Home Coefficient functionals for Sakaguchi-type-Starlike functions subordinated to the three-leaf function
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Coefficient functionals for Sakaguchi-type-Starlike functions subordinated to the three-leaf function

  • Gangadharan Murugusundaramoorthy , Alina Alb Lupas , Alhanouf Alburaikan and Sheza M. El-Deeb EMAIL logo
Published/Copyright: June 13, 2025
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Demonstratio Mathematica
From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

A challenging part of studying geometric function theory is figuring out the sharp boundaries for coefficient-related problems that crop up in the Taylor-Maclaurin series of univalent functions. The objective of this article is to define the families of Sakaguchi-type starlike functions with respect to symmetric points based on q -operator and to investigate the precise boundaries for a range of issues, including the first three initial coefficient estimates, Fekete-Szegö type and the Zalcman inequalities by subordinating to the function of the three leaves. Additionally, we discussed initial coefficients and Fekete-Szegö type inequalities for functions of the form ℱ − 1 and z ℱ ( z ) and 1 2 log ℱ ( z ) z linked with the function of the three leaves.

Keywords: analytic functions; subordination; three-leaf function; convolution; coefficient inequalities; Fekete-Szegő functional; Krushkal inequality
MSC 2010: 30C45; 30C50

1 Introduction, definitions, and preliminaries

Let A represent the family of functions of the form

(1.1) ℱ ( z ) = z + ∑ t ≥ 2 c t z t ( z ∈ U d ) .

which are regular in open unit disc U d = { z ∈ C : ∣ z ∣ < 1 } . Also let S be the subfamily A that consists of holomorphic functions in U d of the type (1.1). If the function h ∈ A is given by

(1.2) h ( z ) ≔ z + ∑ t ≥ 2 d t z t , z ∈ U d ,

then ℱ and h Hadamard (or convolution) product is given by

( ℱ ∗ h ) ( z ) ≔ z + ∑ t ≥ 2 c t d t z t , z ∈ U d .

Let P stand for the family of regular, positive-real part functions p ( z ) in U d and that have the form

(1.3) p ( z ) = 1 + ∑ t = 1 ∞ b t z t ( z ∈ U d ) .

We say that g 1 is subordinated to g 2 for two functions g 1 , g 2 ∈ A , and that g 1 ≺ g 2 symbolically if there exists an analytic function w with the properties ∣ w ( z ) ∣ ≤ ∣ z ∣ and w ( 0 ) = 0 such that for z ∈ U d , g 1 ( z ) = g 2 ( w ( z ) ) . Additionally, in the event where g 2 ∈ S , the condition becomes

g 1 ≺ g 2 ⇔ g 1 ( 0 ) = g 2 ( 0 ) and g 1 ( U d ) ⊂ g 2 ( U d ) .

For 0 < q < 1 , the q -derivative operator [1,2,3,4] for ℱ ∗ h is defined by

D q ( ℱ ∗ h ) ( z ) ≔ D q z + ∑ t ≥ 2 c t d t z t = ( ℱ ∗ h ) ( z ) − ( ℱ ∗ h ) ( q z ) z ( 1 − q ) = 1 + ∑ t ≥ 2 [ t ] q c t d t z t − 1 , z ∈ U d ,

where

(1.4) [ t ] q ≔ 1 − q t 1 − q = 1 + ∑ j = 1 t − 1 q j , [ 0 ] q ≔ 0 .

According to El-Deeb et al. [3] (also see [4]), for α > − 1 and 0 < q < 1 , we let ℋ h α , q : A → A as follows:

ℋ h α , q ℱ ( z ) ∗ ℐ q , α + 1 ( z ) = z D q ( ℱ ∗ h ) ( z ) , z ∈ U d ,

where

ℐ q , α + 1 ( z ) ≔ z + ∑ t ≥ 2 [ α + 1 ] q , t − 1 [ t − 1 ] q ! z t , z ∈ U d .

A quick calculation reveals that

(1.5) ℋ h α , q ℱ ( z ) ≔ z + ∑ t ≥ 2 [ t ] q ! [ α + 1 ] q , t − 1 c t d t z t ( α > − 1 , 0 < q < 1 , z ∈ U d ) .

