Estimates and Radii of Convexity in Some Classes of Regular Functions

F. F. MAIYER, M. G. TASTANOV, A. A. UTEMISSOVA*, N. M. TEMIRBEKOV,

D. S. KENZHEBEKOVA

Department of Mathematics and Physics,

Kostanay Regional University named after. A. Baitursynuly,

Kostanay,

KAZAKHSTAN

*Corresponding Author

Abstract: - A class

),,,(



aCn

is being introduced regular in the circle

 

1:  zzE

functions

)(zf

, satisfying

the condition

,,|))()1)(1((|/1 Ezaazfzz nn 









where

1,2/1,10,0,  na



. Class

),,,(



aCn

generalizes various subclasses of close-to-convex functions, including functions which are convex in a certain

direction and functions with limited rotation. Estimates of the derivative and logarithmic derivative of the

function

   

, , ,

n

f z C a

  



are found, and also the radii of the convexity of the class

),,,(



aCn

. The case is

also considered when the function

)(zf

has gaps in the expansion in a row. Similar results are formulated for

the class

),,,(



aTn

of functions

)(zF

, satisfying the condition

,,|)/)()1)(1((|/1 EzaazzFzz nn 





which

generalizes classes of typically real and close-to-starlike functions. All results are accurate. With the

appropriate selection of parameter values of

, , , , an

  

both new and previously published results are obtained.

Key-Words: - geometric theory of functions, estimates of regular functions, radii of convexity, close-to-convex

functions, typically real functions, close-to-starlike functions.

1 Introduction

Let



denotes a class of functions

...,)( 2

210  zazccz



regular in the circle

 

:1E z z

,

)( 0

c

n



– function class

)(z



with decomposition

.1...,)( 1

10  

nzazccz n

n



Function class

)(zf

, normalized by the condition

,01)0(')0(  ff

with decomposition of the form

12

12 (1)( ) , 1, ,

nn

f z z a z a z n z E



     

we denote by

,

n

N

and

NN 

1

. Let

*0,, SSS

and

K

denote, respectively, the classes of univalent,

convex, starlike and close-to-convex functions

.)( Nzf

Close-to-convex functions are introduced in the

works [1], [2], where it is established that the

function

N)(zf

is a univalent in

E

if and only if

there exists a function

0

)( Szg 

such that

()

Re 0, . (2)

()

fz zE

gz











Each function

)(zg

corresponds to its class

g

K

close-to-convex functions

)(zf

, satisfying

condition (2), and

.)( 0

SzgKK g

In this

case, the functions

)(zf

class

g

K

to a certain

extent, they inherit the properties of the function

)(zg

. Of these, classes are often explored

321 ,, FFF

close-to-convex functions

N)(zf

, satisfying the

conditions

)3(,0))()1Re((: 2

1



zfzF

)4(,0))()1Re((: 2

2



zfzF

)5(0))()1Re((:

3



zfzF

or their generalizations, while the classes

21,FF

characterized by an original geometric property of

the area

)(Ef

– convexity in the direction of the

Received: September 18, 2023. Revised: April 19, 2024. Accepted: June 19, 2024. Published: July 19, 2024.

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.47

F. F. Maiyer, M. G. Tastanov,

A. A. Utemissova, N. M. Temirbekov, D. S. Kenzhebekova

E-ISSN: 2224-2880

446

Volume 23, 2024

imaginary axis, [3], [4] (for the class

1

F

) and

convexity in the positive direction of the real axis

[5] (for the class

2

F

), what is related to the

simplicity of the geometric properties of convex

functions

,

1

ln

2

1

)(

1z

z

zg 





,

1

)(

2z

z

zg 



)1ln()(

3zzg 

,

on the basis of which these classes are built.

Works [3], [4] marked the beginning of a whole

series of studies of the class

1

F

of functions convex

in the direction of the imaginary, as well as various

generalizations of this class. This includes, for

example, articles [6], [7]. In [6], the class

1

F

is

investigated, in [7] – class

)(





C

functions

N)(zf

, satisfying the condition

2

Re [ (1 ) ( )] 0,0 1, ,

i

e z f z R





    

where both the distortion, growth and coverage

theorems and coefficient estimates are investigated.

At

0,1 



this condition sets the class

1

F

, and

at

0



a simple parametric transition from the

class

1

F

to the class of functions with limited

rotation, is performed which is set using the

condition

0)]([Re 

zfei



and it is well known as a

classic sign of univalentness from [8], [9].

In [10], [11] is introduced a subclass

K)K



(

functions of close-to-convex order



, satisfying the

condition

 

()

arg 0 1 (6)

( ) 2

f ' z π

γ , g z S , γ ,z E

g' z     

which aroused great interest due to such a visual

geometric property as reachability from outside the

area

)(Ef

solution angles

 



1

. By analogy,

subclasses were also introduced

,)(



1

F

,)(



2

F

)(



3

F

functions satisfying, respectively, the

conditions

,z'farg 2

))()((







2

z-1

2

))()(( π

γz'farg 

2

z-1

and

.z'farg 2

))()((





z-1

Almost simultaneously, in articles [12], [13], the

angular reachability of two classes of functions

generalizing classes was investigated

21 FF ,

. At the

same time, in [12] emphasis is placed on obtaining

distortion and growth theorems, as well as finding

the radii of convexity, and in the article [13] the

geometric properties of the domain of values of

functions from the introduced classes are

investigated.

