satisfies the condition

 zf''( z )  zf''( z ) R 1 + −α  > β , z ∈ . f'( z ) f'( z )  

(5)

Indeed it follows from (4) and (5) that

f ∈  (α , β ) ⇔ zf' ∈ P (α , β ). Remark

1.

It

is

of

interest

to

note



zf ' ( z ) (1 − λ ) f ( z ) + λ zf ' ( z )

(10)

Theorem 3. A function f ( z ) of the form (1) is in  (λ , α , β ) if

∑ n=2 n[n(1 + β ) − (α + β )(1 + nλ − λ )] | an |

that

≤ 1 − α .‍

P (λ , α , β ) be the subclass of  consisting of functions of the form (1) and satisfying the analytic criterion   zf ' ( z ) R −α   (1 − λ ) f ( z ) + λ zf ' ( z )   

≤ 1− α.

(6)

 (α , 0) = (α ) and P (α , 0) =  * (α ). Motivated by above definitions we define the following subclasses of  due to Murugusundaramoorthy and Magesh [16]. For 0 ≤ λ < 1, 0 ≤ α < 1 and β ≥ 0, we let

(9)



(11)

Theorem 4. A function f ( z ) of the form (2) is in  (λ , α , β ) if and only if ∞

∑ n=2 n‍ [n(1 + β ) − (α + β )(1 + nλ − λ )] | an | ≤ 1− α. We

recall

here

a

generalized

Bessel

ω ( p, b, c) = ω defined in [2] and given by

(12) function

ω ( z ) = ω ( p , b, c ) (7)

− 1 , z ∈ ,



= ∑ n =0

(−1) n c n z b + 1  2  ) n !Γ( p + n + 2

2n + p

(13)



and also, let  (λ , α , β ) be the subclass of  consisting of functions of the form (1) and satisfying the analytic criterion

which is the particular solution of the second order linear homogeneous differential equation

 f ' ( z ) + zf''( z )  f ' ( z ) + zf''( z ) R ' −α  > β ' − 1 , z ∈ . (8)  f ( z ) + λ zf''( z )  f ( z ) + λ zf''( z )  

where b, p, c ∈ , which is natural generalization of Bessel’s equation. The differential equation (14) permits the study of Bessel function, modified Bessel function, spherical Bessel function and modified spherical Bessel functions all together. Solutions of (14) are referred to as the generalized Bessel function of order p. The particular solution given by (13) is called the generalized Bessel function of the first kind of order p. Although the series defined above is convergent everywhere, the function ω p,b,c is generally not univalent in  . It is of interest to

We further let P (λ , α , β ) = P (λ , α , β )∩

and

 (λ , α , β ) =  (λ , α , β )∩ . Suitably specializing the parameters we note that 1. P (0, α , β ) = P (α , β ) [6] 2. P (0, 0, β ) = P ( β ) [24] 3. P (0, α ,1) = P (α ) [6] 4. P (λ , α , 0) =  * (λ , α ) [1] 5. P (0, α , 0) =  * (α ) [21] 6.  (0, α , β ) =  (α , β ) [6] 7.  (0, 0, β ) =  ( β ) [23] 8.  (0, α ,1) =  (α ) [6]

9.  (λ , α , 0) =  (λ , α ) [1] 10.  (0, α , 0) =  (α ) [21]. Now we state the following characterization properties for the classes P (λ , α , β ), P (λ , α , β ),  (λ , α , β ) and  (λ , α , β ) due to Murugusundaramoorthy and Magesh [16].

z 2ω ''( z ) + bzω '( z ) + [cz 2 − p 2 + (1 − b)]ω ( z ) = 0 (14)

note that when b= c= 1 we re-obtain the Bessel function of the first kind ω p ,1,1 = J p , and for c = −1, b = 1 the function ω p ,1,−1 becomes the modified Bessel function

 p Now, we consider the function u p ,b,c ( z ) defined by the transformation

b +1 p  u p= ,b , c ( z ) 2 Γ  p + z 2  

−p 2 ω

p ,b , c (

z ),

1 = 1. By using well known Pochhammer symbol (or the shifted factorial) defined, in terms of the familiar Gamma function, by

Turkish Journal of Analysis and Number Theory

(a ) n :=

3

[1 + β − λ (α + β )]u' p (1) + (1 − α )[u p (1) − 1] ≤ 2(1 − α ). (18)

Γ(a + n) Γ(a )

(n = 0), 1  = a (a + 1)(a + 2) (a + n − 1) (n ∈=  : {1, 2,3,…}). 