We define a new operator by utilizing ℋ h α , q , as follows:

D h , δ α , q , 0 ℱ ( z ) = ℋ h α , q ℱ ( z ) D h , δ α , q , 1 ℱ ( z ) = δ z 3 ( ℋ h α , q ℱ ( z ) ) ‴ + ( 1 + 2 δ ) z 2 ( ℋ h α , q ℱ ( z ) ) ″ + z ( ℋ h α , q ℱ ( z ) ) ′ D h , δ α , q , n ℱ ( z ) = δ z 3 ( D h , δ α , q , n − 1 ℱ ( z ) ) ‴ + ( 1 + 2 δ ) z 2 ( D h , δ α , q , n − 1 ℱ ( z ) ) ″ + z ( D h , δ α , q , n − 1 ℱ ( z ) ) ′ = z + ∑ t ≥ 2 t 2 n ( δ ( t − 1 ) + 1 ) n [ t ] q ! [ α + 1 ] q , t − 1 c t d t z t = z + ∑ t ≥ 2 ρ t c t z t ( n ∈ N 0 , α > − 1 , 0 < q < 1 , z ∈ U d ) ,

where

(1.6) ρ t = t 2 n ( δ ( t − 1 ) + 1 ) n [ t ] q ! [ α + 1 ] q , t − 1 d t .

We may simply confirm that the following relationships apply for all ℱ ∈ A based on the relation defined in (1.5).

(1.7) (i) [ α + 1 ] q D h , δ α , q , n ℱ ( z ) = [ α ] q D h , δ α + 1 , q , n ℱ ( z ) + q μ z D q ( D h , δ α + 1 , q , n ℱ ( z ) ) , z ∈ U d ;

(1.8) (ii) ℛ h , δ α , 0 ℱ ( z ) ≔ lim q → 1 − D h , δ α , q , 0 ℱ ( z ) = z + ∑ t ≥ 2 t ! ( α + 1 ) t − 1 c t d t z t , z ∈ U d .

Remark 1.1

Fixing particular coefficients for d t , we illustrate the special cases for the operator D h , δ α , q , n as follows:

  1. Fixing d t = 1 and n = 0 , we obtain D h , δ α , q , 0 ≡ ℬ q μ defined by Srivastava et al. [5] as follows:

    (1.9) ℬ q α ℱ ( ς ) ≔ z + ∑ t ≥ 2 [ t ] q ! [ α + 1 ] q , t − 1 c t z t , ( α > − 1 , 0 < q < 1 , z ∈ U d ) ;

  2. Taking d t = ( − 1 ) t − 1 Γ ( ρ + 1 ) 4 t − 1 ( t − 1 ) ! Γ ( t + ρ ) , ρ > 0 and n = 0 , we obtain the operator as follows:

    (1.10) N ρ , q α ℱ ( ς ) ≔ z + ∑ t ≥ 2 ( − 1 ) t − 1 Γ ( ρ + 1 ) 4 t − 1 ( t − 1 ) ! Γ ( t + ρ ) ⋅ [ t ] q ! [ α + 1 ] q , t − 1 c t z t = z + ∑ t ≥ 2 ϕ t c t z t ,

    where

    (1.11) ϕ t ≔ ( − 1 ) t − 1 Γ ( ρ + 1 ) [ t ] q ! 4 t − 1 ( t − 1 ) ! Γ ( t + ρ ) [ α + 1 ] q , t − 1 ; ( ρ > 0 , α > − 1 , 0 < q < 1 , z ∈ U d )

    defined in [6,7].

  3. Assuming d t = m + 1 m + t μ , μ > 0 , m ≥ 0 and n = 0 , we have a linear operator

    (1.12) ℳ t , q μ , α ℱ ( ς ) ≔ ς + ∑ t ≥ 2 m + 1 m + t μ ⋅ [ t ] q ! [ α + 1 ] q , t − 1 c t z t , z ∈ U d ;

    defined and studied extensively in [8,9].

  4. For d t = σ t − 1 ( t − 1 ) ! e − σ , σ > 0 and n = 0 , we obtain the q -analogue of Poisson operator as follows:

    (1.13) ℐ q α , σ ℱ ( ς ) ≔ ς + ∑ t ≥ 2 σ t − 1 ( t − 1 ) ! e − σ ⋅ [ t ] q ! [ α + 1 ] q , t − 1 c t z t , z ∈ U d .

    defined by El-Deeb et al. [3].

Pommerenke [10,11] introduced the Hankel determinant H q , t ( ℱ ) , where the parameters q , t ∈ N = { 1 , 2 , 3 , … } for function ℱ ∈ S of the form (1.1) as follows:

(1.14) H q , t ( ℱ ) = c t c t + 1 … c t + q − 1 c t + 1 c t + 2 … c t + q ⋮ ⋮ … ⋮ c t + q − 1 c t + q … c t + 2 q − 2 .

