EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
2025, Vol. 18, Issue 2, Article Number 5823
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Dual Approach to the Generalization of Extended
Bessel Function

Syed Ali Haider Shah1, Hafsa1, Ahmad Alooaily2, Gauhar Rahman3,∗,
Yasser Elmasry4, Salma Haque2, Nabil Mlaiki2

1 Department of Mathematics, University of Sargodha, Sargodha, Pakistan
2 Department of Mathematics and Sciences, Prince Sultan University. Riyadh, 11586
Saudi Arabia
3 Department of Mathematics and Statistics, Hazara University, Mansehra, Pakistan
4 Department of Mathematics- Faculty of Science- King Khalid University, P.O. Box 9004,
Abha 61466, Saudi Arabia.

Abstract. In this paper, we will discuss the geometrical interpretation of generalized Bessel func-
tion, which is defined as:

kHξ,b(z) = z . khξ,b(z) = z +

∞∑
r=1

(−b)r zr+1

r! 4r kr (ξ)r,k

where ξ = v + k ∈ (0,+∞), k ∈ R+, v > −k, b ∈ R. The generalization of Pochammer’s
symbol in the form of inequality:

(q)r,k > q(q + β)r−1

for q > 0, k ∈ R+, 0 ≤ β ≤ β0 =
√
2 ≃ 1.4142..., r ∈ N\{1, 2}, which is proved by using the

generalization of Lemma [1]. This has been proved by many authors by using different methods.
Using this inequality to analyse the order of starlikeness and convexity. We will prove this Lemma
by the same technique used by Zayed and Bulboaca (partial derivative and two-variable extremum
technique). We will give the geometrical interpretation of Generalized Bessel k-function for dif-
ferent values of k. Providing some examples for better understanding of the reader regarding our
approach.

2020 Mathematics Subject Classifications: 30C45, 33C10, 33B15

Key Words and Phrases: Univalent, Starlikeness, Convexity, Pochammer, Gamma Function,
Generalized Bessel function

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v18i2.5823

Email addresses: ali.bukhari78699@gmail.com (S. A. H. Shah), hafsarehman3830@gmail.com (Hafsa)

https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0)



S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5823 2 of 26

1. Introduction

Mathematical Functions that we call Special Functions are very important in different
fields like Applied Mathematics, Physics, Economics, Engineering, Statistics etc. Its a fact
that Special Functions have an imperative space in the solution of many problems. Due
to the unique properties of special functions, these functions play crucial role in solving
differential equations used as mathematical models in different fields. It is linked strongly
with analytic functions series expansion for real and complex variables. Lots of work has
been done in this field of mathematics such as Gamma Functions, Pochammer, Bessel
functions, Hypergeometric Functions, Zeta Function etc.

Here we restrict to the study of Bessel functions which also have important role in
mathematical modeling. Friedrich Wilhelm Bessel was the first who used Bessel functions
to study three body motion. Bessel functions are usually used in solutions of cylindrical
coordinates Boundary value problems. Bessel functions are originated as solution of Bessel
equation [2], that is,

z2w′′(z) + zw′(z) + (z2 − ν2)w = 0,

where ν is the order of Bessel Equation.

Bessel functions are of various kinds. The Bessel function of 1st kind is defined by:

Jν(z) =
∞∑
ν=0

(−1)ν

Γ(ν + k + 1)Γ(k + 1)

(z
2

)ν+2k
.

Bessel function of 2nd kind is also reffered as Weber Function or Neuman Function,
defined as:

Yν(x) =
J−ν(z)cos(νπ)− J−ν(z)

sin(νπ)
.

Here is a 3rd kind named as Hankel Function defined as:

H(1)
ν (z) = Jν(z) + iYν(z)

H(2)
ν (z) = Jν(z)− iYν(z).

For more details about Bessel functions see [2].

Extending the Bessel functions, we have the generalized Bessel functions [3] defined
as:

τJ1
ν (λz) = e−τzx

ν
2 Jν(λ

√
z)

(λ, τ ∈ R+)

τJ2
ν (λz) = eτzx

−ν
2 Jν(λ

√
z).



S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5823 3 of 26

Cesarano and Assante [4] introduced the two-index cylinder generalized Bessel function

Jn,ν(z) =
∞∑

s=−∞
Jn−s(z)Jν−s(z)Js(z).

Bessel function has a wide variety of applications. It is commonly used in the solution
of physical 2nd order differential equation problems [5]. Parand and Nikarya [6] attempted
to solve fractional differential equations using Bessel function. Faisal et.al. [7] gives the
solution of Schrodinger equation in a cylindrical function using Bessel function. More
applications of Bessel functions can be seen in [8, 9].

Researchers are currently working in area of generalized Bessel functions including k,
(s, k) , (p, k), q-Bessel functions and also properties of Bessel, modified Bessel functions
and generalized Bessel functions. One can refer for the recent researches on generalized
Special functions to [10–16].

Univalent function is defined as the function whose domain is meromorphic and injec-
tive. In other words, the function g : D → C∗ is univalent in D if and only if it is analytic
in domain except at most one pole and

g(x1) ̸= g(x2), (x1, x2 ∈ D,x1 ̸= x2).

For more details about univalent functions see [17].

Let S be a set of univalent (meromorphic and injective domain) functions g in U where
U := {y ∈ C : |y| < 1} with g(0) = 0 and g′(0) = 1. Power expansion series [1] for such
functions is of the form:

g(x) =
∞∑
k=1

gkx
k. (1)

The class S is compact (locally bounded and closed) having the most classic example
is Koebe Function, defined as:

k(y) = y(1− y)−2 =
1

4

[(
1 + y

1− y

)2

− 1

]
=

∞∑
k=1

kyk.

Here’s involve the square of Cayley transformation y → 1+y
1−y ∈ S as it is normalized.

A function is known as starlike if its mapping from U onto domain D is starlike. If
the domain is convex, we can simply say it a convex function. Having a convex domain
means that if line segment that joins any two points must lie in domain [18].



S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5823 4 of 26

Let S∗ and K be subclass of S where domain is starlike and convex with respect to
origin respectively. Moreover it is known [19, 20] that g(z) ∈ A is starlike if and only if

Re

(
zg′(z)

g(z)

)
> 0, z ∈ U (2)

where A be the set of analytic (having complex derivatives) functions.

