Subclasses of Multivalent Functions of Complex Order Related to Sigmoid Function

Subclasses of Multivalent Functions of Complex Order Related to Sigmoid Function

Gagandeep Singh

¹

and Gurcharanjit Singh

²

1

Department of Mathematics,

Majha College for Women, Tarn-Taran (Punjab), INDIA.

2

Department of Mathematics,

Guru Nanak Dev University College, Chungh, Tarn-Taran (Punjab), INDIA.

email:[email protected], [email protected].

(Received on: January 17, 2018) ABSTRACT

In this paper, the authors investigate the initial coefficient bounds for some new subclasses of multivalent functions related to Sigmoid function. Also the relevant connections to Fekete-Szegö inequality and Hankel determinant for these classes are briefly discussed. Our results serve as a new generalization in this direction.

Mathematics Subject Classification: 30C45, 33E99

Keywords: Analytic functions, Starlike function, Convex function, Subordination,

Sigmoid function, Fekete- Szegö problem.

1. INTRODUCTION AND PRELIMINARIES

The theory of special functions is significantly important to scientists and engineers.

Though not with any specific definition but its applications extend to physics, computer etc.

Recently, the theory of special functions has been overshadowed by other fields like real analysis, functional analysis, algebra, topology and differential equations.

There are various special functions but we shall concern with one of the activation function known as sigmoid function or simple logistic function. Activation function is an information process that is inspired by the biological nervous system such as brain processes information. It comprises of large number of highly interconnected processing element (neurons) working together to solve a specific task. The function works in similar way the brain does, it learns by examples and cannot be programmed to solve a specific task.

The sigmoid function of the form

 

_z

z e

h

_

  1

1 (1.1)

is differentiable and has the following properties:

 It outputs real numbers between 0 and 1.

 It maps a very large input domain to a small range of outputs.

 It never loses information because it is a one-to-one function.

 It increases monotonically.

The four properties above shows that sigmoid function is very useful in geometric function theory.

Let A

_p

denote the class of functions of the form

  ,

2















n

p n p n

p

a z

z z

f (1.2)

which are analytic and p-valent in the open unit disc ^{E }  ^{z z :  1}  ^.

Let U be the class of bounded functions

  ,

1









n n n

z c z

w (1.3) which are regular in the unit disc and satisfying the conditions

(0) 0

w  and ^{w z }   ^{1 in E.}

For functions f and g analytic in E, we say that f is subordinate to g , denoted by f  g , if there exists a Schwarz function w   z  U , ^{w z}   analytic in E with w (0)  0 and

  ¹

w z  in E, such that ^{f z}   ^{ g w z}     . It follows from Schwarz lemma that

  z g   z z E  f   0 g   0

f     _and f   E  g   E (see detail in

⁵

).

Let    z be an analytic function with positive real part in E such that    0  1 and

  0  0

  and maps E onto a region starlike with respect to 1 and symmetric with respect to the real axis.

Fekete and Szegö in 1933 gave the sharp bound for the functional a

₃

  a

₂²

_{for the} functions in the class S of univalent functions when  is real. The determination of the sharp bounds for the functional a

3

  a

2²

is known as the Fekete-Szegö problem and this has been investigated by several authors for different subclasses of univalent functions.

In 1976, Noonan and Thomas

⁸

stated the qth Hankel determinant for q  1 and n  1 as

 

2 2 1

1

1 1

...

...

...

...

...

...

...

...

...

...



















q n q

n n

q n n

n

q

a a

a

a a

a n H

.

This determinant has also been considered by several authors. Easily, one can observe that the Fekete and Szegö functional is ^H

₂

  ¹ .

For q  2 and n  2 ,  

4 3

3 2 2 2

a a

a

H a

is the second Hankel determinant.

For q = 3 and n = 1,

 

5 4 3

4 3 2

3 2 1 3

1

a a a

a a a

a a a

H



is called the third Hankel determinant.

Obviously   ¹

4 2 3 4 5 3 2²

^.

