Front. Appl. Math. Stat. Frontiers in Applied Mathematics and Statistics Front. Appl. Math. Stat. 2297-4687 Frontiers Media S.A. 10.3389/fams.2026.1777576 Original Research On classes of non-univalent functions influenced by the multiplicative derivative Breaz Daniel 1 Conceptualization Data curation Formal analysis Funding acquisition Investigation Methodology Project administration Resources Software Validation Visualization Writing – original draft Writing – review & editing Karthikeyan Kadhavoor R. 2 * Conceptualization Data curation Formal analysis Investigation Methodology Resources Software Supervision Validation Visualization Writing – original draft Writing – review & editing Mohankumar Dharmaraj 3 Conceptualization Data curation Formal analysis Investigation Methodology Resources Software Validation Visualization Writing – original draft Writing – review & editing Department of Mathematics, “1 Decembrie 1918” University of Alba Iulia, Alba Iulia, Romania Department of Applied Mathematics and Science, National University of Science and Technology, Muscat, Oman Department of Mathematics for Innovation, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences (SIMATS), Thandalam, Chennai, India Correspondence: Kadhavoor R. Karthikeyan, karthikeyan@nu.edu.om 18 02 2026 2026 12 1777576 29 12 2025 31 01 2026 04 02 2026 Copyright © 2026 Breaz, Karthikeyan and Mohankumar. 2026 Breaz, Karthikeyan and Mohankumar https://creativecommons.org/licenses/by/4.0/ This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

In this article, we extend studies on the class of starlike functions, motivated by the definition of the multiplicative derivative. In this direction, we define a new class by combining characterizations of starlike functions and a recently studied subclass of analytic functions. Furthermore, we have defined a new family of analytic functions motivated by the so-called theta-starlike function. Here, we discussed in detail the problems and repercussions of employing a non-Newtonian derivative in studies of various subclasses of analytic functions. Fekete-Szegő inequality and bounds of initial coefficients are our main results. Although our main results have numerous applications, here we highlight the results associated with the conic regions as applications of the main results.

 convex function Fekete-Szegő inequality multiplicative derivative quantum derivative starlike function subordination The author(s) declared that financial support was not received for this work and/or its publication. section-at-acceptance Mathematical Physics Introduction

Denote by A the class of analytic functions in 𝕌 = {z ∈ ℂ:|z| < 1} having a Taylor series expansion of the form

J(z)=z+∑n=2∞χnzn,z∈𝕌.

We let S to denote the class of functions in A which are one-one. Let P denote the family consisting of functions K analytic on 𝕌 with K(0)=1 and Re[K(z)]>0. Geometrically defined function classes of starlike and convex functions (see Refs. [1, 2]) which we will denote here by S* and C satisfies the following respective differential inclusion

As(z)=zJ′(z)J(z)∈P and Ac(z)=1+zJ′′(z)J′(z)∈P.

The well-known classes namely close-to-convex (C*) (see Ref. [3]) and quasi-convex (QC) is known to satisfy the condition

J′(z)G′(z)∈P and (zJ′(z))′G′(z)∈P,  (G∈C;z∈𝕌).

Mocanu class (see Ref. [4]) denoted by M(γ)(0≤γ≤1) combined the expressions of starlike and convex function, which was defined by writing the linear combination of zJ′(z)J(z) and (zJ′(z))′J′(z). Mocanu class apart from generalizing the classes S* and C, is of significant interest in the study of various subclasses of S*. Similar idea was used to extend the class S* and C, by using a combination of powers of zJ′(z)J(z) and (zJ′(z))′J′(z). Precisely, Lewandowski et al. [5, Theorem 1, pg. 54] established the following inclusion

Re[(zJ′(z)J(z))γ((zJ′(z))′J′(z))1-γ]>0 ⇒ J∈S*,(∀γ∈ℝ;J∈A).

Functions J∈A satisfying the differential inequality in Equation 2 is known as gamma starlike function or alpha-logarithmically convex and here lγ will used to denote the class of alpha-logarithmically convex. The class lγ was mainly motivated by the well-known inclusion relationship induced by the differential inequality

Re{[m(z)+zm′(z)m(z)]γ[m(z)]1-γ}>0 ⇒Re{m(z)}>0,(z∈𝕌).