Taking it as of order η, we denote it S∗(η) satisfies the condition:

Re

(
zg′(z)

g(z)

)
> η, z ∈ U, 0 ≤ η ≤ 1. (3)

Also convex function is characterized as g(z) ∈ A satisfies the condition:

1 +Re

(
zg′′(z)

g′(z)

)
> 0, z ∈ U. (4)

Similarly for order η, we can say

1 +Re

(
zg′′(z)

g′(z)

)
> η, z ∈ U, 0 ≤ η ≤ 1. (5)

As S∗(η) ⊂ S∗(0) =: S∗ ⊂ S , K(η) ⊂ K(0) =: K ⊂ S and K(0) ⊂ S∗(0) ⊂ S. For
β < 0, K(η) ⊈ S∗.

Teodor and Hanaa [1] proved the inequality

(a)m > a(a+ α)m−1, a > 0, 0 ≤ α ≤
√
2

by partial derivative and variable extremum method for m ∈ N\{1, 2} and used the in-
equality to examine the order of convexity as well as starlikeness of generalized Bessel
function. Many other researchers [1, 21] have proved its classical form. Teodor and Hanaa
[1] gave the graphical representations of generalized Bessel function, defined by:

Wξ,a = x+
∞∑

m=1

(−a)m

4m(1)m(ξ)m
xm+1, x ∈ U

where
(
ξ = c+ d+2

2

)
/∈ {0,−1,−2, ...} and Pochhammer’s symbol defined by:

(a)p = (a)(a+ 1)(a+ 2)...(a+ p− 1).

In this research paper, we discussed the starlikeness and convexity of order η for nor-
malized form of generalized Bessel function. Also proved the following inequality by partial
derivative and variable extremum method:

(q)r,k > q(q + β)r−1.



S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5823 5 of 26

We also discussed some special cases and examples related to starlikeness and convexity
of normalized form of generalized Bessel function and also discussed starlikeness and con-
vexity conditons by using Silverman’s theorem.

A well-known homogenous differential equation given explicitly by [22]

z2y′′(z) + zy′(z) +
1

k2
(bz2k − v2)y(z) = 0,

gives solution as generalized Bessel k-function, with k ∈ R+ and v > −k.

The generalized Bessel function is defined as:

kWr,b(z) =
∞∑
r=0

(−b)r

r!Γk(rk + v + k)

(z
2

)2r+ v
k
, k ∈ R+, v > −k, b ∈ R. (6)

Γk stands for k-Gamma defined as:

Γk(a) =

∫ ∞

0
ta−1e

−tk

k dt, Re(a) > 0.

Some basic properties of k-Gamma functions [23] are:

Γk(a) = k
a
k
−1Γ

(a
k

)
Γk(a) = aΓk(a)

Γk(k) = 1.

Observe that if k = 1 and b = 1 , the function reduced to classic Bessel function Jv

Jv(z) = 1Wv,1(z) =
∞∑
r=0

(−1)r

r!Γ(r + v + 1)

(z
2

)2r+v
.

If k = 1 and b = −1 , the function reduced to modified Bessel function Iv

Iv(z) = 1Wv,−1(z) =

∞∑
r=0

1

r!Γ(r + v + 1)

(z
2

)2r+v
.

k-digamma function [24] is defined as the logarithmic derivative of k-Gamma function,
which is given as:

ψk(a) =
∂

∂a
log Γk(a) =

Γ′
k(a)

Γk(a)
. (7)



S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5823 6 of 26

Some basic properties of k-digamma function [24, 25] are:

ψk(a+ k) = ψk(a) +
1

a

ψk(a) =
ln k

k
+

1

k
ψ
(x
k

)
.

The function z →k Wv,b(z) /∈ A. So, we define a function originating from kWv,b(z) as:

khv,b(z) =
(
2
√
k
) v

k
Γk(v + k) z−

v
2k kWv,b(

√
z

k
)

=
(
2
√
k
) v

k
Γk(v + k) z−

v
2k

 ∞∑
r=0

(−b)r
( √

z

2
√
k

)2r+ v
k

r! Γk(rk + v + k)


=

∞∑
r=0

(−b)r zr

r! 4r kr (v + k)r,k

=

∞∑
r=0

(−b)r zr

r! 4r kr (ξ)r,k
, (8)

where ξ = v + k /∈ {0,−1,−2, ...}.

2. Main Results

Keeping the above representations in mind, the normalized form of khξ,b(z) is defined
as:

Definition 1. For k ∈ R+, v > −k, b ∈ R, the normalized form of khξ,b(z) is given
by:

kHξ,b(z) = z . khξ,b(z) = z +

∞∑
r=1

(−b)r zr+1

r! 4r kr (ξ)r,k
, (9)

where ξ = v + k ∈ (0,+∞).

We will be in need of the following Lemma in our research. At first, this was proved
in classical form in [21], then proved by Bulboaca and Zayed in [1] by partial derivative
and variable extremum technique. Now we are proving this Lemma for k-Pochammer by
the technique as Bulboaca and Zayed. It is shown for r ∈ N\{1, 2}, q > 0, k ∈ R+, 0 ≤
β ≤

√
2, the inequality satisfied as follows:

Lemma 1. If q > 0, k ∈ R+, 0 ≤ β ≤ β0 =
√
2 ≃ 1.4142... , and r ∈ N\{1, 2}, the results

will sharp
(q)r,k > q(q + β)r−1. (10)



S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5823 7 of 26

Proof. Take fk : (0,+∞)× [3,+∞) → R be defined by:

fk(q, rk) =
Γk(q + rk)

Γk(q + k)
(q + β)1−r − 1, (11)

where 0 ≤ β ≤ 2. By simple computation,

∂

∂r
(fk(q, rk)) =

1

Γk(q + k)

[
∂

∂r

(
Γk(q + rk) (q + β)1−r − 1

)]
=

1

Γk(q + k)

[
Γk(q + rk)

∂

∂r
(q + β)1−r + (q + β)1−r ∂

∂r
Γk(q + rk)

]
=

(q + β)1−r

Γk(q + k)
[−Γk(q + rk) ln(q + β) + Γk(q + rk) ψk(q + rk)] ,

where ψk is k-digamma function defined in eq. 7.

∂

∂r
(fk(q, rk)) =

Γk(q + rk) (q + β)1−r

Γk(q + k)
[ ψk(q + rk)− ln(q + β) ]

=
Γk(q + rk) (q + β)1−r

Γk(q + k)
Gk(q, rk), (12)

where (q, rk) ∈ (0,∞)× [3,∞) and

Gk(q, rk) = ψk(q + rk)− ln(q + β). (13)

Using

Γk(z) =

+∞∫
0

tz−1 e−
tk

k dt

implies that Γk(q + k) > 0, Γk(q + rk) > 0 for all (q, rk) ∈ (0,∞)× [3,∞).