2 3 4 2 3

3

a a a a a a a a a a a

H      

A function f   z  A

_p

is said to be in the class f   z  S

b^*, p

  ^ _if

 

  1   ,  .

1

1 1 z p N z E

z f

z f z p

b   





 

  

  

A function f   z  A

_p

is said to be in the class f   z  C

_b,p

   if it satisfies

 

    ,  .

1 1

1 1 z p N z E

z f

z f z bp

b   





 





 



  

The classes S

_b^*_{, p}

   _and C

_b,p

   were studied in

¹

. For b = 1 we have the classes

  

*

S

p

_and C

_p

   (see [2]) and for p = b = 1 the classes reduced to the classes S

^*

   _and

  ^

C which were earlier introduced and investigated in

⁴

. These classes become the classes of starlike and convex functions respectively when

  .

1 1

z z z



 



Also for p = 1 and  

z z z



  1

 1 , the classes S

_b^*_{, p}

   _and C

_b,p

   reduces to the

classes S

^*

  b _and C   b which were investigated in

⁷

and

¹³

.

Motivated by above defined classes, we introduce the following subclasses of p- valent analytic functions of complex order related to sigmoid functions.

Definition 1. For b  C . Let the class M

p_,

 b , 

m_,n

 denote the subclass of A

_p

consisting of functions of the form (1.2) and satisfying the following condition

   

   1    ⁰

1

²

 





 

 



 

 

  p

z f z

f z

z f z z f z p b





 ,

for 0    1 _and 

_m,n

  z is a simple logistic sigmoid activation function.

Definition 2. For b  C . Let the class G

p_,

 b , 

m_,n

 denote the subclass of A

_p

consisting of functions of the form (1.2) and satisfying the following condition

 

 

 

  ⁰

1

²

 





 

  

 

 p

z f

z f z z f

z f z

p b  _,

for 0    1 _and 

_m,n

  z is a simple logistic sigmoid activation function.

Recently, various authors as Abiodun

⁹

, Murugusundramoorthy et al.

⁶

, Olatunji et al.

¹¹

, and Olatunji

¹⁰

have studied sigmoid function for different classes of analytic and univalent functions.

In the present work, we obtained few coefficient bounds for the classes M

p_,

 b , 

m_,n



and G

_p,

 b , 

_m,n

 and the relevant connection with Fekete-Szegö theorems and Hankel determinant.

To prove our result we shall make use of the following lemmas:

Lemma 1.[12] If a function p  P is given by p   z  1  p

₁

z  p

₂

z

²

 ...  z  E  , then p

_k

 , 2 k  N where P is the family of all functions analytic in E for which p   0  1 and Re  p   z   0  z  E  .

Lemma 2.[3] Let h be the sigmoid function defined in (1.1) and

       

! , 1 2

1 1 2

1 1

m

n

n n

m m

m

n z z

h

z  





 



 

 





  









then    z  P , z  1 _where    z is a modified sigmoid function.

Lemma 3.[3] Let

       

! , 1 2

1 1 2

1 1

,

m

n

n n

m m

m n

m

z

z n h

z  





 



 

 





  









then 

_m,n

  z  2 . Lemma 4.[3] If

  1 ,

1













n n n

z c z

where  

! , 2

1

¹

c n

n n





 _then c

_n

 , 2 n  N and the result is sharp for each n.

2. INITIAL COEFFICIENTS

Theorem 2.1 If f   z  A

_p

of the form (1.2) is belonging to M

p_,

 b , 

m_,n

 , then

 

 

 1  ^,

2

1 1

1

p

p b

a

_p

p









 



(2.1)

 

 

 

 1 1  ^,

8

1 1

1 2 2

2

h

p p b

a

_p

p 







 







(2.2)

 

 

 

 

²

2 2

3

3

1 2 2 1

24

1

1 p b h

p p b

a

_p

p  







 







(2.3) and

 

 

 

  3 ^.

