International Journal of Analysis and Applications

Volume 16, Number 4 (2018), 462-471

URL: https://doi.org/10.28924/2291-8639

DOI: 10.28924/2291-8639-16-2018-462

CONVEXITY OF INTEGRAL OPERATORS INVOLVING DINI FUNCTIONS

SADDAF NOREEN, MUHEY U DIN AND MOHSAN RAZA∗

Department of Mathematics, Government College University Faisalabad, Pakistan

∗Corresponding author: mohsan976@yahoo.com

Abstract. In this article, we are mainly interested to find some covexity properties for certain families of

integral operators involving Dini functions in the open unit disc. The main tool in the proofs of our results

are some functional inequalities of Dini functions. Some particular cases involving Dini functions are also a

part of our investigations.

1. INTRODUCTION

Let A denote the class of functions f of the form

f(z) = z +

∞∑
n=2

anz
n, (1.1)

which are analytic in the open unit disc U = {z : |z| < 1} and S denote the class of all functions which

are univalent in U. Let S∗ (α) , C (α) and K(α) denote the classes of starlike, convex and close-to-convex

functions of order α and are defined as:

S∗ (α) =
{
f : f ∈A and Re

(
zf′ (z)

f (z)

)
> α, z ∈U, α ∈ [0, 1)

}
,

C (α) =
{
f : f ∈A and Re

(
1 +

zf′′ (z)

f′ (z)

)
> α, z ∈U, α ∈ [0, 1)

}
and

K(α) =
{
f : f ∈A and Re

(
zf′ (z)

g (z)

)
> α, z ∈U, α ∈ [0, 1) , g ∈S∗

}
.

Received 2018-01-31; accepted 2018-04-06; published 2018-07-02.

2010 Mathematics Subject Classification. 30C45, 33C10, 30C20, 30C75.

Key words and phrases. analytic functions; convex functions; integral operators; Bessel functions; Dini functions.

c©2018 Authors retain the copyrights

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

462

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-16-2018-462


Int. J. Anal. Appl. 16 (4) (2018) 463

It is clear that

S∗ (0) = S∗, C (0) = C and K(0) = K.

Special functions have great importance in pure and applied mathematics. The widely use of these functions

have attracted many researchers to work on the different directions. Geometric properties of special func-

tions such as Hypergeometric functions, Bessel functions, Struve functions, Mittage-Lefller functions, Wright

functions and some other related functions is an ongoing part of research in geometric function theory. We

refer for some geometric properties of these functions [1, 2, 5, 6, 11] and references therein.

The Bessel function of the first kind Jv is defined by

Jv(z) =

∞∑
n=1

(−1)n

n!Γ(v + n + 1)

(z
2

)2n+v
, (1.2)

where Γ stands for Euler gamma function. It is a particular solution of the second order linear homogeneous

differential equation

z2w′′(z) + zw′(z) + (z2 −v2)w(z) = 0,

where v ∈ C. For some details see [2, 13]. Bessel functions are indispensable in many branches of pure and

applied mathematics. Thus, it is important to study their properties in many aspects. Recently Baricz

et al [3] studied the close-to-convexity of Dini functions and some monotonicity properties and functional

inequalities for the modified Dini function are discussed in [4]. Further some geometric properties of Dini

functions are studied in [3, 4, 7]. Now, we consider the normalized Dini functions qυ : U → C defined as

qυ (z) = 2
υΓ (υ + 1) z1−

υ
2

(
(1 −υ) Jυ

(√
z
)

+
√
zJ′υ

(√
z
))

= z +

∞∑
n=1

(−1)n (2n + 1) Γ (υ + 1)
4nn!Γ (υ + n + 1)

zn+1, z ∈U. (1.3)

The Pochhammer (or Appell) symbol, defined in terms of Euler’s gamma functions is given as (x)n =

Γ(x + n)/Γ(x) = x(x + 1)...(x + n− 1).

