Coefficient problems for classes Hq (ᵠ) and Lq (ᵠ) - UMS INSTITUTIONAL REPOSITORY

(1)

Coefficient problems for classes H

q

( )  and

q

( )

L 

Aini Janteng^1*, and Andy Pik Hern Liew¹

1Faculty of Science and Natural Resources, Universiti Malaysia Sabah, 88400 Kota Kinabalu, Sabah, Malaysia

Abstract. A class of analytic functions is denoted by M. Furthermore, SM includes of analytic, normalized and univalent functions. The main -subclasses of S are starlike functions, S^ and convex functions, C . Recently, many mathematicians studied about the q-derivative operator.

Inspired by the ideas from some previous works, we introduce another two new subclasses of M. The coefficient problems in particular the upper bounds of the Fekete-Szegö (F-S) functional for these subclasses were obtained.

1 Introduction

In 1908, Jackson [1] was initiated the ideas of application of q-calculus. Later, in [2], the authors started to apply these ideas to geometric function theory.

First, we consider the functions ^g

( )

^ of the form

( )

2 k k k

g   ^ a

=

= +



(1) These functions are analytic in the open unit disc 𝑈 = {𝛿 ∈ ℂ: |𝛿| < 1}. The class of functions of the form (1) is denoted by M

.

From ([1], [2]), they defined

( ( ) ) ( ) ( ) ( )

^, ^1, ^{0, 0} ^1,

q 1

g q g

D g q q

q

 



= −    

− (2)

( ( )

0

)

' 0

( )

Dq g =g . Since we have

( )

2 k k k

g   ^ a

=

= +



, then we can deduce the following result:

( ( ) )

¹

2 2

1 [ ] ,

k k

q q k q k

k k

D g  D  ^ a ^ k a ⁻

= =

 

 

=  +



 = +



(3)

*Corresponding author: [email protected]

(2)

where 1 [ ] 1

k q

q

= −

− and note that q approaches 1, then [ ]k_q approaches k. For an example, let a function ^s

( )

^ ⁼^k^ ^,

( ( ) ) ( ) ( )

q q 1

kq k

D s D k k

q

 



= = − =

− ^,

(4)

( ( ) )

( ) ( ) ( )

1 1

lim _q lim '

q D s  q k k s 

→ = → = =

(5)

where s' is the ordinary derivative.

Now, we consider analytic functions m and n, we say that m is subordinate to n, written as 𝑚 ≺ 𝑛, if there exists an analytic function h, ^h

( )

⁰ ⁼⁰^and^|^h

( )

^ ^|^¹ such that

( ) ( ( ) )

m  =n w  . If mS, then 𝑚 ≺ 𝑛 if and only if ^m

( )

⁰ ⁼ⁿ

( )

⁰ ^and^{m U}

( )

^^{n U}

( )

^.

An earlier time, certain researchers discussed about the coefficient estimate which is F-S functional for various type of subclasses of M. After that, some of them started to study the subclasses associated the q -derivative operator (see [3-7]). For example, in [3], they discussed about the bounds of F-S functional of starlike and convex functions regarding the q-derivative. While in [4], the authors found the F-S functional for the function belongs to the class of symmetric and conjugate points. Paper [5] shows the results from some others interesting subclasses, while the authors in [6] discussed q-starlike and q-convex functions of complex order. Lastly, in [7], the authors studied about bi-univalent functions with q- derivative operator.

In this paper, we introduce the subclasses Hq

( )

 and Lq

( )

 of the class M for 0 1 which as follows

( ) ( )

( )

( ( ) )

( )

2

: Re ^q ^q ^q , 0,

q

q D D g

H g M D g U

g g

  

 

   

 

   

   

=   +      , (6)

( ) ( )

( )

( ( ) )

( )

1

: Re ^q 1 ^q ^q , 0,

q

qD D g

L g M D g U

g g

   

 

   

 

     −  

      

     

=     +       . (7) We remark that

( ) ( )

1

lim _q

q H  H 

→ = ,

( ) ( )

lim1 _q

q L  L 

→ = .

