Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 2 (2024) 

 

197 
https://internationalpubls.com 

On a Certain Subclass of Analytic Functions Defined by Bessel 

Functions 

 
Katterapalle Sridevi𝟏, Pasunoori Srinivasulu𝟐 

1 Department of Mathematics, Dr.B.R.Ambedkar Open University, 

Hyderabad - 500 033, T.S, India. 

e-mail  : sridevidrk18@gmail.com 
2 Department of Mathematics, Dr.B.R.Ambedkar Open University, 

Hyderabad - 500 033, T.S, India. 

e-mail  : srinivasrgukt1203@gmail.com 

 

Article History: 

Received: 22-01-2024 

Revised: 08-04-2024 

Accepted: 29-04-2024 

Abstract: In this work, we introduce and investigate a new subclass of analytic 

functions in the open unit disc 𝑈 with negative coefficients. The object of the 

present paper is to determine the coefficient estimates, extreme points, integral 

means inequalities and   subordination results for this class. 

 Keywords: analytic function, uniformly starlike function, coefficient estimate, 

subordination. 

1.Introduction: 

Let 𝐴 be the class of functions 𝑓 normalized by 

                          𝑓(𝑧) = 𝑧 +
∑ 𝑎𝑛

∞
𝑛=2 𝑧𝑛                                       (1.1)                                  

and 𝑇 denote the class of functions in the form of 

𝑓(𝑧) = 𝑧 − ∑ 𝑎𝑛

∞

𝑛=2

𝑧𝑛, (𝑎𝑛 ≥ 0)        (1.2), 

 

 

which are analytic in the open unit disk 𝑈 = {𝑧: 𝑧 ∈ 𝒞 and |𝑧| < 1}. This subclass was given 

in  . Let 𝑇∗(𝛼) and 𝐶(𝛼) be indicate starlike and convex functions of order 𝛼, (0 ≤ 𝛼 < 1), 
respectively. The classes 𝑈𝐶𝑉(𝛼, 𝜎) consists of uniform 𝜎 −convex functions of order 𝛼 and 

𝑆𝑃(𝛼, 𝜎) consists parabolic 𝜎 − starlike functions of order 𝛼, −1 < 𝛼 ≤ 1, 𝜎 ≥ 0, generalizes 

the class 𝑈𝐶𝑉 and 𝑆𝑃 respectively, were given in  such that           

𝑈𝐶𝑉(𝛼, 𝜎) = {𝑓 ∈ 𝐴: 𝑅𝑒 {1 +
𝑧𝑓″(𝑧)

𝑓′(𝑧)
− 𝛼} > 𝜎 {

𝑧𝑓″(𝑧)

𝑓′(𝑧)
} , 𝑧 ∈ 𝑈}          (1.3) 

and 

𝑆𝑃(𝛼, 𝜎) = {𝑓 ∈ 𝐴: 𝑅𝑒 {
𝑧𝑓′(𝑧)

𝑓(𝑧)
− 𝛼} > 𝜎 {

𝑧𝑓′(𝑧)

𝑓(𝑧)
− 1} , 𝑧 ∈ 𝑈}.              (1.4) 

                It is obvious from (1.3) and (1.4) that 𝑓 ∈ 𝑈𝐶𝑉(𝛼, 𝜎) if and only if 𝑧𝑓′(𝑧) ∈
𝑆𝑃(𝛼, 𝜎).   Some interesting situations of the class of starlike and convex of order 𝛼 

associated with Bessel functions (as hypergeometric function), finding condition on the triple 

𝑝, 𝑏 and 𝑐 such that the function 𝑢𝑝,𝑏,𝑐 is starlike and convex of order 𝛼 and finding 

conditions on the parameters for which the Gaussian hypergeometric functions belong to the 

various classes of functions have discussed in the references [1,2,4,9,10] . Let us take into 

consideration second order linear homogenous differential equation ( see [3] ). 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 2 (2024) 

 

198 
https://internationalpubls.com 

𝑧2𝜔″(𝑧) + 𝑏𝑧𝜔′(𝑧) + [𝑐𝑧2 − 𝑝2 + (1 − 𝑏)𝑝]𝜔(𝑧) = 0,   (𝑝, 𝑏, 𝑐 ∈ 𝒞).            (1.5) 

As a particular solition of (1.5)  generalized Bessel function of the first kind of order 𝑝, is 

defined in  as following: 

𝜔(𝑧) = 𝜔𝑝,𝑏,𝑐(𝑧) = ∑
(−1)𝑛𝑐𝑛

𝑛! 𝛤 (𝑝 + 𝑛 +
𝑏 + 1

2 )

∞

𝑛=0

(
𝑧

2
)

2𝑛+𝑝

,  𝑧 ∈ 𝐶,              (1.6) 

where 𝛤 stands for the Euler gamma function and 𝜏 = 𝑝 +
𝑏+1

2
∉ 𝑍0 = {0, −1, −2, ⋯ }. 

