Coefficient estimates and Fekete-Szegö problem for some classes of univalent functions generalized to a complex order

Mihai Aron

doi:10.1515/dema-2024-0086

Article Open Access

Coefficient estimates and Fekete-Szegö problem for some classes of univalent functions generalized to a complex order

Mihai Aron

Published/Copyright: February 17, 2025

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From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

In this article, we determine coefficient estimates and study the Fekete-Szegö problem for classes Λ n ( b ) and Λ n c ( b ) , where b is a nonzero complex order. In addition, we determine estimates for logarithmic coefficients of functions within these classes. Moreover, we extend our study to include estimates for logarithmic coefficients of functions in the classes F n ( b ) and S n c ( b ) , studied by Kanas and Darwish.

Keywords: coefficient estimates; differential operator; Fekete-Szegö problem; starlike and convex functions of complex order; logarithmic coefficients

MSC 2010: 30C45; 30C50; 30C55

1 Preliminaries

Let us denote by A the family of all normalized analytic functions of the form f ( z ) = z + ∑ n = 2 ∞ a n z n on the unit disc U . Also, let us consider S as the subset of A containing all normalized analytic and univalent functions on U .

Fekete and Szegö [1] established the following result: for any f ( z ) = z + a 2 z 2 + a 3 z 3 + … ∈ S and any λ ∈ ( 0 , 1 ] , the following estimate holds:

∣ a 3 − λ a 2 2 ∣ ≤ 1 + 2 e − 2 λ 1 − λ .

Furthermore, this inequality is sharp in the sense that for any λ ∈ ( 0 , 1 ] , there exists a function f ∈ S such that equality is attained.

The Fekete-Szegö problem consists of determining similar estimates for the coefficients in certain subclasses of S .

The functional Φ λ ( f ) = a 3 − λ a 2 2 = 1 6 ( f ‴ ( 0 ) − 3 λ ⁄ 2 [ f ″ ( 0 ) ] 2 ) represents various geometric quantities. Kanas and Darwish [2] observed that Φ 1 ( f ) = S f ( 0 ) ⁄ 6 , where S f = ( f ″ ⁄ f ′ ) ′ − ( f ″ ⁄ f ′ ) 2 ⁄ 2 is the Schwarzian derivative of the function f .

Furthermore, fix z ∈ U and let us consider the Koebe transform of f :

f z ( ζ ) = f ζ + z 1 + z ¯ ζ − f ( z ) ( 1 − ∣ z ∣ 2 ) f ′ ( z ) = ζ + b 2 ( z ) ζ 2 + b 3 ( z ) ζ 3 + … , ζ ∈ U .

This function belongs to S and one deduces that [3]

b 2 ( z ) = 1 2 ( 1 − ∣ z ∣ 2 ) f ″ ( z ) f ′ ( z ) − z ¯ , b 3 ( z ) = 1 6 ( 1 − ∣ z ∣ 2 ) 2 f ‴ ( z ) f ′ ( z ) − z ¯ ( 1 − ∣ z ∣ 2 ) f ″ ( z ) f ′ ( z ) + z ¯ 2 .

Now, it is easy to observe that

Φ 1 ( f z ) = 1 6 ( 1 − ∣ z ∣ 2 ) 2 S f ( z ) .

Moreover, Kanas and Darwish [2] noted that if we consider the n th root transform [ f ( z n ) ] 1 n = z + c n + 1 z n + 1 + c 2 n + 1 z 2 n + 1 + … , then c n + 1 = a 2 n , c 2 n + 1 = a 3 n + n − 1 2 n 2 a 2 2 and, thus,

Φ λ ( f ) = n ( c 2 n + 1 − μ c n + 1 2 ) , μ = λ n + n − 1 2 .

In addition, Φ λ exhibits the following behavior concerning rotation and dilation:

Φ λ ( e − i θ f ( e i θ z ) ) = e 2 i θ Φ λ ( f ) , θ ∈ R ; Φ λ ( r − 1 f ( r z ) ) = r 2 Φ λ ( f ) , 0 < r < 1 .

For a real number α ∈ [ 0 , 1 ) , we denote S ⋆ ( α ) and S c ( α ) the classes of starlike and convex univalent functions of order α , respectively, i.e.,

S ⋆ ( α ) = f ∈ A : Re z f ′ ( z ) f ( z ) > α on U

and

S c ( α ) = f ∈ A : Re 1 + z f ″ ( z ) f ′ ( z ) > α on U .

Note that the conditions from the definitions of these classes imply that all of these functions are univalent.

The notions of starlikeness and convexity of order α ∈ [ 0 , 1 ) were extended to a complex order by Nasr and Aouf [4], Wiatrowski [5], and Nasr and Aouf [6].

Notice that the sets S ⋆ ( 0 ) = S ⋆ and S c ( 0 ) = K represent the classes of standard starlike and convex univalent functions, respectively.

Kumar et al. [7] introduced the class F n ( b ) of functions f ∈ S as follows:

F n ( b ) = f ∈ S : D n f ( z ) z ≠ 0 , Re 1 + 1 b z ( D n f ) ′ ( z ) D n f ( z ) − 1 > 0 on U ,

where b is a nonzero complex number and D n f represents the Ruscheweyh derivative of order n ∈ N 0 of the function f [8]:

D n f ( z ) = z ( z n − 1 f ( z ) ) ( n ) n ! , n ∈ N 0 .

Note that, F 0 ( b ) and F 1 ( b ) represent the classes of starlike and convex functions of complex order b , respectively, mentioned earlier. Moreover, if b ∈ ( 0 , 1 ] , then F 0 ( b ) = S ⋆ ( 1 − b ) and F 1 ( b ) = S c ( 1 − b ) .

Kanas and Darwish investigated the Fekete-Szegö problem concerning the class F n ( b ) in their work [2]. As a generalization of the class of convex functions of complex order b , Kanas and Darwish [2] introduced the class

S n c ( b ) = f ∈ S : Re 1 + 1 b z ( D n f ) ″ ( z ) ( D n f ) ′ ( z ) > 0 on U ,

and investigated the Fekete-Szegö problem for this class, where b is a nonzero complex number. Clearly, S 0 c ( b ) = F 1 ( b ) and if b ∈ ( 0 , 1 ] , then S 0 c ( b ) = S c ( 1 − b ) .

The following operator was introduced by Aron in [9]:

Definition 1.1

The differential operator A n , n ∈ N 0 = { 0 , 1 , … } , is defined as follows:

A 0 f ( z ) = f ( z ) , A 1 f ( z ) = A f ( z ) = z 2 f ′ ( z ) f ( z ) , f ∈ A , f ( z ) ≠ 0 , z ∈ U \ { 0 } , A n + 1 f ( z ) = A ( A n f ) ( z ) , n ∈ N 0 , if there exist A k f , k ≤ n and A n f ( z ) ≠ 0 , z ∈ U \ { 0 } .