From the inspiration of the principle of subordination and q-derivative of a function gM , we now give the definitions. Let P be the class of all function 

.

This class is analytic and univalent in U .  is convex with ^

( )

⁰ ⁼¹^and^Re

(

^{  }

( ) )

⁰^for^ ^^U ^.

Function gHq

( )

 if it satisfies

𝛿𝐷_𝑞g(𝛿)

g(𝛿) +^𝛼𝑞𝛿²^𝐷_g^𝑞^(𝐷^𝑞^g^(𝛿))

(𝛿) ≺ 𝜑(𝛿), (𝜑 ∈ 𝑃). (8) Function gLq

( )

 if it satisfies

(^𝛿𝐷_g^𝑞_(𝛿)^g^(𝛿) )

𝛼

( 1 +^𝛿𝑞𝐷^𝑞_g^(𝐷_(𝛿)^𝑞^g^(𝛿)))

1−𝛼

≺ 𝜑(𝛿), (𝜑 ∈ 𝑃). (9)

(3)

We remark that

i.

( ) ( )

lim1 _q

q H  H 

→ = and

( ) ( )

lim1 _q

q L  L 

→ = ,

ii.

1

lim 1

q 1

q H  H



→

 + 

  =

 

 − 

  and

1

lim 1

q 1

q L  L



→

 + 

  =

 

 − 

  .

Now, we may find the bounds of the F-S functional for gHq

( )

 and gLq

( )

 . Before that, the following lemma is needed.

2 Lemma

Lemma 2.1. [8] If 𝑝(𝛿) = 1 + 𝑐1𝛿 + 𝑐2𝛿²+ ⋯ is a function with positive real part in U and  is a complex number, then

 

2

2 1 1 ; 2 1

c −c  − . (10)

The result is sharp for functions given by

( )

¹

p  1 



= +

− ^and

( )

2²

1 p z 1 



= +

− .

3 Results

First, we determine the bound of F-S functional for gHq

( )

 .

Theorem 3.1. Consider gHq

( )

 with 𝜑(𝛿) = 1 + 𝐵₁𝛿 + 𝐵₂𝛿²+ ⋯ ∈ 𝑃. Then

( ) ( ) ( )

( )

2 1 2 1 1

3 2 2

1

1 [3] [2]

max 1 ,

1 [3] [2] 1 [2] 1 [2]

q q

q q q q

B B B B

a a

q B q q

 

   

 

 + 

 

−  +  + + − + 

.

(11)

Proof.

If gHq

( )

 , then

( ) ( ) ( ( ) )

( ) ( ( ) )

2

q q

q q D D g

D g w

g g

  

 

 

 ⁺  ⁼ (12)

Define the function ^p

( )

^ ^by

𝑝(𝛿) =^1+𝑤(𝛿)

1−𝑤(𝛿)= 1 + 𝑝₁𝛿 + 𝑝₂𝛿²+ ⋯ (13) Note that w is a Schwarz function, thus ^p

( )

 is in the class P. Let

𝑑(𝛿) =^𝛿𝐷^𝑞^g^(𝛿)

g(𝛿) +^𝛼𝑞𝛿²^𝐷^𝑞^(𝐷^𝑞^g^(𝛿))

g(𝛿) = 1 + 𝑑1𝛿 + 𝑑2𝛿²+ ⋯ (14)

From equations (12), (13) and (14), we get

( ) ( )

( )

1 1 d p

p

  



 − 

 

=  +  . (15)

(4)

Since

𝑝(𝛿)−1 𝑝(𝛿)+1=¹

2⌊𝑝₁𝛿 + (𝑝₂−^𝑝¹²

2) 𝛿²+ (𝑝₃+^𝑝¹²

4 − 𝑝₁𝑝₂) 𝛿³+ ⋯ ⌋ (16)

thus, we get

𝜑 (^{𝑝(𝛿)−1}

𝑝(𝛿)+1) = 1 +¹

2𝐵₁𝑝₁𝛿 + [¹

2𝐵₁(𝑝₂−^𝑝¹²

2) +¹

4𝐵₂𝑝₁²] 𝛿²+ ⋯ (17)