By analogy with condition (2) for the class  of

close-to-convex functions, in [14] introduced the

class  close-to-starlike functions 󰇛󰇜 

satisfying the condition

()

Re 0,

()

Fz

gz 

where 󰇛󰇜

   . The subclasses of the  class are the

classes 

 

 

 of close-to-starlike functions

󰇛󰇜, respectively satisfying the conditions:

2

((1 ) ( ) / ) 0,Re z F z z

2

((1 ) ( )/ ) 0,Re z F z z

((1 ) ( )/ ) 0.Re z F z z

There is a simple relationship between classes

1 2 3

, , F F F

and

* * *

1 2 3

, ,CS CS CS

, which is expressed

by the ratio:

     

*, 1,2,3

kk

f z C F z zf z CS k



    

Classes

* * *

1 2 3

, ,CS CS CS

were studied in articles

[15], [16], [17], [18], [19] from the point of view of

estimating the coefficients and finding the radii of

star formation of these classes relative to various

subclasses of the class

*

S

.

Thus, based on the above-described literature

review, the issue of studying the extreme properties

of various

n

N

subclasses remains relevant at

present.

In this paper, a fairly wide class of regular

functions is introduced and in this class, irreducible

distortion theorems and exact convexity radii are

found and also, as a consequence, similar results

are obtained in other subclasses of close-to-convex

functions. In addition, in the class of functions

introduced in the article, exact theorems on

distortions and coverings, as well as starlikeness

radii, generalizing classes of typically real and

close-to-starlike functions, are obtained.

2 Problem Formulation

The purpose of this article is to introduce a

sufficiently wide class of regular functions with

decomposition of the form (1), including the

subclasses of close-to-convex functions listed in the

introduction with obvious geometric properties, in

order to conduct their study from a unified point of

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.47

F. F. Maiyer, M. G. Tastanov,

A. A. Utemissova, N. M. Temirbekov, D. S. Kenzhebekova

E-ISSN: 2224-2880

447

Volume 23, 2024

view. As an application of the main results for

close-to-convex functions, using the ratio

   

F z zf z





, these results are transferred to

subclasses of close-to-starlike functions,

complementing or generalizing some of the known

results.

Definition. We will assume that the function

)(zf

belongs to the class

 

,,a,,Cn



,, 0



1 1/2,1,0 na



, if and if

N)(zf

satisfies

the condition

 

1/

(1 )(1 ) ( ) , (7)

nn

z z f z a a z E







    

in doing so

1. (8)

11 n







It follows from (7) that

0))()1)(1Re(( /1 









zfzz nn

for all

21/a 

,

which in terms of dependence of functions can be

written as

γ

nn

z

(z)f)zδ)(zλ(

















 1

1

11 

. That is

   

, , , , , ,

nn

C a C

     



and the condition is

fulfilled

.,

2

|))()1)(1((arg| Ezzfzz nn 









Therefore, the functions

)(zf

from

 



,,, a

n

C

is univalent and close-to-convex of the order



,

since satisfy condition (6) with convex in

E

function

.

)1)(1(

(z)

0



z

nn δtλt

dt

g

Indeed,

1

() (1 )(1 )

nn

gz zz





and therefore

()

Re Re Re 1

( ) 1 1

11

.

nn

g'' z λnz δnz λn δn

z,

g' z λδ

-λz - δn

zE

      





Due to condition (8), so



Szg )(

.

We note also that the classes

,)(



1

F

,)(



2

F

)(



3

F

, as well as

)(





C

at

0



, are subclasses of

the class

),,,(



aCn

.

Consider the function

0: f E C

, which is

defined by the formula

   

 

  

0

1

111

z

nn

at dt

fz a a t tt













 





where

, , , , an

  

are fixed real parameters

nN

,

1

, 0, 0 1, , 1

2

an

  

    

and the inequality

(8).

Then

 

1/

((1 )(1 ) ( ))

nn

z z f z w z





  



,

where

 

1/

((1 )(1 ) ( ))

nn

z z f z w z





  



–

mapping the disk

E

to a disk

 

: w w a a

.

Therefore,

0()fz

satisfies condition (7). Hence,

 

0( ) , , ,

n

f z C a

  



and

 

, , ,

n

Ca

   



. This

function will be one of the extremal functions in

Theorems 1 and 2. Other examples of functions

1()fz

and

2()fz

from

 

, , ,

n

Ca

  

can be found

in the proofs of Theorems 1 and 2.

The main research method of the article is a

fairly easy-to-use and at the same time effective

method of subordination of regular functions. They

say that the regular function

()z



in

E

is

subordinate to the function

0()z



and write

   

0

zz



, if there exists a function

     

, 0 0, 1zz

  



in

E

such that

   

 

0

zz

  



. In the case of a univalent

function

0

φ ( )z

, the subordination ratio

   

0

zz



means that

   

0

EE





and

   

0

00





, which implies the embedding of

closed regions

 

0n

z r z r



  

at

01r

for the function

 

0

( (0))

n

z



R

.

In the works of many authors, the proof of

various estimates related to the function

()z



is

based directly on the use of the relation

   

 

0

zz

  



using estimates for

 

z



.

Another approach to obtaining estimates of the

function

()z



is to use the embedding ratio of

closed regions

 

0n

z r z r



  

. Hence,

based on the geometric properties of the set

 

0n

zr





, estimates of the real, imaginary part,

argument and modulus of the function

()z



are

easily derived. Adjacent to this approach is the

finding of estimates

()z





based on the properties

of the inner radius of the region.

