Borel series approach to subclasses of univalent analytic functions

1 Introduction

Discrete probability laws are commonly employed to model random phenomena whose outcomes form a countable set. Such distributions are fundamental in mathematical analysis, statistics, and computational sciences, where they assist in understanding probabilistic structures of discrete events. Among these, the Borel distribution, introduced by mile Borel, has proved useful in applications related to reliability modeling and lifetime analysis. Its ability to capture rare-event behavior makes it suitable for studying discrete stochastic mechanisms arising in applied contexts.

The behavior of the Taylor series coefficients is often relied upon in the study of univalent analytic functions. In particular, unique challenges in establishing growth, distortion, and geometric properties are presented by functions with negative coefficients. A natural and systematic framework for defining subclasses of such functions is offered by the Borel series, with explicit control over coefficients provided through its probabilistic structure. Sharper analytic estimates are facilitated by this approach, and the introduction of novel subclasses that extend classical results in geometric function theory is also enabled. By employing the Borel distribution series, probabilistic methods are bridged with analytic function theory, and new avenues for both theoretical exploration and applications are opened.

The subclasses defined via the Borel series have a wide range of applications. Their structured coefficients facilitate the derivation of coefficient bounds, growth and distortion estimates, and Fekete–Szegő type inequalities, which are fundamental in geometric function theory. The probabilistic nature of the Borel series also enables the study of random analytic functions, providing a connection between stochastic processes and analytic function theory. Moreover, these subclasses contribute to complex dynamics and iteration theory, particularly in conformal mapping and the analysis of mapping distortions. The geometric properties considered here may also provide a theoretical framework for conformal mapping and related problems in applied analysis. Any direct applications to specific fluid-mechanical or engineering models, however, require further investigation.

Let A denote the family of functions analytic in the open unit disk 𝕌 = {z ∈ ℂ:|z| < 1} and represented by the expansion

F(z)=z+∑n=2∞cnzn.(1)

Let S be the subclass of A consisting of functions that are univalent in 𝕌. Furthermore, following Robertson [1], denote by S⋆(δ) and C(δ) the subclasses of S consisting of functions that are starlike and convex of order δ, respectively, where 0 ≤ δ < 1. The cases S⋆(0)=S⋆ and C(0)=C correspond to the classical starlike and convex families. Functions of the form

F(z)=z-∑n=2∞cnzn, (cn>0),(2)

defined in 𝕌 are referred to as analytic functions with negative coefficients. Let T denote the subclass of S consisting of functions represented by Equation 2. This family was first introduced by Silverman [2] and later extended to the subclasses ST⋆(δ) and CT(δ), which consist of functions starlike and convex of order δ, respectively.

Let M(λ,η,θ) be the subclass of A consisting of functions F(z) satisfying

where 0 ≤ λ ≤ 1, 0 < η ≤ 1, and 0 ≤ θ < 1 for all z ∈ 𝕌. This class was studied earlier by Darus [3].

Example 1.1. To illustrate the class introduced by Darus [3], consider

F(z)=z-az2, a>0.

Then

F′(z)=1-2az,

and hence

zF′(z)-F(z)=-az2.

Moreover,

λzF′(z)+(1-θ)F(z)=λz(1-2az)+(1-θ)(z-az2)=(1+λ-θ)z-a(1+2λ-θ)z2.

Therefore,

For |z| = r < 1,

Consequently,

*Thus, a sufficient condition for F to belong to the Darus class is*

a<η(1+λ-θ)1+η(1+2λ-θ).

For example, let

λ=12, θ=12, η=12,

and choose

a=18.

Then

F(z)=z-18z2

and, for |z| < 1,

Hence

F(z)=z-18z2

belongs to the class introduced by Darus [3].

Several subclasses of analytic and univalent functions associated with Borel distribution series have also been investigated in recent studies [4–10]. These studies demonstrate the usefulness of distribution-generated convolution operators in deriving coefficient estimates and geometric properties of analytic functions. The present construction is related to these Borel- and distribution-based subclasses, but differs through the simultaneous incorporation of the parameters ς, τ, ϑ, and k, which provide additional flexibility in the associated geometric condition.

To address this gap, we introduce the class 𝕃kμ(ς,τ,ϑ), defined through the Borel distribution convolution operator together with the parameters ς, τ, ϑ, and k.

A discrete random variable X is said to follow a Borel distribution if

P(X=r)=(μr)r-1e-μrr!, r=1,2,…,

where μ > 0 is a real parameter.

Motivated by this distribution, Wanas and Khuttar [10] introduced the generating function

𝔅(μ,z)=z+∑n=2∞[μ(n-1)]n-2e-μ(n-1)(n-1)!zn,(z∈𝕌,0<μ≤1).(4)

An application of the ratio test confirms that the above series converges for all z ∈ ℂ.

Example 1.2. To illustrate the Borel distribution series introduced by Wanas and Khuttar [10], consider the parameter value

μ=1.

Then

ψn=(n-1)n-2e-(n-1)(n-1)!,

and therefore

𝔅(1,z)=z+e-1z2+e-22z3+2e-36z4+⋯ =z+0.3679z2+0.0677z3+0.0166z4+⋯ .

