Fekete-Szegö functional for a class of non-Bazilevic functions related to quasi-subordination

Syed Ghoos Ali Shah; Sa’ud Al-Sa’di; Saqib Hussain; Asifa Tasleem; Akhter Rasheed; Imran Zulfiqar Cheema; Maslina Darus

doi:10.1515/dema-2022-0232

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Fekete-Szegö functional for a class of non-Bazilevic functions related to quasi-subordination

Syed Ghoos Ali Shah , Sa’ud Al-Sa’di , Saqib Hussain , Asifa Tasleem , Akhter Rasheed , Imran Zulfiqar Cheema and Maslina Darus

Published/Copyright: May 25, 2023

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From the journal Demonstratio Mathematica Volume 56 Issue 1

Abstract

In this article, we study the Fekete-Szegö functional associated with a new class of analytic functions related to the class of bounded turning by using the principle of quasi-subordination. We derived the coefficient estimates including the classical Fekete-Szegö inequality for functions belonging to this class. We also improved some existing results.

Keywords: analytic function; bounded turning; principle of quasi-subordination; Fekete-Szegö functional

MSC 2010: 30C45; 30C50

1 Introduction

Let A be the class of functions f of the form:

(1) f ( z ) = z + ∑ ∞ n = 2 a n z n ,

which are analytic in the open unit disk U = { z ∈ C : ∣ z ∣ < 1 } , normalized by f ( 0 ) = 0 and f ′ ( 0 ) = 1 . We also denote the class P of functions φ analytic in U , such that

φ ( 0 ) = 1 and ℜ ( φ ( z ) ) > 0 , ( z ∈ U ) .

For two functions f and g , we say that f is subordinate to g written as f ( z ) ≺ g ( z ) , ( z ∈ U ) , if there exists a Schwarz function w , analytic in U with w ( 0 ) = 0 and ∣ w ( z ) ∣ < 1 , such that f ( z ) = g ( w ( z ) ) . Further, if the function g is univalent in U , then we have the following equivalence relation:

(2) f ( z ) ≺ g ( z ) if and only if f ( 0 ) = g ( 0 ) and g ( U ) ⊃ f ( U ) .

For two analytic functions f and g , the function f is quasi-subordinate to the function g (written as: f ( z ) ≺ q g ( z ) ), if there exists an analytic function ψ ( z ) with ∣ ψ ( z ) ∣ ≤ 1 , such that

f ( z ) = ψ ( z ) ( g ( w ( z ) ) ) , ( z ∈ U ) .

The concept of quasi-subordination was given by Robertson [1]. It can be observed that if ψ ( z ) ≡ 1 ( z ∈ U ) , then quasi-subordination coincides with usual subordination. For the Schwarz function w ( z ) given by w ( z ) = z , the quasi-subordination becomes the majorization. In this case:

(3) f ( z ) ≺ q g ( z ) ⇒ f ( z ) = ψ ( z ) g ( z ) ⇒ f ( z ) ≪ g ( z ) , ( z ∈ U ) .

MacGregor [2] introduced the concept of majorization. For t ≠ 1 and ∣ t ∣ ≤ 1 and for some α ( 0 ≤ α < 1 ), Owa et al. [3] introduced a Sakaguchi-type class S ∗ ( α , t ) of functions f of form (1), which satisfies the condition:

(4) ℜ ( 1 − t ) z f ′ ( z ) f ( z ) − f ( t z ) > α , ( z ∈ U ) .

The class of non-Bazilevic functions was introduced by Obradovic [4] as:

(5) ℜ f ′ ( z ) z f ( z ) 1 + ρ > 0 , ( z ∈ U ; 0 < ρ < 1 ) .

For z ∈ U , 1 ≠ t ∈ C , ∣ t ∣ ≤ 1 , and ρ ≥ 0 , by using the aforementioned concept of quasi-subordination, Sharma and Raina [5] defined as:

(6) ς q ρ ( φ , t ) if such that f ′ ( z ) ( 1 − t ) z f ( z ) − f ( t z ) ρ − 1 ≺ q [ φ ( z ) − 1 ] .

Definition 1

[6] For 1 ≠ t ∈ C , ∣ t ∣ ≤ 1 , ρ ≥ 0 , 0 ≤ α ≤ 1 , f ∈ ℑ q ρ ( φ , t , α ) if and only if

(7) ( 1 − α ) f ( z ) z + α f ′ ( z ) ( 1 − t ) z f ( z ) − f ( t z ) ρ − 1 ≺ q [ φ ( z ) − 1 ] .