For 0 ≤ γ ≤ 1, arithmetic and geometric mean of two numbers Θ and Λ, are defined respectively as follows:

Aγ(Θ,Λ)=(1-γ)Θ+γΛ;Gγ(Θ,Λ)=ΘγΛ1-γ.

By letting Θ=zJ′(z)J(z) and Λ=(zJ′(z))′J′(z) in the above epressions, Aγ(Θ, Λ) reduces to Mocanu class M(γ)(0≤γ≤1) whereas Gγ(Θ, Λ) reduces to the class lγ. Sivaprasad Kumar and Priyanka Goel [6] obtained differential subordination implications involving the harmonic mean.

In a study by Ma and Minda [7], they obtained the coefficient inequalities of the classes S*(𝔎) and C(𝔎) which satisfy the respective condition

As(z)≺𝔎(z) and Ac(z)≺𝔎(z),

where 𝔎(z)∈P satisfies few geometrical condition and is of the form

𝔎(z)=1+κ1z+κ2z2+κ3z3+⋯,  (κ1≠0;z∈𝕌).

Motivated by Ma and Minda [7], several authors restudied the subclasses of S by restricting the differential inequalities given in Equation 4 to a conic region. By expressing a certain analytic characterization listed in Equation 4, for example As(z)≺𝔎(z) (≺ denotes subordination) and choosing 𝔎(z) to map 𝕌 on to a conic regions like cardioid [8, 9], petal shaped [10], crescent shaped [11, 12], leaf-like region in the right-half of the complex plane [13], various interesting families of functions were introduced studied by several authors. For 0 < δ ≤ 1, β ≥ 0 and s ∈ [−1, 1]\{0}, we let Lδ,βk(s;z) to denote

Lδ,βk(s;z)=1+δze2skzk+βs2kz2k.

The function Lδ,βk(s;z) arose in a study associated with the Limaçon domain [14]. The function Lδ,βk(s;z) does not always map the unit disc onto the right-half plane. For a choice of the parameters δ = β = 1, s=12 and k = 9, we can see that the image of 𝕌 under Lδ,βk(s;z) lies in the left-half of the w-plane (see Figure 1A). In Figure 1B, we illustrate that for lots of fixed parameters Lδ,βk(s;z)∈P.

(A) The image of L1,19(1/2;z) with domain as unit disc; (B) The image of L0.7,15(1/2;z) with domain as unit disc.

Panel A shows a polar plot with nine blue petals outlined over concentric red circles, corresponding to delta equals beta equals one, s equals one divided by square root of two, and k equals nine. Panel B presents a polar plot with five light green petals and overlaid gridlines, parameters are delta equals zero point seven, beta equals one, s equals one divided by square root of two, and k equals five. Short introduction to multiplicative calculus

Multiplicative derivative of a function of a real variable has been a useful tool in modeling problems that involve proportional changes. Expressing the numerator of the difference quotient in the form J(x+h)/J(x) and taking 1/h to the power of this fraction, we enter the world of multiplicative calculus when the limit h → 0 is taken in this resulting expression. Explicitly, for x ∈ ℝ

J*(x)=limh→0(J(x+h)J(x))1h=limh→0eln J(x+h)-ln J(x)h           =elimh→0ln J(x+h)-ln J(x)h=e[ln J(x)]′,

where (.)′ denotes the classical derivative. Whereas the well-known quantum derivative of a function J defined on a subset of ℝ is defined by

DqJ(x)=J(qx)-J(x)(q-1)x, (x≠0),DqJ(x)=limx→0DqJ(x),

where qx and x should be in the domain of J and Dq is q-difference operator. Yener and Emiroglu [15] introduced the q-analogs of the multiplicative calculus, which later was known to be the q-multiplicative calculus. Precisely,

Dq*J(x)=(J(qx)J(x))1(q-1)x=eln J(qx)-ln J(x)(q-1)x.

In their study [15], Yener and Emiroglu established that Dq*J(x)=eqDqln J(x) which we disagree that it would hold for all q-differentiable positive function. Refer to Remark 1 presented in Ref. [16] for details, where the authors have discussed comprehensively. Quantum derivative [17] and the multiplicative derivative [18] of a function of a complex variable are also analogous to the functions of a real variable.

 Objectives

Since most geometrically defined subclasses of S have analytic characterizations involving derivatives. We intend to replace the classical derivative with a multiplicative derivative in the analytic characterization of various subclasses of S. However, we realize that existing literature is not equipped to address such a swap, as functions in class S are defined to vanish at z = 0. Here, we study a class of functions motivated by the definition of the multiplicative derivative.