By setting values for q and t , the Hankel determinants for various orders can be obtained. For instance, if q = 2 , t = 1

∣ H 2,1 ( ℱ ) ∣ = ∣ c 3 − c 2 2 ∣ , where c 1 = 1 .

Note that H 2,1 ( ℱ ) = c 3 − c 2 2 , is the classical Fekete-Szegö function. Furthermore, when q = 2 and t = 2 , the second Hankel determinant is expressed as follows:

∣ H 2,2 ( ℱ ) ∣ = ∣ c 2 c 4 − c 3 2 ∣ .

Numerous authors looked into the best possible value of the upper bound for ∣ H 2,1 ( ℱ ) ∣ and ∣ H 2,2 ( ℱ ) ∣ for different subclasses of class A (see [12–17] for details). Recall that there are many uses for Hankel determinant, including discrete inverse scattering, discretization of certain integral equations from mathematical physics, and linear filtering theory [18]. Several scholars have recently examined the subclasses of starlike functions in the recent past by defining as follows:

(1.15) S ∗ ( ϒ ) = ℱ ∈ A : z ℱ ′ ( z ) ℱ ( z ) ≺ ϒ ( z ) ,

where ϒ ( z ) = ( 1 + z ) ⁄ ( 1 − z ) . Lately by varying ϒ in (1.15), then some subclasses of S whose image domains have some interesting geometrical configurations have been extensively studied for initial coefficient bounds, and Hankel inequalities in the literature we listed few as follows:

  1. Cho et al. [19] fixed ϒ ( z ) = 1 + sin z and Mendiratta et al. [20] considered ϒ ( z ) = e z and discussed the class S on certain geometric properties and radii problems.

  2. Sharma et al. [21] for ϒ ( z ) = 1 + 4 3 z + 2 3 z 2 a pedal shaped domain and Wani and Swaminathan [22] fixed ϒ ( z ) = 1 + z − 1 3 z 3 , which maps U d onto the interior of the 2-cusped kidney-shaped region and examined applicability for certain subclasses S in the general coefficient problem.

  3. Raina and Sokól [23] developed ϒ ( z ) = z + 1 + z 2 , which maps U d to crescent shaped region and also assuming ϒ ( z ) = 1 + z , which is bounded by lemniscate of Bernoulli in right half plan they found the initial Taylor coefficients for subclasses S and discussed various geometrical inequalities.

Recently, by selecting specific ϒ in (1.15) those related to Bell numbers, the shell-like curve associated with Fibonacci numbers, conic domain functions, and rational functions, among others numerous subclasses many subclasses of starlike functions were introduced and discussed in [24–26].

Gandhi [27] recently defined the class of starlike functions linked to three Leaf functions, i.e.,

S 3 ℒ ∗ = ℱ ∈ A : z ℱ ′ ( z ) ℱ ( z ) ≺ 1 + 4 5 z + 1 5 z 4 ; z ∈ U d .

Sakaguchi [28] popularized the S s ∗ family of star functions by publishing the family of star functions in 1959 S s ∗ of starlike function. Later Das and Singh [29] developed a family of convex functions K s with symmetry elements in 1977. In two articles, they provided the following description:

S s ∗ = ℱ ∈ S : Re 2 z ℱ ′ ( z ) ℱ ( z ) − ℱ ( − z ) > 0 ; z ∈ U d , K s = ℱ ∈ S : Re 2 ( z ℱ ′ ( z ) ) ′ ( ℱ ( z ) − ℱ ( − z ) ) ′ > 0 ; z ∈ U d .

Sakaguchi further stated in the same study that the families of convex and odd starlike functions are contained in the class S s ∗ , which is a subfamily of the set C of close-to-convex functions performs in relation to the source. After that, a number of mathematicians presented a plethora of novel subfamilies of univalent functions with regard to symmetric points and examined issues of the coefficient kind (a few of which are shown in [30–34]. Influenced by the books and references mentioned above referenced and therein. Now we present a new class SST 3 ℒ in this article as defined follows:

(1.16) SST h , δ , 3 ℒ α , q , n = ℱ ∈ A : 2 z ( D h , δ α , q , n ℱ ( z ) ) ′ D h , δ α , q , n ℱ ( z ) − D h , δ α , q , n ℱ ( − z ) ≺ 1 + 4 5 z + 1 5 z 4 ; z ∈ U d

and examine for coefficient bounds, Fekete-Szegö inequality and Zalcman conjecture for ℱ ∈ SST h , δ , 3 ℒ α , q , n .