Thus, the Borel distribution series determines the coefficients of the associated analytic generating function explicitly. This illustrates how the parameter μ enters the operator used in the present investigation.

We further define the associated function

G(μ,z)=2z-𝔅(μ,z)=z-∑n=2∞[μ(n-1)]n-2e-μ(n-1)(n-1)!zn,

(z∈𝕌).(5)

Using this function, we introduce a linear convolution operator 𝔅μ:A→A defined by

𝔅μF(z)=G(μ,z)*F(z)=z-∑n=2∞ψncnzn, (cn≥0),(6)

where

ψn=[μ(n-1)]n-2e-μ(n-1)(n-1)!.(7)

Linearity of the operator 𝔅_μ. The operator 𝔅_μ is linear. Indeed, let

F(z)=z+∑n=2∞anzn, H(z)=z+∑n=2∞bnzn,

and let α, β ∈ ℂ. Then

αF(z)+βH(z)=(α+β)z+∑n=2∞(αan+βbn)zn.

By the definition of 𝔅_μ,

𝔅μ(αF+βH)(z)=(α+β)z-∑n=2∞ψn(αan+βbn)zn=α(z-∑n=2∞ψnanzn)+β(z-∑n=2∞ψnbnzn)=α𝔅μF(z)+β𝔅μH(z).

Hence 𝔅_μ is a linear operator on A.

Motivated by earlier studies [see [5–9, 11, 12, 15, 16]], we now introduce a new subclass of analytic functions associated with the convolution operator 𝔅_μ. This subclass is denoted by 𝕃kμ(ς,τ,ϑ) and is defined as follows.

Definition 1.1. *Let 0 ≤ ς ≤ 1, 0 ≤ τ < 1, 0 ≤ ϑ < 1, k ≥ 0, and 0 < μ ≤ 1. Let F∈T be of the form*

F(z)=z-∑n=2∞cnzn, cn≥0,

and let

𝔅μF(z)=z-∑n=2∞ψncnzn,

where

ψn=[μ(n-1)]n-2e-μ(n-1)(n-1)!.

*We say that F belongs to the class*

𝕃kμ(ς,τ,ϑ)

if

ℜ{1+z(𝔅μF(z))′-𝔅μF(z)ςz(𝔅μF(z))′+(1-ϑ)𝔅μF(z)}

>k|z(𝔅μF(z))′-𝔅μF(z)ςz(𝔅μF(z))′+(1-ϑ)𝔅μF(z)|+τ

*for all z ∈ 𝕌, where*

𝕌={z∈ℂ:|z|<1}.

Research gap, novelty, and role of the parameters. Several subclasses of analytic and univalent functions associated with Borel distribution series have been investigated in recent studies. These works demonstrate that distribution-generated series and their associated convolution operators provide a useful framework for deriving coefficient estimates and studying geometric properties of analytic functions. However, there remains scope for more flexible parameterized subclasses in which the Borel distribution operator is combined with additional parameters governing the underlying geometric condition.

To address this gap, we introduce the class 𝕃kμ(ς,τ,ϑ), defined through the Borel distribution convolution operator together with the parameters ς, τ, ϑ, and k. The proposed formulation provides a unified parameterized framework for analytic functions with negative coefficients. In contrast with previously studied distribution-based subclasses, the present construction emphasizes the combined effect of the Borel convolution structure and the parameterized geometric condition, thereby allowing the influence of the individual parameters on the coefficient and geometric properties to be investigated systematically.

The parameter μ determines the coefficients of the Borel distribution series and consequently controls the contribution of the Borel convolution operator. The parameter ς modifies the differential expression appearing in the defining condition, while ϑ affects its normalization. The parameter τ determines the real-part threshold in the defining inequality, whereas k controls the strength of the associated conic-type restriction. Thus, these parameters provide a flexible mechanism for generating different subclasses within the proposed Borel-series framework.

The significance of the proposed construction is demonstrated by the range of results obtained for the class, including the coefficient characterization, growth and distortion estimates, extreme-point description, radii of starlikeness and convexity, and partial-sum properties. In addition, a Fekete–Szegő inequality is established as a nontrivial coefficient application beyond the direct consequences of the basic coefficient characterization. Hence, the present study extends the Borel-series approach by combining the distribution-based convolution structure with a flexible parameterized geometric condition and by developing several associated coefficient and geometric results.

The aim of the present work is to explore various geometric properties of functions belonging to the class 𝕃kμ(ς,τ,ϑ). In particular, we establish coefficient bounds for functions in this class and use these results to Fekete–Szegő inequality and derive growth and distortion theorems. Furthermore, we identify the extreme points and determine the radii of starlikeness and convexity associated with 𝕃kμ(ς,τ,ϑ). We begin with the investigation of coefficient estimates.