Recently, Deniz et al. [8], Din et al. [9], Din et al. [10] and Srivastava et al. [12] have obtained sufficient

conditions for the univalence of certain families of integral operators defined by Bessel, Dini, Struve and

Mittage-Leffler functions respectively. The families of integral operators are defined below:

Fα1,...,αn,ζ (z) =


ζ

z∫
0

tζ−1
n∏
i=1

(
fi(t)

t

) 1
αi

dt




1/ζ

, (1.4)

Gξ,n (z) =


(nξ + 1)

z∫
0

n∏
i=1

(fi (t))
ξ
dt




1/(nξ+1)

, (1.5)

Hδ1,...,δn,µ (z) =


µ

z∫
0

tµ−1
n∏
i=1

(f′i(t))
δi
dt




1/µ

(1.6)



Int. J. Anal. Appl. 16 (4) (2018) 464

and

Qλ (z) =


λ

z∫
0

tλ−1
(
ef(t)

)λ
dt




1/λ

. (1.7)

In this paper, we are mainly interested in the convexity of the integral operators involving Dini function

qv. These integral operators are defined as

Fv1,...,vn,α1,...,αn (z) =

z∫
0

n∏
i=1

(
qvi(t)

t

)αi
dt (1.8)

and

Hv1,...,vn,δ1,...,δn (z) =

z∫
0

n∏
i=1

(
q′vi(t)

)δi
dt. (1.9)

Now we prove some functional inequalities which are useful in establishing our main results.

Lemma 1.1. Let v ∈ R and consider the normalized Dini function qυ : U → C, defined by

qυ (z) = 2
υΓ (υ + 1) z1−

υ
2

(
(1 −υ) Jυ

(√
z
)

+
√
zJ′υ

(√
z
))
,

where Jυ is the Bessel function of first kind. Then the following inequalities hold for all z ∈U.

(i)
∣∣∣zq′υ(z)qυ(z) − 1∣∣∣ ≤ 3v+64v2+5v−2, v > −5+√578 ,

(ii)
∣∣∣zq′′υ(z)q′υ(z)

∣∣∣ ≤ 3v+62v2+v−4, v > −1+√334 .
Proof. (i) By using the well known triangle inequality with the equality

Γ (v + 1)

Γ (v + n + 1)
=

1

(v + 1) (v + 2) · · ·(v + n)
=

1

(v + 1)n
, n ∈ N

and the inequality

4nn! (v + 2)n−1 ≥
4

3
n (2n + 1) (v + 2)

n−1
, n ∈ N,

we obtain

∣∣∣∣q′υ (z) − qυ (z)z
∣∣∣∣ ≤ 34 (v + 1)

∞∑
n=1

(
1

v + 2

)n−1
=

3 (v + 2)

4 (v + 1)
2
.

Furthermore, if we use the reverse triangle inequality and the inequality

4nn! (v + 2)n−1 ≥
1

3
(2n + 1) (v + 2)

n−1
, n ∈ N,



Int. J. Anal. Appl. 16 (4) (2018) 465

then we get ∣∣∣∣qυ (z)z
∣∣∣∣ =

∣∣∣∣∣1 +
∞∑
n=1

(−1)n (2n + 1) Γ (υ + 1)
4nn!Γ (υ + n + 1)

∣∣∣∣∣
≥ 1 −

3

4 (v + 1)

∞∑
n=0

(
1

v + 2

)n−1

=
4 (v + 1)

2 − 3 (v + 2)
4 (v + 1)

2
.

By combining the above inequalities, we get∣∣∣∣zq′υ (z)qυ (z) − 1
∣∣∣∣ ≤ 3v + 64v2 + 5v − 2, v > −5 +

√
57

8
.

(ii) By using the well known triangle inequality with equality

Γ (v + 1)

Γ (v + n + 1)
=

1

(v + 1) (v + 2) · · ·(v + n)
=

1

(v + 1)n
, n ∈ N

and the inequality

4nn! (v + 2)n−1 ≥
4

3
(2n + 1) (v + 2)

n−1
, n ∈ N,

we have

|zq′′υ (z)| ≤

∣∣∣∣∣
∞∑
n=1

n (n + 1) (2n + 1) Γ (υ + 1)

4nn!Γ (υ + n + 1)

∣∣∣∣∣
≤

3

2 (v + 1)

∞∑
n=1

(
1

v + 2

)n−1
=

3 (v + 2)

2 (v + 1)
2
.