2 Initial bounds for ℱ ∈ SST h , δ , 3 ℒ α , q , n

We recall the following lemmas, which are required for the proofs of our main findings.

Lemma 2.1

[35] Let p ∈ P and be given as in (1.3). Then, for x and σ with ∣ x ∣ ≤ 1 , ∣ σ ∣ ≤ 1 , such that

(2.1) 2 b 2 = b 1 2 + x ( 4 − b 1 2 ) ,

(2.2) 4 b 3 = b 1 3 + 2 ( 4 − b 1 2 ) b 1 x − b 1 ( 4 − b 1 2 ) x 2 + 2 ( 4 − b 1 2 ) ( 1 − ∣ x ∣ 2 ) σ .

Lemma 2.2

If p ∈ P and be assumed as in (1.3), then

(2.3) ∣ b t + k − μ b t b k ∣ ≤ 2 , f o r 0 ≤ μ ≤ 1

(2.4) ∣ b t ∣ ≤ 2 for t ≥ 1 ,

(2.5) ∣ b 2 − ζ b 1 2 ∣ ≤ 2 max { 1 , ∣ 2 ζ − 1 ∣ } , ζ ∈ C

(2.6) ∣ J b 1 3 − K b 1 b 2 + ℒ b 3 ∣ ≤ 2 ∣ J ∣ + 2 ∣ K − 2 J ∣ + 2 ∣ J − K + ℒ ∣ .

We note that the inequalities (2.3), (2.4), and (2.6) in the mentioned earlier can be found in [11] and (2.5) are from [13], and also (2.6) was evaluated in [36,37].

Lemma 2.3

[38] Let m, t, l, and r with 0 < m < 1 , 0 < r < 1 and

8 r ( 1 − r ) [ ( mt − 2 l ) 2 + ( m ( r + m ) − t ) 2 ] + m ( 1 − m ) ( t − 2 rm ) 2 ≤ 4 m 2 ( 1 − m ) 2 r ( 1 − r ) .

If p ∈ P and be given as in (1.3), then

l b 1 4 + r b 2 2 + 2 m b 1 b 3 − 3 2 t b 1 2 b 2 − b 4 ≤ 2 .

Theorem 2.1

Let ℱ ∈ SST 3 ℒ and be given by (1.1). Then

(2.7) ∣ c 2 ∣ ≤ 2 5 ρ 2 ,

(2.8) ∣ c 3 ∣ ≤ 2 5 ρ 3 ,

(2.9) ∣ c 4 ∣ ≤ 1 5 ρ 4 ,

and

(2.10) ∣ c 5 ∣ ≤ 1 5 ρ 5 ,

where ρ t ( t ∈ { 2,3,4,5 } ) are defined by (1.6).

Proof

Let ℱ ∈ SST h , δ , 3 ℒ α , q , n . Then, (1.16) can be expressed in Schwarz function w as follows:

(2.11) 2 z ( D h , δ α , q , n ℱ ( z ) ) ′ D h , δ α , q , n ℱ ( z ) − D h , δ α , q , n ℱ ( − z ) = 1 + 4 5 w ( z ) + 1 5 ( w ( z ) ) 4 ( z ∈ U d ) .

Also, if p ∈ P , then it can be stated using the Schwarz function w as follows:

p ( z ) = 1 + b 1 z + b 2 z 2 + b 3 z 3 … = 1 + w ( z ) 1 − w ( z ) ,

equivalently,

(2.12) w ( z ) = p ( z ) − 1 p ( z ) + 1 = b 1 z + b 2 z 2 + b 3 z 3 + … 2 + b 1 z + b 2 z 2 + b 3 z 3 + … .

From (2.11), we easily obtain

(2.13) 2 z ( D h , δ α , q , n ℱ ( z ) ) ′ D h , δ α , q , n ℱ ( z ) − D h , δ α , q , n ℱ ( − z ) = 1 + 2 ρ 2 c 2 z + 2 ρ 3 c 3 z 2 + ( 4 ρ 4 c 4 − 2 ρ 2 ρ 3 c 2 c 3 ) z 3 + ( 4 ρ 5 c 5 − 2 c 3 2 ρ 3 2 ) z 4 + … .

By using the series expansion (2.12), we have

(2.14) 1 + 4 5 w ( z ) + 1 5 ( w ( z ) ) 4 = 1 + 2 5 b 1 z + 2 5 b 2 − 1 5 b 1 2 z 2 + 1 10 b 1 3 − 2 5 b 2 b 1 + 2 5 b 3 z 3 + − 3 80 b 1 4 + 3 10 b 1 2 b 2 − 2 5 b 3 b 1 − 1 5 b 2 2 + 2 5 b 4 z 4 + … .