2 Coefficient estimates

Theorem 2.1. *Let 0 ≤ ς ≤ 1, 0 ≤ τ < 1, 0 ≤ ϑ < 1, k ≥ 0, and 0 < μ ≤ 1. A function*

F(z)=z-∑n=2∞cnzn, cn≥0,

belongs to the class 𝕃kμ(ς,τ,ϑ) if and only if

∑n=2∞T^n(ς,τ,ϑ,k,μ)cn≤(1-τ)(1+ς-ϑ),(8)

where

T^n(ς,τ,ϑ,k,μ)=[(k+1)(n-1)+(1-τ)(1+ςn-ϑ)]ψn,(9)

and

ψn=[μ(n-1)]n-2e-μ(n-1)(n-1)!.

Proof: Put

G(z)=𝔅μF(z)=z-∑n=2∞ψncnzn.

Then

G′(z)=1-∑n=2∞nψncnzn-1,

and hence

zG′(z)-G(z)=-∑n=2∞(n-1)ψncnzn.

Moreover,

ςzG′(z)+(1-ϑ)G(z)=ςz-∑n=2∞ςnψncnzn +(1-ϑ)z-∑n=2∞(1-ϑ)ψncnzn=(1+ς-ϑ)z-∑n=2∞(ςn+1-ϑ)ψncnzn.

Therefore

q(z):=zG′(z)-G(z)ςzG′(z)+(1-ϑ)G(z)

=-∑n=2∞(n-1)ψncnzn-11+ς-ϑ-∑n=2∞(ςn+1-ϑ)ψncnzn-1.(10)

Assume first that

F∈𝕃kμ(ς,τ,ϑ).

For 0 < r < 1, define

S(r)=∑n=2∞(n-1)ψncnrn-1,

and

B(r)=∑n=2∞(ςn+1-ϑ)ψncnrn-1.

Since c_*n* ≥ 0 and ψ_*n* > 0,

q(r)=-S(r)1+ς-ϑ-B(r)

and hence

The defining condition gives

1-S(r)1+ς-ϑ-B(r)>kS(r)1+ς-ϑ-B(r)+τ.

Therefore

1-τ>(k+1)S(r)1+ς-ϑ-B(r),

and consequently

(k+1)S(r)<(1-τ)[1+ς-ϑ-B(r)].

Thus

(k+1)S(r)+(1-τ)B(r)<(1-τ)(1+ς-ϑ).

Substituting the definitions of S(r) and B(r), we obtain

∑n=2∞[(k+1)(n-1)+(1-τ)(1+ςn-ϑ)]ψncnrn-1 <(1-τ)(1+ς-ϑ).

Letting r → 1⁻, we obtain

∑n=2∞T^ncn≤(1-τ)(1+ς-ϑ),

where

T^n=[(k+1)(n-1)+(1-τ)(1+ςn-ϑ)]ψn.

Hence the coefficient condition is necessary.

Conversely, assume that

∑n=2∞T^ncn≤(1-τ)(1+ς-ϑ).(11)

Let |z| = r < 1. From Equation 10 and the triangle inequality,

On the other hand, by Equation 11,

(k+1)∑n=2∞(n-1)ψncnrn-1 +(1-τ)∑n=2∞(ςn+1-ϑ)ψncnrn-1≤∑n=2∞T^ncn≤(1-τ)(1+ς-ϑ).

Hence

(k+1)∑n=2∞(n-1)ψncnrn-1≤(1-τ)

[1+ς-ϑ-∑n=2∞(ςn+1-ϑ)ψncnrn-1].

Using Equation 12, it follows that

(k+1)|q(z)|≤1-τ.

Therefore

ℜ{1+q(z)}-k|q(z)|-τ≥1-|q(z)|-k|q(z)|-τ=1-τ-(k+1)|q(z)|≥0.

For F(z) = z, we have q(z) ≡ 0 and, since 0 ≤ τ < 1,

ℜ{1+q(z)}=1>τ.

For every nontrivial function F, the inequality above is strict for |z| < 1. Hence

ℜ{1+q(z)}>k|q(z)|+τ,

and consequently

F∈𝕃kμ(ς,τ,ϑ).

This completes the proof. □

Theorem 2.2. *Let 0 ≤ ς ≤ 1, 0 ≤ τ < 1, 0 ≤ ϑ < 1, k ≥ 0, and 0 < μ ≤ 1. If*

F(z)=z-∑n=2∞cnzn

belongs to the class 𝕃kμ(ς,τ,ϑ), then

cn≤(1-τ)(1+ς-ϑ)T^n(ς,τ,ϑ,k,μ), n=2,3,…,(13)

where

T^n(ς,τ,ϑ,k,μ)=[(k+1)(n-1)+(1-τ)(1+ςn-ϑ)]ψn,(14)

and

ψn=[μ(n-1)]n-2e-μ(n-1)(n-1)!.

Moreover, the estimate is sharp.

Proof: Since

F∈𝕃kμ(ς,τ,ϑ),

Theorem 2.1 gives

∑j=2∞T^j(ς,τ,ϑ,k,μ)cj≤(1-τ)(1+ς-ϑ).(15)

Since

cj≥0 and T^j(ς,τ,ϑ,k,μ)>0,

each term in the sum is nonnegative. Hence, for any fixed n ≥ 2,

T^n(ς,τ,ϑ,k,μ)cn≤(1-τ)(1+ς-ϑ).