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.47

F. F. Maiyer, M. G. Tastanov,

A. A. Utemissova, N. M. Temirbekov, D. S. Kenzhebekova

E-ISSN: 2224-2880

448

Volume 23, 2024

3 Problem Solution

3.1 Distortion Theorem and Radius of

Convexity of Class

 

,,,

n

Cλ δ a γ



Theorem 1. Let

),,,()(



aCzf n



. Then when

1 rz

there are accurate estimates

(1 ) 1 ()

( 1) (1 )(1 )

(1 ) 1 (9)

( 1) (1 )(1 )

γ

nn

γ

nn

a - r f ' z

a a - r λr δr

ar ,

a a r λr δr







 













( ) (2 1) (10)

( ) (1 )( ( 1) )

11

nn

fz λnz δnz γ a r

z.

f z r a a r

λz δz

 

  

  



Proof. In terms of dependence, condition (7)

means that

0

(1 )

( ) (1 )(1 ) ( ) ( ) (11)

( 1) ,

γ

nn az

zλz δz f' z z a a z





    







where

 

)1(

1

)( 



aa

za

zw

at

21/a 

displays a circle

E

on the circle

 

.aw-aw: 

due to dependence, the

inclusion of areas is performed

)()( 0rzrz 



at any

,0 1rr

, that is,

taking into account the type of area

)(

0E



in the

circle

rz 

inequalities are fulfilled

(1 ) ()

( 1)

(1 )

(1 )(1 ) '( ) (12)

( 1)

nn

ar z

a a r

ar

z z f z a a r













    













Due to

0,0 



, then when

1 rz

we

have

).1)(1()1)(1()1)(1( nnnnnn rrzzrr





Combining this estimate with the estimate (12),

we come to (9).

To prove the estimate (10), we use lemma 1,

which follows by replacing

 

()z p z







from the

results from [20] for the class of functions

1

( ) 1 , 1, ,

nn

pz с z с z n z E



     

satisfying

the condition

( ) , 1/ 2p z a a a  

.

Lemma 1. If

)(z



from

)1(

n

R

satisfies the

condition

,,10,2/1,))(( /1 Ezaaaz 

 

then when

1 rz

there is an accurate

assessment

(2 1) (13)

(1 )( ( 1) )'

n

nn

'(z) a nr

z(z) r a a r







  

which is achieved for the function

)()( 0n

zz





at

the point

rz n1

, where

.

)1(

)(

0





















zaa

za

z

Due to

)()( 0zz





, then considering that

N)(zf

, that of (13) when

1n

we get the

estimate (10):

.

))1()(1(

)12(

11

)(

)('

raar

ra

z

nz

z

nz

zf

z

zn

n























Let us now prove that estimates (9) and (10)

cannot be improved.

For the function

),,,(

)1)((

)1(

)(

0







aC

ttz

dt

taa

ta

zf n

nn

z





















we have

0

(1 ) 1

( ) ,

( 1) (1 )(1 )

() (2 1) .

(1 )( ( 1) )

11

()

nn

az

fz a a z zz

fz nz nz a z

zz a a z

zz

fz





  





 



     













Therefore, the right estimate in (9) is achieved

for the function

0()fz

at the point

rz 

, and the

estimate (10) is achieved for the function

)(

0zf

at

the point

zr

, because

.

))1()(1(

)12(

11)(

)(

0

0raar

ra

δz

δ nz

λz

λnz

zf

zn

n

'

''

























rz

If

n

– odd and

zr

, that

nn rz 

.

Therefore, in the left estimate (9) for the function

)(

0zf

at the point

zr

the equal sign is reached

again, since

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.47

F. F. Maiyer, M. G. Tastanov,

A. A. Utemissova, N. M. Temirbekov, D. S. Kenzhebekova

E-ISSN: 2224-2880

449

Volume 23, 2024

 

  

011

() 111

zr nn

ar

fz a a r rr















 



Let now

n

– even.

If

Ν.)12(21062  ,...,kk,...,,,n

Then

1

n

i

and for

irz

we have

rizrriz nnnn  ,

.

Therefore, for the function

),,,(

)1)(1(

)1(

)(

0

1







aC

tt

dt

itaa

ita

zf n

nn

z





















at the point

irz

we get

1(1 ) 1

() ( 1) (1 )(1 )

(1 ) 1 ,

( 1) (1 )(1 )

z ir nn

nn

z ir

a iz

fz a a iz zz

ar

a a r rr











 



 













which proves that the left estimate (9) is not

improved when

N.)12(2  ,kkn

If

N41284  k,...,k,...,,,n

, that

1

n

i

and for

riz n

n2



we have

.

2nnnn rriz  

Besides,

.

2

222

rririizi n

n





Therefore , at the point

riz n

n2



or the function

),,,(

)1)(1(

)1(

)(

02

2







aC

tt

dt

tiaa

tia

zf n

nn

z

n

























we get

2

2(1 ) 1

() (1 )(1 )

( 1)

(1 ) 1 ,

( 1) (1 )(1 )

nn

zi

n

n n n z i r

n

nn

a i z

fz zz

a a i z

ar

a a r rr





















 













which proves the unimprovability of the left

estimate (15) in this case as well.

The theorem is proved.