By using this operator, the author introduces the class Λ n of functions f ∈ A , which satisfies the condition

Re A n + 1 f ( z ) z > 1 2 .

and obtains inclusion properties between these classes and coefficient estimates.

Note that, all of these classes are subclasses of S [9].

Inspired by the research of Kanas and Darwish, and the class defined by Kumar et al., we introduce the classes Λ n ( b ) and Λ n c ( b ) . We investigate the coefficient estimates and solve the Fekete-Szegö problem for these classes.

Definition 1.2

Let b be a nonzero complex number. We say that f ∈ S belongs to Λ n ( b ) if

(1) Re 1 + 1 b A n + 1 f ( z ) z − 1 > 0 on U .

Because of the definition of A n , it is easy to see that f ∈ Λ n ( b ) if and only if A n f ∈ F 0 ( b ) . By using this remark, we define the class Λ n c ( b ) as the class of functions f ∈ S such that A n f ∈ F 1 ( b ) = S 0 c ( b ) :

Definition 1.3

Let b be a nonzero complex number. We say that f ∈ S belongs to Λ n c ( b ) if

(2) Re 1 + 1 b z ( A n f ) ″ ( z ) ( A n f ) ′ ( z ) > 0 on U .

Related to the aforementioned definitions, it is easy to see that Λ 0 ( b ) = F 0 ( b ) . Also, if b ∈ ( 0 , 1 ] , we observe that Λ 0 ( b ) = F 0 ( b ) = S ⋆ ( α ) , where α = 1 − b .

Furthermore, Λ 0 c ( b ) = S 0 c ( b ) = F 1 ( b ) , and if b ∈ ( 0 , 1 ] , all of these classes coincide with the class S c ( α ) , where α = 1 − b .

Remark 1.4

If b ∈ ( 0 , 1 ⁄ 2 ] , the condition f ∈ S in the definition of the class Λ n ( b ) can be replaced by f ∈ A .
If b ∈ ( 0 , 1 ] , the condition f ∈ S in the definition of the class Λ n c ( b ) can be replaced by f ∈ A .

The first part of this remark shows that Λ n ( 1 ⁄ 2 ) = Λ n .

Proof of Remark 1.4

For the first part (i), let β = 1 − b ∈ [ 1 ⁄ 2 , 1 ) . In [10], we find the following implication for functions f ∈ A :

Re z f ′ ( z ) f ( z ) > β ⇒ Re f ( z ) z > 1 3 − 2 β .

Since, the interval [ 1 ⁄ 2 , 1 ) is invariant under the function β ↦ 1 3 − 2 β (Figure 1), we obtain Λ n ( b ) ⊆ S ⋆ ( 1 ⁄ 2 ) ⊆ S .

In part (ii), let α = 1 − b ∈ [ 0 , 1 ) . In [11], the author obtains (as a particular case) that if f ∈ A and α ∈ [ 0 , 1 ) , then

(3) Re 1 + z f ″ ( z ) f ′ ( z ) > α ⇒ Re z f ′ ( z ) f ( z ) > δ ( α ) ,

where

δ ( α ) = 2 α − 1 2 ( 1 − 2 1 − 2 α ) , α ∈ [ 0 , 1 ) \ { 1 ⁄ 2 } , 1 2 ln 2 , α = 1 ⁄ 2 .

Since, δ ( [ 0 , 1 ) ) = [ 1 ⁄ 2 , 1 ) (Figure 1) we obtain Λ n c ( b ) ⊆ S ⋆ ( 1 ⁄ 2 ) ⊆ S .□

$Figure 1 The functions y = 1 ⁄ ( 3 ‒ 2 x ) y=1/\left(3‒2x) and y = δ ( x ) y=\delta \left(x) .$

Figure 1

The functions y = 1 ⁄ ( 3 ‒ 2 x ) and y = δ ( x ) .

2 Main results

Theorem 2.1

Let f ( z ) = z + a 2 z 2 + a 3 z 3 + … ∈ S and also, let A n f ( z ) = z + a 2 ( n ) z 2 + a 3 ( n ) z 3 + … . Then

(4) a 2 ( n ) = a 2 , a 3 ( n ) = 2 n ( a 3 − a 2 2 ) + a 2 2 .

Proof

Clearly, for n = 0 , the relations (4) are verified. Next, let n ∈ N and suppose that the relations from (4) are true for the function A n − 1 f . Since A n f ( z ) = z 2 ( A n − 1 f ) ′ ( z ) A n − 1 f ( z ) , we obtain

z + 2 a 2 ( n − 1 ) z 2 + 3 a 3 ( n − 1 ) z 3 + … z + a 2 ( n − 1 ) z 2 + a 3 ( n − 1 ) z 3 + … = 1 + a 2 ( n ) z + a 3 ( n ) z 2 + … ,

and it follows that a 2 ( n ) = a 2 ( n − 1 ) = a 2 and 3 a 3 ( n − 1 ) = a 3 ( n − 1 ) + a 2 2 + a 3 ( n ) , or equivalent

□ a 3 ( n ) = 2 ( 2 n − 1 ( a 3 − a 2 2 ) + a 2 2 ) − a 2 2 = 2 n ( a 3 − a 2 2 ) + a 2 2 .

Let P denote the Carathéodory class of analytic functions p in U such that p ( 0 ) = 1 and Re p ( z ) > 0 on U .

For further estimates, we need the following:

Theorem 2.2

[2] Let p ( z ) = 1 + c 1 z + c 2 z 2 + … be a function in the class P . Then

(5) ∣ c k ∣ ≤ 2 , k ≥ 1 .

These estimates are sharp for a function p 1 ( z ) = 1 + τ z 1 − τ z , τ ∈ C , ∣ τ ∣ = 1 .

Furthermore,

(6) c 2 − c 1 2 2 ≤ 2 − ∣ c 1 ∣ 2 2 .

This estimate is sharp for a function p 2 ( z ) = 1 + z q ( z ) 1 − z q ( z ) , where q ( z ) = τ 1 ⁄ τ 2 + z τ 2 ¯ + τ 1 ¯ z , τ 1 , τ 2 ∈ C , ∣ τ 1 ∣ < 1 = ∣ τ 2 ∣ .

Theorem 2.3

Let b be a nonzero complex number and f ( z ) = z + a 2 z 2 + a 3 z 3 + … ∈ Λ n ( b ) , then

(7) ∣ a 2 ∣ ≤ 2 ∣ b ∣ ,

(8) ∣ a 3 ∣ ≤ ∣ b ∣ 2 n max { 1 , ∣ 1 − 2 b + 2 n + 2 b ∣ } ,

and

(9) a 3 − 2 n + 1 − 1 2 n + 1 a 2 2 ≤ ∣ b ∣ 2 n .