Therefore,

cn≤(1-τ)(1+ς-ϑ)T^n(ς,τ,ϑ,k,μ), n=2,3,….

This proves the required coefficient estimate.

To show sharpness, consider the function

Fn(z)=z-(1-τ)(1+ς-ϑ)T^n(ς,τ,ϑ,k,μ)zn, n≥2.(16)

For this function,

cn=(1-τ)(1+ς-ϑ)T^n(ς,τ,ϑ,k,μ)

and

cj=0, j≠n.

Consequently,

∑j=2∞T^jcj=T^ncn=(1-τ)(1+ς-ϑ).

Thus equality is attained in Equation 15, and hence in the coefficient estimate. Therefore the estimate is sharp. □

Example 2.1. To illustrate the class 𝕃kμ(ς,τ,ϑ), consider

ς=0.5, τ=0.2, ϑ=0.3, k=1, μ=1,

and the function

F(z)=z-2e7z2.(17)

For μ = 1, we have

ψ2=e-1,

and hence

𝔅μF(z)=z-27z2.

Consequently,

(𝔅μF(z))′=1-47z,

so that

z(𝔅μF(z))′-𝔅μF(z)=-27z2.

Furthermore,

ςz(𝔅μF(z))′+(1-ϑ)𝔅μF(z)=0.5z(1-47z)+0.7(z-27z2)=1.2z-1735z2.

Therefore,

q(z):=z(𝔅μF(z))′-𝔅μF(z)ςz(𝔅μF(z))′+(1-ϑ)𝔅μF(z)=-10z42-17z.

For |z*| = r < 1*,

Hence

ℜ{1+q(z)}-k|q(z)|-τ≥1-(k+1)|q(z)|-τ>1-2(25)-0.2=0.

Thus

ℜ{1+q(z)}>k|q(z)|+τ, z∈𝕌.

Therefore

F(z)=z-2e7z2∈𝕃11(0.5,0.2,0.3).

Moreover,

T^2=3.36e-1,

and

T^22e7=0.96=(1-τ)(1+ς-ϑ),

showing that equality is attained in the coefficient estimate of Theorem 2.2.

Theorem 2.3 (Fekete–Szegö Inequality). Let

F(z)=z-∑n=2∞cnzn

belong to the class 𝕃kμ(ς,τ,ϑ), where

0≤ς≤1, 0≤τ<1, 0≤ϑ<1, k≥0, 0<μ≤1.

Put

C=(1-τ)(1+ς-ϑ),

and define

A=T^2(ς,τ,ϑ,k,μ), B=T^3(ς,τ,ϑ,k,μ).

Then, for every real number λ,

The estimate is sharp.

Proof: By Theorem 2.1,

∑n=2∞T^ncn≤C, C=(1-τ)(1+ς-ϑ).

Since c_*n* ≥ 0, we have in particular

Ac2+Bc3≤C,(19)

where

A=T^2(ς,τ,ϑ,k,μ)>0

and

B=T^3(ς,τ,ϑ,k,μ)>0.

From Equation 19,

0≤c2≤CA,

and

0≤c3≤C-Ac2B.(20)

We consider first the case λ ≥ 0.

Since c₃ ≥ 0,

c3-λc22≤c3≤CB.(21)

On the other hand,

λc22-c3≤λc22≤λC2A2.(22)

Therefore,

Now suppose that λ < 0, and write

λ=-|λ|.

Then

c3-λc22=c3+|λ|c22≥0.

Using Equation 20,

c3+|λ|c22≤C-Ac2B+|λ|c22

=:Φ(c2), 0≤c2≤CA.(24)

Since

Φ(c2)=CB-ABc2+|λ|c22

is a convex quadratic function, its maximum on the closed interval

[0,CA]

is attained at one of the endpoints. Therefore,

Φ(c2)≤max{Φ(0),Φ(CA)}.

But

Φ(0)=CB,

while

Φ(CA)=|λ|C2A2.

Hence

Combining Equations 23 and 25, we obtain

To prove sharpness, consider the two functions

F3(z)=z-CBz3

and

F2(z)=z-CAz2.

For F₃,

c2=0, c3=CB,

and hence

For F₂,

c2=CA, c3=0,

and therefore

Thus the larger of the two quantities is attained, and the estimate is sharp. □

3 Growth and distortion theorems

In this section, we establish growth and distortion results for functions belonging to the class 𝕃kμ(ς,τ,ϑ). As a consequence, a covering property for this class is also obtained.

Theorem 3.1 (Growth Theorem). Let

F(z)=z-∑n=2∞cnzn

belong to the class 𝕃kμ(ς,τ,ϑ), where

0≤ς≤1, 0≤τ<1, 0≤ϑ<1, k≥0, 0<μ≤1.

Let

C=(1-τ)(1+ς-ϑ).