That is, we aim to study the behavior of the class when two differential characterizations are combined by a certain relation and subordinated to a general function. The deviation that we present here is mainly the geometrical impact that it would create when subordinated by domains enclosed by special curves. The other main results are the coefficient inequalities, which would help us understand the geometric and algebraic properties.

 Motivation

We now provide a brief introduction to studies in univalent function theory that involve the exponential function. Arango et al. in their study [19] introduced the so-called α-exponentially convex functions, which consist of a function J∈A such that eαJ(z) is convex univalent in 𝕌. Precisely for α ∈ ℂ\{0}, a function J∈A is in ε(α) if and only if

Re(1+zJ′′(z)J′(z)+αzJ′(z))>0, (z∈𝕌).

In a study by Ponnusamy et al. [20], they obtained several interesting properties of α-exponentially convex functions. Recently, Sharma et al. [21] introduced α-exponentially bi-convex functions and obtained the coefficient inequalities. Sunil Verma et al. [22] studied the family of functions J∈A satisfying the respective characterizations

Re(zJ′(z)eαJ(z)G(z))>0, (z∈𝕌;α∈ℂ\{0}),

and

|zJ′(z)eαJ(z)G(z)-1|<1, (z∈𝕌;α∈ℂ\{0}),

where G(z) satisfies Re(G(z)z)>0. Karthikeyan et al. [23] obtained the coefficient inequalities of functions J∈A satisfying the condition

1+[z2J′′(z)J(z)+zJ′(z)J(z)-(zJ′(z)J(z))2]ezJ′(z)J(z)-1≺𝔎(z),

where 𝔎∈P is defined as in Equation 5. Replacing J′(z) with ez2J′(z)J(z) in As(z), recently Karthikeyan and Murugusundaramoorthy in their study [24] presented a class R(𝔎) of functions J∈A which satisfies

zez2J′(z)J(z)J(z)≺𝔎(z),

where 𝔎∈P is defined as in Equation 5. Further, Karthikeyan et al. [25] introduced and studied a new class Mγ(λ,𝔎) consisting of functions in A which satisfies the condition

(1-λ)(ezJ′(z)J(z)-1)γ+λez2J′(z)J(z)(J(z)z)γ-1≺𝔎(z),(J∈A;z∈𝕌),

where γ ≥ 0, λ∈C such that ℜ(λ) > 0 and 𝔎(z)∈P has a power series representation of the form shown in Equation 5.

 A new family of analytic functions

Motivated by previous studies [19–25], we now define the following.

Definition 1.1. Let KCσ(𝔎) denote the class of functions in J∈A satisfying the conditions

(ezJ′(z)J(z)-1)σ(zezJ′(z)J(z)-1J(z))1-σ≺𝔎(z),

where 0 ≤ σ ≤ 1, and 𝔎∈P is defined as shown in Equation 5.

Motivated by studies on the arithmetic mean, we will now introduce the following.

Definition 1.2. Let RKσ(𝔎) denote the class of functions in J∈A satisfying the conditions

(1-λ)zJ′(z)J(z)+λ(zez2J′(z)J(z)J(z))σ(zezJ′(z)J(z)-1J(z))1-σ≺𝔎(z),

where 0 ≤ σ, λ ≤ 1, and 𝔎∈P is defined as shown in Equation 5.

The functions belonging to the classes KCσ(𝔎) and RKσ(𝔎) need not be univalent, which would be evident once we obtain the coefficient inequalities. Here we will omit the discussion on the exceptional cases of the classes KCσ(𝔎) and RKσ(𝔎). However, we will discuss these details after presenting our main results.

 Coefficient inequalities and Fekete-Szegő inequalities of the <inline-formula><mml:math id="M96"><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant="script">KC</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mi>σ</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>𝔎</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula> and <inline-formula><mml:math id="M97"><mml:msub><mml:mrow><mml:mrow><mml:mi mathvariant="script">RK</mml:mi></mml:mrow></mml:mrow><mml:mrow><mml:mi>σ</mml:mi></mml:mrow></mml:msub><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mrow><mml:mi>𝔎</mml:mi></mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow></mml:math></inline-formula>

We will need the following inequality for the class P.