At

0



from theorem 1 follows

Corollary 1. If the function

)(zf

satisfies the

condition

aazfzn





/1

))(')1((

,

then at

1 rz

there are accurate estimates

nn r

raa

ra

zf

r

raa

ra











































1

)1(

)('

1

)1(

,

.

))1()(1(

)12(

1

)(

raar

ra

z

nz

zf

zn

n

















We consider the limiting case of corollary 1

when

a

.

Corollary 2. Where

),,0,()(



 n

Czf

, so

)(zf

satisfies the condition

Ezzfzn ,

2

))(')1arg((





.

Then at

1 rz

accurate estimates are made

,

1

)('

1

nn r

r

zf

r











































.

1

2

1

)(

2

r

z

nz

zf

zn

n















Corollary 2 at

1,2 



n

obtained in [6],

when

10,1,2 



n

– in [12], and for the case

,1,10,2 



n

coincides with a special case

of grades for a class

)(





C

at

0



from [7].

Note also that when

 an ,1



corollary 2

gives estimate for the class

).(

3



F

At





,

a

the following result follows

from Theorem 1.

Corollary 3. If

   

, , ,

n

f z C a

  



, so

 

fz

satisfies the condition

,

2

))(')1arg(( 2





 zfzn

,Ez 

then when

1 rz

accurate estimates are

made

22 )1(

1

)('

)1(

1

nn r

r

zf

r











































,

2

( ) 2 2

( ) 1 1

n

f z nz r

zf z z r





 



.

We note that when

1



n

from corollary 3,

the estimate for the class are obtained

)(

2



F

.

Theorem 2. The exact radius of convexity

0

r

of

class

),,,(



aCn

is determined as the only root of

the equation on the interval

;1)0(

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.47

F. F. Maiyer, M. G. Tastanov,

A. A. Utemissova, N. M. Temirbekov, D. S. Kenzhebekova

E-ISSN: 2224-2880

450

Volume 23, 2024

(2 1)

1 0 (14)

(1 )( ( 1) ) 1 1

nn

y a r nr nr

r a a r r r





   

    

Proof. Let's check that in the circle

0

rz 

the

convexity condition is satisfied

0

)(

Re1





zf

zf

z

.

Due to the estimate (10) we get

,

))1()(1(

)12(

11

)(

Re raar

ra

z

nz

z

nz

zf

zn

n































from where for all

rzz ,

we find

( ) (2 1)

1 Re 1

( ) (1 )( ( 1) )

min Re min Re

11

nn

z r z r

f z a r

zf z r a a r

nz nz

zz







 

   

  





   

   

   

Because for any

10,  tt

, equality is fulfilled

,

111

min tr

tnr

t

tn

t

tn

rz

r

























that

( ) (2 1)

1 Re 1

( ) (1 )( ( 1) )

.

11

nn

f z a r

zf z r a a r

nr nr

rr





 

   

  





Therefore, if

);10(

0 rr

– the root of equation

(14), then in the circle

0

rz 

the convexity

condition is met.

We now show that equation (14) has a single

root

0

r

on the interval

;1)0(

. We denote

0

(2 1)

( ) 1 , ( ) .

1 1 (1 )( ( 1) )

nn

nr nr a r

r M r

r r r a a r

  





   

    

Because

)(

0r



decreases by

;1)0(

from

1)0(

0



before 󰇛󰇜, in doing so

]1;0[)1(

0



in

virtue of (8),

)(rM

increases by

;1)0(

from 0

before



, then on

;1)0(

there is a single point

0

r

, in which

)()(

0rMr 



, that was what needed to be

proved.

We show that the radius of the convexity cannot

be improved.

For odd

n

we consider the extremal function

),,,(

)1)(1(

)1(

)(

0







aC

tt

dt

taa

ta

zf n

nn

z





















At the point

rz 

, where

0

rr 

, considering that

nn rz 

, we get

''

0

'

0

(2 1)

1(1 )( ( 1) )

()

1()

11

(2 1)

1 0.

(1 )( ( 1) ) 1 1

nn

zr

az

z a a z

fz

zfz nz nz

zz

a r nr nr

r a a r r r





  









  

  







    

    







That is, for odd

n

the radius of convexity is

accurate.

If

n

– even, then, as in proving the accuracy of the

left estimate (9), we consider two cases.

1) Let

N)....12(2,...10,6,2  kkn

. Then

1

n

i

and

nn rz 

for

irz

. For the function

),,,(

)1)(1(

)1(

)(

0

1







aC

tt

dt

itaa

ita

zf n

nn

z





















we have

,

11

))1()(1(

)12(

1

)(

1'

1

''

1n

n

z

nz

z

nz

izaaiz

iza

zf

z





















and at the point   , where   , we find

0

11

))1()(1(

)12(

1

)(

1'

01

''

1

















n

irzz

nz

z

nz

raar

ra

zf

z







Therefore, in this case, the radius of convexity

cannot be improved.

2) Let

N.,...,4,...,12,8,4  kkn

we denote

n

i

2





. Then

1

n

i

,

1

n



and for

rz





we

get

rzrz nn  1

,



. Because for the function

),,,(

)1)(1()1(

)1(

)( 01

1

2







aC

tt

dt

taa

ta

zf n

nn

z























we have:

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.47

F. F. Maiyer, M. G. Tastanov,

A. A. Utemissova, N. M. Temirbekov, D. S. Kenzhebekova

E-ISSN: 2224-2880

451

Volume 23, 2024

n

z

nz

zaaiz

za

zf

z















 



1))1()(1(

)12(

1

)(

11

1

'

2

''

2

Because of this, at the point

rz





, where

0

rr 

,

find

,0

11

))1()(1(

)12(

1

)(

1'

2

''

2

















n

rz r

nr

r

nr

raar

ra

zf

z









which proves that the radius of convexity is

unimprovable in this case as well.