For different choices of parameters involved in definition (1), we have the following classes of analytic functions:

For ψ ( z ) ≡ 1 , the class ℑ q ρ ( φ , t , α ) becomes class ℑ ρ ( φ , t , α ) .
For α = 1 , the class ℑ q ρ ( φ , t , α ) is equivalent to class ς q ρ ( φ , t ) .
For φ ( z ) = 1 + z 1 − z ( z ∈ U ) , the class ℑ q ρ ( φ , t , α ) reduces to class studied by Nunokwa et al. [7] (see [8–13]).

Motivated by the aforementioned work, we define the following class.

Definition 2

For z ∈ U , 1 ≠ t ∈ C ; ∣ t ∣ ≤ 1 ; ρ ≥ 0 ; 0 ≤ α ≤ 1 , and f ∈ ℑ q ρ ( φ , s , t , α ) if and only if

(8) ( 1 − α ) f ′ ( z ) + α f ′ ( z ) ( s − t ) z f ( s z ) − f ( t z ) ρ − 1 ≺ q [ φ ( z ) − 1 ] .

Special cases

For s = 1 and α = 1 in ℑ q ρ ( φ , s , t , α ) , we have the class ς q ρ ( φ , t ) studied by Sharma and Raina [5].
For α = 0 and ρ = 0 in ℑ q ρ ( φ , s , t , α ) , we have the class ℛ q ( φ ) , f ′ ( z ) − 1 ≺ q [ φ ( z ) − 1 ] .

The study of functional made up of combinations of the coefficients of the original function is a typical problem in geometric function theory. Initially in 1933, Fekete and Sezgö [14] (see also [15–22]) obtained a sharp bound of the functional ∣ a 3 − μ a 2 2 ∣ for univalent functions f ∈ A of form (1) with real μ ( 0 ≤ μ ≤ 1 ) . Since then, the problem of finding the sharp bounds for this functional of any compact family of function f ∈ A has with any complex μ is generally known as the classical Fekete-Szegö problem. Many authors have studied the Fekete-Szegö problem for several subclasses of analytic functions in detail [23–40].

The primary goal of this article is to determine the coefficient estimates including Fekete-Szegö inequality the functions belonging the class ℑ q ρ ( φ , s , t , α ) . Improving our results, we use the following lemmas:

Lemma 1

[37] Let the Schwarz function w ( z ) be given by:

(9) w ( z ) = w 1 z + w 2 z 2 + w 3 z 3 + ⋯ ( z ∈ U ) ,

then

(10) ∣ w 1 ∣ ≤ 1 a n d ∣ w 2 − κ w 1 2 ∣ ≤ 1 + ( ∣ κ ∣ − 1 ) ∣ w 1 ∣ 2 ≤ max { 1 , ∣ κ ∣ } ( κ ∈ C ) .

Remark 1

Let the function f ∈ A be of form (1), then it can be observed that
(11) f ( s z ) − f ( t z ) s − t = z + ∑ n = 2 ∞ δ n a n z n ( z ∈ U ) ,
where
(12) δ n = s n − t n s − t ( n ∈ N ) .
Therefore, for ρ ≥ 0 , we find that
(13) ( s − t ) z f ( s z ) − f ( t z ) ρ − 1 = 1 − ρ δ 2 a 2 z + ρ ( 1 + ρ ) 2 δ 2 2 a 2 2 − ρ δ 3 a 3 z 2 + ⋯ .
We also suppose that the function φ ( z ) ∈ P is of the form:
(14) φ ( z ) = 1 + c 1 z + c 2 z 2 + ⋯ ( c 1 > 0 , z ∈ U ) ,
and the ψ ( z ) , analytic in U , is of the form:
(15) ψ ( z ) = b 0 + b 1 z + b 2 z 2 + ⋯ ( z ∈ U ) .
Throughout this article, we assume that ρ and t are such that
(16) ρ δ n ≠ n ( z ∈ U ) and ρ δ n < n , ( t ∈ R , n = 2 , 3 , … ) .

2 Main results

Theorem 1

Let the function f ∈ A be in the class ℑ q ρ ( φ , s , t , α ) . Then,

(17) ∣ a 2 ∣ ≤ c 1 ∣ 2 − α ρ δ 2 ∣ ,

and for some μ ∈ C

(18) ∣ a 3 − μ a 2 2 ∣ ≤ c 1 ∣ 3 − α ρ δ 2 ∣ max 1 , c 1 L − c 2 c 1 ,

where

(19) L = μ ( 3 − α ρ δ 3 ) ( 2 − α ρ δ 2 ) 2 − α ρ ( 4 − ( 1 + ρ ) δ 2 ) δ 2 2 ( 2 − α ρ δ 2 ) 2 ,

and δ n ( n ∈ N ) is given by (12).