Furthermore, if we use the reverse triangle inequality and the inequality

4nn! (v + 2)n−1 ≥
2

3

(
2n2 + 3n + 1

)
(v + 2)

n−1
, n ∈ N,

then we get

|q′υ (z)|

∣∣∣∣∣≥ 1 −
∞∑
n=1

(2n + 1) (n + 1) Γ (υ + 1)

4nn!Γ (υ + n + 1)

∣∣∣∣∣
≥ 1 −

3

2 (v + 1)

∞∑
n=1

(
1

v + 2

)n−1
=

2v2 + v − 4
4 (v + 1)

2

By combining the above inequalities, it can be easily obtained∣∣∣∣zq′′υ (z)q′υ (z)
∣∣∣∣ ≤ 3v + 62v2 + v − 4, v > −1 +

√
33

4
.

�



Int. J. Anal. Appl. 16 (4) (2018) 466

2. Convexity of Integral Operators Defined by Generalized Dini Functions

The main objective of this paper is to give convexity properties of integral operators involving Dini

function. The main results are given as follows.

Theorem 2.1. Let v1, . . . ,vn >
−5+

√
57

8
, where n ∈ N. Let qυi : U → C be defined as

qυi (z) = 2
υiΓ (υ + 1) z1−

υi
2

(
(1 −υi) Jυi

(√
z
)

+
√
zJ′υi

(√
z
))
. (2.1)

Suppose that v = min{v1, . . . ,vn} and α1, . . . ,αn be positive real numbers such that these numbers satisfy

the following inequality

0 ≤ 1 −
3v + 6

4v2 + 5v − 2

n∑
i=1

αi < 1.

Then, the function Fv1,...,vn,α1,...,αn defined by (1.8), is in the class C (β), where

β = 1 −
3v + 6

4v2 + 5v − 2

n∑
i=1

αi.

Proof. We observe that qυi,∀i = 1, 2, · · ·n are such that qυi (0) = q′υi (0) − 1 = 0. It is also clear that

Fv1,...,vn,α1,...,αn ∈ A. That is Fv1,...,vn,α1,...,αn (0) = F ′v1,...,vn,α1,...,αn (0) − 1 = 0. On the other hand, it is

easy to see that

F ′v1,...,vn,α1,...,αn (z) =

n∏
i=1

(
qvi(z)

z

)αi
. (2.2)

Differentiating logarithmically, we get

zF ′′v1,...,vn,α1,...,αn (z)

F ′v1,...,vn,α1,...,αn (z)
=

n∑
i=1

αi

(
zq′vi (z)

qvi (z)
− 1
)
. (2.3)

This implies that

Re

{
1 +

zF ′′v1,...,vn,α1,...,αn (z)

F ′v1,...,vn,α1,...,αn (z)

}
=

n∑
i=1

αiRe

(
zq′vi (z)

qvi (z)

)
+

(
1 −

n∑
i=1

αi

)
.

Now, by using the assertion (i) of Lemma 1.1 for each vi, where i = 1, 2, · · ·n, we obtain

Re

{
1 +

zF ′′v1,...,vn,α1,...,αn (z)

F ′v1,...,vn,α1,...,αn (z)

}
≥

n∑
i=1

αi

(
1 −

3vi + 6

4v2i + 5vi − 2

)
+

(
1 −

n∑
i=1

αi

)

= 1 −
n∑
i=1

αi
3vi + 6

4v2i + 5vi − 2
.

Consider the function φ :
(
−5+

√
57

8
,∞
)
→ R defined as

φ (x) =
3x + 6

4x2 + 5x− 2

is decreasing function such that

3vi + 6

4v2i + 5vi − 2
≤

3v + 6

4v2 + 5v − 2
, ∀i = 1, 2, · · ·n.



Int. J. Anal. Appl. 16 (4) (2018) 467

Therefore

Re

{
1 +

zF ′′v1,...,vn,α1,...,αn (z)

F ′v1,...,vn,α1,...,αn (z)

}
> 1 −

3v + 6

4v2 + 5v − 2

n∑
i=1

αi.