Equality holds in (7) and (9) if A n + 1 f ( z ) = z + b z [ p 1 ( z ) − 1 ] , and in (8) if A n + 1 f ( z ) = z + b z [ p 2 ( z ) − 1 ] , where p 1 and p 2 are the extremal functions in Theorem 2.2.

Proof

Let f ∈ Λ n ( b ) . From the definition of the class Λ n ( b ) , we observe that there exists a function p ( z ) = z + c 1 z + c 2 z 2 + … ∈ P such that A n + 1 f ( z ) z = 1 + b [ p ( z ) − 1 ] , which implies

1 + a 2 ( n + 1 ) z + a 3 ( n + 1 ) z 2 + … = 1 + b c 1 z + b c 2 z 2 + … , z ∈ U .

Now, it is easy to see that a 2 ( n + 1 ) = b c 1 and a 3 ( n + 1 ) = b c 2 . In view of (4), we obtain

(10) a 2 = b c 1 , a 3 = b 2 n + 1 ( c 2 − b c 1 2 ) + b 2 c 1 2 .

Taking into account (16) and Theorem 2.2, we obtain

∣ a 2 ∣ = ∣ b c 1 ∣ ≤ 2 ∣ b ∣

and

∣ a 3 ∣ = ∣ b ∣ 2 n + 1 ∣ c 2 − b c 1 2 + 2 n + 1 b c 1 2 ∣ = ∣ b ∣ 2 n + 1 c 2 − c 1 2 2 + c 1 2 2 ( 1 − 2 b + 2 n + 2 b ) ≤ ∣ b ∣ 2 n + 1 2 − ∣ c 1 ∣ 2 2 + ∣ c 1 ∣ 2 2 ∣ 1 − 2 b + 2 n + 2 b ∣ = ∣ b ∣ 2 n + 1 2 + ∣ c 1 ∣ 2 2 ( ∣ 1 − 2 b + 2 n + 2 b ∣ − 1 ) ≤ ∣ b ∣ 2 n + 1 max { 2 , 2 + 2 ∣ 1 − 2 b + 2 n + 2 b ∣ − 2 } = ∣ b ∣ 2 n max { 1 , ∣ 1 − 2 b + 2 n + 2 b ∣ } .

Moreover,

a 3 − 2 n + 1 − 1 2 n + 1 a 2 2 = ∣ b ∣ 2 n + 1 ∣ c 2 − b c 1 2 + 2 n + 1 b c 1 2 − ( 2 n + 1 − 1 ) b c 1 2 ∣ = ∣ b ∣ 2 n + 1 ∣ c 2 ∣ ≤ ∣ b ∣ 2 n ,

as desired.

Examining the proof, the equalities are attained in (7) and (9) if A n + 1 f ( z ) z = 1 + b [ p 1 ( z ) − 1 ] , and in (8) if A n + 1 f ( z ) z = 1 + b [ p 2 ( z ) − 1 ] .□

Remark 2.4

The inequalities (8) and (9) are special cases of the Fekete-Szegö problem for the class Λ n ( b ) , which provides sharp estimates for the functional ∣ a 3 − μ a 2 2 ∣ for the cases μ = 0 and μ = 2 n + 1 − 1 2 n + 1 , respectively.

Moreover, setting b = 1 ⁄ 2 in the aforementioned results, we obtain the results from [9].

Next, we present the Fekete-Szegö type inequality for the class Λ n ( b ) with the complex number μ :

Theorem 2.5

Let b be a nonzero complex number and f ( z ) = z + a 2 z 2 + a 3 z 3 + … ∈ Λ n ( b ) . Then for μ ∈ C ,

(11) ∣ a 3 − μ a 2 2 ∣ ≤ ∣ b ∣ 2 n max { 1 , ∣ 1 − 2 b + 2 n + 2 b ( 1 − μ ) ∣ } .

For each value of μ , there exists a function in Λ n ( b ) such that equality in (11) holds.

Proof

By using the notations from the previous proof and taking into account Theorem 2.2, we obtain

∣ a 3 − μ a 2 2 ∣ = ∣ b ∣ 2 n + 1 ∣ c 2 − b c 1 2 + 2 n + 1 b c 1 2 − 2 n + 1 μ b c 1 2 ∣ = ∣ b ∣ 2 n + 1 c 2 − c 1 2 2 + c 1 2 2 [ 1 − 2 b + 2 n + 2 b ( 1 − μ ) ] ≤ ∣ b ∣ 2 n + 1 2 − ∣ c 1 ∣ 2 2 + ∣ c 1 ∣ 2 2 ∣ 1 − 2 b + 2 n + 2 b ( 1 − μ ) ∣ = ∣ b ∣ 2 n + 1 2 + ∣ c 1 ∣ 2 2 [ ∣ 1 − 2 b + 2 n + 2 b ( 1 − μ ) ∣ − 1 ] ≤ ∣ b ∣ 2 n + 1 max { 2 , 2 + 2 ∣ 1 − 2 b + 2 n + 2 b ( 1 − μ ) ∣ − 2 } = ∣ b ∣ 2 n max { 1 , ∣ 1 − 2 b + 2 n + 2 b ( 1 − μ ) ∣ } .

The proof shows that equality is attained in the first case by choosing c 1 = 0 and c 2 = 2 . In this case, the extremal function can be chosen such that it satisfies

A n + 1 f ( z ) z = 1 + b 1 + z 2 1 − z 2 − 1 = 1 + 2 b z 2 1 − z 2 .

For the second case, equality is attained by choosing c 1 = c 2 = 2 and the extremal function can be chosen such that

□ A n + 1 f ( z ) z = 1 + 2 b z 1 − z .

The next result is a particular case of the previous theorem for b and μ being real numbers:

Theorem 2.6

Let b > 0 and f ( z ) = z + a 2 z 2 + a 3 z 3 + … ∈ Λ n ( b ) . Then for μ ∈ R ,

∣ a 3 − μ a 2 2 ∣ ≤ b 2 n [ 1 − 2 b + 2 n + 2 b ( 1 − μ ) ] , if μ ≤ 2 n + 1 − 1 2 n + 1 , b 2 n , if 2 n + 1 − 1 2 n + 1 ≤ μ ≤ 2 n + 1 − 1 + 1 b 2 n + 1 , b 2 n [ 2 b − 1 + 2 n + 2 b ( μ − 1 ) ] , if μ ≥ 2 n + 1 − 1 + 1 b 2 n + 1 .

For each value of μ , there exists a function in Λ n ( b ) such that equality holds.