Proof

Suppose that the function f ∈ ℑ q ρ ( φ , s , t , α ) is given by (1). Then, by using the concept of quasi-subordination with Schwarz functions w ( z ) given by (9) and for the analytic function ψ ( z ) defined in (15), it follows that

(20) ( 1 − α ) f ′ ( z ) + α f ′ ( z ) ( s − t ) z f ( s z ) − f ( t z ) ρ − 1 = ψ ( z ) [ φ ( w ( z ) ) − 1 ] .

Moreover, in view of (14), it is easy to see that

(21) ψ ( z ) [ φ ( z ) − 1 ] = ( b 0 + b 1 z + b 2 z 2 + ⋯ ) ( c 1 w 1 z + ( c 1 w 2 + c 2 w 1 2 ) z 2 + ⋯ ) = b 0 c 1 w 1 z + [ b 0 ( c 1 w 2 + c 2 w 1 2 ) + b 1 c 1 w 1 ] z 2 + ⋯ .

Since f ∈ A by using (1) together with (13) and after some computation, we obtain

(22) ( 1 − α ) f ′ ( z ) + α f ′ ( z ) ( s − t ) z f ( s z ) − f ( t z ) ρ − 1 = ( 2 a 2 − α ρ δ 2 a 2 ) z + 3 a 3 + α ρ 1 + ρ 2 δ 2 − 2 δ 2 a 2 2 − α ρ δ 3 a 3 z 2 + ⋯

Now, by putting the values from (21), (22) in (20) and equating the coefficients of z and z 2 , we have

(23) a 2 ( 2 − α ρ δ 2 ) = b 0 c 1 w 1

and

(24) a 3 ( 3 − α ρ δ 3 ) − α ρ 2 − ( 1 + ρ ) 2 δ 2 δ 2 a 2 2 = b 0 ( c 1 w 2 + c 2 w 1 2 ) + b 1 c 1 w 1 .

Since ∣ b 0 ∣ ≤ 1 and ∣ c 1 ∣ ≤ 1 , we find from (23) that

(25) ∣ a 2 ∣ ≤ c 1 ∣ 2 − α ρ δ 2 ∣ .

Now, (24) and (25) yield

a 3 ( 3 − α ρ δ 3 ) = α ρ 2 − ( 1 + ρ ) 2 δ 2 δ 2 b 0 2 c 1 2 w 1 2 ( 2 − α ρ δ 2 ) 2 + b 0 ( c 1 w 2 + c 2 w 1 2 ) + b 1 c 1 w 1 ,

which implies that

(26) a 3 = c 1 ( 3 − α ρ δ 3 ) b 1 w 1 + b 0 w 2 + c 2 c 1 + α ρ ( 4 − ( 1 + ρ ) δ 2 ) b 0 c 1 δ 2 2 ( 2 − α ρ δ 2 ) 2 w 1 2 .

Also, for some μ ∈ C , we find from (25) and (26) that

(27) a 3 − μ a 2 2 = c 1 ( 3 − α ρ δ 3 ) b 1 w 1 + ( w 2 + c 2 c 1 w 1 2 ) b 0 − c 1 L w 1 2 b 0 2 ,

where L is given by (19). Since the function ψ defined in (19) is analytic and bounded in U , by using a result recorded in Nehari [38], we have

(28) ∣ b 0 ∣ ≤ 1 and b 1 = ( 1 − b 0 2 ) x 2 ,

for some x ( ∣ x ∣ ≤ 1 ) . Thus, upon substituting the values of b 1 from (28) into (27), we obtain

(29) a 3 − μ a 2 2 = c 1 ( 3 − α ρ δ 3 ) x w 1 + w 2 + c 2 c 1 w 1 2 b 0 − ( c 1 L w 1 2 + x w 1 ) b 0 2 .

For b 0 = 0 in (29), it follows that

∣ a 3 − μ a 2 2 ∣ ≤ c 1 ∣ 3 − α ρ δ 3 ∣ .