As β > 0, then by eq 13, the sign of ∂
∂rfk(q + rk) is same as of Gk(q, rk).

As it is well known by [26], for x > 0 and 0 < k ≤ 1, we have

1

k
lny − 1

y
< ψk(y) <

1

k
lny. (14)

Using eq. 14 in eq. 13, we get

Gk(q, rk) ≥ 1

k
ln(q+rk)− 1

q + rk
− ln(q+β) = ln

(q + rk)
1
k

q + β
− 1

q + rk
, β > 0, r ≥ 3.

Gk(q, rk) ≥ ln
(q + 3k)

1
k

q + β
− 1

q + 3k
= Ik(q), (15)



S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5823 8 of 26

where Ik(q) = ln (q+3k)
1
k

q+β − 1
q+3k .

By taking derivative of Ik(q),

I ′k(q) =
1

(q+3k)
1
k

q+β

[
d

dq

(
(q + 3k)

1
k

q + β

)]
+

1

(q + 3k)2

=
q + β

(q + 3k)
1
k

[
(q + β) 1

k (q + 3k)
1
k
−1 − (q + 3k)

1
k

(q + β)2

]
+

1

(q + 3k)2

=
1

k(q + 3k)
− 1

q + β
+

1

(q + 3k)2

=
1

q + 3k

[
1

k
− q + 3k

q + β
+

1

q + 3k

]
, q > 0, 0 ≤ β ≤ 2. (16)

And we have the following equivalences:

I ′k(q) < 0 ⇔ q + 3k

q + β
>

1

k
+

1

q + 3k
⇔ β <

k(q + 3k)2

q + 4k
− q (17)

because

inf

{
k(q + 3k)2

q + 4k
− q : q > 0

}
= 2.

If β ≤ 2 , the inequality I ′k(q) < 0 is satisfied, for all q > 0. Hence function Ik(q) is
decreasing strictly on (0,+∞),

⇒ Ik(q) > lim
x→+∞

Ik(y) = 0, q > 0. (18)

By combining eqs. 12 and 15

∂

∂r
(fk(q, rk)) > 0, (q, rk) ∈ (0,∞)× [3,∞)

which implies fk(q, rk) is strictly increasing function on r ∈ [3,+∞), q > 0.

fk(q, rk) ≥ fk(q, 3k) =
Γk(q + 3k)

Γk(q + k)
(q + β)−2 − 1

=
Γk(q + 3k)

Γk(q + k)(q + β)2
− 1



S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5823 9 of 26

=
(q + 2k)(q + k)Γk(q + k)

Γk(q + k)(q + β)2
− 1

=
(q + 2k)(q + k)

(q + β)2
− 1. (19)

Since (q+2k)(q+k)
(q+β)2

− 1 > 0 under conditions q > 0, β > 0, Therefore, we have

(q + 2k)(q + k) > (q + β)2

|q + β| <
√
(q + 2k)(q + k)

β <
√
(q + 2k)(q + k)− q = gk(q). (20)

Taking derivative of gk(q) =
√

(q + 2k)(q + k)− q gives

g′k(q) =
1

2
[(q + 2k)(q + k)]−

1
2
d

da
[(q + 2k)(q + k)]− 1

=
(q + 2k)(q + k)

2
√

(q + 2k)(q + k)
− 1.

By easy computation, we have

g′k(q) =
k2

2
√

(q + 2k)(q + k)(2q + 3k + 2
√
(q + 2k)(q + k))

> 0, q > 0, β ∈ R+

gk is strictly increasing on (0,+∞) , hence

lim
x→0+

gk(q) =
√
2k < gk(q) < lim

x→+∞
gk(q) =

3k

2
. (21)

Consequently if β ≤
√
2

fk(q, rk) > 0, q > 0, r ≥ 3, k ∈ R+. (22)

Left side of inequality 14 could be improved as:

inf

{
ψk(y)−

1

k
ln(y) +

1

y
: y > 0

}
= 0.

As y 7−→ ψk(y)− 1
k ln(y) +

1
y is decreasing and positive on (0,+∞) and by inequality 14,

we have

lim
y→+∞

(
ψk(y)−

1

k
ln(y) +

1

y

)
= 0. (23)

Using eqs. 15 , 19 and 21, we conclude that the value
√
2k is maximum possible to holds

the inequality 22 for sharp results that completes the proof.



S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5823 10 of 26

3. Starlikeness and Convexity of Order η

In this section, the following Theorems provide the starlikeness and convexity of order
η with improved results for generalized Bessel function kHξ,b.

Theorem 1. Assume ξ > 0 and let b ∈ C∗, k ∈ R+ with

0 < |b| < 4kξ

1 + ξ
=: b∗. (24)

If

η ≤ 1− |b|
ξ(4k − |b|)− |b|

=: η∗ (25)

then kHξ,b ∈ S∗(η).

Proof. By considering k-form of the condition in [20], we have∣∣∣∣z(kHξ,b(z))
′

kHξ,b(z)
− 1

∣∣∣∣ < 1− η, η ≤ 1. (26)

Taking

∣∣∣∣(kHξ,b(z))
′ − kHξ,b(z)

z

∣∣∣∣ =
∣∣∣∣∣∣∣∣
(
z +

∞∑
r=1

(−b)r zr+1

r! 4r kr (ξ)r,k

)′

−
z +

∞∑
r=1

(−b)r zr+1

r! 4r kr (ξ)r,k

z

∣∣∣∣∣∣∣∣
=

∣∣∣∣∣∣∣∣1 +
∞∑
r=1

(r + 1)(−b)r zr

r! 4r kr (ξ)r,k
−
z

[
1 +

∞∑
r=1

(−b)r zr

4r kr r! (ξ)r,k

]
z

∣∣∣∣∣∣∣∣
=

∣∣∣∣∣
∞∑
r=1

(r + 1) (−b)r zr

4r kr r! (ξ)r,k
−

∞∑
r=1

(−b)r zr

4r kr r! (ξ)r,k

∣∣∣∣∣
=

∣∣∣∣∣
∞∑
r=1

(r) (−b)r zr

4r kr r! (ξ)r,k

∣∣∣∣∣ < sup
θ∈2π

∣∣∣∣∣
∞∑
r=1

(r) (−b)r eiθr

4r kr r! (ξ)r,k

∣∣∣∣∣
≤

∞∑
r=1

(r) |b|r

4r kr r! Γk(ξ+rk)
Γk(ξ)