Then, for 0 < |z*| = r < 1*,

r-CM0(r)≤|F(z)|≤r+CM0(r),(26)

where

M0(r)=supn≥2rnT^n(ς,τ,ϑ,k,μ).(27)

Proof: Since

F(z)=z-∑n=2∞cnzn,

we have, for |z| = r < 1,

and

By Theorem 2.1,

∑n=2∞T^ncn≤C.(29)

Therefore,

∑n=2∞cnrn=∑n=2∞T^ncnrnT^n≤(supn≥2rnT^n)∑n=2∞T^ncn≤Csupn≥2rnT^n=CM0(r).(30)

Substituting Equations 30 into 28, we obtain

r-CM0(r)≤|F(z)|≤r+CM0(r).

This completes the proof.

Example 3.1. To illustrate the growth estimate in Theorem 3.1, consider the parameter values

ς=0.5, τ=0.2, ϑ=0.3, k=1, μ=1,

and let

r=0.4.

Then

C=(1-τ)(1+ς-ϑ)=(0.8)(1+0.5-0.3)=0.96.

*For n = 2*,

ψ2=e-1,

and hence

T^2=[(k+1)(2-1)+(1-τ)(1+2ς-ϑ)]ψ2=[2+(0.8)(1+2(0.5)-0.3)]e-1=3.36e-1≈1.2361.

*For r = 0.4, we have*

M0(0.4)=supn≥2(0.4)nT^n.

*For the present parameter values, direct evaluation of the terms shows that the maximum is attained at n = 2. Therefore*,

M0(0.4)=(0.4)2T^2=0.161.2361≈0.12944.

Consequently,

CM0(0.4)=0.96(0.12944)≈0.12426.

By Theorem 3.1,

0.4-0.12426≤|F(z)|≤0.4+0.12426, |z|=0.4.

Thus,

0.27574≤|F(z)|≤0.52426, |z|=0.4.

This numerical example illustrates the growth estimate obtained for the proposed class.

Theorem 3.2 (Distortion Theorem). Let

F(z)=z-∑n=2∞cnzn

belong to the class 𝕃kμ(ς,τ,ϑ), where

0≤ς≤1, 0≤τ<1, 0≤ϑ<1, k≥0, 0<μ≤1.

Let

C=(1-τ)(1+ς-ϑ).

Then, for 0 < |z*| = r < 1*,

1-CM1(r)≤|F′(z)|≤1+CM1(r),(31)

where

M1(r)=supn≥2nrn-1T^n(ς,τ,ϑ,k,μ).(32)

Here

T^n(ς,τ,ϑ,k,μ)=[(k+1)(n-1)+(1-τ)(1+ςn-ϑ)]ψn.

Proof: Since

F′(z)=1-∑n=2∞ncnzn-1,

we have, for |z| = r < 1,

and

By Theorem 2.1,

∑n=2∞T^ncn≤C.(34)

Therefore,

∑n=2∞ncnrn-1=∑n=2∞T^ncnnrn-1T^n≤(supn≥2nrn-1T^n)∑n=2∞T^ncn≤Csupn≥2nrn-1T^n=CM1(r).(35)

Substituting Equation 35 into Equation 33, we obtain

1-CM1(r)≤|F′(z)|≤1+CM1(r).

This completes the proof. □

Example 3.2. To illustrate the distortion estimate in Theorem 3.2, consider the parameter values

ς=0.5, τ=0.2, ϑ=0.3, k=1, μ=1,

and let

r=0.4.

Then

C=(1-τ)(1+ς-ϑ)=(0.8)(1+0.5-0.3)=0.96.

*For n = 2, we have*

ψ2=[1(2-1)]0e-11!=e-1,

and therefore

T^2=[(k+1)(2-1)+(1-τ)(1+2ς-ϑ)]ψ2=[2+(0.8)(1+2(0.5)-0.3)]e-1=3.36e-1≈1.2361.

*For r = 0.4, we compute*

M1(0.4)=supn≥2n(0.4)n-1T^n.

*For the present parameter values, direct evaluation of the terms shows that the maximum is attained at n = 2. Hence*

M1(0.4)=2(0.4)T^2=0.81.2361≈0.64721.

Consequently,

CM1(0.4)=0.96(0.64721)≈0.62132.

By Theorem 3.2, we therefore obtain

1-0.62132≤|F′(z)|≤1+0.62132, |z|=0.4.

Thus,

0.37868≤|F′(z)|≤1.62132, |z|=0.4.

This numerical example illustrates the distortion estimate obtained for the proposed class.

4 Extreme points

In this section, we describe the extreme points of the closed convex hull associated with the class 𝕃kμ(ς,τ,ϑ) and provide a characterization in terms of convex combinations of suitably chosen extremal functions.

Theorem 4.1 (Extreme Points). Let

C=(1-τ)(1+ς-ϑ).

Define

F1(z)=z

*and, for n = 2, 3, …*,

Fn(z)=z-CT^n(ς,τ,ϑ,k,μ)zn,(36)

where

T^n(ς,τ,ϑ,k,μ)=[(k+1)(n-1)+(1-τ)(1+ςn-ϑ)]ψn.

Then a function

F(z)=z-∑n=2∞cnzn

belongs to 𝕃kμ(ς,τ,ϑ) if and only if it can be represented in the form

F(z)=∑n=1∞λnFn(z),(37)

where

λn≥0, ∑n=1∞λn=1.