For the case b 0 ≠ 0 , we consider

g ( b 0 ) = x w 1 + w 2 + c 2 c 1 w 1 2 b 0 − ( c 1 L w 1 2 + x w 1 ) b 0 2 ,

which is a polynomial in b 0 and is, therefore, analytic with ∣ b 0 ∣ ≤ 1 , and the maximum of g ( b 0 ) is attained at b 0 = e i θ ( 0 ≤ θ < 2 π ) . We find that

max 0 ≤ θ < 2 π ∣ g ( b 0 ) ∣ = ∣ g ( 1 ) ∣ ( b 0 = e i θ )

and

(30) ∣ a 3 − μ a 2 2 ∣ ≤ c 1 ∣ 3 − α ρ δ 3 ∣ w 2 − c 1 L − c 2 c 1 w 1 2 .

Finally, by using Lemma 1, we have assertion (18).□

It is worth mentioning that for specializing the parameters, we obtain the number of important corollaries as:

Corollary 1

[5] For α = 1 , s = 1 , we have

∣ a 2 ∣ ≤ c 1 ∣ 2 − ρ δ 2 ∣ ,

and for some μ ∈ C ,

∣ a 3 − μ a 2 2 ∣ ≤ c 1 ∣ 3 − ρ δ 3 ∣ max 1 , c 1 L − c 2 c 1 ,

where

L = μ ( 3 − α ρ δ 3 ) ( 2 − α ρ δ 2 ) 2 − α ρ ( 4 − ( 1 + ρ ) δ 2 ) δ 2 2 ( 2 − α ρ δ 2 ) 2 ,

and δ n ( n ∈ N ) is given by (12).

Corollary 2

For α = 1 , s = 1 , and t = 0 , the class ℑ q ρ ( φ , s , t , α ) reduces to S q ∗ ( φ ) , and we have

∣ a 2 ∣ ≤ c 1 ,

and for some μ ∈ C ,

∣ a 3 − μ a 2 2 ∣ ≤ c 1 2 max 1 , c 2 c 1 + ( 1 − 2 μ ) c 1 .

Corollary 3

For α = 0 , the class ℑ q ρ ( φ , s , t , α ) reduces to class ℛ q ( φ ) , then

∣ a 2 ∣ ≤ c 1 2 ,

and for some μ ∈ C ,

∣ a 3 − μ a 2 2 ∣ ≤ c 1 3 max 1 , c 2 c 1 − 3 μ 4 c 1 .

Corollary 4

For α = 1 2 and s = 1 , we have

∣ a 2 ∣ ≤ 2 c 1 ∣ 4 − ρ δ 2 ∣ ,

and for some μ ∈ C ,

∣ a 3 − μ a 2 2 ∣ ≤ 2 c 1 ∣ 6 − ρ δ 3 ∣ max 1 , c 1 L − c 2 c 1 ,

where

L = μ ( 3 − α ρ δ 3 ) ( 2 − α ρ δ 2 ) 2 − α ρ ( 4 − ( 1 + ρ ) δ 2 ) δ 2 2 ( 2 − α ρ δ 2 ) 2 .

Corollary 5

For ψ ( z ) ≡ 1 ,

∣ a 2 ∣ ≤ c 1 ∣ 2 − α ρ δ 2 ∣ ,

and for some μ ∈ C ,

∣ a 3 − μ a 2 2 ∣ ≤ c 1 ∣ 3 − α ρ δ 3 ∣ max 1 , c 1 L − c 2 c 1 ,

where

L = μ ( 3 − α ρ δ 3 ) ( 2 − α ρ δ 2 ) 2 − α ρ ( 4 − ( 1 + ρ ) δ 2 ) δ 2 2 ( 2 − α ρ δ 2 ) 2 .

Remark 2

For t = 0 , α ≠ 1 , and ρ = 1 , we obtain the following result:

∣ a 2 ∣ ≤ c 1 ∣ 2 − α ∣ ,

and for some μ ∈ C ,

∣ a 3 − μ a 2 2 ∣ ≤ c 1 ∣ 3 − α ∣ max 1 , c 2 c 1 + α − ( 3 − α ) μ ( 2 − α ) 2 c 1 .

Remark 3

For t = 0 , α ≠ 1 , and ρ = 0 , we obtain the following result:

∣ a 2 ∣ ≤ c 1 2 ,

and for some μ ∈ C ,

∣ a 3 − μ a 2 2 ∣ ≤ c 1 3 max 1 , c 2 c 1 − 3 μ c 1 4 .