4 2 3 1

192

1 1

3 2

2 2 2

4

p b h

p p b

a

_p

p  







 





 (2.4)

Proof. As f   z  M

p_,

 b , 

m_,n

 , therefore

   

      p p   z

z f z

f z

z f z z f z

p b

_m,n

2

1

1   





 

 



 

 

 





 (2.5)

Where   ...

20160 779 64

1 240

1 24

1 2

1 1

^{3 5 6 7}

,

      



_mn

z z z z z z (2.6)

Using (2.6), (2.5) can be expanded as

             

 

           

 ^{1 1 1 1 1 1 2 ...} 

240 ...

1 24

1 2 1

...

3 1

4 2

1 3 1

1 2 1

3 3 2

2 1

5 3

4 4 3

3 2

2 1





















 





 

   







































z a p z

a p z

a p p

z z z bp

z a p z

a p z

a p z

a p

p p

p

p p

p p

















(2.7)

Equating the coefficients of z , z

²

, z

³

and z

⁴

in (2.7), we obtain

 

 

 ¹  ^,

2

1 1

1

p

p a

_p

pb









 



(2.8)

 

 

 

 1 1  ^,

8

1

2

1

2

2

 



 



p

p b

a

_p

p



 (2.9)

 

 

 

  ^ _



 

 







 



3

1 2 2 1

24

1

1

^{2 2}

3

b p p

p a

_p

pb



 (2.10)

and

 

 

 

  3 ^.

4 2 3 1

192

1

1

^{2 2}

2 2

4







 

 







 



b p p

p b

a

_p

p



 (2.11)

Results (2.1), (2.2), (2.3) and (2.4) can be easily obtained from (2.8), (2.9), (2.10) and (2.11) respectively.

For   0 in Theorem 2.1, the following result is obvious:

Corollary 2.1 If f   z  M

p_,0

 b , 

m_,n

 , then 2 ,

1

b

a

_p

 p , 8

2 2

2

b a

_p

 p

3 1 2 24

2 2

3

 



b b p

a

_p

p and .

3 4 2 192

2 2 2 2

4

 



b b p a

_p

p

For   1 Theorem 2.1 gives the following result:

Corollary 2.2 If f   z  M

p_,1

 b , 

m_,n

 , then

 1  ^,

2

2

1

p

b a

_p

p

 



 2  ^,

8

3 2

2

p

b a

_p

p

 



  3

1 2 3

24

2 2 2

3



 



b p p b

a

_p

p and

  3 ^.

4 2 4

192

2 2 2

3

4



 



b p p b

a

_p

p

Putting p  1 in Theorem 2.1, we obtain the following result:

Corollary 2.3 If f   z  M

_

 b , 

m_,n

 , then

 1  ^,

2

2



 b 

a

 1 2  

8

2

3

 b 

a _,

  3

1 2 3 1 24

2

4



 b  b

a  ^and

  3 ^.

4 2 4 1 192

2 2

5



 b  b

a 

Theorem 2.2 If f   z  A

_p

of the form (1.2) is belonging to G

p_,

 b , 

m_,n

 , then

 

 1 1  ^,

1

2



 



p p

b a

_p

p

 (2.12)

  

     

^1,

2 2

2

4 2 1 2 1 1 m

p p p

p

b

a

_p

p 









 



  (2.13)

  

          

²

2 2

3

3

1 1 1

2 1 2

3 2 3

8 m

p p p

p

b p p

p b

a

_p

p  















 



   (2.14)

and

  

               

3



1



1

 

^.

1 3

1 1 1

2 1 2 3 2 3

1 4

3 4

16 ³

2 2 2

2

4 m

p p p

p p

p b p p

p p

p b

a_p p 



 







 





















 

     

(2.15)

Proof. Since f   z  G

p_,

 b , 

m_,n

 , therefore

 

 

 

  p p   z

z f

z f z z

f z f z

p b

_m,n

1

2



 





 

  

 

  (2.16)

where   ...

