THIRD HANKEL DETERMINANT FOR A SUBCLASS OF ALPHA CONVEX FUNCTIONS

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THIRD HANKEL DETERMINANT FOR A SUBCLASS OF

ALPHA CONVEX FUNCTIONS

Gagandeep Singh

Department of Mathematics, M.S.K. Girls College, Bharowal(Tarn-Taran), Punjab,

India

Gurcharanjit Singh

Department of Mathematics, Guru Nanak Dev University College,

Chungh(Tarn-Taran), Punjab, India

Abstract

:

In this paper a sharp upper bound of third Hankel determinant

H

3

 

1

for the functions belonging to a subclass of alpha convex functions is established. By giving the particular values to alpha, it is easy to obtain the upper bound of

 

1

3

H

for starlike and convex functions.

Mathematics Subject classification

:

30C45

Keywords

:

Analytic functions, Starlike functions, Convex functions, Alpha convex functions, Third Hankel determinant.

1. Introduction

Let A be the class of analytic functions of the form

 











2

k k k

z

a

z

f

(1.1)

(2)

By S we denote the class of functions

f

 

z



A

and univalent in E.

M

 denote the class of functions

f

 

z



A

and satisfying the condition

 



  

 

0 ,

0

1 ,

.

1 Re

2

E

z

f

z

f

z

f

z

f

z

_

_

_

























_



(1.2)

The class

M

_ is the subclass of alpha-convex functions studied by Singh [11]. Also

M

0



S

, the class of starlike functions and

M

₁



K

, the class of convex functions.

For the complex sequence

a

_n

,

a

_n_₁

,

a

_n_₂_,

...,

the Hankel matrix, named after Herman Hankel(1839-1873), is the infinite matrix whose

 

i

,

j

th entry

a

_ij is defined by

a

ij



a

nij2



i

,

j

,

n



N



.

The qth Hankel matrix (q ϵ N \ {1} ) is by definition, the following q x q square sub matrix:

.

...

2 2 1

1

1 1













  

 

  

q n q

n n

q n n

n

a

We observe that the Hankel matrix has constant positive slopping diagonals whose entries also satisfy:

a

ij



a

i1,j1



i



N

\



1 ;

j



N



.

This also describes the Hankel matrix without reference to a particular sequence. The determinant of the _qth_{Hankel matrix,}

usually denoted by

 

,

...

2 2 1

1

1 1

  

 

  



q n q

n n

q n n

n

q

a

n

H

is called the qthHankel determinant. In the particular cases

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H

₂

 

1 

a

₃



a

₂2 _and

H

₂

 

2 

a

₂

a

₄



a

₃2

.

In this paper, we consider the Hankel determinant in the case q = 3 and n = 1,

 

1 .

5 4 3

4 3 2

3 2 1 3

a

H



 

2

H

and

H

₃

 

1

are respectively called second and third Hankel determinants.

For

f



S

,

a

₁



1

so that,

 



 

2



2 3 5 3 2 4 4 2 3 4 2 3

3

1 a

a

H









and by using the triangle inequality, we have

 

1

₃ ₂ ₄ ₃2 ₄ ₂ ₃ ₄ ₅ ₃ ₂2

.

3

a

H













(1.3)

Second Hankel determinant for various classes has been extensively studied by various authors including Singh[11,12],

Mehrok and Singh[8] and Janteng et al.[3,4,5]. But Third Hankel determinant has been studied by some of the Researchers

including Babalola [1] and Shanmugam et al.[10].

For our discussion in this paper, we consider the third Hankel determinant and obtain an upper bound to the functional

 

1

3

H

for the functions in the class

M

_. Results due to Babalola [1] follows as special cases.

2. Preliminary Results

Let P be the family of all functions p analytic in E for which

Re



p

 

z





0

and

p

 

z



1 

p

1

z



p

2

z

2



...

for

z



E

.

Lemma 2.1.[9] If

p



P , then

p

k



2 

k



1 ,

2 ,

3 ,...



.

Lemma 2.2.[6,7] If

p



P , then

2 

4

12



,

2

1

2

p

x

p







(4)

for some x and z satisfying

x



1 ,

z



1

and

p

1



 

0 ,

2

. Lemma 2.3.[2] If

p



P , then































.

2

1

2 ,

2

0

2 ,

0

1

2

2 1 2



if

p

Lemma 2.4.[11] If

f

 

z



M

_, then







.

3

1

2 3 4

2

_







a

3. Main Results

Theorem 3.1 If

f



M

_, then

,

1

2



_

_

a

,

2

1

3

_





a

1

3 

4



_

a

and

.

4

1

5



_



a

Proof. Since

f

 

z



M

_, then there exists

p



P

such that



 

  

 

.

1

2

z

p

z

f

z

f

z

f

z

f

z

_















(3.1)

Equating coefficients in (3.1) yields

,

1

1 2



_

_

p

(5)



 



,

2

1

2

1

2

2 1 2

3



_





_



p

a

(3.3)



 



,

3

1

6

3

1

2

3

1

3

3 1 2 1 3

4



_

_



_

_



_

_

p

a

_(3.4)

4 

1

4 

 

4

1

4 

 

8

1

4 

 

3

1

4 



24 

1

4 



4 1 3 1 2 2 2 2 1 4

5



_



_



_



_



_

p

a

(3.5)

and the results follow by triangle inequality and using Lemma 2.1.

Theorem 3.2 If

f



M

_, then





.

1

0

3

1

3

2

0

2

4 3 2























if

a

Proof. From equations (3.2),(3.3) and (3.4), we obtain













1 

1

2 

1

3  

3

1

3 

.