Theorem 2

Let 1 ≠ b ∈ C , ∣ b ∣ ≤ 1 , and 0 ≤ α ≤ 1 . If the function f ∈ ℑ q ρ ( φ , s , t , α ) is of the form given by (1) and satisfies the following majorization condition:

(31) ( 1 − α ) f ′ ( z ) + α f ′ ( z ) ( s − t ) z f ( s z ) − f ( t z ) ρ − 1 ≪ [ φ ( z ) − 1 ] ,

then

∣ a 2 ∣ ≤ c 1 ∣ 2 − α ρ δ 2 ∣ .

Also for some μ ∈ C

∣ a 3 − μ a 2 2 ∣ ≤ c 1 ∣ 3 − α ρ δ 2 ∣ max 1 , c 1 L − c 2 c 1 ,

where L is defined in (19) and δ n is defined in (12). The result is sharp.

Proof

By using the concept of majorization and Theorem 1, we have w ( z ) ≡ z in (9). This implies that

w 1 = 1 and w n = 0 ( n = 2 , 3 , … ) .

Thus, if we make use of (25) and (27), we obtain

∣ a 2 ∣ ≤ c 1 ∣ 2 − α ρ δ 2 ∣

and

(32) a 3 − μ a 2 2 = c 1 ( 3 − α ρ δ 3 ) b 1 w 1 + ( w 2 + c 2 c 1 w 1 2 ) b 0 − c 1 L w 1 2 b 0 2 .

Substituting the values of b 1 from (28) into (22), we have

(33) a 3 − μ a 2 2 = c 1 ( 3 − α ρ δ 3 ) x + c 2 c 1 b 0 − ( c 1 L + x ) b 0 2 .

If b 0 = 0 , then

(34) ∣ a 3 − μ a 2 2 ∣ = c 1 ∣ 3 − α ρ δ 3 ∣ .

Also for the case b 0 ≠ 0 , we consider

H ( b 0 ) = x + c 2 c 1 b 0 − ( c 1 L + x ) b 0 2 ,

which is a polynomial in b 0 and is, therefore, analytic in ∣ b 0 ∣ ≤ 1 , and the maximum of ∣ H ( b 0 ) ∣ is attained at b 0 = e i θ ( 0 ≤ θ < 2 π ) . We thus find that

max 0 ≤ θ < 2 π ∣ H ( b 0 ) ∣ = ∣ H ( 1 ) ∣ ,

and therefore, we have

(35) ∣ a 3 − μ a 2 2 ∣ ≤ c 1 ∣ 3 − α ρ δ 3 ∣ c 1 L − c 2 c 1 .

Now, for inequality (35) together with (34), we have the result asserted by Theorem 2.

We now find the bounds of the Fekete-Szegö functional ∣ a 3 − μ a 2 2 ∣ when μ and t are the real numbers. We first obtain the following result for the class ℑ q ρ ( φ , s , t , α ) .□

Corollary 6

Let the function f ∈ ℑ q ρ ( φ , s , t , α ) be of form (1). Then, for real values μ and b ,

∣ a 3 − μ a 2 2 ∣ ≤ c 1 ∣ 3 − α ρ δ 3 ∣ c 2 c 1 + c 1 ( ρ ( 4 α − α ( 1 + ρ ) δ 2 ) δ 2 − 2 μ ( 3 − α ρ δ 3 ) ) 2 ( 2 − α ρ δ 2 ) 2 ( μ ≤ ν ) c 1 ∣ 3 − α ρ δ 3 ∣ ( ν ≤ μ ≤ ν + 2 γ ) c 1 ∣ 3 − α ρ δ 3 ∣ c 1 ( 2 μ ( 3 − α ρ δ 3 ) ) − ( ρ ( 4 α − α ( 1 + ρ ) δ 2 ) δ 2 ) − 2 ( 2 − α ρ δ 2 ) 2 − c 2 c 1 ( γ + ν < μ < ν + 2 γ ) ,

where

(36) ν = α ρ ( ( 4 − ( 1 + ρ ) δ 2 ) δ 2 ) 2 ( 3 − α ρ δ 3 ) − ( 2 − α ρ δ 2 ) 2 ( 3 − α ρ δ 3 ) 1 c 1 − c 2 c 1

and

(37) γ = ( 2 − α ρ δ 2 ) 2 c 1 ( 3 − α ρ δ 3 ) .

Proof

For real values of μ and b and by working in a similar way in Theorem 1, we obtain the result asserted by Corollary 6 by taking the following three cases.

□ c 1 L − c 2 c 1 ≤ − 1 , c 1 L − c 2 c 1 ≤ 1 , and c 1 L − c 2 c 1 ≥ − 1 .