, |z| ≤ 1

≤ Γk(ξ + k)

ξ

∞∑
r=1

|b|r (r)

4r kr Γk(r + k)Γk(ξ + rk)
. (27)

Letting the function

σk(a) =
a

Γk(a+ k)Γk(ξ + ak)
, r ∈ N. (28)



S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5823 11 of 26

By easy computation

σk(r + 1)− σk(r) =
r + 1

Γk(r + 1 + k)Γk(ξ + (r + 1)k)
− r

Γk(r + k)Γk(ξ + rk)

=
r + 1− (r)(r + k)(ξ + rk)

(r + k)(ξ + rk)Γk(r + k)Γk(ξ + rk)

=
1 + r − rk(r2 − ξ)− r2(k2 + ξ)

Γk(r + k + 1)Γk(ξ + rk + k)
(29)

which implies that
σk(r + 1)− σk(r) < 0, r ∈ N. (30)

Hence the function is strictly decreasing, so:

a

Γk(a+ k)Γk(ξ + ak)
≤ σk(1) =

1

Γk(1 + k)Γk(ξ + k)
.

By the inequality 27, we get:∣∣∣∣(kHξ,b(z))
′ − kHξ,b(z)

z

∣∣∣∣ ≤ Γk(ξ + k)

ξ

∞∑
r=1

|b|r

4r kr Γk(1 + k)Γk(ξ + k)

=
1

ξ

∞∑
r=1

(
|b|
4k

)r

=
|b|

ξ(4k − |b|)

⇒
∣∣∣∣(kHξ,b(z))

′ − kHξ,b(z)

z

∣∣∣∣ ≤ |b|
ξ(4k − |b|)

, z ∈ U, k ∈ R+. (31)

Now by assuming the above inequality |b|
ξ(4k−|b|) > 0, equivalence to 0 < |b| < 4k which

holds according to 35 and the assumption

0 < |b| < 4kξ

1 + ξ
(32)

The case |b| = 0 is the trivial case gives the identity function.

Now taking the other part, we have∣∣∣∣kHξ,b(z)

z

∣∣∣∣ =
∣∣∣∣∣1 +

∞∑
r=1

(−b)r zr

r! 4r kr (ξ)r,k

∣∣∣∣∣ .
By using triangular inequality and the modulus theorem:∣∣∣∣kHξ,b(z)

z

∣∣∣∣ > 1− sup
θ∈2π

∣∣∣∣∣
∞∑
r=1

(−b)r eiθr

4r kr r! (ξ)r,k

∣∣∣∣∣



S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5823 12 of 26

≥ 1−
∞∑
r=1

|b|r

4r kr r! (ξ)r,k

= 1−
∞∑
r=1

|b|r Γk(ξ)

4r kr Γk(ξ + rk) Γk(r + k)

= 1− Γk(ξ + k)

ξ

∞∑
r=1

|b|r

4r kr Γk(ξ + rk) Γk(r + k)
.

As we know that 1
Γk(ξ+rk) Γk(r+k) is strictly decreasing, we have∣∣∣∣kHξ,b(z)

z

∣∣∣∣ > 1− Γk(ξ + k)

ξ

∞∑
r=1

|b|r

4r kr Γk(ξ + k) Γk(1 + k)

= 1− 1

ξ

∞∑
r=1

(
|b|
4k

)r

= 1− |b|
ξ(4k − |b|)

=
ξ(4k − |b|)− |b|
ξ(4k − |b|)

(33)

where
ξ(4k − |b|)− |b|
ξ(4k − |b|)

> 0. (34)

Above equation holds because ξ > 0 and

|b| < min

{
4;

4kξ

1 + ξ

}
=

4kξ

1 + ξ
. (35)

Since ∣∣∣∣z(kHξ,b(z))
′

kHξ,b(z)
− 1

∣∣∣∣ = ∣∣∣∣(kHξ,b(z))
′ − kHξ,b(z)

z

∣∣∣∣ ∣∣∣∣ z

kHξ,b(z)

∣∣∣∣
<

|b|
ξ(4k − |b|)

× ξ(4k − |b|)
ξ(4k − |b|)− |b|

<
|b|

ξ(4k − |b|)− |b|
≤ 1− η

which gives

η ≤ 1− |b|
ξ(4k − |b|)− |b|

. (36)

Finally from inequality 26, it is proved that kHξ,b ∈ S∗(η).



S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5823 13 of 26

Theorem 2. Assume ξ > 1
2 and let b ∈ C∗, k ∈ R+ with

0 < |b| < 4kξ

2 + ξ
=: bc. (37)

If

η ≤ 1− 2|b|
ξ(4k − |b|)− 2|b|

=: η∗ (38)

then kHξ,b ∈ K(η).

Proof. By considering k-form of the condition in [20], we have∣∣∣∣z(kHξ,b(z))
′′

(kHξ,b(z))′

∣∣∣∣ < 1− η, η ≤ 1. (39)

Taking the part:

∣∣z(kHξ,b(z))
′′∣∣ = ∣∣∣∣∣z

(
z +

∞∑
r=1

(−b)r zr+1

r! 4r kr (ξ)r,k

)′′∣∣∣∣∣
=

∣∣∣∣∣z
(
1 +

∞∑
r=1

(r + 1)(−b)r zr

r! 4r kr (ξ)r,k

)′∣∣∣∣∣
=

∣∣∣∣∣z
( ∞∑

r=1

(r + 1) (r) (−b)r kr zr−1

4r kr r! (ξ)r,k

)∣∣∣∣∣
=

∣∣∣∣∣
∞∑
r=1

(r + 1) (r) (−b)r zr

4r kr r! (ξ)r,k

∣∣∣∣∣ .
By using maximum modulus theorem:

∣∣z(kHξ,b(z))
′′∣∣ = ∣∣∣∣∣

∞∑
r=1

(r + 1)(r)(−b)r zr

4rkrr!(ξ)r,k

∣∣∣∣∣
< sup

θ∈2π

∣∣∣∣∣
∞∑
r=1

(r + 1)(r)(−b)r eiθr

4r krr!(ξ)r,k

∣∣∣∣∣
≤

∞∑
r=1

(r + 1)(r)|b|r

4rkrr!Γk(ξ+rk)
Γk(ξ)