Proof

In view of the aforementioned proof, we have

(12) ∣ a 3 − μ a 2 2 ∣ ≤ b 2 n + 1 2 + ∣ c 1 ∣ 2 2 [ ∣ 1 − 2 b + 2 n + 2 b ( 1 − μ ) ∣ − 1 ] .

If μ ≤ 2 n + 1 − 1 2 n + 1 < 2 n + 1 − 1 + 1 b 2 n + 1 , it follows that 1 − 2 b + 2 n + 2 b ( 1 − μ ) ≥ 1 . Taking into account (12) and Theorem 2.2, we obtain

∣ a 3 − μ a 2 2 ∣ ≤ b 2 n + 1 { 2 + 2 [ 1 − 2 b + 2 n + 2 b ( 1 − μ ) ] − 2 } = b 2 n [ 1 − 2 b + 2 n + 2 b ( 1 − μ ) ] .

Now, let 2 n + 1 − 1 2 n + 1 ≤ μ ≤ 2 n + 1 − 1 + 1 b 2 n + 1 or equivalent − 1 ≤ 1 − 2 b + 2 n + 2 b ( 1 − μ ) ≤ 1 . Hence, in (12), the maximum value is attained for c 1 = 0 , i.e.,

∣ a 3 − μ a 2 2 ∣ ≤ b 2 n .

Finally, μ ≥ 2 n + 1 − 1 + 1 b 2 n + 1 follows that 1 − 2 b + 2 n + 2 b ( 1 − μ ) ≤ − 1 and thus,

∣ a 3 − μ a 2 2 ∣ ≤ b 2 n + 1 { 2 + 2 [ 2 b − 1 + 2 n + 2 b ( μ − 1 ) ] − 2 } = b 2 n [ 2 b − 1 + 2 n + 2 b ( μ − 1 ) ] .

The extremal function can be chosen for the second case as in the first case of the extremal function from the previous theorem (i.e., c 1 = 0 , c 2 = 2 ). For the first and third cases, the function can be chosen as in the second case of the previous theorem (i.e., c 1 = c 2 = 2 ).□

It is important to mention that by setting n = 0 , all the results for the class Λ 0 ( b ) coincide with the results obtained by Kanas and Darwish [2] for the specific class F 0 ( b ) .

In the next part of this section, we construct similar theorems for the class Λ n c ( b ) .

Theorem 2.7

Let b be a nonzero complex number and f ( z ) = z + a 2 z 2 + a 3 z 3 + … ∈ Λ n c ( b ) , then

(13) ∣ a 2 ∣ ≤ ∣ b ∣ ,

(14) ∣ a 3 ∣ ≤ ∣ b ∣ 3 ⋅ 2 n max { 1 , ∣ 1 − b + 3 ⋅ 2 n b ∣ } ,

and

(15) a 3 − 3 ⋅ 2 n − 1 3 ⋅ 2 n a 2 2 ≤ ∣ b ∣ 3 ⋅ 2 n .

Equality holds in (13) and (15) if z ( A n f ) ″ ( z ) ( A n f ) ′ ( z ) = b [ p 1 ( z ) − 1 ] , and in (14) if z ( A n f ) ″ ( z ) ( A n f ) ′ ( z ) = b [ p 2 ( z ) − 1 ] .

Proof

Let f ∈ Λ n c ( b ) , then there exists a function p ( z ) = z + c 1 z + c 2 z 2 + … ∈ P such that

z ( A n f ) ″ ( z ) ( A n f ) ′ ( z ) = b [ p ( z ) − 1 ] ,

which follows

2 a 2 ( n ) z + 6 a 3 ( n ) z 2 + … = b c 1 z + ( b c 2 + 2 b c 1 a 2 ( n ) ) z 2 + … , z ∈ U .

Therefore, a 2 ( n ) = b c 1 2 and a 3 ( n ) = b 6 ( c 2 + b c 1 2 ) and, in view of (4), we obtain

(16) a 2 = b c 1 2 , a 3 = b 3 ⋅ 2 n + 1 c 2 − b c 1 2 2 + b 2 c 1 2 4 .

Taking into account (16) and Theorem 2.2, we have

∣ a 2 ∣ = ∣ b c 1 ∣ 2 ≤ ∣ b ∣

and

∣ a 3 ∣ = ∣ b ∣ 3 ⋅ 2 n + 1 c 2 − b c 1 2 2 + 3 ⋅ 2 n b c 1 2 2 = ∣ b ∣ 3 ⋅ 2 n + 1 c 2 − c 1 2 2 + c 1 2 2 ( 1 − b + 3 ⋅ 2 n b ) ≤ ∣ b ∣ 3 ⋅ 2 n + 1 2 − ∣ c 1 ∣ 2 2 + ∣ c 1 ∣ 2 2 ∣ 1 − b + 3 ⋅ 2 n b ∣ = ∣ b ∣ 3 ⋅ 2 n max { 1 , ∣ 1 − b + 3 ⋅ 2 n b ∣ } .

Finally,

a 3 − 3 ⋅ 2 n − 1 3 ⋅ 2 n a 2 2 = ∣ b ∣ 3 ⋅ 2 n + 1 c 2 − b c 1 2 2 + 3 ⋅ 2 n b c 1 2 2 − ( 3 ⋅ 2 n − 1 ) b c 1 2 2 = ∣ b ∣ 3 ⋅ 2 n + 1 ∣ c 2 ∣ ≤ ∣ b ∣ 3 ⋅ 2 n .

The proof shows that equalities are attained in these estimates if the conditions specified in the theorem statement, are satisfied.□

Once again, the inequalities (14) and (15) are particular cases of the Fekete-Szegö problem for the class Λ n c ( b ) . The following result presents the Fekete-Szegö inequality for the class Λ n c ( b ) and the complex number μ :

Theorem 2.8

Let b be a nonzero complex number and f ( z ) = z + a 2 z 2 + a 3 z 3 + … ∈ Λ n c ( b ) . Then for μ ∈ C ,

(17) ∣ a 3 − μ a 2 2 ∣ ≤ ∣ b ∣ 3 ⋅ 2 n max { 1 , ∣ 1 − b + 3 ⋅ 2 n b ( 1 − μ ) ∣ } .

For each value of μ , there exists a function in Λ n c ( b ) such that equality in (17) holds.

Proof

In this case, taking into account Theorem 2.2, we have

∣ a 3 − μ a 2 2 ∣ = ∣ b ∣ 3 ⋅ 2 n + 1 c 2 − b c 1 2 2 + 3 ⋅ 2 n b c 1 2 2 − 3 ⋅ 2 n μ b c 1 2 2 = ∣ b ∣ 3 ⋅ 2 n + 1 c 2 − c 1 2 2 + c 1 2 2 [ 1 − b + 3 ⋅ 2 n b ( 1 − μ ) ] ≤ ∣ b ∣ 3 ⋅ 2 n + 1 2 − ∣ c 1 ∣ 2 2 + ∣ c 1 ∣ 2 2 ∣ 1 − b + 3 ⋅ 2 n b ( 1 − μ ) ∣ ≤ ∣ b ∣ 3 ⋅ 2 n max { 1 , ∣ 1 − b + 3 ⋅ 2 n b ( 1 − μ ) ∣ } .