Theorem 3

Let the function f ( z ) ∈ ℑ q ρ ( φ , s , t , α ) , then for real values of μ and b ,

(38) ∣ a 3 − μ a 2 2 ∣ + ( μ − ν ) ∣ a 2 2 ∣ ≤ c 1 ∣ 3 − α ρ δ 3 ∣ ( ν < μ ≤ ν + γ )

and

(39) ∣ a 3 − μ a 2 2 ∣ + ( ν + 2 γ − μ ) ∣ a 2 2 ∣ ≤ c 1 ∣ 3 − α ρ δ 3 ∣ ( γ + ν < μ < ν + 2 γ ) ,

where ν and γ are given by (36) and (37), respectively, and δ n is given by (12).

Proof

Suppose that the function f ∈ ℑ q ρ ( φ , s , t , α ) be in the form is given by (1). If ν < μ ≤ ν + γ , then by using (25) and (30), we have

∣ a 3 − μ a 2 2 ∣ + ( μ − ν ) ∣ a 2 2 ∣ ≤ c 1 ∣ 3 − α ρ δ 3 ∣ ∣ w 2 ∣ − c 1 ( 3 − α ρ δ 3 ) ( 2 − α ρ δ 2 ) 2 ( μ − ν − γ ) ∣ w 1 2 ∣ + c 1 ( 3 − α ρ δ 3 ) ( 2 − α ρ δ 2 ) 2 ( μ − ν ) ∣ w 1 2 ∣ , ≤ c 1 ∣ 3 − α ρ δ 3 ∣ ∣ w 2 ∣ + c 1 ( 3 − α ρ δ 3 ) ( 2 − α ρ δ 2 ) 2 γ ∣ w 1 2 ∣ , ≤ c 1 ∣ 3 − α ρ δ 3 ∣ ( ∣ w 2 ∣ − ( − 1 ) ∣ w 1 2 ∣ ) .

Hence, by applying Lemma 1, we obtain

∣ a 3 − μ a 2 2 ∣ + ( μ − ν ) ∣ a 2 2 ∣ ≤ c 1 ∣ 3 − α ρ δ 3 ∣ .

If γ + ν < μ < ν + 2 γ , then we again make use of (25) and (30), in conjunction with Lemma 1, thus we obtain

∣ a 3 − μ a 2 2 ∣ + ( ν + 2 γ − μ ) ∣ a 2 2 ∣ ≤ c 1 ∣ 3 − α ρ δ 3 ∣ ∣ w 2 ∣ − c 1 ( 3 − α ρ δ 3 ) ( 2 − α ρ δ 2 ) 2 ( μ − ν − γ ) ∣ w 1 2 ∣ + c 1 ( 3 − α ρ δ 3 ) ( 2 − α ρ δ 2 ) 2 ( ν + 2 γ − μ ) ∣ w 1 2 ∣ ,

so that, by using Lemma 1, we obtain

∣ a 3 − μ a 2 2 ∣ + ( ν + 2 γ − μ ) ∣ a 2 2 ∣ ≤ c 1 ∣ 3 − α ρ δ 3 ∣ .

Hence, we have established the result asserted by Theorem 3.□

3 Conclusion

Using the concept of quasi-subordination, we have systematically studied coefficient estimates and Fekete-Szegö functional for a new class of analytic functions related to non-Bazilevic functions and bounded turning. We also pointed out many new and already existing corollaries by assigning by specializing the parameters.

Acknowledgement

The authors would like to thank the anonymous referees for their comments and suggestions, which helped greatly improve the manuscript.

Funding information: None declared.
Author contributions: These authors contributed equally to this work. All authors read and approved the final manuscript.
Conflict of interest: The authors state no conflict of interest.
Ethical approval: The conducted research is not related to either human or animal use.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during this study.

References

[1] M. S. Robertson, Quasi subordination and coefficient conjectures, Bull. Amer. Math. Soc. 76 (1970), 1–9. 10.1090/S0002-9904-1970-12356-4Search in Google Scholar

[2] T. H. MacGregor, Majorization by univalent functions, Duke Math. J. 34 (1967), 95–102. 10.1215/S0012-7094-67-03411-4Search in Google Scholar

[3] S. Owa, T. Sekine, and R. Yamakawa, On Sakaguchi-type functions, Appl. Math. Comput. 187 (2007), 356–361. 10.1016/j.amc.2006.08.133Search in Google Scholar

[4] M. Obradovic, A class of univalent functions, Hokkaido Math. J. 2 (1988), 329–335. Search in Google Scholar