Theorem 2 is proved.

At

2,0  n



theorem 2 implies

Corollary 4. Accurate radius of convexity

0

r

function class

N,)(zf

satisfying the condition

,,2/1,10,10,))()1(( /1'2 Ezaaazfz 

 

is determined as the only root of the equation on

the interval

)1;0(

.0)1()12())1()(1)(1( 22  rraraarz



At

1



,

a

from Corollary 4 we obtain the

radius of convexity of the class

)(

1



F

from [12],

and for

1



,

a

– class

1

F

radius of convexity

from [6].

3.2 Distortion Theorems and Radii of

Convexity in the Case When

( ) n

fz N

Theorem 3. Let the function

)(zf

with a

decomposition of the form (1) belongs to the class

),,,(



aCn

. Then at

10,  rrz

there are

exact estimates

(1 ) 1 ()

( 1) (1 )(1 )

(1 ) 1 (15)

( 1) (1 )(1 )

n

n n n

n

n n n

ar fz

a a r r r

ar

a a r r r









   



   













()

( ) 1 1

(2 1) (16)

(1 )( ( 1) )

nn

n

nn

f z nz nz

zf z z z

a nr

r a a r





   





  

and the exact radius of convexity

0

r

class

),,,(



aCn

is determined as the only root of the

equation on the interval

)1;0(

(2 1)

1(1 )( ( 1) )

0 (17)

11

n

nn

a nr

r a a r

nr nr

rr









  

  



The proof of Theorem 3 is carried out similarly

to the proofs of Theorems 1 and 2, only Lemma 1

is applied for the case

1n

taking into account the

fact that

n

zf N)(

and therefore

)(z



from (11)

has a decomposition of the form

Eznzczcz n

n

n 

,1...,1)( 1

1



.

Corollary 5. Let a function

)(zf

belong to the

class

 

.10,

2

'arg;)(),,0,0( 

















zfzfC nn N

Then at

10,  rrz

there are exact estimates

,))1/()1(()('))1/()1((



nnnn rrzfrr 

n

r

nr

zf

z2

1

2

)(











and the exact radius

0

r

of the convexity class

),,0,0(





n

C

is determined by the formula

n

nnr /122

0)1(





.

Corollary 5 follows from Theorem 3 for

.,0  a



This result for

1



coincides with the estimates

and the radius of convexity

12

0r

lass of

functions with limited rotation

,,0)(Re Ezzf 



obtained in [21].

At

2,1,0  n



and at

0



the

following corollaries follow from Theorem 3.

Corollary 6, [22]. Accurate radius of the

convexity

0

r

of function class

2

)( Nzf

, satisfying

the condition

Ezaaazfz 



,2/1,)()1( 2

, is

determined as the only root of the equation on the

interval

)1;0(

.0)15()45()1( 246  ararara

Corollary 7. Accurate radius of the convexity

0

r

of function class

2

)( Nzf

, satisfying the

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.47

F. F. Maiyer, M. G. Tastanov,

A. A. Utemissova, N. M. Temirbekov, D. S. Kenzhebekova

E-ISSN: 2224-2880

452

Volume 23, 2024

condition

Ezaaazf 

,2/1,))(/1



is

determined by the formula

2

0

1/

1 (2 1) (1 (2 1) ) 4 ( 1) , 1;(18)

2(1 )

(1 ) , 1.

n

a n a n a a a

ra

na







       















At

1



this result was obtained in [22].

3.3 Generalization of Classes of Typically

Real and Close-To-Starlike Functions

In [23] introduced typically real functions, that is,

the functions

,1)0(,0)0(),(



FFzF

which takes

real values at points

)1;1(z

and at other points of

the circle

E

satisfy the condition

.0Im)(Im  zzF

In [23] he also obtained a membership criterion

)(zF

to class

T

of typically real functions:

2

1

( ) Re ( ) 0, .(19)

z

F z T F z z E

z





   





By analogy with the class

),,,,(



aCn

,1,2/1,10,0,  na



we introduce a class

),,,(



aTn

functions

N,)(zF

satisfying the

condition

1/

(1 )(1 ) ( ) , ,(20)

nn

zz

F z a a z E

z







   





In doing so

1. (21)

11n







Comparing (7) and (20), we get that:

( ) ( , , , ) ( ) ( ) ( , , , )(22)

nn

f z C a F z zf z T a

     



   

and at

 an ,1,2,0



class

),,,(



aTn

is transformed into a class

T

of

typically real functions.

Wherein,

)1,,0,1(),,0,1( 22  TTaT



for all

.10,2/1 



a

On the other hand, when

 a,1,0



we obtain the class of close-to-starlike functions

introduced in [14] using the condition

.,0

)(

Re Ez

z

zF 

Wherein,

),,0,0(



aTn

is a

subclass of the class of close-to-starlike functions

for all

.10,2/1 



a

We note also that if the

class functions

),,,(



aCn

are univalent, then the

class functions

),,,(



aTn

generally speaking,

they are not.

Relation (22), taking into account the equality

1

)(

)()( 









zF

zFz

zf

zfz

makes it easy to transfer all

results for a class

),,,(



aCn

per class

),,,(



aTn

.