The equality is attained in the first case by choosing c 1 = 0 and c 2 = 2 , and in the second case by choosing c 1 = c 2 = 2 .□

We mention that by setting n = 0 , all the results for the class Λ 0 c ( b ) coincide with the results obtained by Kanas and Darwish [2] for the specific classes S 0 c ( b ) and F 1 ( b ) .

2.1 Logarithmic coefficients of the classes Λ n ( b ) and Λ n c ( b )

For a function f ∈ S , the logarithmic coefficients γ k of f are defined by the following series expansion [12–14]:

(18) log f ( z ) z = 2 ∑ k = 1 ∞ γ k z k .

In [9], the author studies these coefficients for the classes Λ n , noting that the operator A f can be defined in terms of log f ( z ) z :

(19) A f ( z ) z = 1 + z log f ( z ) z ′

and

A n + 1 f ( z ) z = 1 + z log A n f ( z ) z ′ .

Taking into account (19), (18), and Theorem 2.1 [9,12–14], we obtain

(20) γ 1 = a 2 2 , γ 2 = 1 2 a 3 − a 2 2 2 .

Theorem 2.9

Let b be a nonzero complex number and f ∈ Λ n ( b ) such that log f ( z ) z = 2 ∑ k = 1 ∞ γ k z k . Then

(21) ∣ γ 1 ∣ ≤ ∣ b ∣ ,

(22) ∣ γ 2 ∣ ≤ ∣ b ∣ 2 n + 1 max { 1 , ∣ 1 − 2 b + 2 n + 1 b ∣ } ,

and

(23) γ 2 − 2 n − 1 2 n γ 1 2 ≤ ∣ b ∣ 2 n + 1 .

Equality holds in (21) and (23) if A n + 1 f ( z ) = z + b z [ p 1 ( z ) − 1 ] , and in (22) if A n + 1 f ( z ) = z + b z [ p 2 ( z ) − 1 ] .

Proof

By using the notations from the proof of Theorem 2.3, we obtain

(24) γ 1 = b c 1 2 , γ 2 = b 2 n + 2 ( c 2 − b c 1 2 ) + b 2 c 1 2 4 ,

and applying Theorem 2.2, we have

∣ γ 1 ∣ = ∣ b c 1 ∣ 2 ≤ ∣ b ∣ ,

∣ γ 2 ∣ = ∣ b ∣ 2 n + 2 ∣ c 2 − b c 1 2 + 2 n b c 1 2 ∣ = ∣ b ∣ 2 n + 2 c 2 − c 1 2 2 + c 1 2 2 ( 1 − 2 b + 2 n + 1 b ) ≤ ∣ b ∣ 2 n + 2 2 − ∣ c 1 ∣ 2 2 + ∣ c 1 ∣ 2 2 ∣ 1 − 2 b + 2 n + 1 b ∣ ≤ ∣ b ∣ 2 n + 1 max { 1 , ∣ 1 − 2 b + 2 n + 1 b ∣ } .

Also,

γ 2 − 2 n − 1 2 n γ 1 2 = ∣ b ∣ 2 n + 2 ∣ c 2 − b c 1 2 + 2 n b c 1 2 − ( 2 n − 1 ) b c 1 2 ∣ = ∣ b ∣ 2 n + 2 ∣ c 2 ∣ ≤ ∣ b ∣ 2 n + 1 .

It is easy to see that the equalities are attained if f satisfies the extremal conditions from Theorem 2.3.□

The inequalities (22) and (23) are particular cases of the following results:

Theorem 2.10

Let b be a nonzero complex number and f ∈ Λ n ( b ) such that log f ( z ) z = 2 ∑ k = 1 ∞ γ k z k . Then for μ ∈ C ,

(25) ∣ γ 2 − μ γ 1 2 ∣ ≤ ∣ b ∣ 2 n + 1 max { 1 , ∣ 1 − 2 b + 2 n + 1 b ( 1 − μ ) ∣ } .

For each value of μ , there exists a function in Λ n ( b ) such that equality in (25) holds.

Proof

Theorem 2.2 implies

∣ γ 2 − μ γ 1 2 ∣ = ∣ b ∣ 2 n + 2 ∣ c 2 − b c 1 2 + 2 n b c 1 2 − 2 n μ b c 1 2 ∣ = ∣ b ∣ 2 n + 2 c 2 − c 1 2 2 + c 1 2 2 [ 1 − 2 b + 2 n + 1 b ( 1 − μ ) ] ≤ ∣ b ∣ 2 n + 2 2 − ∣ c 1 ∣ 2 2 + ∣ c 1 ∣ 2 2 ∣ 1 − 2 b + 2 n + 1 b ( 1 − μ ) ∣ ≤ ∣ b ∣ 2 n + 1 max { 1 , ∣ 1 − 2 b + 2 n + 1 b ( 1 − μ ) ∣ } .

The first equality is attained for c 1 = 0 , c 2 = 2 , and the second equality is attained for c 1 = c 2 = 2 .□

Remark 2.11

Setting b = 1 ⁄ 2 in Theorems 2.9 and 2.10, we obtain the results from [9].

Next, we construct similar results for the class Λ n c ( b ) .

Theorem 2.12

Let b be a nonzero complex number and f ∈ Λ n c ( b ) such that log f ( z ) z = 2 ∑ k = 1 ∞ γ k z k . Then

(26) ∣ γ 1 ∣ ≤ ∣ b ∣ 2 ,

(27) ∣ γ 2 ∣ ≤ ∣ b ∣ 3 ⋅ 2 n + 1 max 1 , 1 − b + 3 2 2 n b ,

and

(28) γ 2 − 3 ⋅ 2 n − 2 3 ⋅ 2 n γ 1 2 ≤ ∣ b ∣ 3 ⋅ 2 n + 1 .

Equality holds in (26) and (28) if z ( A n f ) ″ ( z ) ( A n f ) ′ ( z ) = b [ p 1 ( z ) − 1 ] , and in (27) if z ( A n f ) ″ ( z ) ( A n f ) ′ ( z ) = b [ p 2 ( z ) − 1 ] .