6

2

6

2

3

1

2

1

2

3 ₃

1 2 2 1 2 4 3

2

_



















a

p

a

(3.6)

Substituting for

p

₂ and

p

₃ from Lemma 2.2 and letting

p

₁



p

, we get















 

















.

3

1

6

1

4

3

1

12

4

3

1

2

1

6

4

5

3

1

3

1

2

1

4

3

1

2 3 2 2 2 2 2 2

4 3

2

_





























a

p

px

p

px

p

x

z

a

Since

p



p

1



2

by using Lemma 2.1, we may assume that

p



 

0 ,

2

. Then using triangle inequality and

z



1

with

x





, we obtain















 



























3

1

12

4

2

3

1

6

4

3

1

2

1

6

4

5

3

1

3

1

2

1

4

3

1

2 3 2 2 2 2 2

4 3 2

_





















a

p

a



F

 



.

Then

 



_

_

 

_

_





_





_



.

3

1

6

4

2

3

1

2

1

6

4

5

3

1

2 2 2























p

(6)

Note that

F



 





F



 

1 

0 .

Then there exists

p





 

0 ,

2

such that

F



 





0

for

p



(

p



,

2 ]

and

F



 





0

otherwise. Then for

p



(

p



,

2 ]

,

F

 





F

 

1 .

But

 

_



_



_{ }

_

_



3

1

2

1

2

3

1

2

1

4

3

1

3 2 2









p

F



G

 

p

.

If





0 ,

we have

G

 

p



p



2 .

Otherwise

G

 

p

is maximum at ₂ 2

6

4

3

1 







p

and is given by

 











.

6

4

3

1

3

1

2

1

3

4

3

1

2

2 2 2







p

G

If

p



[

0 ,

p



]

, then

F

 





F

 

0 ,

that is

 

_



_



_



_



_





3

1

6

4

3

1

2

1

4

3

1

2 3 2













p

F



G

 

p

.

Now

G

 

p

turns at p = 0 or





₂



4

3

1

9

2

1

4 









p

with its maximum at p = 0.

Hence





.

3

1

3

2

4 3

2

a



a



_



a

For





0

, Theorem 3.2 agree with the following result due to Babalola [1].

Corollary 3.2.1 If

f

 

z



S

, then

a

2

a

3



a

4



2 .

For





1

, Theorem 3.4 gives the following result due to Babalola [1].

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.

6

1

4 3

2

a



a



a

Theorem 3.3 If

f



M

_, then

.

2

1

2 2

3

_







a

Proof. Since

f

 

z



M

_, then using equations (3.2) and (3.3) we obtain















2

2 2

1 2

2 2 3

1

2

1

2

1

2

1

2 

















a

p

a











1 

2 .

2

1

2

1

2

1

₁2

2 2 2

p















Using Lemma 2.3, with







1 

2 ,

2

1

2

0

2 2

















we have

.

2

1

2 2

3

_







a

Theorem 3.4 If

f



M

_, then

 





 



.

1

0

4

1

3

1

2

1

3

199

445

224

32

0

16

1

2 3 2

3























if

H

Proof. Using Lemma 2.4 and Theorems 3.1, 3.2 and 3.3 in inequality (1.3), the above result can be easily obtained.

For





0

, Theorem 3.4 agree with the following result due to Babalola [1].

Corollary 3.4.1 If

f

 

z



S

, then

H

3

 

1 

16 .

For





1

, Theorem 3.4 gives the following result due to Babalola [1].

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 

.

24

15

1

3



H

4. References

[1]

_{K. O. Babalola, On}

 

₁

3

H

Hankel determinant for some classes of univalent functions, Inequality

Theory and Applications, 6(2010), 1-7.

[2]

_{K. O. Babalola and T. O. Opoola, On the coefficients of certain analytic and univalent functions,}

Advances in Inequalities for Series, (Edited by S. S. Dragomir and A. Sofo) Nova Science Publishers, (2006),

5-17.

[3]

Aini Janteng, Suzeini Abdul Halim and Maslina Darus, Coefficient inequality for a function whose

derivative has a positive real part, J. Ineq. Pure Appl. Math., 7(2) (2006), 1-5, Art. 50.

[4]

_{Aini Janteng, Suzeini Abdul Halim and Maslina Darus, Hankel determinant for starlike and}

convex functions, Int. J. Math. Anal., 1(13) (2007), 619-625.

[5]

_{Aini Janteng, Suzeini Abdul Halim and Maslina Darus (2006), Hankel determinant for functions}

starlike and convex with respect to symmetric points, J. Quality Measurement and Anal., 2(1), 37-43.

[6]

R. J. Libera and E-J. Zlotkiewiez, Early coefficients of the inverse of a regular convex function, Proc.

Amer. Math. Soc., 85(1982), 225-230.

[7]

_{R. J. Libera and E-J. Zlotkiewiez, Coefficient bounds for the inverse of a function with}

derivative in P, Proc. Amer. Math. Soc., 87(1983), 251-257.

[8]

B. S. Mehrok and Gagandeep Singh, Estimate of second Hankel determinant for certain classes of

analytic functions, Scientia Magna, 8(3)(2012), 85-94.

[9]

Ch. Pommerenke, Univalent functions, Göttingen: Vandenhoeck and Ruprecht., 1975.

[10]

G. Shanmugam, B. Adolf Stephen and K. O. Babalola, Third Hankel determinant for





starlike

functions, Gulf Journal of Mathematics, 2(2)(2014), 107-113.

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