Though the series given in (1.6) is convergent everywhere, the function 𝜔𝑝,𝑏,𝑐 is not univalent 

in 𝑈. Specially, choosing 𝑏 = 𝑐 = 1 in (1.6), we get Bessel function of the first kind of order 

𝑝 given in  as 

                         𝐽𝑝(𝑧) = ∑
(−1)𝑛

𝑛! 𝛤(𝑝 + 𝑛 + 1)

∞

𝑛=0

(
𝑧

2
)

2𝑛+𝑝

, 𝑧 ∈ 𝐶.                                     (1.7) 

Choosing 𝑏 = 1 and 𝑐 = −1 in (1.6), we get the modified Bessel function of the first kind 

order of 𝑝 given in  as 

                                                𝐼𝑝(𝑧) = ∑
1

𝑛!𝛤(𝑝+𝑛+1)
∞
𝑛=0 (

𝑧

2
)

2𝑛+𝑝

, 𝑧 ∈ 𝐶.                               (1.8) 

Further choosing 𝑏 = 2 and 𝑐 = 1 in (1.6), the functions 𝜔𝑝,𝑏,𝑐 reduces to √2
𝑗𝑝

√𝜋
, where 𝑗𝑝 is 

the spherical Bessel function of the first kind of order 𝑝, given in  as 

𝑗𝑝(𝑧) = √
𝜋

2
 ∑

(−1)𝑛

𝑛! 𝛤 (𝑝 + 𝑛 +
3
2)

∞

𝑛=0

(
𝑧

2
)

2𝑛+𝑝

, 𝑧 ∈ 𝐶.                                   (1.9) 

The function 𝜗𝑝,𝑏,𝑐 is defined in  as 

                𝜗𝑝,𝑏,𝑐(𝑧) = 2𝑝𝛤 (𝑝 +
𝑏 + 1

2
) 𝑧1−

𝑝
2𝜔𝑝,𝑏,𝑐(√𝑧)                                                         (1.10) 

 

in terms of generalized Bessel function 𝜔𝑝,𝑏,𝑐. By the help of Pochhammer symbol, Gamma 

function is defined as 

and we get 𝜗𝑝,𝑏,𝑐 given in (1.10)  as 

𝜗𝑝,𝑏,𝑐(𝑧) = 𝑧 + ∑
(−𝑐)𝑛

4𝑛(𝜏)𝑛𝑛!

∞

𝑛=1

,                                                                      (1.11) 

where 𝜏 = 𝑝 +
𝑏+1

2
∉ 𝑍0 and 𝑁 = {1,2,3, ⋯ }.     We will write 𝜗𝜏,𝑐(𝑧) = 𝜗𝑝,𝑏,𝑐(𝑧) for 

convenience. Now, we consider 𝑆𝜏
𝑐 operator given as 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 2 (2024) 

 

199 
https://internationalpubls.com 

𝑆𝜏
𝑐𝑓(𝑧) = 𝜗𝜏,𝑐(𝑧) ∗ 𝑓(𝑧) = 𝑧 + ∑

(−𝑐)𝑛𝑎𝑛+1

4𝑛(𝜏)𝑛𝑛!

∞

𝑛=1

𝑧𝑛+1

= 𝑧 + ∑
(−𝑐)𝑛−1𝑎𝑛

4𝑛−1(𝜏)𝑛−1(𝑛 − 1)!