Proof

By using the assumptions in the proof of Theorem 2.7 and applying (16) and (20), we obtain

(29) γ 1 = b c 1 4 , γ 2 = b 3 ⋅ 2 n + 2 c 2 − b c 1 2 2 + b 2 c 1 2 16 ,

and by in view of Theorem 2.2, we obtain

∣ γ 1 ∣ = ∣ b c 1 ∣ 4 ≤ ∣ b ∣ 2 ,

∣ γ 2 ∣ = ∣ b ∣ 3 ⋅ 2 n + 2 c 2 − b c 1 2 2 + 3 ⋅ 2 n b c 1 2 4 = ∣ b ∣ 3 ⋅ 2 n + 2 c 2 − c 1 2 2 + c 1 2 2 1 − b + 3 2 2 n b ≤ ∣ b ∣ 3 ⋅ 2 n + 2 2 − ∣ c 1 ∣ 2 2 + ∣ c 1 ∣ 2 2 1 − b + 3 2 2 n b ≤ ∣ b ∣ 3 ⋅ 2 n + 1 max 1 , 1 − b + 3 2 2 n b ,

and

γ 2 − 3 ⋅ 2 n − 2 3 ⋅ 2 n γ 1 2 = ∣ b ∣ 3 ⋅ 2 n + 2 c 2 − b c 1 2 2 + 3 ⋅ 2 n b c 1 2 4 − ( 3 ⋅ 2 n − 2 ) b c 1 2 4 = ∣ b ∣ 3 ⋅ 2 n + 2 ∣ c 2 ∣ ≤ ∣ b ∣ 3 ⋅ 2 n + 1 .

The equalities are attained if f satisfies the extremal conditions from Theorem 2.7.□

Next, we generalize the inequalities (27) and (28) to the following result:

Theorem 2.13

Let b be a nonzero complex number and f ∈ Λ n c ( b ) such that log f ( z ) z = 2 ∑ k = 1 ∞ γ k z k . Then for μ ∈ C ,

(30) ∣ γ 2 − μ γ 1 2 ∣ ≤ ∣ b ∣ 3 ⋅ 2 n + 1 max 1 , 1 − b + 3 2 2 n b ( 1 − μ ) .

For each value of μ , there exists a function in Λ n c ( b ) such that equality in (30) holds.

Proof

By applying Theorem 2.2 again, we obtain

∣ γ 2 − μ γ 1 2 ∣ = ∣ b ∣ 3 ⋅ 2 n + 2 c 2 − b c 1 2 2 + 3 ⋅ 2 n b c 1 2 4 − 3 ⋅ 2 n μ b c 1 2 4 = ∣ b ∣ 3 ⋅ 2 n + 2 c 2 − c 1 2 2 + c 1 2 2 1 − b + 3 2 2 n b ( 1 − μ ) ≤ ∣ b ∣ 3 ⋅ 2 n + 2 2 − ∣ c 1 ∣ 2 2 + ∣ c 1 ∣ 2 2 1 − b + 3 2 2 n b ( 1 − μ ) ≤ ∣ b ∣ 3 ⋅ 2 n + 1 max 1 , 1 − b + 3 2 2 n b ( 1 − μ ) .

The first equality is attained for c 1 = 0 , c 2 = 2 , and the second equality is attained for c 1 = c 2 = 2 .□

2.2 Logarithmic coefficients of the classes F n ( b ) and S n c ( b )

In [2], the authors show that if f ( z ) = z + a 2 z 2 + a 3 z 3 + … ∈ F n ( b ) , then there exists a function p ( z ) = 1 + c 1 z + c 2 z 2 + … ∈ P such that z ( D n f ) ′ ( z ) D n f ( z ) = 1 + b [ p ( z ) − 1 ] and, thus,

(31) a 2 = b c 1 n + 1 , a 3 = b ( n + 1 ) ( n + 2 ) ( c 2 − b c 1 2 ) .

Theorem 2.14

Let b be a nonzero complex number and f ∈ F n ( b ) such that log f ( z ) z = 2 ∑ k = 1 ∞ γ k z k . Then

(32) ∣ γ 1 ∣ ≤ ∣ b ∣ n + 1 ,

(33) ∣ γ 2 ∣ ≤ ∣ b ∣ ( n + 1 ) ( n + 2 ) max 1 , 1 + 2 b − b ( n + 2 ) n + 1 ,

and

(34) γ 2 − n n + 2 γ 1 2 ≤ ∣ b ∣ ( n + 1 ) ( n + 2 ) .

Equality holds in (32) and (34) if z ( D n f ) ′ ( z ) D n f ( z ) = 1 + b [ p 1 ( z ) − 1 ] , and in (33) if z ( D n f ) ′ ( z ) D n f ( z ) = 1 + b [ p 2 ( z ) − 1 ] .

Proof

In view of (31) and (20), we obtain

γ 1 = b c 1 2 ( n + 1 ) , γ 2 = b 2 ( n + 1 ) ( n + 2 ) ( c 2 + b c 1 2 ) − b 2 c 1 2 4 ( n + 1 ) 2 .

By applying Theorem 2.2, we obtain

∣ γ 1 ∣ = ∣ b c 1 ∣ 2 ( n + 1 ) ≤ ∣ b ∣ n + 1 ,

∣ γ 2 ∣ = ∣ b ∣ 2 ( n + 1 ) ( n + 2 ) c 2 − c 1 2 2 + c 1 2 2 1 + 2 b − b ( n + 2 ) n + 1 ≤ ∣ b ∣ ( n + 1 ) ( n + 2 ) max 1 , 1 + 2 b − b ( n + 2 ) n + 1 ,

and

γ 2 − n n + 2 γ 1 2 = ∣ b ∣ 2 ( n + 1 ) ( n + 2 ) ∣ c 2 ∣ ≤ ∣ b ∣ ( n + 1 ) ( n + 2 ) .

The equalities in (32) and (34) are attained for ∣ c 1 ∣ = ∣ c 2 ∣ = 2 . Therefore, we can choose f such that

z ( D n f ) ′ ( z ) D n f ( z ) = 1 + b [ p 1 ( z ) − 1 ] .

The equality in (32) is attained when the equality in (6) holds. Thus, we can choose f such that

□ z ( D n f ) ′ ( z ) D n f ( z ) = 1 + b [ p 2 ( z ) − 1 ] .

Theorem 2.15

Let b be a nonzero complex number and f ∈ F n ( b ) such that log f ( z ) z = 2 ∑ k = 1 ∞ γ k z k . Then for μ ∈ C ,

(35) ∣ γ 2 − μ γ 1 2 ∣ ≤ ∣ b ∣ ( n + 1 ) ( n + 2 ) max 1 , 1 + 2 b − b ( n + 2 ) n + 1 ( 1 + μ ) .

For each value of μ , there exists a function in F n ( b ) such that equality in (35) holds.

Proof

Theorem 2.2 implies

∣ γ 2 − μ γ 1 2 ∣ = ∣ b ∣ 2 ( n + 1 ) ( n + 2 ) c 2 − c 1 2 2 + c 1 2 2 1 + 2 b − b ( n + 2 ) n + 1 ( 1 + μ ) ≤ ∣ b ∣ ( n + 1 ) ( n + 2 ) max 1 , 1 + 2 b − b ( n + 2 ) n + 1 ( 1 + μ ) .