20160 779 64

1 240

1 24

1 2

1 1

^{3 5 6 7}

,

      



_mn

z z z z z z (2.17)

Using (2.17) in (2.16), it yields

                    



1 ...



24 ...

1 2 1

...

4 3 4 3

2 3 2

1 2 1 1 1

3 3 2 2 1 3

4 4 3

3 2

2 1







 







  



















































z a z a z a z

z bp

z a p p z

a p p z

a p p z a p p p

p

p p p

p p

p

p   





(2.18)

Equating the coefficients of z , z

²

, z

³

and z

⁴

in (2.18), we obtain

 

 1 1  ^,

1

2



 



p p

a

_p

pb

 (2.19)

  

 2 1 2   1  1   ^,

4

2 2

2

     



p p p p

b a

_p

p



 (2.20)

  

           ^ _



 

 















 



3

1 1 1

2 1 2

3 2 3

8

2 2

3

p p p p

b p p

p a

_p

pb





 (2.21)

and

  

                ³  ¹  ¹   ^.

1 3

1 1 1

2 1 2 3 2 3

1 4

3 4

16

2 2 2

2

4















 

 



 

 





















 

 p p pp pp

b p p

p p

p b a_p p









 (2.22)

Results (2.12),(2.13),(2.14) and (2.15) can be easily obtained from (2.19),(2.20),(2.21) and (2.22) respectively.

For   0 in Theorem 2.2, the following result is obvious:

Corollary 2.4 If f   z  G

p_,0

 b , 

m_,n

 , then 2 ,

1

b a

_p

 p

8

2 2 2

b a

_p

 p _,

3 1 2 24

2 2

3

 



b b p

a

_p

p and .

3 4 2 192

2 2 2 2

3

 



b b p a

_p

p

For   1 , Theorem 2.2 gives the following result:

Corollary 2.5 If f   z  G

p_,1

 b , 

m_,n

 , then

 

 1 1  ^,

1

2



 



p p

b

a

_p

p

  

 2 1 2   1  1   ^,

4

2 2

2

     



p p p p

b

a

_p

p

  

           3

1 1 1

2 1 2

3 2 3

8

2 2

3

















 



p p p p

b p p

p b a

_p

p

and

  

                ³  ¹  ¹   ^.

1 3

1 1 1

2 1 2 3 2 3

1 4

3 4 16

2 2 2

2

4

   





 

 





















 



p p p p p p

b p p

p p

p b

a

_p

p

Putting p  1 in Theorem 2.2, we obtain the following result:

Corollary 2.6 If f   z  G

_

 b , 

m_,n

 , then

 1 2  ^,

2

2



  b

a

 1 2  1 3  ^,

8

2

3

   b  

a

     3

1 3 1 2 1 2 4 1 24

2

4







 





 b b

a

and        3  1 2  ^.

1 3 1 2 1 6 2 4 1 3

1 5 1 64

2 2

5

       





 

 







 b  b

a

3. FEKETE-SZEGÖ INEQUALITY

Theorem. 3.1 If f   z  A

_p

of the form (1.2) is belonging to M

p_,

 b , 

m_,n

 , then

 

 

 

 

 

   ^{ } 

_4.

2 2

2 2 2

1

2

1 1

1 1

2 1 1

4

1

1 p h

p p p

p b

a p

a

_{p p}

   











 



_



 







  (3.1)

Proof. From (2.7) and (2.8), we have

 

 

 

 

 

   1 ^ 1 ^  .

1 1

2 1 1

4

1

1

²

2 2

2 2

1

2







 

   











 



_



p

p p p

p b

a p

a

_{p p}

 







  (3.2)

Hence (3.1) can be easily obtained from (3.2)

For p  1 , Theorem 3.1 gives the following result:

Corollary 3.1 If f   z  M

_

 b , 

m_,n

 , then

 

 

 1 2  ^.

