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Coefficient Inequalities for Certain Univalent Analytic Starlike And

Convex Functions in Leaf Like Domain Through Jackson Q-

Derivative Operator

¹Sanjay Issar, ²Hemlata

¹Department of Mathematics, Government Dungar College, Bikaner-334001, India

²Department of Mathematics, Tantia University, Sriganganagar-335002, India

DOI : https://doi.org/10.51583/IJ L T EMAS.2025.1411000098

Received: 07 December 2025; Accepted: 14 December 2025; Published: 22 December 2025

ABSTRACT

Two subclasses of starlike and convex functions analytic in the unit open disk using q-derivative operator have

been investigated in the present paper. The necessary and sufficient condition for the function belonging to these

classes have been obtained. We further examine various properties, such as the Hadamard product and the quasi-

Hadamard product. The coefficient estimates for the function belonging to these classes are also found.

Mathematics Subject Classification 2020: 30C45, 30C50

Keywords: Starlike, Convex, Hadamard product (or convolution), quasi-Hadamard product, Subordination, q-

derivative operator.

INTRODUCTION

Let A be the class of functions of the form

,

(1.1)

which are analytic in the unit open disc U = {z ∈ C : |z| < 1}. Let S be the sub class of A consisting of univalent

functions. Let Ω be the class of analytic functions ω satisfying the conditions ω(0) = 0, |ω(z)| < 1 for all z ∈ U.

Let f and g be two analytic functions in U, then function f is said to be subordinate to g if there exists an analytic

function w ∈ Ω such that f(z) = g(w(z)) (z ∈ U). We denote this subordination by f ≺ g. In particular, if the

function g is univalent in U, then the above subordination is equivalent to f(0) = g(0) and f(U) ⊆ g(U).

Quantum calculus, often referred to as q-calculus, offers a natural extension of classical calculus by incorporating

a parameter q, where 0 < q < 1 and without the notion of limit, thereby broadening the range of traditional

analytical methods. In recent years, the field of q-calculus has drawn significant attention from researchers due

to its strong connections with physics, quantum mechanics, and Geometric Function Theory (GFT). The

application of q-calculus was initiated by Jackson ([13], [14]), who first investigated its uses and effectively

introduced the q-derivative and q-integral operators.

Recently in the field of Geometric Function Theory many function classes have been introduced with the help

of q-derivative and q-integral operators and investigated by a number of researchers including ([1],

[2],[3],[6],[38], [26], [29], [34],[35] ,[36],[37],[39],).

The purpose of this article is to introduce and study two subclasses of univalent functions by applying q-

derivative operator in conjunction with the principle of subordination.

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Jackson q-derivative with 0 < q < 1, of a function f is defined as ([11],[13],[14]).

(z = 0)

̸

(0)

(z = 0),

provided that f^′(0) exists.

For f given by (1.1), we can easily obtain that

∞

D_qf(z) = 1 + ^∑[n]_qa_nzⁿ⁻¹,

n=2

where

.

It is easy to see that as q → 1⁻, [n]_q→ n and D_qf(z) = f^′(z).

Also, we have the followingq-derivative rules

The q − integral of the function f is defined as (see [13])

,

(1.2)

provided that the series converges. Here we observe that

and

.

as q → 1⁻.

A function f ∈ A is starlike if f(U) is starlike with respect to origin and it is convex if f(U) is convex. Analytically

in terms of subordination, a function f ∈ A is starlike and convex if and only if the subordination relations

and

for z ∈ U respectively hold. Ma and Minda[21] introduced and studied the following two

subclasses S^∗(ϕ) and K(ϕ) of starlike and convex functions respectively in term of subordination relations:

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and

,

′ where ϕ is analytic and normalized by ϕ(0) = 1 and ϕ

(0) > 0 with ℜϕ(z) > 0 in U. Seoudy and Aouf [29] further generalized these classes by using the q-derivative

operator in the following manner:

and

.

Recently, several Ma - Minda type subclasses of starlike and convex functions have been introduced and studied

by

considering

different

image

domains

ϕ(U).

For

some

examples,

we

can

see([7],[15],

[17],[30],[31],[18],[32],[41]). Motivated by the work of Kumar and C¸etinkaya [17] , in the present paper we

consider two subclasses

) and K_q(ψ) of starlike and convex functions respectively, which are associated with the analytic function

.