The first equality is attained for c 1 = 0 , c 2 = 2 , and the second equality is attained for c 1 = c 2 = 2 .□

We note that all results regarding the logarithmic coefficients of classes F 0 ( b ) and Λ 0 ( b ) coincide, as expected.

Next, let f ( z ) = z + a 2 z 2 + a 3 z 3 + … ∈ S n c ( b ) , then there exists p ( z ) = 1 + c 1 z + c 2 z 2 + … ∈ P such that

z ( D n f ) ″ ( z ) ( D n f ) ′ ( z ) = b [ p ( z ) − 1 ] .

If we denote D n f ( z ) = z + A 2 z 2 + A 3 z 3 + … , we obtain

A 2 = b c 1 2 , A 3 = b 6 ( c 2 + b c 1 2 ) .

The coefficients A 2 and A 3 can be written in terms of a 2 and a 3 as follows [2,8]:

A 2 = ( n + 1 ) a 2 , A 3 = ( n + 1 ) ( n + 2 ) 2 a 3 ,

and now, we obtain

(36) a 2 = b c 1 2 ( n + 1 ) , a 3 = b 3 ( n + 1 ) ( n + 2 ) ( c 2 + b c 1 2 ) .

Theorem 2.16

Let b be a nonzero complex number and f ∈ S n c ( b ) such that log f ( z ) z = 2 ∑ k = 1 ∞ γ k z k . Then

(37) ∣ γ 1 ∣ ≤ ∣ b ∣ 2 ( n + 1 ) ,

(38) ∣ γ 2 ∣ ≤ ∣ b ∣ 3 ( n + 1 ) ( n + 2 ) max 1 , 1 + 2 b − 3 b ( n + 2 ) 4 ( n + 1 ) ,

and

(39) γ 2 − 5 n + 2 3 ( n + 2 ) γ 1 2 ≤ ∣ b ∣ 3 ( n + 1 ) ( n + 2 ) .

Equality holds in (37) and (39) if z ( D n f ) ″ ( z ) ( D n f ) ′ ( z ) = b [ p 1 ( z ) − 1 ] , and in (38) if z ( D n f ) ″ ( z ) ( D n f ) ′ ( z ) = b [ p 2 ( z ) − 1 ] .

Proof

Taking into account (36) and (20), we obtain

γ 1 = b c 1 4 ( n + 1 ) , γ 2 = b 6 ( n + 1 ) ( n + 2 ) ( c 2 + b c 1 2 ) − b 2 c 1 2 16 ( n + 1 ) 2 .

So, Theorem 2.2 implies

∣ γ 1 ∣ = ∣ b c 1 ∣ 4 ( n + 1 ) ≤ ∣ b ∣ 2 ( n + 1 ) ,

∣ γ 2 ∣ = ∣ b ∣ 6 ( n + 1 ) ( n + 2 ) c 2 − c 1 2 2 + c 1 2 2 1 + 2 b − 3 b ( n + 2 ) 4 ( n + 1 ) ≤ ∣ b ∣ 3 ( n + 1 ) ( n + 2 ) max 1 , 1 + 2 b − 3 b ( n + 2 ) 4 ( n + 1 ) ,

and

γ 2 − 5 n + 2 3 ( n + 2 ) γ 1 2 = ∣ b ∣ 6 ( n + 1 ) ( n + 2 ) ∣ c 2 ∣ ≤ ∣ b ∣ 3 ( n + 1 ) ( n + 2 ) .

It is easy to see again that the equalities are attained in these estimates for the extremal functions from the statement of the theorem.□

Theorem 2.17

Let b be a nonzero complex number and f ∈ S n c ( b ) such that log f ( z ) z = 2 ∑ k = 1 ∞ γ k z k . Then for μ ∈ C ,

(40) ∣ γ 2 − μ γ 1 2 ∣ ≤ ∣ b ∣ 3 ( n + 1 ) ( n + 2 ) max 1 , 1 + 2 b − 3 b ( n + 2 ) 4 ( n + 1 ) ( 1 + μ ) .

For each value of μ , there exists a function in S n c ( b ) such that equality in (40) holds.

Proof

Also, Theorem 2.2 implies

∣ γ 2 − μ γ 1 2 ∣ = ∣ b ∣ 6 ( n + 1 ) ( n + 2 ) c 2 − c 1 2 2 + c 1 2 2 1 + 2 b − 3 b ( n + 2 ) 4 ( n + 1 ) ( 1 + μ ) ≤ ∣ b ∣ 3 ( n + 1 ) ( n + 2 ) max 1 , 1 + 2 b − 3 b ( n + 2 ) 4 ( n + 1 ) ( 1 + μ ) .

The first equality is attained for c 1 = 0 , c 2 = 2 , and the second equality is attained for c 1 = c 2 = 2 .□

As a final remark, we note that, as expected, all the estimates regarding the logarithmic coefficients of classes S 0 c ( b ) , F 1 ( b ) , and Λ 0 c ( b ) coincide.

3 Concluding remarks

In [9], the author introduces the class Λ n of normalized analytic functions that satisfy the following geometric condition:

Re A n + 1 f ( z ) z > 1 2 ,

which implies that these functions are univalent. In this article, we generalize this class to the class Λ n ( b ) of normalized univalent functions that satisfy the following condition:

Re 1 + 1 b A n + 1 f ( z ) z − 1 > 0 ( b ∈ C \ { 0 } ) .

On the other hand, we define the class Λ n c ( b ) of normalized univalent functions that satisfy the condition:

Re 1 + 1 b z ( A n f ) ″ ( z ) ( A n f ) ′ ( z ) > 0 ( b ∈ C \ { 0 } ) ,

thus generalizing the class of convex functions of complex order b , which coincides with the class Λ 0 c ( b ) .

In this article, we extend the results obtained by Aron [9] for the class Λ n to more general results, defining and studying the same properties for the classes Λ n ( b ) and Λ n c ( b ) . The technique used to construct these classes is similar to the one employed by Kanas and Darwish [2], so the idea is to obtain results of the same type as [2] for these newly defined classes. It is important to mention that the classes Λ n ( b ) and F n ( b ) are generalizations in different ways of the class S ⋆ ( α ) , in the sense that for b ∈ ( 0 , 1 ] , we have Λ 0 ( b ) = F 0 ( b ) = S ⋆ ( 1 − b ) . Also, the classes Λ n c ( b ) and S n c ( b ) are generalizations in different ways of the class S c ( α ) , in the sense that for b ∈ ( 0 , 1 ] , we have Λ 0 c ( b ) = S 0 c ( b ) = S c ( 1 − b ) .