Proof. Since

z (2 − u p ( z )) = z−∑

(

∞ n = 2 ( m)

−c n −1 ) 4 zn ‍ n − ( 1)! n −1

(19)

according to Theorem 2, we must show that we can express u p ,b,c ( z ) as ∞

u p ,b , c ( z ) = ∑ n = 0

(−c / 4) n b +1   p+  2 n 

 zn    ‍ ,  n! 

(15)



∞ [‍ n(1 + β ) − (α n=2

≤ (1 − α ) Now,

b +1  where  p +  ≠ 0, −1, −2, …. This function is analytic 2   on  and satisfies the second-order linear differential equation 4 z u''( z ) + 2(2 p + b + 1) zu'( z ) + czu ( z ) = 0. 2



(

−c

≤∑

 (c, m ) :  → 

) n −1

∞ ‍ n=2

n −1

( ) (n − 1)[1 + β − λ (α + β )] 4  +(1 − α )  ( m) ( n − 1)!   n −1 (



−c

)n



∞ ∞ = [1 + β − λ (α + β )] n 1 = ‍n 4 + (1 − α ) n 1 ‍ = ( m) n ( n)!

defined by

 (c, m) f ( z ) = zu p ,b,c ( z )*f ( z ) ∞

−c

∞ 4 [ n(1 + β ) − (α + β )(1 + nλ − λ )] n=2 ( m) n −1 ( n − 1)!

Now, we considered the linear operator

= z + ∑ n=2

−c n −1 ) 4 + β )(1 + nλ − λ )] (20) (m) n −1 (n − 1)! (

(16)

(−c / 4) n −1 an z n (m) n −1 (n − 1)!

(



−c

)n



∞ ∞ 4 ‍ = [1 + β − λ (α + β )] n 1 = + (1 − α ) n 1 ‍ = ( m) n ( n − 1)!

−c

−c

)n 4 ( m) n ( n)! (

(

n

) 4 ( m) n ( n)!

−c

n +1

−c

n +1

( ) ( ) 1 + β  ∞ ∞ 4 4 = + (1 − α )∑ n 1 ‍ ‍ ‍ (b + 1) ∑   1 = = n where m = p+ ≠ 0 . For convenience throughout in − ( + ) λ α β ( ) ( )! ( ) m n m   n +1 n +1 ( n + 1)! 2 −c  ( −c ) n  the sequel, we use the following notations ( )   ∞ 4 = [1 + β − λ (α + β )] 4 ∑ n =1 ‍   b +1 m ( m + 1) n ( n)!  u p ,b,c= u p , m= p + .  2  

and for if c < 0, m > 0(m ≠ 0, −1, −2, …) let, ∞

z (2 − u p ( z )) = z − ∑ n=2



(−c / 4) n −1 n z ‍. (m) n −1 (n − 1)!

(17)

The generalized Bessel function is a recent topic of study in Geometric Function Theory (e.g. see the work of [2,3,4,5] and [14]. Motivated by results on connections between various subclasses of analytic univalent functions by using hypergeometric functions (see [8,13,15,22,25]) and by work of Baricz [2,3,4,5], we obtain necessary and sufficient condition for the function z (2 − u p ( z )) to belong to the classes P (λ , α , β ) and  (λ , α , β )

2. Main Results and Their Consequences Lemma 1. [5] If b, p, c ∈ and m ≠ 0, −1, −2,... then

(

4

(

)

n +1

−c

) = [1 + β − λ (α + β )] 4 u p +1 (1) + (1 − α )[u p (1) − 1] m = [1 + β − λ (α + β )]u' p (1) + (1 − α )[u p (1) − 1].

But the last expression is bounded above by 1 − α if and only if (18) holds. Remark 2. In particular when c = −1 and b = 1, the condition 18 becomes

[1 + β − λ (α + β )] p +1 (1)  2 p − 2 Γ( p + 1)   ≤ 1 − α , (21)  +2(1 − α )  p (1)  which

is

necessary

and

sufficient

condition

for

z (2 − ζ p ( z1/2 )) to be in P (λ , α , β ) where

the function u p satisfies the recursive relation

4mu' p ( z ) = −cu p +1 ( z ) for all z ∈  and Theorem 5. If c0

then

Theorem 6. If c < 0 and m > 0 then z (2 − u p ( z )) is in

 (λ , α , β ) if and only if

4

Turkish Journal of Analysis and Number Theory

1 + β − λ (α + β )  u ' p (1) + 3 + 2 β − α − 2λ (α + β )  u ' p (1)

(22)

+ (1 − α ) u p (1) ≤ 2(1 − α ).