∞

𝑛=2

𝑧𝑛 = 𝑧 + ∑ 𝐸

∞

𝑛=2

(𝑐, 𝜏, 𝑛)𝑎𝑛𝑧𝑛

where 𝐸(𝑐, 𝜏, 𝑛) =
(−𝑐)𝑛−1

4𝑛−1(𝜏)𝑛−1(𝑛 − 1)!
,  𝜏 = (𝑝 +

𝑏 + 1

2
) ≠ 0, −1, −2, ⋯.                     (1.12)

 

              For 𝛼 ≥ 0, 0 ≤ 𝛽 < 1, we set 𝑆𝜏
𝑐(𝛼, 𝛽) be the subclass of 𝐴 consisting of functions 

of the form (1.1) and satisfy 

𝑅𝑒 (
𝑆𝜏

𝑐𝑓(𝑧)

𝑧
) ≥ 𝛼 |(𝑆𝜏

𝑐𝑓(𝑧))′ −
𝑆𝜏

𝑐𝑓(𝑧)

𝑧
|

+ 𝛽                                                                               (1.13) 

where 𝑆𝜏
𝑐𝑓(𝑧) is given by (1.12). 

We further let 𝑇𝑆𝜏
𝑐(𝛼, 𝛽) = 𝑆𝜏

𝑐(𝛼, 𝛽) ∩ 𝑇.               

In this paper, we obtain coefficient inequalities, extreme points, integral means inequalities 

for the functions in the class 𝑇𝑆𝜏
𝑐(𝛼, 𝛽) and also subordination results for the class of function 

𝑓 ∈ 𝑆𝜏
𝑐(𝛼, 𝛽). 

2. Coefficient Estimates 

Theorem 2.1.  The function 𝑓 defined by (1.1) is in the class 𝑆𝜏
𝑐(𝛼, 𝛽) if 

                         ∑ [1 + 𝛼(𝑛 − 1)]∞
𝑛=2 𝐸(𝑐, 𝜏, 𝑛)|𝑎𝑛| ≤ 1 −

𝛽,                                                                     (2.1)  

where  𝛼 ≥ 0,0 ≤ 𝛽 < 1 and 𝐸(𝑐, 𝜏, 𝑛) is given by (1.12). 

Proof. It suffices to show that 

𝛼 |(𝑆𝜏
𝑐𝑓(𝑧))′ −

𝑆𝜏
𝑐𝑓(𝑧)

𝑧
| − 𝑅𝑒 {

𝑆𝜏
𝑐𝑓(𝑧)

𝑧
− 1} ≤ 1 − 𝛽.

We have

𝛼 |(𝑆𝜏
𝑐𝑓(𝑧))′ −

𝑆𝜏
𝑐𝑓(𝑧)

𝑧
| − 𝑅𝑒 {

𝑆𝜏
𝑐𝑓(𝑧)

𝑧
− 1}

≤ 𝛼 |
∑ (𝑛 − 1)∞

𝑛=2 𝐸(𝑐, 𝜏, 𝑛)𝑎𝑛𝑧𝑛

𝑧
| + |

∑ 𝐸∞
𝑛=2 (𝑐, 𝜏, 𝑛)𝑎𝑛𝑧𝑛

𝑧
|

≤ 𝛼 ∑(𝑛 − 1)

∞

𝑛=2

𝐸(𝑐, 𝜏, 𝑛)|𝑎𝑛| + ∑ 𝐸

∞

𝑛=2

(𝑐, 𝜏, 𝑛)|𝑎𝑛|

= ∑[1 + 𝛼(𝑛 − 1)]

∞

𝑛=2

𝐸(𝑐, 𝜏, 𝑛)|𝑎𝑛|.

 

The last expression is bounded above by (1 − 𝛽) if 

∑[1 + 𝛼(𝑛 − 1)]

∞

𝑛=2

𝐸(𝑐, 𝜏, 𝑛)|𝑎𝑛| ≤ 1 − 𝛽 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 2 (2024) 

 

200 
https://internationalpubls.com 

and the proof of theorem is completed.  

In the following theorem, we obtain necessary and sufficient conditions for functions in 

𝑇𝑆𝜏
𝑐(𝛼, 𝛽). 

 

Theorem 2.2.  For 𝛼 ≥ 0,0 ≤ 𝛽 < 1, a function 𝑓 of the form  (1.2)   to be in the class 

𝑇𝑆𝜏
𝑐(𝛼, 𝛽) if and only if  

∑[1 + 𝛼(𝑛 − 1)]

∞

𝑛=2

𝐸(𝑐, 𝜏, 𝑛)|𝑎𝑛| ≤ 1 − 𝛽. 