Consequently, the extreme points of the closed convex hull of 𝕃kμ(ς,τ,ϑ) are precisely the functions

F1(z)=z

and

Fn(z)=z-(1-τ)(1+ς-ϑ)T^n(ς,τ,ϑ,k,μ)zn, n=2,3,….(38)

Proof: Suppose first that F has the representation

F(z)=∑n=1∞λnFn(z),

where

λn≥0, ∑n=1∞λn=1.

Since F₁(z) = z, we have

F(z)=λ1z+∑n=2∞λn(z-CT^nzn)=(∑n=1∞λn)z-∑n=2∞CλnT^nzn=z-∑n=2∞CλnT^nzn.

Hence

cn=CλnT^n, n≥2.(39)

Therefore,

∑n=2∞T^ncn=C∑n=2∞λn=C(1-λ1)≤C.

By Theorem 2.1,

F∈𝕃kμ(ς,τ,ϑ).

Conversely, suppose that

F(z)=z-∑n=2∞cnzn∈𝕃kμ(ς,τ,ϑ).

By Theorem 2.1,

∑n=2∞T^ncn≤C.

Define

λn=T^ncnC, n=2,3,…,(40)

and set

λ1=1-∑n=2∞λn.(41)

Since

∑n=2∞λn=1C∑n=2∞T^ncn≤1,

we have

λ1≥0.

Moreover,

λn≥0 (n≥2)

and

∑n=1∞λn=1.

Using Equation 40,

CλnT^n=cn,

and therefore

∑n=1∞λnFn(z)=λ1z+∑n=2∞λn(z-CT^nzn)=z-∑n=2∞cnzn=F(z).

Thus Equation 37 holds.

Finally, each F_*n* corresponds to the case in which

λn=1

and all other λ_*j* = 0. Hence these functions are the extreme points of the closed convex hull of the class. □

Corollary 4.1. The extreme points of the closed convex hull of the class 𝕃kμ(ς,τ,ϑ) are precisely the functions

F1(z)=z

and

Fn(z)=z-(1-τ)(1+ς-ϑ)T^n(ς,τ,ϑ,k,μ)zn, n=2,3,…,

where

T^n(ς,τ,ϑ,k,μ)=[(k+1)(n-1)+(1-τ)(1+ςn-ϑ)]ψn,

and

ψn=[μ(n-1)]n-2e-μ(n-1)(n-1)!.

We investigate the radii of starlikeness and convexity associated with the class 𝕃kμ(ς,τ,ϑ).

5 Radius of starlikeness and convexity

The radii of starlikeness and convexity for functions belonging to the class 𝕃kμ(ς,τ,ϑ) are determined in the following results.

Theorem 5.1 (Radius of Starlikeness). Let

F(z)=z-∑n=2∞cnzn

belong to the class 𝕃kμ(ς,τ,ϑ), where

0≤ς≤1, 0≤τ<1, 0≤ϑ<1, k≥0, 0<μ≤1.

Let 0 ≤ δ < 1, and put

C=(1-τ)(1+ς-ϑ).

*Then F is starlike of order δ in the disk*

where

Rδ=sup{0<r<1:Csupn≥2(n-δ)rn-1T^n(ς,τ,ϑ,k,μ)≤1-δ},(42)

and

T^n(ς,τ,ϑ,k,μ)=[(k+1)(n-1)+(1-τ)(1+ςn-ϑ)]ψn,

with

ψn=[μ(n-1)]n-2e-μ(n-1)(n-1)!.

Proof: For |z| = r < 1, we have

zF′(z)F(z)=1-∑n=2∞ncnzn-11-∑n=2∞cnzn-1.

Hence

zF′(z)F(z)-δ=(1-δ)-∑n=2∞(n-δ)cnzn-11-∑n=2∞cnzn-1.(43)

A sufficient coefficient condition for F to be starlike of order δ in |z| < r is

∑n=2∞(n-δ)cnrn-1≤1-δ.(44)

By Theorem 2.1,

∑n=2∞T^ncn≤C.(45)

Therefore,

∑n=2∞(n-δ)cnrn-1=∑n=2∞(T^ncn)(n-δ)rn-1T^n≤(supn≥2(n-δ)rn-1T^n)∑n=2∞T^ncn

≤Csupn≥2(n-δ)rn-1T^n.(46)

Thus, whenever

Csupn≥2(n-δ)rn-1T^n≤1-δ,

condition Equation 48 is satisfied. Consequently,

ℜ{zF′(z)F(z)}>δ, |z|<r.

Hence F is starlike of order δ in |z| < r.

Taking the supremum over all admissible r ∈ (0, 1) gives the radius R_δ stated in Equation 42. □

Theorem 5.2 (Radius of Convexity). Let

F(z)=z-∑n=2∞cnzn

belong to the class 𝕃kμ(ς,τ,ϑ), where

0≤ς≤1, 0≤τ<1, 0≤ϑ<1, k≥0, 0<μ≤1.

Let 0 ≤ ξ < 1, and put

C=(1-τ)(1+ς-ϑ).