Proof. In view of Theorem 4,we need to show that



∞ ‍[n(1 + β ) − (α n=2

−c n −1 ) 4 + β )(1 + nλ − λ )] (m) n −1 (n − 1)! (

≤ (1 − α ).

−c −c   ( ) n +1 ( ) n +1  (α + β )   ∞ ∞ 4 4 − ‍ + ∑n 0 ‍   ∑ n 0= = (m) n +1 (n)! (m) n +1 (n + 1)!  (1 − λ )     −c ( ) n +1 ∞ 4 = [1 + β − λ (α + β )]∑ n =0 ‍ (m) n +1 (n − 1)! +[3 + 2 β − α − 2λ (α + β )]∑

Now



∞ ‍n[ n(1 + n=2

≤∑

∞ ‍ (n + n=0

(

β ) − (α + β )(1 + nλ − λ )]

−c 4

)

+(1 − α )∑

n −1

( m) n −1 ( n − 1)! (

−c

) n +1

 ( n + 2)(1 + β )  4 2)   (m) (n + 1)! (1 ( 2) )( ) − + n + − + λ λ α β   n +1

= [1 + β − λ (α + β )]∑

−(α + β )(1 − λ )∑

∞ ‍ (n + n =0

∞ ‍ (n + n =0

( 2) (

2)

2

−c 4

) n +1

( m) n +1 ( n + 1)! −c 4

) n +1

( m) n +1 ( n + 1)!

−c 1 −c 1   ( )n+ ( )n+  ∞  ∞ 4 4 ‍ ( n + 1) 2 ‍ + 2∑ n 0 ‍ ( n + 1) ∑ n 0= ( m) n +1 ( n + 1)! ( m) n +1 ( n + 1)!     −c   ( ) n +1 + ∞  4 ‍  ∑ n = 0 ( m) ( n + 1)!    n +1 −c   ( ) n +1  ∞  4 ‍ ∑ n = 0 ( n + 1) ( m) n +1 ( n + 1)!   −(α + β )(1 − λ )   −c   ( ) n +1 + ∞  4  ∑ n = 0 ( m) ( n + 1)!‍    n +1 −c   ( ) n +1  ∞  4 ∑ n = 0 ( n + 1)  ( m) n +1 ( n)! 1 + β   =   λ ( α β ) − c − c − +   ( ) n +1 ( ) n +1  ∞  +2 ∞  4 4 + ∑n 0 ‍ =  ∑ n 0= ( m) n +1 ( n)! ( m) n +1 ( n + 1)!    −c n +1 −c   ( ) n +1  ∞ (4)  ∞ 4 ‍ −(α= + β )(1 − λ ) ∑ n 0= + ∑n 0 ( ) ( )! ( ) ( 1)! + m n m n n +1 n +1     −c   ( ) n +1  ∞  4 ‍ ∑ n = 0  ( m) n +1 ( n − 1)! 1 + β   =   ( ) − c − c − λ α + β   ( ) n +1 ( ) n +1  ∞  +3 ∞  4 4 + ∑n 0 ‍ =  ∑ n 0=  ( ) ( )! ( ) ( 1)! m n m n +   n +1 n +1

(

∞ n = 0 ( m)

−c ( ) n +1 ∞ 4 ‍ n = 0 ( m) n +1 ( n)!

−c n +1 ) 4 n +1 ( n + 1)!

−c 2 −c   ( ) n −1  1 + β  (4) ∞  4 =  ‍ ∑  = 0 n  −λ (α + β )  m(m + 1)  (m + 2) n −1 (n − 1)!    −c −c n    3 + 2 β − α  ( 4 ) ∞  ( 4 ) +   ∑  n = 0 ( m + 1) ( n)! n  −2λ (α + β )  m     −c ( ) n +1 ∞ 4 ‍ + (1 − α )∑ n =0 (m) n +1 (n + 1)! −c 2 ) 4 u p + 2 (1) = 1 + β − λ (α + β )  m ( m + 1) (

 −c    4 + 3 + 2 β − α − 2λ (α + β )    u p +1 (1) m +(1 − α )[u p (1) − 1] = [1 + β − λ (α + β )]u'' p (1) +[3 + 2 β − α − 2λ (α + β )]u' p (1) +(1 − α )[u p (1) − 1]. By a simplification, we see that the last expression is bounded above by (1 − α ) if and only if (22) holds.