1.5

1

0.5

a

−

− −

1.5 1 0.5

0.5 1 1.5 2 2.5 3 3.5 4

−

0.5

−

1

−

1.5

−2

The function ψ maps an open unit disc onto a simply connected bounded region in the right - half of the complex

plane. Analytically these classes are defined as:

(1.3)

and

. (1.4)

From (1.3) and (1.4), we find that

.

(1.5)

Let f and g be two analytic functions of the form

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and

,

then the Hadamard product (or convolution) of f and g is defined as

.

By taking g(z) to be

and

, it is easy to see that

,

(1.6)

,

(1.7)

(1.8)

(1.9)

proceeding as above, we can obtain the following

and

.

Further, let T be the subclass of analytic functions with negative coefficients of the form

0)

(1.10)

(1.11)

defined in U. For the function f defined by (1.10) and

0,b_n≥ 0), the quasi -Hadamard product of f and g is defined as

.

As above, we can define a quasi-Hadamard product of more than two functions. The quasi-Hadamard product

of two or more functions has recently been defined and used by Owa ([23],-[24]), Kumar([19]-[20]), Sekine

[28], Aouf [4], Frasin and Aouf [10], Hossen [12], Darwish [8] and El-Ashwah et al. [9].

In the next section we give characterizations for defined subclasses of q-starlike and q-convex functions with

the help of Hadamard products. For each of these subclasses, we first find a function g depending on each

concerning class

) and

K_q(ψ), such that

= 0 is both necessary and sufficient for f to be in

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and K_q(ψ). Further we use these findings to determine the coefficients estimates for a function belonging to

) and K_q(ψ). For some recent similar studies on various classes of analytic functions, one can find in ([5],

[16],,[22],[25],[27], [29], [33] , [37], [40]) and the references cited therein.

Then we establish certain results concerning the quasi-Hadamard product of functions belonging in the classes

) and K_q(ψ) analogous to the results due to Kumar ([19],[20]) and Sekine [28].

Hadamard product properties

We assume throughout this paper that 0 < q < 1, and θ ∈ [0,2π).

Theorem 1. The function f ∈ T defined by (1.10) is in the class K_q(ψ) if and only if

= 0

(2.1)

for all

, where θ ∈ [0,2π) and is also true for L = 1.

Proof. Suppose that the function f ∈ K_q(ψ), then we have

(2.2)

Due to analyticity of the function

, we have D_qf(z) = 0 which

̸

is equivalent to the fact that (2.1) holds for L = 1. In view of (2.2)

where ω ∈ Ω is a Schwarz function, hence

.

(2.3)

Using (1.8)and (1.9), we obtain

.

or

where L is given by (2.1). Using

the identity zD_qf(z).g(z) = f(z).zD_qg(z), we get

,

which leads to (2.1).

For only if part of the theorem, suppose that f satisfies the condition (2.1). Since it is shown here that assumption

(2.1) is equivalent to (2.3), so we have

.

(2.4)

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Suppose that

and

. The relation (2.4) means that ϕ(U) ∩ ψ(∂U) = ∅.

Thus, the simply connected domain ϕ(U) is included in a connected component of C\ψ(∂U). Therefore, using the

fact that ϕ(0) = ψ(0) together with the univalence of the function

, it follows that ϕ(z) ≺ ψ(z).

Hence f ∈ K_q(ψ), which complete the proof of Theorem 1.

By using the technique as given in Srivastava and Zayed [37], we prove the following convolution condition for

the subclass

Theorem 2. The function f ∈ T defined by (1.10) is in the class

if and only if

= 0

(2.5)

for all

, where θ ∈ [0,2π) and is also true for L = 1 .

Proof. It follows from (1.5) that

) if and only if

.

Then according to the Theorem 1, the function f belongs to

) if and only if

(2.6)

where

.

From (1.2), we have

,

and therefore

.

(2.7)

Using the identity

and (2.7) in relation (2.6), we get the desired result (2.5).

Theorem 3. The function f ∈ T defined by (1.10) is in the class K_q(ψ) if and only if

= 0

(2.8)

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Proof. If f ∈ T given by (1.10), then from Theorem 1, we have f ∈ K_q(ψ) if and only if (2.1) holds. Since

it follows that

,

where

and so (2.1) may be written as

.

which completes the proof.

Theorem 4. The function f ∈ T defined by (1.10) is in the class

if and only if

= 0

(2.9)

Proof. Since

so

(2.10)

In view of (2.5) and (2.10), a simple calculation provides the desired result. Theorem 5. If the function f ∈ T

defined by (1.10) satisfies the inequality

(a_n≥ 0,a₁> 0),

(2.11)

then f ∈ K_q(ψ).