, |z| ≤ 1

=
Γk(ξ + k)

ξ

∞∑
r=1

|b|r(r)(r + 1)

4rkrΓk(r + k)Γk(ξ + rk)
. (40)

Letting the function

ϱk(a) =
(a)(a+ 1)

Γk(a+ k)Γk(ξ + ak)
, r ∈ N. (41)



S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5823 14 of 26

By easy computation

ϱk(r + 1)− ϱk(r) =
(r + 1)(r + 2)

Γk(r + 1 + k)Γk(ξ + (r + 1)k)
− (r)(r + 1)

Γk(r + k)Γk(ξ + rk)

=
r + 1

Γk(r + k)Γk(ξ + rk)

[
r + 2

(r + k)(ξ + rk)
− r

]
=

r + 1

Γk(r + k)Γk(ξ + rk)

[
r + 2− (r)(r + k)(ξ + rk)

(r + k)(ξ + rk)

]
= −

r2
(
ξ + k2

)
+ ξkr + kr3 − r − 2

Γk(r + 1 + k)Γk(ξ + rk + k)

(42)

which implies that:
ϱk(r + 1)− ϱk(r) < 0, r ∈ N. (43)

Hence the function is decreasing strictly, so:

(a)(a+ 1)

Γk(a+ k)Γk(ξ + ak)
≤ ϱk(1) =

2

Γk(1 + k)Γk(ξ + k)
.

By the inequality 40, we get:

∣∣z(kHξ,b(z))
′′∣∣ < Γk(ξ + k)

ξ

∞∑
r=1

2 |b|r

4r kr Γk(1 + k)Γk(ξ + k)

=
2

ξ

∞∑
r=1

(
|b|
4k

)r

=
2|b|

ξ(4k − |b|)∣∣z(kHξ,b(z))
′′∣∣ < 2|b|

ξ(4k − |b|)
, z ∈ U, k ∈ R+. (44)

Now by assuming the above inequality 44 is greater than 0, gives

0 < |b| < 4k, (45)

which holds because of our assumption. The case |b| = 0 is the trivial case gives the
identity function.

Now the other part, we have

∣∣(kHξ,b(z))
′∣∣ = ∣∣∣∣∣

(
z +

∞∑
r=1

(−b)r zr+1Γk(ξ)

kr 4r Γk(r + k) Γk(ξ + rk)

)′∣∣∣∣∣



S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5823 15 of 26

=

∣∣∣∣∣1 +
∞∑
r=1

(−b)r zr (r + 1)Γk(ξ + k)

ξ kr 4r Γk(r + k) Γk(ξ + rk)

∣∣∣∣∣ .
By using triangular inequality and the modulus theorem:∣∣(kHξ,b(z))

′∣∣ > |eiθr| − sup
θ∈2π

∣∣∣∣∣
∞∑
r=1

(r + 1) (−b)r eiθr Γk(ξ + k)

4r kr Γk(r + k) Γk(ξ + rk)

∣∣∣∣∣
≥ 1− Γk(ξ + k)

ξ

∞∑
r=1

(r + 1) |b|r

4r kr Γk(ξ + rk) Γk(r + k)
.

As we know that 1
Γk(ξ+rk) Γk(r+k) is strictly decreasing, we have

∣∣(kHξ,b(z))
′∣∣ > 1− Γk(ξ + k)

ξ

∞∑
r=1

2 |b|r

4r kr Γk(ξ + k) Γk(1 + k)

= 1− 2

ξ

∞∑
r=1

(
|b|
4k

)r

= 1− 2|b|
ξ(4k − |b|)

=
ξ(4k − |b|)− 2|b|

ξ(4k − |b|)
(46)

where
ξ(4k − |b|)− 2|b|

ξ(4k − |b|)
> 0. (47)

Above equation holds because ξ ≥ 0 and

|b| < min

{
4;

4kξ

2 + ξ

}
=

4kξ

2 + ξ
. (48)

Since ∣∣∣∣z(kHξ,b(z))
′′

(kHξ,b(z))′

∣∣∣∣ = ∣∣z(kHξ,b(z))
′′∣∣ ∣∣∣∣ 1

(kHξ,b(z))′

∣∣∣∣
<

2|b|
ξ(4k − |b|)

× ξ(4k − |b|)
ξ(4k − |b|)− 2|b|

<
2|b|

ξ(4k − |b|)− 2|b|
≤ 1− η

which gives

η ≤ 1− 2|b|
ξ(4k − |b|)− 2|b|

. (49)

Finally from inequality 39, it is proved that kHξ,b ∈ K(η).



S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5823 16 of 26

4. Examples

This section illustrates some examples to check convexity and starlikeness of different
orders:

Example 1. Taking ξ = 2.2565, b = 0.0253, η = 0 and k = 0.1. To check 0.1H2.2565,0.0253 ∈
S∗(0) or 0.1H2.2565,0.0253 ∈ K(0) , we will check the sufficient conditions for starlikeness
and convexity, that is:
For ξ > 0 and let b ∈ C∗, k ∈ R+ with

0 < |b| < 4kξ

1 + ξ
=: b∗. (50)

If

η ≤ 1− |b|
ξ(4k − |b|)− |b|

=: η∗ (51)

then kHξ,b ∈ S∗(η). Above parameter values satisfy the condition;

0 < |0.0253| < 4× 0.1(2.2565)

2.2565 + 1
= 0.277169.

and

0 ≤ 1− |0.0253|
2.2565(4× 0.1− |0.0253|)− |0.0253|

= 0.969154.

Both conditions satisfied, hence 0.1H2.2565,0.0253 ∈ S∗(0).
Now check the conditions for convexity: For ξ > 1

2 and let b ∈ C∗, k ∈ R+ with

0 < |b| < 4kξ

2 + ξ
=: bc. (52)

If

η ≤ 1− 2|b|
ξ(4k − |b|)− 2|b|

=: η∗ (53)

then kHξ,b ∈ K(η). Above parameter values satisfy the condition;

0 < |0.0253| < 4× 0.1(2.2565)

2.2565 + 2
= 0.212052.

and

0 ≤ 1− 2|0.0253|
2.2565(4(0.1)− |0.0253|)− 2|0.0253|

= 0.936345.

Both conditions satisfied, hence 0.1H2.2565,0.0253 ∈ K(0).