From the equations (14) and (17), we obtain

1 1 1

1

d =2B p (18)

and

2 1 2

2 1 2 2 1

1 1

2 2 4

d B p p B p

 

 

=  − + ^. (19)

Next, we can get

𝛿𝐷𝑞g(𝛿)

g(𝛿) +^𝛼𝑞𝛿²^𝐷^𝑞^(𝐷^𝑞^g(𝛿))

g(𝛿) = 1 + (1 + 𝛼[2]_𝑞)𝑞𝑎₂𝛿 + {(1 + 𝛼[3]_𝑞)[2]_𝑞𝑎₃− (1 + 𝛼[2]_𝑞)𝑎₂²}𝑞𝛿²+ ⋯ (20) Then from (14), we get

d1 = +

(

1 [2]_q

)

qa2 (21) and d2 = +

(

1 [3]_q

)

q[2]_qa3− +

(

1 [2]_q

)

qa2² (22) or equivalently

(

¹ ¹

)

2 2 1 [2]

a B p

q  q

= + (23)

and

( ) ( ) ( )( )

2 2 2 2

1 1 2 1 1 1

3 2 1 [3] [2]_q _q 2 2 4 1 [3] [2]_q _q 4 2 1 [3]_q 1 [2] [2]_q _q

B p B p B p

a p

q  q  q  

 

 

= +  −  + + + + + . (24)

Therefore

( ) ( )

2 1 2

3 2 2 1

2 1 [3] [2]_q _q

a a B p vp

 q

− =  −

+ (25)

where

( ) ( )

( )

2 1 1

2 1

1 [3] [2]

1

2 2 2 1 [2] 2 1 [2]

q q

B B B

v B q q

 

 

= − − + +

+ + ^. (26)

By using Lemma 2.1, then the result is followed. This finalises the proof of Theorem 3.1.

Similar approaches are applied to the class Lq

( )

 and the result as follows.

(5)

Theorem 3.2. Consider gLq

( )

 with 𝜑(𝑧) = 1 + 𝐵1𝛿 + 𝐵2𝛿²+ ⋯ ∈ 𝑃. Then

₃ ₂² ¹ ² ₂ ¹ ₂ ₂³

3 1 2 3 2

max 1 , [2]

[2] [2] 2

q

q q

B B tB s

a a

qs B q s s qs

 ^^^  ^^^

−   + + 

(27) where

(

¹

) (

^{1 [2]}²

) (

^{1 [2]}

) (

^[2]

)

2 ^q ^q ^q

t   q q

 ⁻  

= − + + − − and sk = + −

(

¹ 

)

^{[ ]}k q.

References

1. F.H. Jackson, Tran. of the R. Soc. of Ed. 46, 253 – 281 (1908)

2. A. Aral, V.G. Ravi, P. Agarwal, Application of q-calculus in operator theory (Springer, New York, 2013)

3. H. Aldweby, M. Darus, Adv. St. in Con. Math. 26, 21 – 26 (2016) 4. S.O. Olatunji, H. Dutta, Proy. J. of Math. 37, 627 – 635 (2018)

5. A. Janteng, A.P.H. Liew, R. Omar, App. Math. Sc. 14, 481 – 488 (2020) 6. T.M. Seoudy, M.K. Aouf, J. of Math In. 10, 135 – 145 (2016)

7. S. Elhaddad, M. Darus, MDPI Math. 8 (2020)

8. W.C. Ma, D. Minda, A unified treatment of some special classes of univalent functions, in Proceeding of the Conference on Complex Analysis (Tianjin), 157 – 169, Internat.

Press, Cambridge, MA (1992)