Proof. Suppose 𝑓(𝑧) of the form (1.2)  is in the class 𝑇𝑆𝜏
𝑐(𝛼, 𝛽). Then 

𝑅𝑒 {
𝑆𝜏

𝑐𝑓(𝑧)

𝑧
} − 𝛼 |(𝑆𝜏

𝑐𝑓(𝑧))′ −
𝑆𝜏

𝑐𝑓(𝑧)

𝑧
| ≥ 𝛽.

Equivalently 

𝑅𝑒 [1 − ∑ 𝐸

∞

𝑛=2

(𝑐, 𝜏, 𝑛)|𝑎𝑛|𝑧𝑛−1] − 𝛼 [∑(𝑛 − 1)

∞

𝑛=2

𝐸(𝑐, 𝜏, 𝑛)𝑎𝑛𝑧𝑛−1] ≥ 𝛽.

 

Letting 𝑧 to be real values and as |𝑧| → 1, we have 

1 − ∑ 𝐸

∞

𝑛=2

(𝑐, 𝜏, 𝑛)|𝑎𝑛| − 𝛼 ∑(𝑛 − 1)

∞

𝑛=2

𝐸(𝑐, 𝜏, 𝑛)|𝑎𝑛| ≥ 𝛽 

which implies 

∑[1 + 𝛼(𝑛 − 1)]

∞

𝑛=2

𝐸(𝑐, 𝜏, 𝑛)|𝑎𝑛| ≤ 1 − 𝛽, 

 

where 𝛼 ≥ 0, 0 ≤ 𝛽 < 1, 𝐸(𝑐, 𝜏, 𝑛) is given by (1.12) and the sufficiency follows from 

Theorem 2.1. 

Corollary 2.3.  If 𝑓 ∈ 𝑇𝑆𝜏
𝑐(𝛼, 𝛽) then 

 

                                                       𝑎𝑛 ≤
1−𝛽

[1+𝛼(𝑛−1)]𝐸(𝑐,𝜏,𝑛)
.  

 

Equality holds for the function 

 

                                               𝑓(𝑧) = 𝑧 −
1−𝛽

[1+𝛼(𝑛−1)]𝐸(𝑐,𝜏,𝑛)
𝑧𝑛, 

 

 𝛼 ≥ 0,   0 ≤ 𝛽 < 1, 𝐸(𝑐, 𝜏, 𝑛) is given by (1.12). 
3. Extreme Points 

Theorem 3.1.  Let 𝑓1(𝑧) = 𝑧 and 𝑓𝑛(𝑧) = 𝑧 −
1−𝛽

[1+𝛼(𝑛−1)]𝐸(𝑐,𝜏,𝑛)
𝑧𝑛, 𝑛 ≥ 2 for 𝛼 ≥ 0,0 ≤ 𝛽 <

1, 𝐸(𝑐, 𝜏, 𝑛) is given by (1.12) Then 𝑓(𝑧) is in the class 𝐸(𝑐, 𝜏, 𝑛) if and only if it can be 

expressed in the form 

 𝑓(𝑧) = ∑ 𝜆𝑛
∞
𝑛=1 𝑓𝑛(𝑧), where 𝜆𝑛 and ∑ 𝜆𝑛

∞
𝑛=1 = 1. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 2 (2024) 

 

201 
https://internationalpubls.com 

Proof. If 𝑓(𝑧) = ∑ 𝜆𝑛
∞
𝑛=1 𝑓𝑛(𝑧) with  𝜆𝑛 ≥ 0 and ∑ 𝜆𝑛

∞
𝑛=1 = 1. Then 

𝑓(𝑧) = ∑ 𝜆𝑛

∞

𝑛=1

𝑓𝑛(𝑧)

= 𝜆1𝑓1(𝑧) + ∑ 𝜆𝑛

∞

𝑛=2

𝑓𝑛(𝑧)

= (1 − ∑ 𝜆𝑛

∞

𝑛=2

) 𝑧 + ∑ [𝜆𝑛 (𝑧 −
1 − 𝛽

[1 + 𝛼(𝑛 − 1)]𝐸(𝑐, 𝜏, 𝑛)
𝑧𝑛)]

∞

𝑛=2

= 𝑧 − ∑
1 − 𝛽

[1 + 𝛼(𝑛 − 1)]𝐸(𝑐, 𝜏, 𝑛)

∞

𝑛=2

𝑧𝑛.