S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5823 17 of 26

(a) 0.1H2.2565,0.0253 (b) 0.4H2.2565,0.0253 (c) 0.8H2.2565,0.0253

(d) 1H2.2565,0.0253 (e) H2.2565,0.0253

Figures are the illustrations of the Example 1 which show the contour plot of generalized
Bessel k-function for different values of k. It can be observed that the graphs show the
symmetrical behaviour and provide a way to check the accuracy of our results i.e., as k
approaches to 1, the graphical behaviour will approach the classical form of generalized
Bessel function. Exactly at k = 1, the graphs are same.

Example 2. Taking ξ = 2.5, b = 1.2, η = 0.32 and k = 0.6. To check 0.6H2.5,1.2 ∈ S∗(0.32)
or 0.6H2.5,1.2 ∈ K(0.32).

We will check the sufficient conditions for starlikeness and convexity, that is: For ξ > 0
and let b ∈ C∗, k ∈ R+ with

0 < |b| < 4kξ

1 + ξ
=: b∗. (54)

If

η ≤ 1− |b|
ξ(4k − |b|)− |b|

=: η∗ (55)

then kHξ,b ∈ S∗(η). Above parameter values satisfy the condition;

0 < |1.2| < 4× 0.6(2.5)

2.5 + 1
= 1.71429.

Now check for:

0.32 ≤ 1− |1.2|
2.5(4× 0.6− |1.2|)− |1.2|

= 0.333333.



S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5823 18 of 26

Both conditions satisfied, hence 0.6H2.5,1.2 ∈ S∗(0.32).

Now check the conditions for convexity: For ξ > 1
2 and let b ∈ C∗, k ∈ R+ with

0 < |b| < 4kξ

2 + ξ
=: bc. (56)

If

η ≤ 1− 2|b|
ξ(4k − |b|)− 2|b|

=: η∗ (57)

then kHξ,b ∈ K(η). Above parameter values do not satisfy the conditions;

0 < |1.2| < 4× 0.6(2.5)

2.5 + 2
= 1.3333.

0.32 ≰ 1− 2× 2.5

2.5(4× 0.6− |1.2|)− 2|1.2|
= −3.

Hence 0.6H2.5,1.2 /∈ K(0.32).

(f) 0.6H2.5,1.2 (g) 0.8H2.5,1.2 (h) 0.9H2.5,1.2

(i) 1H2.5,1.2 (j) H2.5,1.2

Figures are the illustrations of the Example 2 which show the contour plot of general-
ized Bessel k-function for different values of k. It can be observed that the graphs show
the symmetrical domains with respect to real axis and provide a way to check the accuracy
of our results i.e., as k approaches to 1, the graphical behaviour will approach the classical
form of generalized Bessel function. Exactly at k = 1, the graphs are same.



S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5823 19 of 26

5. Order of Starlikeness and Convexity by Silverman’s Theorem

In this section, the Theorem describes the sufficient condition for starlikeness and
convexity of order η by using the Lemma 1 and a result from by Silverman ([27], Theorem
1).

Theorem 3. Let

kXξ,b =

(
1− |b|

4ξ
− |b|2

32ξ(ξ + k)
− |b|3

16ξ(1 +
√
2)(ξ +

√
2k)(4ξ + 4

√
2ξ + 4

√
2k + 8k − |b|)

)
η

+
|b|
2ξ

+
3|b|2

32ξ(ξ + k)
+

3|b|3

64ξ(1 +
√
2)(ξ +

√
2k)(ξ +

√
2ξ +

√
2k + 2k − |b|)

+
|b|3

16ξ(1 +
√
2)(ξ +

√
2k)(4ξ + 4

√
2ξ + 4

√
2k + 8k − |b|)

− 1

where ξ > 2 and b ∈ C∗ . If there exists η < 1 such that

kXξ,b ≤ 0, (58)

then kHξ,b ∈ S∗(η).

Proof. From Silverman’s [27] well-known result, if f is in form of eq. 1 and satisfies
∞∑
r=2

(r − η)|fk| ≤ 1− η, then f ∈ S∗(η). According to eq. 9, it is sufficient to show:

B1 :=

∞∑
r=2

(r − η)

∣∣∣∣ (−b)r−1

4r−1(1)r−1,k(ξ)r−1,k

∣∣∣∣ ≤ 1− η, ξ > 0, b ∈ C∗

=

∞∑
r=2

(r − η)|b|r−1

4r−1(1)r−1,k(ξ)r−1,k

=
∞∑
r=1

(r + 1− η)|b|r

4r(1)r(ξ)r,k

=
(2− η)|b|

4ξ
+

(3− η)|b|2

8(ξ)2,k(1)2,k
+

∞∑
r=3

(1− η)|b|r

4r(1)r,k(ξ)r,k
+

∞∑
r=3

(r)|b|r

4r(1)r,k(ξ)r,k

=
(2− η)|b|

4ξ
+

(3− η)|b|2

8(ξ)2,k(1)2,k
+ (1− η)

∞∑
r=3

|b|r

4r(1)r,k(ξ)r,k
+

∞∑
r=3

(r)|b|r

4r(1)r,k(ξ)r,k
.

By simple mathematical induction, we have r ≤
(

3
64

)
4r for all r ∈ N\[1, 2]. By using

Lemma, we have

(1)r,k > (1 + β)r−1, r ∈ N\[1, 2], 0 ≤ β ≤
√
2.

Since
max(1 + β)r−1 : 0 ≤ β ≤

√
2 = (1 +

√
2)r−1



S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5823 20 of 26

follows that:
(ξ)r,k ≥ ξ(ξ +

√
2)r−1, ξ > 0. (59)

As by eq. 58, kXξ,b(η) ≤ 0 , we have

B1 ≤
(2− η)|b|

4ξ
+

(3− η)|b|2

32ξ(ξ + k)
+

3|b|
64ξ

∞∑
r=3

|b|r−1

(1 +
√
2)r−1(ξ +

√
2k)r−1

+
(1− η)|b|

4ξ

∞∑
r=3

|b|r−1

4r−1(1 +
√
2)r−1(ξ +

√
2k)r−1

.