Proof. Since

Thus, the inequality (2.11) holds and our result follows from Theorem 3.

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Using similar arguments to those in the proof of Theorem 5, we may also prove the Theorem 6.

Theorem 6. If the function f ∈ T defined by (1.10) satisfies the inequality

∞

^∑(5[n]_q− 2⁾a_n≤ 3a₁

(a_n≥ 0,a₁> 0),

(2.12)

n=2

then

.

Quasi-Hadamard product properties

The quasi-Hadamard product properties of the classes

) and K_q(ψ) are obtained in this section. For this

purpose we need to define the following subclass of analytic functions:

A function f of the form (1.10) belongs to the class

) if and only if

∞

c(

^∑([n]_q5[n]_q− 2⁾a_n≤ 3a₁,

(3.1)

) n=2

where c is a non-negative real number. We note that for a non-negative real number c the class

empty as the function of the form

) is non-

) and for

,

where a₁> 0, λ_n> 0 and

). Further,

1 satisfies the inequality (3.1). Here we note that

if c > n ≥ 0, the containment being proper.

Let the functions of the form

0)

(3.2)

(3.3)

and

∞ gi(z) = b1,i − ∑bn,izn

(b1,i > 0,bn,i ≥ 0)

n=2

be analytic in the unit disc U.

Theorem 7. Let the functions f_i(i = 1,2,···,m) given by (3.2), belong to the class

. Then, the quasi-Hadamard product f₁∗^′f₂∗^′· · · ∗^′f_mbelongs to the class

.

Proof. Here, we need to show that

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.

(3.4)

(3.5)

Since

∞

, we have

∑(5[n]q − 2)an,i ≤ 3a1,i

(a1,i > 0, an,i ≥ 0).

n=2 Therefore

,

which implies

,

(3.6)

Using (3.6) for i = 1,2,· · ·,m − 1 and (3.5) for i = m, we get

∞

m

∞

m−1

∑[([n]q)m−1(5[n]q − 2)∏an,i] = ∑[( ∏ ([n]qan,i)(5[n]q − 2)an,m]

i=1 n=2 i=1

m−1

n=2

∞

≤ ( ∏ a1,i)∑(5[n]q − 2)an,m

i=1

n=2 m

≤ 3^∏a₁,i.

i=1

which completes the proof.

Theorem 8. Let the functions f_i(i = 1,2,· · ·,m) given by (3.2), belong to the the class K_q(ψ). Then, the quasi-

Hadamard product f₁∗^′f₂∗^′· · · ∗^′f_mbelongs to the class Sq(2m−1)(ψ).

Proof. Since f_i(z) ∈ K_q(ψ), we have

(a_1,i> 0, a_n,i≥ 0).

(3.7)

Therefore

,

which implies

,

(3.8)

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Using (3.8) for i = 1,2,· · ·,m − 1 and (3.7) for i = m in the following:

∞

m

∞

m−1

∑[([n]q)2m−1(5[n]q − 2)∏an,i] = ∑[{ ∏ ([n]q)2an,i}[n]q(5[n]q − 2)an,m]

n=2

i=1

n=2 i=1

m−1

∞

≤ ( ∏ a1,i)∑[n]q(5[n]q − 2)an,m

i=1

n=2

.

which is the required condition for f₁∗^′f₂∗^′· · · ∗^′f_mto be in the class S_q^(2m−1)(ψ).

On using the similar arguments as used in the Theorem 7 and Theorem 8 we may obtain the following result:

Theorem 9. Let the functions f_i(i = 1,2,· · ·,m) given by (3.2), belong to the the class K_q(ψ) and the functions g_j(j

= 1,2,· · ·,s) given by (3.3) belong to the class

. Then, the quasi-Hadamard product f₁∗^′f₂∗^′· · · ∗^′f_m∗^′g₁∗^′g₂∗^′· · · ∗^′g_sbelongs to the class

CONCLUSIONS

In this paper, we have used q-calculus to introduce two subclasses

) and K_q(ψ) which are q-analogue of the

classes studied by Kumar and C¸etinkaya [17]. These classes are associated with the analytic function

, which maps an open unit disc onto the leaf shaped bounded region. We investigate several

fundamental aspects of the classes

) and K_q(ψ), including Hadamard products, necessary and sufficient

conditions, and coefficient estimates. Additionally, we establish new results concerning the quasi-Hadamard

products associated with these classes.

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