Lemma 2.1. See Ref. [7] Let p(z)=1+∑n=1∞pnzn∈P. For v to be a complex number, we have the following inequalities

|p2-vp12|≤2max{1;|2v-1|}.

The result is sharp.

Theorem 2.2. If J(z)∈KCσ(𝔎), then we have

|χ2|≤|κ1|σ,|χ3|≤|κ1|(1+σ)max{1;|κ2κ1+κ1(1+σ-σ2)2σ2|},

and for all ν ∈ ℂ

|χ3-νχ22|≤|κ1|(1+σ)max{1;|κ2κ1+κ1(1+σ-σ2)2σ2                         -ν(1+σ)κ1σ2|}.

Proof. As J∈KCσ(𝔎), by Equation 10 we have

(ezJ′(z)J(z)-1)σ(zezJ′(z)J(z)-1J(z))1-σ=𝔎[w(z)].

Thus, let ϑ∈P be of the form ϑ(z)=1+∑n=1∞ϖnzn and defined by

ϑ(z)=1+w(z)1-w(z), z∈𝕌.

On computation, the right hand side of Equation 14 becomes

𝔎[w(z)]=1+ϖ1κ12z+κ12[ϖ2-ϖ122(1-κ2κ1)]z2+⋯.

Expanding left hand side of Equation 14, we have

(ezJ′(z)J(z)-1)σ(zezJ′(z)J(z)-1J(z))1-σ=1+χ2σz+[χ3(1+σ)-χ222(1+σ-σ2)]z2+[χ4(σ+2)+χ2χ3(σ2-2)+χ236(4-σ-3σ2+σ3)]z3+⋯

From Equations 15 and 16, we obtain

χ2=ϖ1κ12σ

and

χ3=κ12(1+σ)[ϖ2-ϖ122(1-κ2κ1)]+χ222(1+σ)(1+σ-σ2),=κ12(1+σ)[ϖ2-ϖ122(1-κ2κ1-κ1(1+σ-σ2)2σ2)].

The bound χ2 can be obtained by using |ϖ1| ≤ 2 in Equations 17 and χ3 can be obtained by using Lemma 2.1.

Now to prove the Fekete-Szegő inequality for the class KCσ(𝔎), we consider

|χ3−νχ22|=|κ12(1+σ)[ϖ2−ϖ122(1−κ2κ1−κ1(1+σ−σ2)2σ2)]−νϖ12κ124σ2|=|κ12(1+σ)[ϖ2−ϖ122(1−κ2κ1−κ1(1+σ−σ2)2σ2+ν(1+σ)κ1σ2)]|.

Now using Lemma 2.1, we obtain the assertion (Equation 13).

Theorem 2.3. If J(z)∈RKσ(𝔎), then we have

|χ2|≤1|1-λ-λσ|[|κ1|+|λσ|], |χ3|≤|κ1||2−λ−2σλ|[max{1;|κ2κ1−κ1(λσ2+2λσ+λ−2)2(1−λ−λσ)2|}    +λσ|2λσ+2λ+σ−3(1−λ−λσ)2|+λσ2|4λ2+4λ2σ−6λ+1|2|κ1|(1−λ−λσ)2],

and for all ν ∈ ℂ

        |χ3-νχ22|≤|κ1|(2-λ-2σλ)[max{1;|κ2κ1     -κ1[λσ2+2λσ(1-2ν)+λ(1-2ν)+4ν-2]2(1-λ-λσ)2|}        +λσ|2λσ(1-2ν)+2λ(1-ν)+σ+4ν-3(1-λ-λσ)2|+λσ2|2λ(2ν-3)+4λ2σ(1-ν)+2λ2(2-ν)+1|2|κ1|(1-λ-λσ)2].

Proof. As J∈RKσ(𝔎), by Equation 11 we have

(1-λ)zJ′(z)J(z)+λ(zez2J′(z)J(z)J(z))σ(zezJ′(z)J(z)-1J(z))1-σ=𝔎[w(z)].