[5] P. Sharma and R. K. Raina, On a Sakaguchi-type class of univalent functions associated with quasi-subordination, Comment. Math. Univ. St. Paul 64 (2015), 59–70. Search in Google Scholar

[6] H. M. Srivastava, S. Hussain, A. Raziq, and M. Raza, The Fekete-Szegö functional for a subclass of analytic functions associated with quasi-subordination, Carpathian J. Math. 34 (2018), 103–113. 10.37193/CJM.2018.01.11Search in Google Scholar

[7] M. Nunokawa, M. Obradovic, and S. Owa, One criterion for univalency, Proc. Amer. Math. Soc. 106 (1989), 1035–1037. 10.1090/S0002-9939-1989-0975653-5Search in Google Scholar

[8] K. S. Charak and S. Sharma, Value distribution and normality of meromorphic functions involving partial sharing of small functions, Turk. J. Math. 41 (2017), 404–411. 10.3906/mat-1511-69Search in Google Scholar

[9] S. G. A. Shah, S. Hussain, A. Rasheed, Z. Shareef, and M. Darus, Application of quasi subordination to certain classes of meromorphic functions, J. Funct. Spaces 2020 (2020), Article ID 4581926. 10.1155/2020/4581926Search in Google Scholar

[10] S. Mahmood, H. M. Srivastava, and S. N. Malik, Some subclasses of uniformly univalent functions with respect to symmetric points, Symmetry 11 (2019), 287. 10.3390/sym11020287Search in Google Scholar

[11] V. Ravichandran, A. Naz, and S. Nagpal, Star-likeness associated with the exponential function, Turk. J. Math. 43 (2019), 1353–1371. 10.3906/mat-1902-34Search in Google Scholar

[12] S. G. A. Shah, S. Noor, M. Darus, W. UlHaq, and S. Hussain, On meromorphic functions defined by a new class of liu-srivastava integral operator, Int. J. Anal. Appl. 18 (2020), no. 6, 1056–1065. Search in Google Scholar

[13] S. Porwal and S. Bulut, Radii problem of certain subclasses of analytic functions with fixed second coefficients, Honam Math. J. 37 (2015), 317–323. 10.5831/HMJ.2015.37.3.317Search in Google Scholar

[14] M. Fekete and G. Szegö, Eine Bemerkung, Uber ungrade schlichet funktionen, J. Math. Soc. 8 (1933), 85–89. 10.1112/jlms/s1-8.2.85Search in Google Scholar

[15] B. Deng, D. Liu, and D. Yang, Meromorphic function and its difference operator share two sets with weight k, Turk. J. Math. 41 (2017), 1155–1163. 10.3906/mat-1509-73Search in Google Scholar

[16] Y. Polatoglu and H. E. Ozkan, Growth and distortion theorems for multivalent Janowski close-to-convex harmonic functions with shear construction method, Turk. J. Math. 37 (2013), 437–444. 10.3906/mat-1012-543Search in Google Scholar

[17] A. Naik, T. Panigrahi, and G. Murugusundaramoorthy, Coefficient estimate for class of meromorphic Bi-BazileviČ type functions associated with linear operator defined by convolution, Jordan J. Math. Stat. 14 (2021), no. 2, 287–305. Search in Google Scholar

[18] S. G. A. Shah, S. Noor, S. Hussain, A. Tasleem, A. Rasheed, and M. Darus, Analytic functions related with starlikeness, Mathematical Problems in Engineering 2021 (2021), Article ID 9924434. 10.1155/2021/9924434Search in Google Scholar

[19] G. Murugusundaramoorthy and H. Dutta, Fekete-Szegö inequality and application of Poisson distribution series for some subclasses of analytic functions related with Bessel functions, Appl. Math. Info. Sci. 16 (2022), no. 2, 305–313. 10.18576/amis/160218Search in Google Scholar

[20] A. Rasheed, S. Hussain, S. G. A. Shah, M. Darus, and S. Lodhi, Majorization problem for two subclasses of meromorphic functions associated with a convolution operator, AIMS Mathematics 5 (2020), no. 5, 5157–5170. 10.3934/math.2020331Search in Google Scholar

[21] M. Çağlar, G. Palpandy, and H. Orhan, The coefficient estimates for a class defined by Hohlov operator using conic domains, TWMS J. Pure Appl. Math. 11 (2020), no. 2, 157–172. Search in Google Scholar

[22] M. Çağlar and H. Orhan, Fekete-Szegö problem for certain subclasses of analytic functions defined by the combination of differential operators. Bol. Soc. Mat. Mex. 27 (2021), no. 41, 1–12. 10.1007/s40590-021-00349-9Search in Google Scholar