The majority of results obtained in this article refer to sharp estimates of the coefficients of functions in the introduced classes (and the Fekete-Szegö problem) and sharp estimates of the logarithmic coefficients of these functions. Furthermore, the technique used to determine sharp estimates for the logarithmic coefficients of the classes Λ n ( b ) and Λ n c ( b ) allows us to establish sharp estimates for the logarithmic coefficients of the classes F n ( b ) and S n c ( b ) studied by Kanas and Darwish [2].

We conclude our study by concluding that there are numerous recent results concerning sharp estimates of the coefficients of functions in certain classes of univalent functions:

Srivastava et al. [15] investigated the Fekete-Szegö problem, bounds for Hankel determinant, and coefficient bounds for a class of analytic functions defined using the Hohlov operator.
Srivastava et al. [16] determined bounds for ∣ a 2 ∣ and ∣ a 3 ∣ and examined the Fekete-Szegö for a class of analytic functions that satisfy a subordination condition associated with the Gegenbauer polynomials.
Two other important functionals (which can be written in terms of coefficients), with an important geometrical role in fluid dynamics (more precisely, in the Hele-Shaw flow problem), are the integral means and the area of the subdisc image, investigated by Srivastava et al. [17], for the family of Janowski type ( j , k ) -symmetric starlike functions.
Another investigation on the Fekete-Szegö problem for a class of analytic functions associated with quasi-subordonation was developed in [18] by Srivastava et al.
It is observed that all the results obtained in this article can be particularized for the class Λ n . However, the author in [9] obtains different types of coefficient estimates, which cannot be generalized for the class Λ n ( b ) (Theorems 7, 8, and 11 in [9]).

Funding information: This work was supported by a grant of the Ministry of Research, Innovation and Digitization, CNCS/CCCDI - UEFISCDI, project number PN-IV-P8-8.1-PRE-HE-ORG-2023-0118, within PNCDI IV.
Author contributions: The author has accepted responsibility for the entire content of this manuscript and approved it for submission.
Conflict of interest: The author declares no conflicts of interest related to this research work.
Ethical approval: The author declares that there is no ethical problem in the production of this article.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References

[1] M. Fekete and G. Szegö, Eine Bemerkung über ungerade schlichte Funktionen, J. Lond. Math. Soc. 8 (1933), 85–89. 10.1112/jlms/s1-8.2.85Search in Google Scholar

[2] S. Kanas and H. E. Darwish, Fekete-Szegö problem for starlike and convex functions of complex order, Appl. Math. Lett. 23 (2010), 777–782. 10.1016/j.aml.2010.03.008Search in Google Scholar

[3] I. Graham and G. Kohr, Geometric Function Theory in One and Higher Dimensions, Marcel Dekker, New York, 2003. 10.1201/9780203911624Search in Google Scholar

[4] M. A. Nasr and M. K. Aouf, Starlike function of complex order, J. Natur. Sci. Math. 25 (1985), 1–12. Search in Google Scholar

[5] P. Wiatrowski, The coefficients of a certain family of holomorphic functions, Zeszyty Naukowe Uniwesytetu Lodzkiego, Seria: Nauki Matematyzno-Przyrodnicze, (1971), no. 2, 75–85.Search in Google Scholar

[6] M. A. Nasr and M. K. Aouf, On convex functions of complex order, Mansoura Sci. Bull. 9 (1982), 565–582. Search in Google Scholar

[7] V. Kumar, S. L. Shukla, and A. M. Chaudhary, On a class of certain analytic functions of complex order, Tamkang J. Math. 21 (1990), no. 2, 101–109. 10.5556/j.tkjm.21.1990.4643Search in Google Scholar

[8] S. Ruschewyh, New criteria for univalent functions, Proc. Amer. Math. Soc 49 (1975), 109–115. 10.1090/S0002-9939-1975-0367176-1Search in Google Scholar

[9] M. Aron, New classes of univalent functions using a new differential operator, Rend. Circ. Mat. Palermo (2) 73 (2024), 1731–1746, DOI: https://doi.org/10.1007/s12215-024-01007-5. 10.1007/s12215-024-01007-5Search in Google Scholar

[10] M. Nunokawa, H. M. Strivastava, N. Tuneski, and B. Jolevska-Tuneska, Some Marx-Strohhäcker type results for a class of multivalent functions, Miskolc Math. Notes 18 (2017), no. 1, 353–364. 10.18514/MMN.2017.1952Search in Google Scholar

[11] G. S. Sălăgean, Subclasses of univalent functions, Lecture Notes in Mathematics, Springer Verlag, Berlin, vol. 1013, 1983, pp. 362–372. 10.1007/BFb0066543Search in Google Scholar

[12] D. Girela, Logarithmic coefficients of univalent functions, Ann. Fenn. Math. 25 (2000), 337–350. Search in Google Scholar

[13] M. Obradovic, S. Ponnusamy, and K. J. Wirths, Logarithmic coefficients and a coeffcient conjecture for univalent functions, Monatsh. Math. 185 (2018), 489–501. 10.1007/s00605-017-1024-3Search in Google Scholar

[14] O. Roth, A sharp inequality for the logarithmic coeffcients of univalent functions, Proc. Amer. Math. Soc. 135 (2007), 2051–2054. 10.1090/S0002-9939-07-08660-1Search in Google Scholar

[15] H. M. Srivastava, T. G. Shaba, G. Murugusundaramoorthy, A. K. Wanas, and G. I. Oros, The Fekete-Szegö functional and the Hankel determinant for a certain class of analytic functions involving the Hohlov operator, AIMS Math. 8 (2022), no. 1, 340–360. 10.3934/math.2023016Search in Google Scholar

[16] H. M. Srivastava, M. Kamali, and A. Urdaletova, A study of the Fekete-Szegö functional and coefficient estimates for subclasses of analytic functions satisfying a certain subordination condition and associated with the Gegenbauer polynomials, AIMS Math. 7 (2022), no. 2, 2568–2584. 10.3934/math.2022144Search in Google Scholar

[17] H. M. Srivastava, A. Prajapati, and P. Gochhayat, Integral means and Yamashitaas conjecture associated with the Janowski type (j,k)-symmetric starlike functions, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 116 (2022), 165, DOI: https://doi.org/10.1007/s13398-022-01310-9. 10.1007/s13398-022-01310-9Search in Google Scholar

[18] 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

Received: 2023-08-21

Revised: 2024-08-13

Accepted: 2024-10-16

Published Online: 2025-02-17

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

Articles in the same Issue

https://doi.org/10.1515/dema-2024-0086

Keywords for this article

coefficient estimates; differential operator; Fekete-Szegö problem; starlike and convex functions of complex order; logarithmic coefficients

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