*Then F is convex of order ξ in the disk*

where

Rξc=sup{0<r<1:Csupn≥2n(n-ξ)rn-1T^n(ς,τ,ϑ,k,μ)≤1-ξ},(47)

and

T^n(ς,τ,ϑ,k,μ)=[(k+1)(n-1)+(1-τ)(1+ςn-ϑ)]ψn,

with

ψn=[μ(n-1)]n-2e-μ(n-1)(n-1)!.

Proof: A sufficient coefficient condition for F to be convex of order ξ in |z| < r is

∑n=2∞n(n-ξ)cnrn-1≤1-ξ.(48)

By Theorem 2.1,

∑n=2∞T^ncn≤C, C=(1-τ)(1+ς-ϑ).(49)

Hence

∑n=2∞n(n-ξ)cnrn-1=∑n=2∞(T^ncn)n(n-ξ)rn-1T^n≤Csupn≥2n(n-ξ)rn-1T^n.

Therefore, if

Csupn≥2n(n-ξ)rn-1T^n≤1-ξ,

then

∑n=2∞n(n-ξ)cnrn-1≤1-ξ,

and consequently

ℜ{1+zF″(z)F′(z)}>ξ, |z|<r.

Thus F is convex of order ξ in |z| < r, and taking the supremum over all such r gives the stated radius. □

6 Partial sums

Motivated by the pioneering investigations of Silverman [13] and Silvia [14] on partial sums of analytic functions, we now study the behavior of partial sums associated with functions in the class 𝕃kμ(ς,τ,ϑ).

Let F be a function of the form

F(z)=z-∑n=2∞cnzn, (cn≥0),(50)

and denote by F_*m* its m-th partial sum, defined by

Fm(z)=z-∑n=2mcnzn, m∈ℕ.(51)

We derive sharp lower bounds for the real parts of the ratios ℜ{Fm(z)F(z)} and ℜ{F(z)Fm(z)} when F∈𝕃kμ(ς,τ,ϑ).

Theorem 6.1 (Partial Sums). Let

F(z)=z-∑n=2∞cnzn∈𝕃kμ(ς,τ,ϑ),

*and let its m-th partial sum be*

Fm(z)=z-∑n=2mcnzn.

Put

C=(1-τ)(1+ς-ϑ).

*For 0 < r < 1, define*

Em(r)=Csupn≥m+1rnT^n(ς,τ,ϑ,k,μ)

and

E(r)=Csupn≥2rnT^n(ς,τ,ϑ,k,μ).

If

E(r)<r

and

ρm(r)=Em(r)r-E(r)<1,(52)

then

ℜ{Fm(z)F(z)}≥1-ρm(r)1+ρm(r), |z|=r.(53)

Proof: Since

F(z)-Fm(z)=-∑n=m+1∞cnzn,

we have, for |z| = r < 1,

By Theorem 2.1,

∑n=2∞T^ncn≤C.

Hence

=Em(r).(54)

On the other hand,

Again, by Theorem 2.1,

∑n=2∞cnrn≤(supn≥2rnT^n)∑n=2∞T^ncn≤E(r).

Therefore

Define

P(z)=Fm(z)F(z).

Then

P(z)-1=Fm(z)-F(z)F(z).

Using Equations 54 and 55, we obtain

The standard estimate

now gives

ℜ{Fm(z)F(z)}≥1-ρm(r)1+ρm(r).

This proves Equation 53. □

Theorem 6.2 (Reciprocal Partial Sums). Let

F(z)=z-∑n=2∞cnzn∈𝕃kμ(ς,τ,ϑ),

and let

Fm(z)=z-∑n=2mcnzn

*be its m-th partial sum. Put*

C=(1-τ)(1+ς-ϑ).

*For 0 < r < 1, define*

Em(r)=Csupn≥m+1rnT^n(ς,τ,ϑ,k,μ)

and

E(r)=Csupn≥2rnT^n(ς,τ,ϑ,k,μ).

If

E(r)<r

and

ρm(r)=Em(r)r-E(r)<1,(57)

then

ℜ{F(z)Fm(z)}≥1-ρm(r)1+ρm(r), |z|=r.(58)

Proof: Since

F(z)-Fm(z)=-∑n=m+1∞cnzn,

we have, for |z| = r < 1,

By Theorem 2.1,

∑n=2∞T^ncn≤C.

Consequently,

=Em(r).(59)

Also,

Using Theorem 2.1 once again,

∑n=2∞cnrn≤Csupn≥2rnT^n=E(r).

Hence

Now define

Q(z)=F(z)Fm(z).

Then

Q(z)-1=F(z)-Fm(z)Fm(z).

By Equations 59 and 60,

Using the standard estimate

we obtain

ℜ{F(z)Fm(z)}≥1-ρm(r)1+ρm(r).

This proves Equation 58. □

Corollary 6.1 (Univalence of Partial Sums). Let

F(z)=z-∑n=2∞cnzn∈𝕃kμ(ς,τ,ϑ),

and let

Fm(z)=z-∑n=2mcnzn

*be its m-th partial sum. Put*

C=(1-τ)(1+ς-ϑ).