[23] S. P. Goyal and P. Goswami, Certain coefficient in equalities for Sakaguchi-type functions and applications to fractional derivatives operator, Acta Univ. Apulensis 19 (2009), 159–166. Search in Google Scholar

[24] M. H. Mohd. and M. Darus, Fekete-Szegö problems for quasi-subordination classes, Abstr. Appl. Anal. 2012 (2012), 1–14, Article ID 192956. 10.1155/2012/192956Search in Google Scholar

[25] A. Motamednezhad and S. Salehian, New subclass of bi-univalent functions by (p,q)-derivative operator, Honam Math. J. 41 (2019), no. 2, 381–390. Search in Google Scholar

[26] V. Ravichandran, A. Gangadharen, and M. Darus, Fekete-Szegö inequality for certain class of Bazelevic functions, Far East J. Math. Sci. 15 (2004), 171–180. Search in Google Scholar

[27] V. Ravichandran, M. Darus, M. H. Khan, and K. G. Subramanian, Fekete-Szegö inequality for certain class of analytic functions, Austral. J. Math. Anal. Appl. 4 (2004), no. 1, 1–7. Search in Google Scholar

[28] V. Ravichandran, M. Bokal, Y. Polotoglu, and A. Sen, Certain subclasses of starlike and convex functions of complex order, Hacetlepe J. Math. Stat. 34 (2005), 9–15. Search in Google Scholar

[29] K. Sakaguchi, On a certain univalent mapping, Math. J. Soc. Japan 11 (1959), 72–75. 10.2969/jmsj/01110072Search in Google Scholar

[30] T. N. Shanmugham and S. Sivasubramanian, On the Fekete-Szegö problem for some subclasses of analytic functions, J. Inequal. Pure Appl. Math. 6 (2005), no. 3, Art 71, 6 pages. Search in Google Scholar

[31] T. N. Shanmugham, C. Ramachandran, and V. Ravichandran, Fekete-Szegö problem for subclasses of starlike function with respect to symmetric points, Bull. Korean Math. Soc. 43 (2006), no. 3, 589–598. 10.4134/BKMS.2006.43.3.589Search in Google Scholar

[32] T. N. Shanmugham, S. Sivasubramanian, and M. Darus, Fekete-Szegö inequality for certain classes of analytic functions, Mathematica 34 (2007), 29–34. Search in Google Scholar

[33] H. M. Srivastava, A. K. Mishra, and M. K. Das, The Fekete-Szegö problem for a subclass of class-to-convex functions, Complex Variables Theory Appl. 44 (2001), 145–163. 10.1080/17476930108815351Search in Google Scholar

[34] H. M. Srivastava and S. Owa, An application of the fractional derivative, Math. Japan 29 (1984), 383–389.Search in Google Scholar

[35] N. Tuneski and M. Darus, Fekete-Szegö functional for non-Bazilevic functions, Acta. Math. Acad. Paedagog. Nyhazi. (N. S) 18 (2002), no. 2, 63–65. Search in Google Scholar

[36] K. T. Wiecław, P. Zaprawa, M. Gregorczyk, and A. Rysak, On the Fekete-Szegö type functionals for close-to-convex functions, Symmetry 11 (2019), no. 12, 1497. 10.3390/sym11121497Search in Google Scholar

[37] F. R. Keogh and E. P. Merkers, A coefficient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc. 20 (1969), 8–12. 10.1090/S0002-9939-1969-0232926-9Search in Google Scholar

[38] Z. Nehari, Conformal Mapping, McGraw-Hill Book Company, New York, 1952. Search in Google Scholar

[39] H. Arikan, M. Çağlar, and H. Orhan, Fekete-Szego problem for subclasses of analytic functions associated with quasi-subordination, Bull. Transilv. Univ.Braşov Ser. III Math. Inform. Phys. 12 (2019), no. 2, 221–232. 10.31926/but.mif.2019.12.61.2.3Search in Google Scholar

[40] H. Orhan, E. Deniz, and M. Çağlar, Fekete Szegö problem for certain subclasses of analytic functions, Demonstr. Math. 45 (2012), no. 4, 835–846. 10.1515/dema-2013-0423Search in Google Scholar

Received: 2021-01-03

Revised: 2023-03-11

Accepted: 2023-04-17

Published Online: 2023-05-25

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/dema-2022-0232

Keywords for this article

analytic function; bounded turning; principle of quasi-subordination; Fekete-Szegö functional

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