2 1 1

4

2 2

2 2

2

3







  





 

 b

a a

Theorem. 3.2 If f   z  A

_p

of the form (1.2) is belonging to G

p_,

 b , 

m_,n

 , then

 

 

 

 

  

 2 1 2  ^.

1 1

1 1

4

^{2 4}

2 2 2

1

2

m

p p

p p p

p b a p

a

_{p p}

 













 



_









  (3.3)

Proof. Using (2.16) and (2.17), we have

 

 

 

 

  

 2 1 2  ^.

1 1

1 1

4

²

2 2 2

1

2







 

 













 



_









 

p p

p p p

p b a p

a

_{p p}

(3.4)

Hence (3.3) can be easily obtained from (3.4)

For p  1 , Theorem 3.2 gives the following result:

Corollary 3.2 If f   z  G

_

 b , 

m_,n

 , then

 

 

 ^{1 3}  ^.

2 2 1 2 1 4

2 2

2 2

2

3







  





 

 b

a a

4. SECOND HANKEL DETERMINANT

Theorem. 4.1 If f   z  A

_p

of the form (1.2) is belonging to M

p_,

 b , 

m_,n

 , then

 

 

      3 4  1 _ 1 _  ^.

1 2 2 1

1 3

1 16

1 1

2 5 2 2 2

2 2 2 2

2 2 3

1

h

p b p b

p p p p

b a p

a

a

_{p p p}





 

 



 

 









 



_





 





  (4.1)

Proof. From (2.7),(2.8) and (2.9), we have

 

 

     

 

 

 

 1 1  ^.

64

1 1

3 1 2 2 1

1 48

1 1

2 4 2

4 2

2 2 2

2 2

2 3

1

 



 

 



 

 









 



_





p

p b

p b

p p

p p b

a p a

a

_{p p p}



 





  (4.2)

Hence (4.1) can be easily obtained from (4.2)

For p  1 , Theorem 4.1 gives the following result:

Corollary 4.1 If f   z  M

_

 b , 

_m,n

 , then

    3 4  1 2  ^.

1 2 3 1 1 3

1

16

²

2 2 2

2 3 4

2

 



 

 

 



 

 



 

 b b b

a a a

Theorem. 4.2 If f   z  A

_p

of the form (1.2) is belonging to G

_p,

 b , 

_m,n

 , then

 

 

^{2 5}

2 2 2

2 3

1

16 1 1 m

p p

b a p

a

a

_{p p p}

 



 



_





 

  (4.3)

where    

  

           ^ _



 

 



















 

3 1 1 1

2 1 2

3 2 3

1

1

^{2 2}

p p p

p

b p p

p p p







  _and

  

 2 1 2 

²

^.

2 2





 

p p

b p

 

Proof. From (2.16),(2.17) and (2.18), we have

 

               

  

 2 1 2   1  1   ^.

16

3 1 1 1

2 1 2

3 2 3

1 1

16

2 2

4 4

2 2 2

2 2

2 3 1













 



 

 



















 



_





p p p

p

b p

p p p

p

b p p

p p

p

b a p

a

a

_{p p p}



 







 

(4.4)

Hence (4.3) can be easily obtained from (4.4).

For p  1 , Theorem 4.2 gives the following result:

Corollary 4.2 If f   z  G

_

 b , 

m_,n

 , then

 

 

     ³  ^{2 6}  ^.

1 2 1 6 2 12 3

2 1 2

1

16

²

2 2

2 2 2

2 3

1

 









 

 

 



 

 









 



_





b b b

a a

a

_{p p p}

5. THIRD HANKEL DETERMINANT

Theorem. 5.1 If f   z  A

_p

of the form (1.2) is belonging to M

p_,

 b , 

m_,n

 , then

 

 

     

 

 

 

  3 ^.

1 2 2 1

24

1 1

1 1

1 16

1 1

6 2

2 2 3

3 3 2

1

p b h

p p pb p

p p b

a p a

a

_{p p p}

 





 

 







 









 



_













 (5.1)

Proof. Using (2.1), (2.2) and (2.3), the result (5.1) is obvious.