Since 0 ≤ 1 − 3v+6
4v2+5v−2

n∑
i=1

αi < 1, therefore Fv1,...,vn,α1,...,αn ∈C (β), where

β = 1 −
3v + 6

4v2 + 5v − 2

n∑
i=1

αi,

which completes the proof. �

By setting α1 = α2 = · · · = αn = α in Theorem 2.1, we obtain the result given below.

Corollary 2.1. Let v1, . . . ,vn >
−5+

√
57

8
, where n ∈ N. Let qυi : U → C be defined as

qυi (z) = 2
υiΓ (υ + 1) z1−

υi
2

(
(1 −υi) Jυi

(√
z
)

+
√
zJ′υi

(√
z
))
. (2.4)

Suppose that v = min{v1, . . . ,vn} and α be positive real numbers such that these numbers satisfy the following

inequality

0 ≤ 1 −
nα (3v + 6)

4v2 + 5v − 2
< 1.

Then, the function Fv1,...,vn,α defined by

Fv1,...,vn,α (z) =

z∫
0

n∏
i=1

(
qvi(t)

t

)α
dt

is in the class C (β1), where

β = 1 −
nα (3v + 6)

4v2 + 5v − 2
.

The next theorem gives convexity properties of the integral operator defined in (1.9). The key tool in the

proof is inequality (ii) of Lemma 1.1.

Theorem 2.2. Let v1, . . . ,vn >
−1+

√
33

4
, where n ∈ N. Let qυi : U → C be defined as

qυi (z) = 2
υiΓ (υ + 1) z1−

υi
2

(
(1 −υi) Jυi

(√
z
)

+
√
zJ′υi

(√
z
))
. (2.5)

Suppose that v = min{v1, . . . ,vn} and δ1, . . . ,δn be positive real numbers such that these numbers satisfy the

following inequality

0 ≤ 1 −
3v + 6

2v2 + v − 4

n∑
i=1

δi < 1.

Then, the function Hv1,...,vn,δ1,...,δn defined by (1.9), is in the class C (γ), where

γ = 1 −
3v + 6

2v2 + v − 4

n∑
i=1

δi.



Int. J. Anal. Appl. 16 (4) (2018) 468

Proof. It can easily be observed that, the operator defined in (1.9) belongs to class A, that is Hv1,...,vn,δ1,...,δn (0) =

H′v1,...,vn,δ1,...,δn (0) − 1 = 0. Differentiating (1.9), we have

H′v1,...,vn,δ1,...,δn (z) =

n∏
i=1

(
q′vi(z)

)δi
.

Differentiating logarithmically, we obtain

zH′′v1,...,vn,δ1,...,δn (z)

H′v1,...,vn,δ1,...,δn (z)
=

n∑
i=1

δi

(
zq′′vi (z)

q′vi (z)

)
.

This implies that

Re

{
1 +

zH′′v1,...,vn,δ1,...,δn (z)

H′v1,...,vn,δ1,...,δn (z)

}
= 1 +

n∑
i=1

δiRe

(
zq′′vi (z)

q′vi (z)

)
.

Now, by using the assertion (ii) of Lemma 1.1 for each vi, where i = 1, 2, · · ·n, we obtain

Re

{
1 +

zH′′v1,...,vn,δ1,...,δn (z)

H′v1,...,vn,δ1,...,δn (z)

}
> 1 −

n∑
i=1

δi

(
3vi + 6

2v2i + vi − 4

)
.

Consider the function ϕ :
(
−1+

√
33

4
,∞
)
→ R defined as

ϕ (x) =
3x + 6

2x2 + x− 4

is decreasing function such that

3vi + 6

2v2i + vi − 4
≤

3v + 6

2v2 + v − 4
, ∀i = 1, 2, · · ·n.

It follows that

Re

{
1 +

zH′′v1,...,vn,δ1,...,δn (z)

H′v1,...,vn,δ1,...,δn (z)

}
> 1 −

3v + 6

2v2 + v − 4

n∑
i=1

δi.