Expanding left hand side of Equation 22, we have

(1-λ)zJ′(z)J(z)+λ(zez2J′(z)J(z)J(z))σ(zezJ′(z)J(z)-1J(z))1-σ                =1+[χ2(1-λ-λσ)+λσ]z+             12[2χ3(2-λ-2σλ)+2χ2λσ(1-σ)           +χ22(λσ2+2λσ+λ-2)+λσ2]z2+⋯

From Equations 15 and 23, we obtain

χ2=1(1-λ-λσ)[ϖ1κ12-λσ],

and

                              χ3=κ12(2-λ-2σλ){[ϖ2             -ϖ122(1-κ2κ1+κ1(λσ2+2λσ+λ-2)2(1-λ-λσ)2)]+λσϖ1[2λσ+2λ+σ-3(1-λ-λσ)2]-λσ2[4λ2+4λ2σ-6λ+1]κ1(1-λ-λσ)2}

The bound χ2 can be obtained by using |ϖ1| ≤ 2 in Equations 24 and 20 can be obtained by using Lemma 2.1 in Equation 25.

Now to prove the Fekete-Szegő inequality for the class RKσ(𝔎), we consider

                |χ3-νχ22|=|κ12(2-λ-2σλ){[ϖ2-ϖ122(1-κ2κ1                                          +κ1(λσ2+2λσ+λ-2)2(1-λ-λσ)2)]              +λσϖ1[2λσ+2λ+σ-3(1-λ-λσ)2]-λσ2[4λ2+4λ2σ-6λ+1]κ1(1-λ-λσ)2}                                 -ν(1-λ-λσ)2[ϖ1κ12-λσ]2|                        =|κ12(2-λ-2σλ){[ϖ2-ϖ122(1-κ2κ1              +κ1[λσ2+2λσ(1-2ν)+λ(1-2ν)+4ν-2]2(1-λ-λσ)2)]              +λσϖ1[2λσ(1-2ν)+2λ(1-ν)+σ+4ν-3(1-λ-λσ)2]-λσ2κ1(1-λ-λσ)2[2λ(2ν-3)+4λ2σ(1-ν)+2λ2(2-ν)+1]}|.

Now using Lemma 2.1, we obtain the assertion provided in Equation 21.

Corollary 2.4. Let J∈R(𝔎). Then,

|χ2|≤1+|κ1| |χ3|≤|κ1|[max{1;|κ2κ1-κ1|}+32|κ1|+2]

and for a complex number ν,

|χ3-νχ22|≤|κ1|[max{1;|κ2κ1-κ1(1-ν)|}+2|1-ν|                                            +12|κ1||3-2ν|].

Corollary 2.5. If J∈S*(𝔎), then,

|χ2|≤|κ1|,  |χ3|≤|κ1|2max{1;|κ2κ1+κ1|}

and for a complex number ν,

|χ3-νχ22|≤|κ1|2max{1;|κ2κ1+κ1(1-2ν)|}.

Remark 2.1. We now enumerate several notable cases of our results.

(1) Letting 𝔎(z)=1+z1-z in Corollary 2.5, we get |χ2| ≤ 2 ([1, pg. 116]), |χ3| ≤ 3 (see Theorem 4.1.5 presented in Ref. [3]) and for ν ∈ ℂ we have |χ3-νχ22|≤max{1;|3-4ν|} (see page 89 of Ref. [3]).

(2) Letting 𝔎(z)=1+ω-ηπilog(1-e2πi((1-η)/(ω-η))z1-z) (ω and η are real numbers such that 0 ≤ η < 1 < ω) in Corollary 2.5, we get the result obtained by Sim and Kwon [26] (also see Ref. [27]) which is given by

    |χ3−νχ22|≤ω−ηπsinπ(1−η)ω−ηmax{1;|12+(1−2ν)ω−ηπi+(12−(1−2ν)ω−ηπi)e2πi1−ηω−η|},(ν∈ℂ).

(3) Letting 𝔎(z)=1+12isinηlog(1+eiηz1+e-iηz) (π2≤η<1<π) in Corollary 2.5, we get the result obtained by Sun et al. [28] (also see Ref. [27]) which is given by

|χ2|≤1, |χ3|≤1 and|χ3-νχ22|≤12max{1,|1                                            -cosη-2ν|}.

(4) Letting 𝔎(z)=z+1+z2 in Corollary 2.5, we get the result of the starlike function associated with the class with a crescent-shaped domain [11, 29, 30] which is given by

|χ2|≤1, |χ3|≤34and|χ3-νχ22|≤12max{1,|32-2ν|}.

It is clear from the above enumeration that numerous classical and new results can be obtained by specializing the parameters involved [12, 13, 16, 27, 31, 32].