Theorem. 5.2 If f   z  A

_p

of the form (1.2) is belonging to G

p_,

 b , 

m_,n

 , then

 

                     3 ^.

1 1 1 2 1 2 3 2 3 2 8 1 2 1 1

8

⁶

2 2 2

3 3 3

2

1 m

p p p

p b p p

p pb p

p p

p

b a p

a

a_{p p p}

 





 

 















 









 



_





     (5.2)

Proof. Using (2.12), (2.13) and (2.14), the result (5.1) is obvious.

Theorem. 5.3 If f   z  A

_p

of the form (1.2) is belonging to M

p_,

 b , 

m_,n

 , then

 

_{1 5 2 6 3 4}

,

3

p h h h h h h

H   

where h

1

, h

2

, h

3

, h

4

, h

5

_and h

₆

are given by (2.2),(2.3),(2.4),(3.1),(4.1) and (5.1) respectively.

Proof. The proof of the result is obvious.

Theorem. 5.4 If f   z  A

_p

of the form (1.2) is belonging to G

p_,

 b , 

m_,n

 , then

 

_{1 5 2 6 3 4}

,

3

p m m m m m m

H   

where m

1

, m

2

, m

3

, m

4

, m

5

_and m

₆

are given by (2.13),(2.14),(2.15),(3.3),(4.3) and (5.2)

respectively.

Proof. The proof of the result is obvious.

REFERENCES

1. R.M. Ali, V. Ravichandran and N. Seenivasagan, Coefficient bounds for p-valent functions, Applied Mathematics and Computation, 187, 35-46 (2007).

2. R.M. Ali, V Ravichandran and K.S Lee, Subclasses of Multivalent Starlike and Convex functions, Bull. Belg. Math. Soc. 16, 385−394 (2009).

3. O. A Fadipe-Joseph, A. T Oladipo and U.A Ezeafulukwe, Modified sigmoid function in univalent function theory, International Journal of Mathematical Sciences and Engineering Application, 7, 313-317 (2013).

4. F. R. Keogh and E. P. Merkes, A Coefficient Inequality for certain class of Analytic functions, Proc. Amer. Math. Soc. 20, 8−12 (1969).

5. S.S. Miller and P. T. Mocanu, Differential subordination, Theory and Applications, Series on Monographs and Text books in Pure and Applied Mathematics, Vol. 225, Mercel Dekker, New York (2000).

6. G. Murugusundaramoorthy and T. Janani, Sigmoid function in the space of univalent λ- pseudo starlike functions, Int. J. of Pure and Applied Mathematics, 101, 33-41 (2015), doi: 10.12732 / ijpam.v101i1.4.

7. M. A. Nasr and M. K. Aouf, Radius of convexity for the class of starlike functions of complex order, Bull. Fac. Sci. Assiut. Univ. Sect. A, 12, 153-159 (1983).

8. J.W. Noonan and D.K. Thomas, On the second Hankel determinant of areally mean p-valent functions, Trans. Amer. Math. Soc. 223(2), 337-346 (1976).

9. Abiodun Tinuoye Oladipo., Coefficient inequality for subclass of analytic univalent functions related to simple logistic activation functions, Stud. Univ. Babes-Bolyai Math., 61, 45-52 (2016).

10. S. Olatunji, Sigmoid Function in the Space of Univalent λ -Pseudo Starlike Function with Sakaguchi Type Functions, Journal of Progressive Research in Mathematics, 7, 1164- 1172 (2016).

11. S. Olatunji, E. Dansu and A. Abidemi, On a sakaguchi type class of analytic functions associated with quasi-subordination in the space of modified sigmoid functions, Electronic Journal of Mathematical Analysis and Applications, 5(1), 97-105 (2017).

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

13. P. Wiatrowski, On the coefficients of some family of holomorphic functions, Zeszyty

Nauk. Uniw. Lódz Nauk. Mat.-Przyrod. (Ser. 2), 39, 75-85 (1970).