3. Inclusion Properties A function f ∈  is said to be in the class τ ( A, B ), , (τ ∈  \ {0}, −1 ≤ B < A ≤ 1) , if it satisfies the inequality

f ' ( z) − 1 ( A − B )τ − B[ f ' ( z ) − 1]

0 If f ∈ τ ( A, B ) and if the inequality

[1 + β − λ (α + β )]u' p (1)  ( A − B) | τ |   ≤ 1−α + (1 − α )[u p (1) − 1] 

5

(24)

−c   ( ) n +1   ∞ 4 ‍ [1 + β − λ (α + β )]∑ n =0 (m) n +1 (n)!   ≤ ( A − B) | τ |   −c   ( ) n +1 +(1 − α ) ∞  4 ‍ ∑ n=0 (m) (n + 1)!   n +1   − c   ( )   4 u p +1 (1)  ≤ ( A − B ) | τ | [1 + β − λ (α + β )] m   (1 )[ (1) 1] + − − α u p   [1 + β − λ (α + β )]u' p (1)  ≤ ( A − B) | τ |   + (1 − α )[u p (1) − 1]  But this last expression is bounded above by 1 − α if and only if (23) holds. Theorem 8. Let c < 0 and m > 0 then

is satisfied, then  (c, m)( f ) ∈  (λ , α , β ) . Proof. Let f be of the form (1) belong to the class

z

(m, c,= z)

τ

 ( A, B ) By virtue of Theorem 4, it suffices to show that

(

−c

) n −1

 n(1 + β )  ∞ 4 ‍n | an | ≤ 1 − α . n = 2  −(α + β )(1 + nλ − λ )  ( m)   n −1 ( n − 1)!

Since

0

is in  (λ , α , β ) if and only if inequality

L (α , β , λ )



∫ ‍ (2 − u p (t ))dt

[1 + β − λ (α + β )]u' p (1) + (1 − α )[u p (1) − 1] ≤ 1 − α . (26) Proof. Since

f ∈ τ ( A, B ) then by Lemma 2 we have

an ≤ ( A − B ) L(α= , β , λ)

τ n



(m, c, z )= z − ∑

∞ n=2

. Hence

−c n −1 ) zn 4 .‍ (m) n −1 (n)!

(

then by Theorem 4 we need only to show that ∞ n[n(1 + β ) − (α n=2

+ β )(1 + nλ − λ )]‍

−c n −1 ) τ 4 ( A − B) (m) n −1 (n − 1)! n (

 ∞  n (1 + β )  ∑ n=2  ‍    − (α + β )(1 + nλ − λ )    ≤ ( A − B) | τ |   ( −c ) n −1    4  (m) (n − 1)!  n −1    ∞ (n − 1)[1 + β − λ (α + β )]  ∑ n=2  ‍   +(1 − α )    ≤ ( A − B) | τ |  −c n −1  ( )  4    ( m) n −1 ( n − 1)!  −c   ( ) n  (25)  ∞ 4 ‍ (1 + β − λ (α + β ))∑ n =1 n (m) n (n)!   ≤ ( A − B) | τ |   −c ( )n   4  +(1 − α )∑ ∞  ‍ =1 ( m) ( n)! n   n Further, proceeding as in Theorem 5

P(α , β , λ ) = ∑

 n(1 + β ) ∞ n n = 2  −(α + β )(1 + nλ 

 −c n −1  ( )   4  ‍  − λ )   (m) n −1 n!     

Now

 −c n −1  (4)  + β + n λ − λ )(1 )]  ‍ ∑  (m) n −1 n!    −c n −1    (4)  ∞ ≤ ∑ n = 2 [n(1 + β ) − (α + β )(1 + nλ − λ )]    (m) n −1 (n − 1)!    ∞ n[n(1 + β ) − (α n=2

Further, proceeding as in Theorem 5 P (α , β , λ ) −c

n +1

−c

n +1

( ) ( ) 1 + β  ∞ ∞ 4 ≤ + − (1 α ) ∑ n 0 (m) 4 (n + 1)!‍  −λ (α + β )  ∑ n 0= = ( m ) ( n )!   n +1 n +1 (

−c

) = [1 + β − λ (α + β )] 4 u p +1 (1) + (1 − α )[u p (1) − 1]. m = [1 + β − λ (α + β )]u' p (1) + (1 − α )[u p (1) − 1]

which is bounded above by 1 − α if and only if (26) holds.

6

Turkish Journal of Analysis and Number Theory

Concluding Remarks If we put c=-1 and b=1 in above theorems we obtain analogous results of (21). Further by taking β=0 and specializing the parameter λ we can state various interesting results (as proved in above theorems) for the various subclasses listed in the introduction.

[10] A.W.Goodman, On uniformly convex functions, Ann.polon.Math., 56, (1991), 87-92.

[11] A.W.Goodman, [12] [13] [14]

Acknowledgement

[15]

The authors thank the referees for their valuable suggestions to improve the paper in present form.

[16]

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