By comparing (2.13) and (2.14), we obtain

(2.15) c 2 = 1 5 ρ 2 b 1 ,

(2.16) c 3 = 1 5 ρ 3 b 2 − 1 10 ρ 3 b 1 2 ,

(2.17) c 4 = 1 20 ρ 4 3 10 b 1 3 − 8 5 b 1 b 2 + 2 b 3 ,

and

(2.18) c 5 = − 1 10 ρ 5 7 160 b 1 4 + 3 10 b 2 2 + b 3 b 1 − 3 20 b 1 2 b 2 − b 4 .

For c 2 , putting (2.4) in (2.15), we have

∣ c 2 ∣ ≤ 2 5 ρ 2 .

For c 3 , simplifying (2.16), we obtain

c 3 = 1 5 ρ 3 b 2 − 1 2 b 1 2 ,

and applying (2.3), we have

∣ c 3 ∣ ≤ 2 5 ρ 3 .

For c 4 , using (2.17), we obtain

∣ c 4 ∣ = 1 20 ρ 4 3 10 b 1 3 − 8 5 b 1 b 2 + 2 b 3 .

By applying (2.6), we obtain

∣ c 4 ∣ ≤ 1 20 ρ 4 2 3 10 + 2 8 5 − 2 3 10 + 2 3 10 − 8 5 + 2 = 1 5 ρ 4 .

For c 5 , application of Lemma 2.3 to (2.18), we obtain

□ ∣ c 5 ∣ ≤ 1 5 ρ 5 .

Next, we find Fekete-Szegö inequality for ℱ ∈ ST h , δ , 3 ℒ α , q , n .

Theorem 2.2

If ℱ ∈ SST h , δ , 3 ℒ α , q , n and is given (1.1), then

(2.19) ∣ c 3 − ζ c 2 2 ∣ ≤ 2 5 ρ 3 max 1 , 2 ∣ ζ ∣ ρ 3 5 ρ 2 2 , ( ζ ∈ C )

where ρ t ( t ∈ { 2 , 3 } ) are defined by (1.6).

Proof

By using (2.15) and (2.16), we can write

∣ c 3 − ζ c 2 2 ∣ = 1 5 ρ 3 b 2 − 1 10 ρ 3 b 1 2 − ζ b 1 2 25 ρ 2 2 .

By rearranging, we have

(2.20) ∣ c 3 − ζ c 2 2 ∣ = 1 5 ρ 3 b 2 − 2 ζ ρ 3 + 5 ρ 2 2 10 ρ 2 2 b 1 2 .

By applying (2.5), we obtain

□ ∣ c 3 − ζ c 2 2 ∣ ≤ 2 5 ρ 3 max 1 , 2 ∣ ζ ∣ ρ 3 5 ρ 2 2 .

Fixing ζ = 1 , then we have the following result:

Corollary 2.1

If ℱ ∈ SST h , δ , 3 ℒ α , q , n , and of the form (1.1), then

(2.21) ∣ c 3 − c 2 2 ∣ ≤ 2 5 ρ 3 max 1 , 2 ρ 3 5 ρ 2 2 .

where ρ t ( t ∈ { 2 , 3 } ) are defined by (1.6).

3 Coefficient inequalities for the function ℱ − 1

Theorem 3.1

If ℱ ∈ SST h , δ , 3 ℒ α , q , n and of the form (1.1). The analytic continuation to U d of the inverse function of ℱ with ∣ w ∣ < r 0 , ( r 0 ≥ 1 4 ) the radius of the Koebe domain, and ℱ − 1 ( w ) = w + ∑ n ≥ 2 ℏ n w n , then

(3.1) ∣ ℏ 2 ∣ ≤ 2 5 ρ 2 ,

(3.2) ∣ ℏ 3 ∣ ≤ 2 5 ρ 3 max 1 , 4 ρ 3 5 ρ 2 2

and then for ℵ ∈ C ,

(3.3) ∣ ℏ 3 − ℵ ℏ 2 2 ∣ ≤ 2 5 ρ 3 max 1 , 4 ρ 3 + 2 ℵ ρ 3 5 ρ 2 2 ,

where ρ t ( t ∈ { 2 , 3 } ) are defined by (1.6).

Proof

If

(3.4) ℱ − 1 ( w ) = w + ∑ t ≥ 2 ℏ t w t ,

it can be seen that

(3.5) ℱ − 1 ( ℱ ( z ) ) = ℱ ( ℱ − 1 ( z ) ) = z .