After simplifications, we get:

B1 =
(2− η)|b|

4ξ
+

(3− η)|b|2

32ξ(ξ + k)
+

3|b|3

64ξ(1 +
√
2)(ξ +

√
2k)((1 +

√
2)(ξ +

√
2k)− |b|)

+
(1− η)|b|3

16ξ(1 +
√
2)(ξ +

√
2k)(4(1 +

√
2)(ξ +

√
2k)− |b|)

=
(2− η)|b|

4ξ
+

(3− η)|b|2

32ξ(ξ + k)
+

3|b|3

64ξ(1 +
√
2)(ξ +

√
2k)(ξ + ξ

√
2 +

√
2k + 2k − |b|)

+
(1− η)|b|3

16ξ(1 +
√
2)(ξ +

√
2k)(4ξ + 4

√
2ξ + 4

√
2k + 8k − |b|)

≤ 1− η. (60)

This completes the proof.

Example 3. Some special cases of Theorem 3 gives the following different situations:

Case-1

For b = 0.1, η = 0.1 and ξ = 2.05166.

(k) 0.8H2.05166,0.1 (l) 0.9H2.05166,0.1

This shows that the behaviour of the graph is same whether we change the values of k.
Exactly at k = 1, the conditions and graph approaches to classical form which gives the
accuracy of our results.



S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5823 21 of 26

Case-2

For b = 1.2 , η = 0.2 and ξ = 2.8566.

(m) 0.8H2.8566,1.2 (n) 0.9H2.8566,1.2

This shows that the behaviour of the graph is same whether we change the values of k.
Exactly at k = 1, the conditions and graph approaches to classical form which gives the
accuracy of our results.

Theorem 4. Let

kYξ,b =

(
1− |b|

2ξ
− 3|b|2

32ξ(ξ + k)
− 3|b|3

64ξ(1 +
√
2)(ξ +

√
2k)(ξ +

√
2ξ +

√
2k + 2k − |b|)

− |b|3

16ξ(1 +
√
2)(ξ +

√
2k)(4ξ + 4

√
2ξ + 4

√
2k + 8k − |b|)

)
η

+
|b|
ξ

+
9|b|2

32ξ(ξ + k)
+

15|b|3

64ξ(1 +
√
2)(ξ +

√
2k)(ξ +

√
2ξ +

√
2k + 2k − |b|)

+
|b|3

16ξ(1 +
√
2)(ξ +

√
2k)(4ξ + 4

√
2ξ + 4

√
2k + 8k − |b|)

− 1

where ξ > 3 and b ∈ C∗ . If η < 1 then

kYξ,b ≤ 0, (61)

Hence kHξ,b ∈ K(η).

Proof. From Silverman’s [27] well-known result, if f is in form of eq. 1 and satisfies
∞∑
r=2

r(r − η)|fk| ≤ 1− η, then f ∈ K(η). According to eq. 9, it is sufficient to show:

B2 :=

∞∑
r=2

r(r − η)

∣∣∣∣ (−b)r−1

4r−1(1)r−1,k(ξ)r−1,k

∣∣∣∣ ≤ 1− η, ξ > 0, b ∈ C∗

=

∞∑
r=2

r(r − η)|b|r−1

4r−1(1)r−1,k(ξ)r−1,k



S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5823 22 of 26

=

∞∑
r=1

(r + 1)(r + 1− η)|b|r

4r(1)r,k(ξ)r,k

=
2(2− η)|b|

4ξ
+

3(3− η)|b|2

16(ξ)2,k(1)2,k
+

∞∑
r=3

(r + 1)(r + 1− η)|b|r

4r(1)r,k(ξ)r,k

=
(2− η)|b|

2ξ
+

3(3− η)|b|2

32ξ(ξ + 1)
+

∞∑
r=3

r2|b|r

4r(1)r,k(ξ)r,k
+ (2− η)

∞∑
r=3

r|b|r

4r(1)r,k(ξ)r,k

+ (1− η)

∞∑
r=3

|b|r

4r(1)r,k(ξ)r,k
.

By simple mathematical induction, we have r2 ≤
(

9
64

)
4r and r ≤

(
9
64

)
4r for all r ∈ N\[1, 2]

. By using Lemma, we have

(1)r,k > (1 + β)r−1, r ∈ N\[1, 2], 0 ≤ β ≤
√
2.

Since
max(1 + β)r−1 : 0 ≤ β ≤

√
2 = (1 +

√
2)r−1

follows that:
(ξ)r,k ≥ ξ(ξ +

√
2)r−1, ξ > 0. (62)

As by eq. 61, kYξ,b(η) ≤ 0 , we have

B2 ≤
(2− η)|b|

2ξ
+

3(3− η)|b|2

32ξ(ξ + k)
+

(9 + 3(2− η))|b|
64ξ

∞∑
r=3

|b|r−1

(1 +
√
2)r−1(ξ +

√
2k)r−1

+
(1− η)|b|

4ξ

∞∑
r=3

|b|r−1

4r−1(1 +
√
2)r−1(ξ +

√
2k)r−1

.

After simplifications, we get:

B2 =
(2− η)|b|

2ξ
+

3(3− η)|b|2

32ξ(ξ + k)
+

3(5− η)

64ξ
.

|b|3

(1 +
√
2)(ξ +

√
2k)(2k +

√
2k + ξ + ξ

√
2− |b|

+
(1− η)|b|3

16ξ(1 +
√
2)(ξ +

√
2k)(8k + 4

√
2k + 4ξ + 4ξ

√
2− |b|

=
(2− η)|b|

2ξ
+

3(3− η)|b|2

32ξ(ξ + k)
+

3(5− η)

64ξ
.

|b|3

(1 +
√
2)(ξ +

√
2k)(2k +

√
2k + ξ + ξ

√
2− |b|

+
(1− η)|b|3

16ξ(1 +
√
2)(ξ +

√
2k)(8k + 4

√
2k + 4ξ + 4ξ

√
2− |b|

≤ 1− η. (63)

This completes the proof, hence kHξ,b ∈ K(η) .

Example 4. Some special cases of Theorem 4 gives the following different situations:

Case-1

For b = 1.2, η = 0.3 and ξ = 5.87545.



S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5823 23 of 26

(o) 0.8H5.87545,1.2 (p) 0.9H5.87545,1.2

This shows that the behaviour of the graph is same whether we change the values of k.
Exactly at k = 1, the conditions and graph approaches to classical form which shows the
accuracy of our results.

Case-2

For b = 1.2, η = 0.4 and ξ = 4.7998.

(q) 0.9H4.7998,1.2 (r) 0.9H4.7998,1.2

This shows that the behaviour of the graph is same whether we change the values of k.
Exactly at k = 1, the conditions and graph approaches to classical form which shows the
accuracy of our results.