Now ∑
[1 + 𝛼(𝑛 − 1)]𝐸(𝑐, 𝜏, 𝑛)

1 − 𝛽

∞

𝑛=2

1 − 𝛽

[1 + 𝛼(𝑛 − 1)]𝐸(𝑐, 𝜏, 𝑛)
𝜆𝑛

= ∑ 𝜆𝑛

∞

𝑛=2

= 1 − 𝜆1 ≤ 1.

 

Then 𝑓 ∈ 𝑇𝑆𝜏
𝑐(𝛼, 𝛽). Conversely suppose that 𝑓 ∈ 𝑇𝑆𝜏

𝑐(𝛼, 𝛽). Then Corollary 2.3 gives 

𝑎𝑛 ≤
1 − 𝛽

[1 + 𝛼(𝑛 − 1)]𝐸(𝑐, 𝜏, 𝑛)
,  𝑛 ≥ 2

set 𝜆𝑛 =
[1 + 𝛼(𝑛 − 1)]𝐸(𝑐, 𝜏, 𝑛)

1 − 𝛽
𝑎𝑛,  𝑛 ≥ 2

where  𝜆𝑛 = 1 − ∑ 𝜆𝑛

∞

𝑛=2

.

 

Then 

𝑓(𝑧) = 𝑧 − ∑ 𝑎𝑛

∞

𝑛=2

𝑧𝑛

= 𝑧 − ∑ 𝜆𝑛

∞

𝑛=2

1 − 𝛽

[1 + 𝛼(𝑛 − 1)]𝐸(𝑐, 𝜏, 𝑛)

= 𝑧 − [1 − ∑ 𝜆𝑛

∞

𝑛=2

] + ∑ 𝜆𝑛

∞

𝑛=2

𝑓𝑛(𝑧)

= 𝜆1𝑓1(𝑧) + ∑ 𝜆𝑛

∞

𝑛=2

𝑓𝑛(𝑧)

= ∑ 𝜆𝑛

∞

𝑛=1

𝑓𝑛(𝑧).

 

The poof of theorem is completed . 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 2 (2024) 

 

202 
https://internationalpubls.com 

4. Integral Means Inequalities 

Definition 4.1.  (Subordination principle) for analytic function 𝑔 and ℎ with 𝑔(0) =
ℎ(0),  𝑔 is said to be subordinate to ℎ, denoted by 𝑔 ≺ ℎ if there exists an analytic function 𝜔 

such that 𝜔(0) = 0, |𝜔(𝑧)| < 1 and 𝑔(𝑧) = ℎ(𝜔(𝑧)), for all 𝑧 ∈ 𝑈. 

Lemma 4.2. [6] If the function 𝑓(𝑧) and 𝑔(𝑧) are analytic in 𝑈 with 𝑔(𝑧) ≺ ℎ(𝑧) then 

 

 ∫ |𝑔(𝑟𝑒𝑖𝜃)|
𝑝2𝜋

0
𝑑𝜃 ≤ ∫ |𝑓(𝑟𝑒𝑖𝜃)|

𝑝2𝜋

0
𝑑𝜃  (0 ≤ 𝑟 < 1, 𝑝 > 0). 

Theorem 4.3.  Suppose 𝑓 ∈ 𝑇𝑆𝜏
𝑐(𝛼, 𝛽), 𝑝 > 0, 𝛼 ≥ 0,0 ≤ 𝛽 < 1 and 𝑓(𝑧) is defined by 

                                                          𝑓2(𝑧) = 𝑧 −
1−𝛽

(1+𝛼)𝐸(𝑐,𝜏,𝑛)
.  

Then for 𝑧 = 𝑟𝑒𝑖𝜃,  0 ≤ 𝑟 < 1, 

                                          

∫ |𝑓(𝑧)|𝑝2𝜋

0
𝑑𝜃 ≤ ∫ |𝑓2(𝑧)|𝑝2𝜋

0
𝑑𝜃                                                             (4.1) 

Proof. For 𝑓(𝑧) = 𝑧 − ∑ |𝑎𝑛|∞
𝑛=2 𝑧𝑛, (4.1) is equivalent to proving that 

∫ |𝑧 − ∑|𝑎𝑛|

∞

𝑛=2

𝑧𝑛|

𝑝2𝜋

0

𝑑𝜃 ≤ ∫ |𝑧 −
1 − 𝛽

1 + 𝛼)𝐸(𝑐, 𝜏, 𝑛)
|

𝑝
2𝜋

0

𝑑𝜃,  (𝑝 > 0). 