From (3.5), we obtain

(3.6) ℱ − 1 z + ∑ t ≥ 2 c t z t = z .

Thus, (3.5) and (3.6) yield

(3.7) z + ( c 2 + ℏ 2 ) z 2 + ( c 3 + 2 c 2 ℏ 2 + ℏ 3 ) z 3 + … = z .

Thus, it is evident from equating the corresponding coefficients of z that

(3.8) ℏ 2 = − c 2 ,

(3.9) ℏ 3 = 2 c 2 2 − c 3 = − ( c 3 − 2 c 2 2 ) .

The estimate for ∣ ℏ 2 ∣ = ∣ c 2 ∣ follows immediately from (2.7). Letting ζ = 2 in (2.19), we obtain the estimate ∣ ℏ 3 ∣ . To find the Fekete-Szegő inequality for the inverse function, consider

∣ ℏ 3 − ℵ ℏ 2 2 ∣ = ∣ 2 c 2 2 − c 3 − ℵ c 2 2 ∣ = ∣ c 3 − ( ℵ − 2 ) c 2 2 ∣ .

Thus, by fixing ζ = ( ℵ − 2 ) in the (2.19), we obtain the desired result.□

4 Coefficient associated with z ℱ ( z )

In this section, we determine the coefficient bounds and Fekete-Szegö problem associated with the function H ( z ) given by

(4.1) H ( z ) = z ℱ ( z ) = 1 + ∑ t = 1 ∞ u t z t ( z ∈ U d ) .

where ℱ ∈ SST h , δ , 3 ℒ α , q , n .

Theorem 4.1

If ℱ of the form (1.1) belongs to SST h , δ , 3 ℒ α , q , n and H ( z ) is given by (4.1). Then

(4.2) ∣ u 1 ∣ ≤ 2 5 ρ 2 ,

(4.3) ∣ u 2 ∣ ≤ 2 5 ρ 3 max 1 , 2 ρ 3 5 ρ 2 2

(4.4) ∣ u 2 − ℘ u 1 2 ∣ ≤ 2 5 ρ 3 max 1 , 2 ( 1 − ℘ ) ρ 3 5 ρ 2 2 , f o r ℘ ∈ C ,

where ρ t ( t ∈ { 2 , 3 } ) are defined by (1.6).

Proof

By routine calculation, we obtain

(4.5) H ( z ) = z ℱ ( z ) = 1 − c 2 z + ( c 2 2 − c 3 ) z 2 + ( c 2 3 + 2 c 2 a 3 − c 4 ) z 3 + … .

Comparing the coefficients of z , z 2 , and z 3 on both sides of (4.1) and (4.5), we obtain

(4.6) u 1 = − c 2 ,

and

(4.7) u 2 = c 2 2 − c 3 = − ( c 3 − c 2 2 ) .

The estimate for ∣ u 1 ∣ = ∣ c 2 ∣ follows immediately from (2.7). The bound for u 2 followed by Corollary 2.1.

For ℘ ∈ C , we obtain

∣ u 2 − ℘ u 1 2 ∣ = ∣ − c 3 + c 2 2 − ℘ c 2 2 ∣ = ∣ c 3 − ( 1 − ℘ ) c 2 2 ∣ .

Thus, by taking ζ = ( 1 − ℘ ) in (2.19), we obtain the desired result.□

5 Initial logarithmic coefficient bounds and Fekete-Szegö problem for the class SST h , δ , 3 ℒ α , q , n

The logarithmic coefficients of a given function ℱ , denoted by γ n ≔ γ n ( ℱ ) , are defined as follows:

(5.1) 1 2 log ℱ ( z ) z = ∑ n = 1 ∞ γ n z n .

The theory of Schlicht functions is significantly impacted by these coefficients in various estimations. In 1985, de-Branges [39] determined that

∑ k = 1 n k ( n − k + 1 ) ∣ γ n ∣ 2 ≤ ∑ k = 1 n n − k + 1 k , for n ≥ 1 ,

and for the particular function ℱ ( z ) = z ⁄ ( 1 − e i θ z ) with θ ∈ R , equality is attained. This inequality is the source of the most general version of the famous Bieberbach-Robertson-Milin conjectures concerning Taylor coefficients of ℱ ∈ S . To learn more about the explanation of de-Brange’s claim, one refer to [40–42]. In terms of conformal mappings, Brennan’s conjecture was addressed by transforming the logarithmic coefficients by Kayumov [43] in 2005. Numerous studies [44–46] that have made substantial progress on logarithmic coefficients. Following section includes a study of logarithmic coefficients. From the aforementioned definition, the logarithmic coefficients for ℱ ∈ S are as given as follows:

(5.2) γ 1 = 1 2 c 2 ,

(5.3) γ 2 = 1 2 c 3 − 1 2 c 2 2 ,

(5.4) γ 3 = 1 2 c 4 − c 2 c 3 + 1 3 c 2 3 .