Conclusions
In our current findings, we have discussed the geometrical interpretation for different val-
ues of k.

(i) The generalization of Pochammer’s symbol in the form of inequality (q)r,k > q(q +
β)r−1 is proved by using the generalization of Lemma [1]

(ii) Convexity and Starlikeness of order η for generalized Bessel function. Theorem 1 in
our paper is the generalized form of theorem 2.1 from [1], and Theorem 2 is gener-
alization of Theorem 2.2 in [1]. It has given the sufficient conditions for finding the



S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5823 24 of 26

order of starlikeness and Convexity respectively.

(iii) Convexity and Starlikeness of order η by Silverman’s theorem for generalized Bessel
function is proved. Theorem 3 in our paper is the extended form of Theorem 4.1
from [1], and Theorem 4 is generalization of Theorem 4.2 in [1]. It gives sufficient
condition on ξ ,b and its proof uses the Lemma 1.

(iv) For the reader’s help, illustrated some examples with graphs to estimate our ap-
proach. There is a comparison between different graphs with different values of
k. By observing them, it is concluded that the behaviour of the graphs is same
whether the values of k changes and approaches to the graph of classical function as
k approaches to 1.

Acknowledgements

The authors extend their appreciation to the Deanship of Research and Graduate
Studies at King Khalid University for funding this work through Large Research Project
under grant number RGP2/588/45.
The authors A. Aloqaily, S. Haque, N. Mlaiki would like to thank Prince Sultan University
for paying the APC and for the support through the TAS research lab.

References

[1] T Bulboacă and H M Zayed. Analytical and geometrical approach to the generalized
bessel function. Journal of Inequalities and Applications, 2024(1):51, 2024.

[2] G Dattoli and A Torre. Theory and applications of generalized bessel functions.
Aracne Rome, 1996.

[3] H Khosravian-Arab, M Dehghan, and M R Eslahchi. Generalized bessel func-
tions: Theory and their applications. Mathematical Methods in the Applied Sciences,
40(18):6389–6410, 2017.

[4] C Cesarano and D Assante. A note on generalized bessel functions. International
Journal of Mathematical Models and Methods in Applied Sciences, 7(6):625–629, 2013.

[5] J Niedziela. Bessel functions and their applications. University of Tennessee-
Knoxville,(Dated: October 29, 2008), 2008.

[6] K Parand and M Nikarya. Application of bessel functions for solving differential and
integro-differential equations of the fractional order. Applied Mathematical Modelling,
38(15-16):4137–4147, 2014.

[7] F A Idris, A L Buhari, and T U Adamu. Bessel functions and their applications: So-
lution to schrödinger equation in a cylindrical function of the second kind and hankel
functions. International Journal Novel Research in Physics Chemistry & Mathemat-
ics, 3:17–31, 2016.



S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5823 25 of 26

[8] Robert Reynolds and Allan Stauffer. Double integral involving the product of the
bessel function of the first kind and modified bessel function of the second kind:
Derivation and evaluation. European Journal of Pure and Applied Mathematics,
15(3):856–863, 2022.

[9] Robert Reynolds and Allan Stauffer. Triple integral involving the bessel-integral
function jiv (z): Derivation and evaluation. European Journal of Pure and Applied
Mathematics, 15(3):916–923, 2022.

[10] B A Frasin and I Aldawish. On subclasses of uniformly spiral-like functions associated
with generalized bessel functions. Journal of Function Spaces, 2019(1):1329462, 2019.

[11] Syed Ali Haider Shah and Shahid Mubeen. Expressions of the laguerre polynomial
and some other special functions in terms of the generalized meijer g-functions. AIMS
Math, 6:11631–11641, 2021.

[12] R M Ali, S K Lee, and S R Mondal. Inequalities on an extended bessel function.
Journal of Inequalities and Applications, 2018:1–22, 2018.

[13] Á Baricz. Generalized bessel functions of the first kind. Lecture Notes in
Mathematics/Springer-Verlag, 2010.

[14] Á Baricz. Functional inequalities for galué’s generalized modified bessel functions.
Journal of Mathematical Inequalities, 1(2):183–193, 2007.

[15] R Ali, M Kamran, and I Nayab. Some results of generalized k-fractional integral
operator with k-bessel function. Turkish Journal of Science, 5(3):157–169, 2020.

[16] H M Srivastava. An introductory overview of bessel polynomials, the generalized
bessel polynomials and the q-bessel polynomials. Symmetry, 15(4):822, 2023.

[17] D Girela. Basic theory of univalent functions. In Conference Paper. Universidad de
Málaga, 2013.

[18] D K Thomas, N Tuneski, and A Vasudevarao. Univalent Functions: A Primer,
volume 69. Walter de Gruyter GmbH & Co KG, 2018.

[19] S R Mondal and A Swaminathan. Geometric properties of generalized bessel func-
tions. Bulletin of the Malaysian Mathematical Sciences Society, 35(1), 2012.

[20] H M Zayed and T Bulboacă. Normalized generalized bessel function and its geometric
properties. Journal of Inequalities and Applications, 2022(1):158, 2022.

[21] Á Baricz and S Ponnusamy. Starlikeness and convexity of generalized bessel functions.
Integral Transforms and Special Functions, 21(9):641–653, 2010.

[22] S R Mondal and M S Akel. Differential equation and inequalities of the generalized
k-bessel functions. Journal of Inequalities and Applications, 2018:1–14, 2018.

[23] R Diaz and E Pariguan. On hypergeometric functions and pochhammer k-symbol.
arXiv preprint math/0405596, 2004.

[24] S Ahmed. Some integrals involving k-gamma and k-digamma function. Journal of
the Egyptian Mathematical Society, 28(1):39, 2020.

[25] E M R A H Yildirim and I N C I Ege. On k-analogues of digamma and polygamma
functions. Journal of Classical Analysis, 13(2):123–131, 2018.

[26] K Nantomah, F Merovci, and S Nasiru. Some monotonicity properties and inequalities
for the (p, k)-gamma function. Kragujevac Journal of Mathematics, 42(2):287–297,
2018.



S. A. H. Shah et al. / Eur. J. Pure Appl. Math, 18 (2) (2025), 5823 26 of 26

[27] H Silverman. Univalent functions with negative coefficients. Proceedings of the Amer-
ican Mathematical Society, 51(1):109–116, 1975.


	Introduction
	Main Results
	Starlikeness and Convexity of Order 
	Examples
	Order of Starlikeness and Convexity by Silverman's Theorem