Theorem 4. Let

).,,,()(



aTzF n

N

Then

at

1 rz

there are exact estimates

(1 ) ()

( 1) (1 )(1 )

(1 ) , (23)

( 1) (1 )(1 )

nn

a r r Fz

a a r r r

a r r

a a r r r







   



   













() 1

( ) 1 1

(2 1) . (24)

(1 )( ( 1) )

nn

F z nz nz

zF z z z

ar

r a a r





   





  

and the exact radius of starlikeness

*

r

of the class

),,,(



aTn

is determined as the only root of

equation (14) on the interval

)1;0(

.

If, in addition to the conditions of the theorem,

the function

)(zF

expands into a series of the form

(1), then we have exact estimates

(1 ) ()

( 1) (1 )(1 )

(1 ) , (25)

( 1) (1 )(1 )

n

n n n

n

n n n

a r r Fz

a a r r r

a r r

a a r r r







   



   













() 1

( ) 1 1

(2 1) . (26)

(1 )( ( 1) )

nn

n

nn

F z nz nz

zF z z z

a nr

r a a r





   





  

and the exact radius of starlikeness

*

r

of the class

),,,(



aTn

is defined as the only root of equation

(17) on the interval

)1;0(

.

Corollary 8. Let

),,,0,()(



aTzF n



that is

)(zF

satisfies the condition

,

)(

)1( /1 aa

z

zF

zn

















.Ez

Then at

1 rz

there are exact estimates

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.47

F. F. Maiyer, M. G. Tastanov,

A. A. Utemissova, N. M. Temirbekov, D. S. Kenzhebekova

E-ISSN: 2224-2880

453

Volume 23, 2024

,

1

)1(

)(

1

)1(

nn r

r

raa

ra

zF

r

raa

ra











































))1()(1(

)12(

1

)1(1

1

)(

raar

ra

z

zn

zF

zn

n



















and the exact starlikeness radius

*

r

of the class

),,0,(



aTn

is determined as the only root of the

equation on the interval

)1;0(

(2 1)

10

(1 )( ( 1) ) 1

( ) ;

(2 1)

10

(1 )( ( 1) ) 1

()

nn

n n n

n

a nr nr

r a a r r

for the case when F z

a r nr

r a a r r

for the case when F z











  

   





  

   













N

N.

At

 an ,1,2,0



class

),,,(



aTn

coincides the class

T

and from

Corollary 8 we obtain the well-known estimates

[23] in the class

T

22

2

21

2

1

)(

,

)1(

)(

)1)(1(

)1(

r

z

zF

z

r

zF

rr

n



















and exact radius of starlikeness











 15(215

2

1

*

r

of class

T

, previously

found in [24].

We note also that when

1,0,2 



n

Corollary 8 the exact radius of starlikeness of the

subclass of typically real functions

,)( 2

NzF

satisfying the condition

,,

)(

)1( 2Ezaa

z

zF

z

as the only root of the equation on the interval

)1;0(

,0)15()45()1( 246  ararara

found in

[22].

At

0



it turns out the class

,,

)(

:)( /1































Ezaa

z

zF



the exact radius

starlikeness of which, in the case

,)( n

zF N

is

determined by formula (18), which coincides with

the result from [22].

From here at

a

we get class

,,

2

)(

arg:)( 









 Ez

z

zF





whose exact radius

of starlikeness is

.)1( /122* n

nnr





At

1



n

is get the radius of starlikeness

.12

*r

from [21] of the class of close-to-

starlike functions

,)( NzF

satisfying the

condition

.,0

)(

Re Ez

z

zF 

Note. Recently, a whole series of articles has

been published, [16], [17], [18], [19], devoted to

finding the radii of starlikeness of various classes

of close-to-starlike functions with respect to some

subclasses of the class

*

S

(Recently, a whole series

of articles has been published, [23], [24], devoted

to finding the radii of starlikeness of various classes

of close-to-starlike functions with respect to some

subclasses of the class:

*()

( ): ,0 1,

()

Fz

S F z Rez z E

Fz







   





ς

starlike functions

)(zf

of order



,

2

*()

( ): 1 1,

()

LFz

S F z z z E

Fz









  











ς

lemniscate

starlike functions and others).

If in Theorems 2-3 instead of the convexity

condition we use the condition

()

1

()

fz

Rez fz









convexity of order

1,,0 



and accordingly

in Theorem 4 – condition

()

Fz

Rez Fz





starlike of

order



, then we obtain the exact radius of

starlikeness of order



of class

),,0,(



n

T

as

the only one on the interval (0;1) root of the

equation:

(2 1)

10

(1 )( ( 1) 1 1

( ) ; (27)

(2 1)

10

(1 )( ( 1) 11

()

n n n

n n n n

n

nn

a nr nr nr

r a a r r r

forthecasewhen F z

a r nr nr

r a a r rr

forthecasewhen F z

  



  















    

    





    

   



N

N.

In special cases, when

NzF )(

, we obtain a

number of well-known results.

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.47

F. F. Maiyer, M. G. Tastanov,

A. A. Utemissova, N. M. Temirbekov, D. S. Kenzhebekova

E-ISSN: 2224-2880

454

Volume 23, 2024

1) For

 an ,0,1,2



we obtain

the star radius [16] of order



class















EzzF

z

zFK ,0)(

1

Re:)( 2

1

as the only root of the equation (0;1) on the interval

0)1()1(2)1( 24 



rrrr

.