Theorem 5.1

Let ℱ ∈ SST h , δ , 3 ℒ α , q , n and using (1.1), then

∣ γ 1 ∣ ≤ 1 5 ρ 2 , ∣ γ 2 ∣ ≤ 1 5 ρ 3 max 1 , ρ 3 5 ρ 2 2 , ∣ γ 3 ∣ ≤ 1 10 ρ 4 ,

where ρ t ( t ∈ { 2 , 3 , 4 } ) are defined by (1.6).

Proof

Applying (2.4), (2.17) in (5.2), (5.3), and (5.4), we obtain

(5.5) γ 1 = 1 10 ρ 2 b 1 ,

(5.6) γ 2 = 1 2 1 5 ρ 3 b 2 − 1 10 ρ 3 b 1 2 − b 1 2 50 ρ 2 2 = 1 10 ρ 3 b 2 − 5 ρ 2 2 + ρ 3 10 ρ 2 2 b 1 2 .

The bounds of γ 1 and γ 2 are clear.

Now to find γ 3 using (5.4), we obtain

γ 3 = 1 40 ρ 4 2 b 3 − 8 5 1 + ρ 4 2 ρ 2 ρ 3 b 1 b 2 + 3 10 + 2 ρ 4 5 ρ 2 ρ 3 + 4 ρ 4 25 ρ 2 2 b 1 3 ,

then

∣ γ 3 ∣ = 1 40 ρ 4 2 b 3 − 8 5 1 + ρ 4 2 ρ 2 ρ 3 b 1 b 2 + 3 10 + 2 ρ 4 5 ρ 2 ρ 3 + 4 ρ 4 25 ρ 2 2 b 1 3 .

By using (2.6) and triangle inequality, we obtain

□ ∣ γ 3 ∣ ≤ 1 10 ρ 4 .

Theorem 5.2

If ℱ ∈ SST h , δ , 3 ℒ α , q , n and using (1.1), then for ℵ ∈ C ,

∣ γ 2 − ℵ γ 1 2 ∣ ≤ 1 5 ρ 3 max 1 , ( 1 + ℵ ) ρ 3 5 ρ 2 2 ,

where ρ t ( t ∈ { 2 , 3 } ) are defined by (1.6).

Proof

From (5.5) and (5.6), we have

∣ γ 2 − ℵ γ 1 2 ∣ = 1 10 ρ 3 b 2 − 5 ρ 2 2 + ρ 3 10 ρ 2 2 b 1 2 − ℵ 100 ρ 2 2 b 1 2 , = 1 10 ρ 3 b 2 − 5 ρ 2 2 + ρ 3 10 ρ 2 2 b 1 2 − ℵ ρ 3 10 ρ 2 2 b 1 2 , = 1 10 ρ 3 b 2 − 5 ρ 2 2 + ( 1 + ℵ ) ρ 3 10 ρ 2 2 b 1 2 .

By using (2.5) and triangle inequality, we have

□ ∣ γ 2 − ℵ γ 1 2 ∣ ≤ 1 5 ρ 3 max 1 , ( 1 + ℵ ) ρ 3 5 ρ 2 2 .

Fixing ℵ = 1 , we obtain the following consequence.

Corollary 5.1

Let ℱ of the form (1.1) belongs to ST h , δ , 3 ℒ α , q , n . Then

∣ γ 2 − γ 1 2 ∣ ≤ 1 5 ρ 3 max 1 , 2 ρ 3 5 ρ 2 2 .

6 Zalcman functional

In 1960, Lawrence Zalcman proposed one of the central hypotheses of geometric function theory, which states that the coefficients of class S hold the inequality,

(6.1) ∣ c t 2 − c 2 t − 1 ∣ ≤ ( t − 1 ) 2 .

The Koebe function k ( z ) = z ( 1 − z ) 2 is widely recognized, and the aforementioned expression represents the equality of rotations. When t = 2 , the widely known Fekete-Szego inequality holds true. Many researchers have examined Zalcman functional [47–49].