*Then F_*m* is univalent in the disk*

where

Rm=sup{0<r<1:Csup2≤n≤mnrn-1T^n(ς,τ,ϑ,k,μ)<1}.(62)

Here

T^n(ς,τ,ϑ,k,μ)=[(k+1)(n-1)+(1-τ)(1+ςn-ϑ)]ψn,

where

ψn=[μ(n-1)]n-2e-μ(n-1)(n-1)!.

Proof: For

Fm(z)=z-∑n=2mcnzn,

we have

Fm′(z)=1-∑n=2mncnzn-1.

Hence, for |z| = r < 1,

By Theorem 2.1,

∑n=2∞T^ncn≤C.

Therefore,

∑n=2mncnrn-1=∑n=2m(T^ncn)nrn-1T^n≤(sup2≤n≤mnrn-1T^n)∑n=2mT^ncn≤Csup2≤n≤mnrn-1T^n.

Thus, whenever

Csup2≤n≤mnrn-1T^n<1,

we have

Consequently,

ℜ{Fm′(z)}≥1-|Fm′(z)-1|>0.

By the Noshiro–Warschawski univalence criterion, F_*m* is univalent in |z| < r. Taking the supremum over all such r gives R_*m*. □

Theorem 6.3 (Starlikeness of Partial Sums). Let

F(z)=z-∑n=2∞cnzn∈𝕃kμ(ς,τ,ϑ),

and let

Fm(z)=z-∑n=2mcnzn

*be its m-th partial sum. Let 0 ≤ η < 1, and put*

C=(1-τ)(1+ς-ϑ).

*Then F_*m* is starlike of order η in the disk*

where

Rm*(η)=sup{0<r<1:Csup2≤n≤m(n-η)rn-1T^n(ς,τ,ϑ,k,μ)≤1-η}.(63)

Here

T^n(ς,τ,ϑ,k,μ)=[(k+1)(n-1)+(1-τ)(1+ςn-ϑ)]ψn,

and

ψn=[μ(n-1)]n-2e-μ(n-1)(n-1)!.

Proof: For

Fm(z)=z-∑n=2mcnzn,

we have

Fm′(z)=1-∑n=2mncnzn-1.

Hence

zFm′(z)Fm(z)=1-∑n=2mncnzn-11-∑n=2mcnzn-1.

Therefore

zFm′(z)Fm(z)-η=(1-η)-∑n=2m(n-η)cnzn-11-∑n=2mcnzn-1.(64)

A sufficient coefficient condition for F_*m* to be starlike of order η in |z| < r is

∑n=2m(n-η)cnrn-1≤1-η.(65)

By Theorem 2.1,

∑n=2∞T^ncn≤C.(66)

Consequently,

∑n=2m(n-η)cnrn-1=∑n=2m(T^ncn)(n-η)rn-1T^n≤(sup2≤n≤m(n-η)rn-1T^n)∑n=2mT^ncn

≤Csup2≤n≤m(n-η)rn-1T^n.(67)

Thus, whenever

Csup2≤n≤m(n-η)rn-1T^n≤1-η,

condition Equation 65 is satisfied. Hence

ℜ{zFm′(z)Fm(z)}>η, |z|<r.

Therefore F_*m* is starlike of order η in |z| < r.

Taking the supremum over all admissible r ∈ (0, 1) gives the radius Rm*(η) in Equation 63. □

7 Conclusion

In this paper, we introduced a parameterized subclass 𝕃kμ(ς,τ,ϑ) of analytic functions with negative coefficients through the Borel distribution convolution operator. A coefficient characterization was established, and several geometric properties of the class were investigated, including growth and distortion estimates, extreme points, radii of starlikeness and convexity, and partial-sum properties. In addition, a Fekete–Szegő inequality was obtained as a nontrivial coefficient application of the proposed framework.

The main significance of the present study lies in the combination of the Borel distribution structure with the additional geometric parameters, which provides a flexible framework for studying coefficient-dependent subclasses of analytic functions. At the same time, the present investigation is primarily theoretical and is restricted to the particular Borel operator, negative-coefficient setting, and parameter ranges considered here. Consequently, the results do not by themselves establish direct applications to specific physical or engineering models.

Several directions remain open for future investigation. These include higher-order coefficient functionals, Hankel and Toeplitz determinants, logarithmic coefficient estimates, bi-univalent and multivalent extensions, and subclasses associated with generalized or other probability distribution series. Such extensions may provide further insight into the interaction between distribution-based operators and geometric properties of analytic functions.

Generative AI statement

The author(s) declared that Generative AI was not used in the creation of this manuscript.

Any alternative text (alt text) provided alongside figures in this article has been generated by Frontiers with the support of artificial intelligence and reasonable efforts have been made to ensure accuracy, including review by the authors wherever possible. If you identify any issues, please contact us.

Publisher's note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Footnotes

Edited by: Lixin Shen, Syracuse University, United States

Reviewed by: Nihad Shehab, Directorate General of Education in Anbar, Iraq

Adel Salim Tayyah, University of Al-Qadisiyah, Iraq