By applying Little wood’s subordination theorem (Lemma 4.2), 

it would be sufficient to show that 

1 − ∑|𝑎𝑛|

∞

𝑛=2

𝑧𝑛−1 ≺ 1 −
1 − 𝛽

1 + 𝛼)𝐸(𝑐, 𝜏, 𝑛)
𝑧.                                                (4.2) 

Setting 

1 − ∑|𝑎𝑛|

∞

𝑛=2

𝑧𝑛−1 ≺ 1 −
1 − 𝛽

1 + 𝛼)𝐸(𝑐, 𝜏, 𝑛)
𝜔(𝑧). 

We have 𝜔(𝑧) =
1+𝛼)𝐸(𝑐,𝜏,𝑛)

1−𝛽
∑ 𝑎𝑛

∞
𝑛=2 𝑧𝑛−1 and and 𝜔(𝑧) is analytic in 𝑈 with 𝜔(0) = 0. 

Moreover it suffices to prove that 𝜔(𝑧) satisfies |𝜔(𝑧)| < 1, 𝑧 ∈ 𝑈. Now 

|𝜔(𝑧)| = |∑
(1 + 𝛼)𝐸(𝑐, 𝜏, 𝑛)

1 − 𝛽

∞

𝑛=2

𝑎𝑛𝑧𝑛−1|

≤ |𝑧| ∑
(1 + 𝛼)𝐸(𝑐, 𝜏, 𝑛)

1 − 𝛽

∞

𝑛=2

|𝑎𝑛|                                                           (4.3)

≤ |𝑧| < 1.

 

Thus is view of the inequality (4.3) the subordination  (4.2) follows, which proves the 

Theorem.  

5. Subordination Results 

Definition 5.1.  (Subordination factor sequence ) A sequence { 𝑏𝑛 }𝑛=2
∞  of complex 

numbers is said to be a  



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 2 (2024) 

 

203 
https://internationalpubls.com 

subordinating sequence if, whenever 𝑓(𝑧) = ∑ 𝑎𝑛
∞
𝑛=2 𝑧𝑛, 𝑎1 = 1 is regular, univalent and 

convex in 𝑈, we  

have ∑ 𝑏𝑛
∞
𝑛=1 𝑎𝑛𝑧𝑛 ≺ 𝑓(𝑧),  𝑧 ∈ 𝑈. 

Theorem 5.2. [11] The sequence { 𝑏𝑛 }𝑛=2
∞  is a subordinating factor sequence if and only if 

 

                                             𝑅𝑒{1 + 2 ∑ 𝑏𝑛
∞
𝑛=1 𝑧𝑛} > 0, 𝑧 ∈ 𝑈. 

Theorem 5.3.  Let 𝑓 ∈ 𝑆𝜏
𝑐(𝛼, 𝛽) and 𝑔(𝑧) any function in the usual class of convex function 

𝐶. Then  

(1 + 𝛼)𝐸(𝑐, 𝜏, 𝑛)

2(1 − 𝛽) + (1 + 𝛼)𝐸(𝑐, 𝜏, 𝑛)
(𝑓 ∗ 𝑔)(𝑧) ≺ 𝑔(𝑧)                                 (5.1) 

 where 𝛼 ≥ 0,0 ≤ 𝛽 < 1 with 𝐸(𝑐, 𝜏, 𝑛) is given by  (1.12) 

                                               𝑅𝑒{𝑓(𝑧)} > −
(1−𝛽)+(1+𝛼)𝐸(𝑐,𝜏,𝑛)

(1+𝛼)𝐸(𝑐,𝜏,𝑛)
,  𝑧 ∈ 𝐸.                              (5.2)  

The constant 
(1+𝛼)𝐸(𝑐,𝜏,𝑛)

2(1−𝛽)+(1+𝛼)𝐸(𝑐,𝜏,𝑛)
 is the best estimate. 