Theorem 6.1

Let ℱ ∈ A belong to SST h , δ , 3 ℒ α , q , n . Then

(6.2) ∣ c 3 2 − c 5 ∣ ≤ 1 5 ρ 5 ,

where ρ 5 is defined by (1.6).

Proof

By using (2.16) and (2.18), we obtain Zalcman function,

∣ c 3 2 − c 5 ∣ = 1 10 ρ 5 ρ 5 10 ρ 3 2 + 7 160 b 1 4 − ρ 5 ρ 3 2 + 3 20 b 1 2 b 2 + b 3 b 1 − 2 ρ 5 5 ρ 3 2 + 3 10 b 2 2 − b 4 .

By using Lemma 2.3, we can obtain the desired result (6.2).□

7 Krushkal inequality for the class SST h , δ , 3 ℒ α , q , n

For a choice of n = 4 , p = 1 , and for n = 5 , p = 1 . Krushkal introduced and proved the inequality

(7.1) ∣ c n p − c 2 p ( n − 1 ) ∣ ≤ 2 p ( n − 1 ) − n p

for the whole class of univalent functions in [50].

Now, in the below theorem for ℱ ∈ SST h , δ , 3 ℒ α , q , n , we will give direct proof of (7.1).

Theorem 7.1

Let ℱ ∈ SST h , δ , 3 ℒ α , q , n . Then

∣ c 4 − c 2 3 ∣ ≤ 1 5 ρ 4 ,

where ρ 4 is defined by (1.6).

Proof

From equations (2.15) and (2.17), we obtain

∣ c 4 − c 2 3 ∣ = 1 20 ρ 4 3 10 − 4 ρ 4 25 ρ 2 2 b 1 3 − 8 5 b 2 b 1 + 2 b 3 .

By using (2.6) to the aforementioned condition, we obtain the desired result.□

Theorem 7.2

Let ℱ of the form (1.1) belongs to SST h , δ , 3 ℒ α , q , n . Then

∣ c 5 − c 2 4 ∣ ≤ 1 5 ρ 5 ,

where ρ 5 is defined by (1.6).

Proof

From equations (2.15) and (2.17), we obtain

∣ c 5 − c 2 4 ∣ = − 1 10 ρ 5 7 160 + 10 ρ 5 625 ρ 2 4 b 1 4 + 3 10 b 2 2 + b 3 b 1 − 3 20 b 1 2 b 2 − b 4

Using Lemma 2.3, we can obtain the necessary result for the last expression.□

8 Conclusion

Obtaining estimates of the coefficients appearing within the analytical univalent function is one of the principal issues inside the discipline of feature principle. The the primary concept at the back of locating the limits of coefficients in certain families of univalent functions is to express their coefficients by Carath’eodory functions with the intention to examine the coefficient inequalities, which can be recognized for the Carath’eodory functions. In our cutting-edge research, we used a novel technique to find out the boundaries of numerous problems related to coefficients, Feketo-Szegö inequalities and Zalcman inequalities. We also chose to use an important observation that was made in a recently published evaluate-cum-expository article by Srivastava ([51], p. 340). This observation noted that the effects for the new or previously mentioned q analogs could easily (and possibly trivially) be translated into corresponding effects for the so-called (p; q)-analogues (with 0 < ∣ q ∣ ≤ 1 ) using some seemingly parametric and argumentative versions of the extra parameter p is redundant. In future, the results related for subclasses of Sakaguchi-type-starlike functions (starlike functions with respect to symmetric point or conjugate symmetric point) defined by subordinating with various domains, namely, limaçon domain, convex functions in one direction, the cosine function, the nephroid domain, Four leaf functions, can be discussed.

, ,

Acknowledgement

The Researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC- 2025).

  1. Funding information: The Researchers would like to thank the Deanship of Graduate Studies and Scientific Research at Qassim University for financial support (QU-APC- 2025).

  2. Author contributions: The author confirmed the sole responsibility for the conception of the study, presented results, and prepared manuscript. All authors equally contributed and approved the final version of the manuscript.

  3. Conflict of interest: The authors declares that there are no conflicts of interest regarding the publication of this paper.

  4. Ethical approval: The conducted research is not related to either human or animals use.

  5. Data availability statement: No data were used to support this study.

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Received: 2024-04-28
Revised: 2024-10-10
Accepted: 2025-03-20
Published Online: 2025-06-13

© 2025 the author(s), published by De Gruyter

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

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https://doi.org/10.1515/dema-2025-0123
Keywords for this article
analytic functions; subordination; three-leaf function; convolution; coefficient inequalities; Fekete-Szegő functional; Krushkal inequality
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