2) For

 an ,0,1



we obtain

a star radius of order



class















EzzF

z

zFF ,0)(

1

Re:)(

3

from [18] as the only root of the equation

,013

2



rr

, on the interval (0;1), that is

 

*3 9 4 1

2

r





  



3) For

 an ,1



we obtain a

star radius of order



class















EzzF

z

zFF ,0)(

)1(

Re:)( 2

4

from [18] as the only root of the equation

014)1( 2



rr

, on the interval (0;1),

that is

2

*23

1

r







.

Other applications of the

 

, , ,

n

Ta

  

class of

almost star-like functions, as well as promising

tasks for further research, are to find estimates of

the coefficients of functions of the

 

, , ,

n

Ta

  

class, as well as the radii of the starlikeness of this

class relative to such subclasses of the

*

S

class as

   

 

**

1 / (1 )S S z z



  

is a highly starlike,

** 2

21

1 log 1

Pz

SS z

















parabolic,

 

**

1

L

S S z

lemniscatic,



* * 2

1S S z z  

π

moon-shaped,

 

**

1 z

car

S S ze

cardioid and

other starlike functions.

4 Conclusion

The article introduces a class

),,,(



aCn

of

functions

)(zf

, satisfying the condition

.,|))()1)(1((|/1 Ezaazfzz nn 









It is

shown that all functions of this class are close-to-

convex and its subclasses are the classes of

functions convex in a certain direction and

functions with bounded rotation. In class

),,,(



aCn

exact distortion theorems and exact

convexity radii are found, which generalize

previously known results. The case when the

function

 

fz

in the expansion in a power series

has omissions of terms.

It is also considered the class

),,,(



aTn

of

functions

)(zF

, associated with class

),,,(



aCn

functions

 

fz

ratio

).()( zfzzF 



Class

),,,(



aTn

is a generalization of classes of

typically real and close-to-starlike functions. Exact

covering theorems and exact radii of starlikeness

are obtained in the class, which generalize

previously published results for typically real and

close-to-starlike functions.

Thus, the authors of the article, based on a

single approach, solved a number of extreme

problems for fairly wide subclasses of close-to-

convex and close-to-starlike functions. At the same

time, along with the generalization of known

results, new results are obtained in particular cases.

In conclusion, we note that the results of the

article admit of a simple generalization if, instead

of conditions (7)-(8), we use conditions of the form

)'7(,,1,))('))1((

1

Ezkaazfz

k

i

n

i









and

1

1. (8')

1

ki

iin











It is known that the solution of many extreme

problems for subclasses of functions

 

fz

from



 is reduced to finding, for       

accurate estimates of the functionals

     

( ) ( )

max , min (28)

zr

zz

z Re z Rez

zz



  



















on the class

 

0n



P

defining the class of functions

 

fz

. In this regard, the application of evaluation

(13) for the class of regular functions

 

z



from

n

R

satisfying the condition

1/

( ( ))z a a







,

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.47

F. F. Maiyer, M. G. Tastanov,

A. A. Utemissova, N. M. Temirbekov, D. S. Kenzhebekova

E-ISSN: 2224-2880

455

Volume 23, 2024

1/ 2, 0 1a



  

, is promising, as it will further

solve a number of extreme problems related to

various subclasses of univalent functions, which in

this article is demonstrated by the example of the

class

),,,(



aCn

.

As directions for further research, we note the

following.

1) Introduction of a unified class

 

0n



P

of

regular functions

 

z



from

0

()

nсR

satisfying the

subordination condition

   

0

zz



,

 

*

0,z



P

and finding accurate estimates of

functionals (28) in this class. Here, the class of

functions

 

0z



is denoted by

*,P

which

conformally map the circle

E

to a region starlike

relative to the point

1w

, belonging to the half-

plane

0Rew 

and symmetrical with respect to the

real axis.

2) Application of the obtained estimates of the

functionals (28) for the study of unified subclasses

of close-to-convex functions, including convex

ones in a certain direction. This includes classes of

functions for which

 

0

()

fz z

gz





,

 

00

n

z



P

, and

()gz

is a convex function.

3) Introduction of new subclasses of double

close-to-convex, close-to-starlike and double close-

to-starlike functions based on the class

 

0

n



P

and

investigation of extreme properties of functions of

these classes using functional estimates (28).

4) Finding estimates of coefficients for

functions of the classes under consideration.

References:

[1] Ozaki S., On the theory of multivalent

functions, Sci. Rep. Tokyo Bunrika Daigaku,

Sect. A. Math. Phys. Chem., Vol. 2, 1935,

pp. 167-188, [Online].

https://www.jstor.org/stable/43700132

(Accessed Date: February 1, 2024).

[2] Kaplan W., Close-to-convex schlicht

functions, Michigan Math. J., Vol. 1(2),

1952, pp. 169-185.

https://doi.org/10.1307/mmj/1028988895.

[3] Robertson M.S., Analytic functions star-like

in one direction, Amer. J. Math., Vol. 58.

№3, 1936, pp. 465-472, [Online].

https://www.jstor.org/stable/2370963

(Accessed Date: March 3, 2024).

[4] Goodman A.W., Univalent functions, Tampa:

Mariner. 1983, 246 p.

[5] Bshouty D., Lyzzaik A., Univalent functions

starlike with respect to a boundary point,

Contemp. Math., Vol. 382, 2005, pp. 83-87.

https://doi.org/10.1016/S0022-

247X(03)00258-0.