Since

0 ≤ 1 −
3v + 6

2v2 + v − 4

n∑
i=1

δi < 1,

therefore Hv1,...,vn,δ1,...,δn ∈ C (γ), where

γ = 1 −
3v + 6

2v2 + v − 4

n∑
i=1

δi,

which completes the proof. �

By setting δ1 = δ2 = δn = δ in Theorem 2.2, we obtain the result given below.

Corollary 2.2. Let v1, . . . ,vn >
−1+

√
33

4
, where n ∈ N. Let qυi : U → C be defined as

qυi (z) = 2
υiΓ (υ + 1) z1−

υi
2

(
(1 −υi) Jυi

(√
z
)

+
√
zJ′υi

(√
z
))
. (2.6)

Suppose that v = min{v1, . . . ,vn} and δ be positive real numbers such that these numbers satisfy the following

inequality

0 ≤ 1 −
nδ (3v + 6)

2v2 + v − 4
< 1.



Int. J. Anal. Appl. 16 (4) (2018) 469

Then, the function Hv1,...,vn,δ defined as

Hv1,...,vn,δ (z) =

z∫
0

n∏
i=1

(
q′vi(t)

)δ
dt

is in the class C (γ1), where

γ1 = 1 −
nδ (3v + 6)

2v2 + v − 4
.

3. Some particular cases of Dini function

By choosing v = 1
2

and v = 3
2

in (1.2), we get the following forms of the normalized Dini function

q1
2

(z) =
3

2

√
z
(
sin
√
z +
√
z cos

√
z
)
,

q3
2

(z) =
3

2
√
z

(
(z − 1) sin

√
z +
√
z cos

√
z
)
.

In particular, the results of the above mentioned theorems are given below.

Corollary 3.1. Let v > −5+
√
57

8
and qυ : U → C be defined as

qυ (z) = 2
υΓ (υ + 1) z1−

υ
2

(
(1 −υ) Jυ

(√
z
)

+
√
zJ′υ

(√
z
))
.

Suppose that α be positive real number such that these numbers satisfy the following inequality

0 ≤ 1 −
α (3v + 6)

4v2 + 5v − 2
< 1.

Then, the function Fv,α defined by

Fv,α (z) =

z∫
0

(
qv(t)

t

)α
dt

is in the class C (β2), where

β2 = 1 −
α (3v + 6)

4v2 + 5v − 2
.

In particular,

(i) if α ≤ 15, then the function F1
2
,α : U → C defined by

F1
2
,α (z) =

z∫
0

(
3
2

(
sin
√
t +
√
t cos

√
t
)

√
t

)α
dt

is in the class C (β3), where β3 = 1 − α15.

(ii)If α ≤ 29
21
, then the function F3

2
,α : U → C defined by

F3
2
,α (z) =

z∫
0

(
3
2

(
(t− 1) sin

√
t +
√
t cos

√
t
)

t

)α
dt

is in the class C (β4), where β4 = 1 − 21α29 .



Int. J. Anal. Appl. 16 (4) (2018) 470

Corollary 3.2. Let v > −1+
√
33

4
and qυ : U → C be defined as

qυ (z) = 2
υΓ (υ + 1) z1−

υ
2

(
(1 −υ) Jυ

(√
z
)

+
√
zJ′υ

(√
z
))
.

Suppose that δ be positive real numbers such that these numbers satisfy the following inequality

0 ≤ 1 −
δ (3v + 6)

2v2 + v − 4
< 1.

Then, the function Hv,δ defined as

Hv,δ (z) =

z∫
0

(q′v(t))
δ
dt

is in the class C (γ2), where

γ2 = 1 −
nδ (3v + 6)

2v2 + v − 4
.

In particular,

(i) if δ ≤ 4
21

, then the function H3
2
,α : U → C defined by

H3
2
,α (z) =

z∫
0

(
q′3

2
(t
)δ
dt

is in the class C (γ3), where γ3 = 1 − 214 δ.

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[13] G. N. Watson. A Treatise on the Theory of Bessel Functions, Second edition, Cambridge University Press, Cambridge,

London and New York, 1944.


	1. INTRODUCTION
	2.  Convexity of Integral Operators Defined by Generalized Dini Functions
	3. Some particular cases of Dini function
	References