Theorem 2.6. If J(z)∈A satisfies the condition

(1-λ)zJ′(z)J(z)+λ(zez2J′(z)J(z)J(z))σ(zezJ′(z)J(z)-1J(z))1-σ                                       ≺1+δze2sz+βs2z2,

with 0 < δ ≤ 1, β ≥ 0 and s ∈ [−1, 1]\{0}, then we have

|χ2|≤[δ+|λσ|]|1−λ−λσ|,                                     |χ3|≤δ|2−λ−2σλ|              [max{1;|2s−δ(λσ2+2λσ+λ−2)2(1−λ−λσ)2|}+λσ|2λσ+2λ+σ−3(1−λ−λσ)2|+λσ2|4λ2+4λ2σ−6λ+1|2δ(1−λ−λσ)2],

and for all ν ∈ ℂ

|χ3−νχ22|≤δ(2−λ−2σλ)[max{1;|2s−δ[λσ2+2λσ(1−2ν)+λ(1−2ν)+4ν−2]2(1−λ−λσ)2|}+λσ|2λσ(1−2ν)+2λ(1−ν)+σ+4ν−3(1−λ−λσ)2|+λσ2|2λ(2ν−3)+4λ2σ(1−ν)+2λ2(2−ν)+1|2δ(1−λ−λσ)2].

Proof. The function is 1+δze2sz+βs2z2 is in P and has a power series expansion of the form

1+δze2sz+βs2z2=1+δz+2δsz2+(2δs2+δsβ2)z3                                 +(4δs33+δs2β)z4+⋯.

Replacing κ1 = δ and κ2 = 2δs in Theorem 2.3, we obtain the assertion of the Theorem 2.6.

Recently, Kumar and Kamaljeet [33] (also see Refs. [34–36]) obtained several interesting results involving a class of functions J∈A satisfying the condition

zJ′(z)J(z)≺1+zez.

Note that the superordinate function of the above expression maps the unit disc onto a cardioid in the right half plane (see Figure 2).

Mapping of |z| < 1 under the transformation 1 + zez.

Contour plot featuring concentric blue level curves and intersecting red radial lines, illustrating a mathematical or complex function overlaid on a Cartesian coordinate grid with axes labeled from zero to approximately three point five.

If we let λ = 0, δ = 1 and s=12, we get the following result obtained by Kumar and Kamaljeet [33].

Corollary 2.7. See Ref. [33] If J(z)∈A satisfies the condition

zJ′(z)J(z)≺1+zez,

then we have

|χ2|≤1, |χ3|≤1 and |χ3-νχ22|≤12max{1;2|1-ν|}. Conclusion

The classes introduced and studied by various authors have primarily concentrated on finding coefficient estimates. Apart from these, the bulk of studies on arithmetic and geometric means focused on finding sufficient conditions for convexity and starlikeness. The classes KCσ(𝔎) and RKσ(𝔎) introduced here are more versatile when compared with studies in the existing literature, since the defined function classes involve exponential function.

What separates the present study from other various well-known studies in this field of research is that the classes KCσ(𝔎) and RKσ(𝔎) involves analytic characterization obtained by classical derivative with a multiplicative derivative. Many authors have not attempted to obtain the coefficient bounds of classes when they involve an exponential function, simply because it is computationally long and tedious.

Another interesting aspect of the class introduced here is that it is not a simple generalization for which the results can be obtained analogously. The results we have obtained here not only unify and generalize well-known results in geometric function theory but also generate very interesting new results (see Refs. [37, 38]).

 Data availability statement

The original contributions presented in the study are included in the article/supplementary material, further inquiries can be directed to the corresponding author.

 Author contributions

DB: Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Resources, Software, Validation, Visualization, Writing – original draft, Writing – review & editing. KK: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Resources, Software, Supervision, Validation, Visualization, Writing – original draft, Writing – review & editing. DM: Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Resources, Software, Validation, Visualization, Writing – original draft, Writing – review & editing.

 Acknowledgments

The authors are grateful to the reviewers for their comments and observations, which helped us to remove numerous mistakes in the first version of the paper.

 Conflict of interest

The author(s) declared that this work was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Edited by: Teodor Bulboaca, Babeș-Bolyai University, Romania

Reviewed by: Sarika Verma, University of Jammu, India

Eduard Stefan Grigoriciuc, Babeș-Bolyai University, Romania