[6] Hengartner W., Schober G., Analytic

functions close to mappings convex in one

direction, Proc. Amer. Math. Soc., Vol. 28,

№2, 1971, pp. 519-524.

[7] Lecko A., Some subclasses of close-to-

convex functions, Annales Polonici

Mathematici, Vol. 58, №1. 1993, pp. 53–64.

[8] Noshiro K., On the theory of schlicht

functions, J. Fac. Sci. Hokkaido Imp. Univ.

Ser. I Math., Vol. 2, № 3, 1934, pp. 129-

155. https://doi.org/10.14492/hokmj/1531209

828.

[9] Warschawski S.E., On the higher derivatives

at the boundary in conformal mapping,

Trans. Am. Math. Soc., 1935. Vol. 38. № 2.

pp. 310-340. https://doi.org/10.1090/S0002-

9947-1935-1501813-X.

[10] Reade M., The coefficients of close-to-

convex functions, Duke Math. J., Vol. 23,

№ 3, 1956, pp. 459-462.

https://doi.org/10.1215/S0012-7094-56-

02342-0.

[11] Renyi A., Some remarks on univalent

functions, Ann. Univ. Mariae Curie-

Sklodowska. Sec. A. 3, 1959, pp. 111-121.

[12] Maiyer F.F., Geometrical properties of some

classes of functions analytic in a circle,

convex in the direction of the imaginary axis,

Science Bulletin of A. Baitursynov Kostanay

State University. Series of natural and

technical sciences, Vol. 6. No. 2, 2002, pp.

48-50.

[13] Lecko A., The class of functions convex in

the negative direction of the imaginary axis

of order (α,β), Journal of the Australian

Mathematical Society, Vol. 29(11), 2002, pp.

641–650.

[14] Reade M., On close-to-convex univalent

functions, Michigan Math. J., Vol. 3. 1955,

pp. 59–62. DOI: 10.1307/mmj/1031710535.

[15] Lecko A., Sim Y.J. Coefficient problems in

the subclasses of close-to-star functions.

Results Math., 2019, Vol. 3, pp. 104.

https://doi.org/10.1007/s00025-019-1030-y.

[16] Khatter K., Lee S.K., Ravichandran V.,

Radius of starlikeness for classes of analytic

functions, arXiv preprint arXiv:2006.11744.

2020. https://arxiv.org/pdf/2006.11744.

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.47

F. F. Maiyer, M. G. Tastanov,

A. A. Utemissova, N. M. Temirbekov, D. S. Kenzhebekova

E-ISSN: 2224-2880

456

Volume 23, 2024

[17] El-Faqeer A.S.A., Mohd M.H., Ravichandran

V., Supramaniam S., Starlikeness of certain

analytic functions, arXiv preprint

arXiv:2006.11734, 2020 arxiv.org.

https://doi.org/10.48550/arXiv.2006.11734

[18] Sebastianc A., Ravichandran V., Radius of

starlikeness of certain analytic functions.

Math. Slovaca, 71. No. 1, 2021, pp. 83–104.

https://doi.org/10.48550/arXiv.2001.06999

[19] Sharma K., Jain N.K., Kumar S. Constrained

radius estimates of certain analytic functions.

arXiv:2305.16210v1 [math.CV] 25 May

2023.

https://doi.org/10.48550/arXiv.2305.16210

[20] Shaffer D.B., Distortion theorems for a

special class of analytic functions, Proc.

Amer. Math. Soc., Vol. 39, № 2, 1973,

pp. 281-287, [Online]. https://www.sci-

hub.ru/10.2307/2039632 (Accessed Date:

February 1, 2024).

[21] MacGregor T., Functions whose derivative

has a positive real part, Trans. Amer. Math.

Soc., Vol. 104, 1962, pp. 532–537.

https://doi.org/10.1090/S0002-9947-1962-

0140674-7.

[22] Chichra P., On the radii of starlikeness and

convexity of certain classes of regular

functions, J. of the Australian Math. Soc.,

Vol. 13, №2, 1972, pp. 208-218.

https://doi.org/10.1017/S1446788700011290.

[23] Rogosinski W., Über Positive Harmonische

Entwicklungen und typisch-reelle

Potenzreihen, Math. Zeitschr., Vol. 35, № 1,

1932, pp. 93-121.

https://doi.org/10.1007/BF01186552

[24] Libera R.J., Some radius of convexity

problems, Duke Math. J., Vol. 31, №1, 1964,

pp.143-158. DOI: 10.1215/S0012-7094-64-

03114-X.

Contribution of Individual Authors to the

Creation of a Scientific Article (Ghostwriting

Policy)

The authors equally contributed in the present

research, at all stages from the formulation of the

problem to the final findings and solution.

Sources of Funding for Research Presented in a

Scientific Article or Scientific Article Itself

No funding was received for conducting this study.

Conflict of Interest

The authors have no conflicts of interest to declare.

Creative Commons Attribution License 4.0

(Attribution 4.0 International, CC BY 4.0)

This article is published under the terms of the

Creative Commons Attribution License 4.0

https://creativecommons.org/licenses/by/4.0/deed.e

n_U

WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2024.23.47

F. F. Maiyer, M. G. Tastanov,

A. A. Utemissova, N. M. Temirbekov, D. S. Kenzhebekova

E-ISSN: 2224-2880

457

Volume 23, 2024