Proof. Let 𝑓 ∈ 𝑆𝜏
𝑐(𝛼, 𝛽) and 𝑔(𝑧) = 𝑧 + ∑ 𝑐𝑛

∞
𝑛=2 𝑧𝑛 ∈ 𝐶. 

Then 

(1 + 𝛼)𝐸(𝑐, 𝜏, 𝑛)

2(1 − 𝛽) + (1 + 𝛼)𝐸(𝑐, 𝜏, 𝑛)
(𝑓 ∗ 𝑔)(𝑧)

=
(1 + 𝛼)𝐸(𝑐, 𝜏, 𝑛)

2(1 − 𝛽) + (1 + 𝛼)𝐸(𝑐, 𝜏, 𝑛)
(𝑧 + ∑ 𝑐𝑛

∞

𝑛=2

𝑎𝑛𝑧𝑛). 

Then by Definition 5.1, the subordination result holds true if {
(1+𝛼)𝐸(𝑐,𝜏,𝑛)

2(1−𝛽)+(1+𝛼)𝐸(𝑐,𝜏,𝑛)
}

𝑛=1

∞

 is a 

subordinating factor sequence with 𝑎1 = 1. 
 

In view of Theorem 5.2, this is equivalent to the following inequality. 

𝑅𝑒 {1 + ∑
(1 + 𝛼)𝐸(𝑐, 𝜏, 𝑛)

(1 − 𝛽) + (1 + 𝛼)𝐸(𝑐, 𝜏, 𝑛)

∞

𝑛=1

𝑎𝑛𝑧𝑛} > 0,  𝑧 ∈ 𝑈.                         (5.3) 

Now for |𝑧| = 𝑟 < 1, we have 

𝑅𝑒 {1 + ∑
(1 + 𝛼)𝐸(𝑐, 𝜏, 𝑛)

2(1 − 𝛽) + (1 + 𝛼)𝐸(𝑐, 𝜏, 𝑛)

∞

𝑛=1

𝑎𝑛𝑧𝑛}

= 𝑅𝑒 {1 +
(1 + 𝛼)𝐸(𝑐, 𝜏, 𝑛)

(1 − 𝛽) + (1 + 𝛼)𝐸(𝑐, 𝜏, 𝑛)
𝑧 +

∑ (1 + 𝛼)∞
𝑛=2 𝐸(𝑐, 𝜏, 𝑛)𝑎𝑛𝑧𝑛

(1 − 𝛽) + (1 + 𝛼)𝐸(𝑐, 𝜏, 𝑛)
}

≥ 1 −
(1 + 𝛼)𝐸(𝑐, 𝜏, 𝑛)

(1 − 𝛽) + (1 + 𝛼)𝐸(𝑐, 𝜏, 𝑛)
𝑟 −

∑ (1 + 𝛼)∞
𝑛=2 𝐸(𝑐, 𝜏, 𝑛)𝑎𝑛𝑟𝑛

(1 − 𝛽) + (1 + 𝛼)𝐸(𝑐, 𝜏, 𝑛)

≥ 1 −
(1 + 𝛼)𝐸(𝑐, 𝜏, 𝑛)

(1 − 𝛽) + (1 + 𝛼)𝐸(𝑐, 𝜏, 𝑛)
𝑟 −

1 − 𝛽

(1 − 𝛽) + (1 + 𝛼)𝐸(𝑐, 𝜏, 𝑛)
𝑟

> 0.

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 31 No. 2 (2024) 

 

204 
https://internationalpubls.com 

Using (2.1) and the fact that 1 + 𝛼(𝑛 − 1)𝐸(𝑐, 𝜏, 𝑛) is increasing function for 𝑛 ≥ 2. 

 

This proves the inequality (5.3)  and hence also the subordination result (5.1) asserted by 

Theorem 5.3. 

 

The inequality (5.2) follows from (5.1) by taking 

𝑔(𝑧) =
𝑧

1 − 𝑧
= 𝑧 + ∑ 𝑧𝑛

∞

𝑛=2

∈ 𝐶. 

Now we consider the function 𝑓(𝑧) = 𝑧 −
1−𝛽

(1+𝛼)𝐸(𝑐,𝜏,𝑛)
𝑧2, where 𝛼 ≥ 0,0 ≤ 𝛽 < 1. 

Clearly 𝐹 ∈ 𝑆𝜏
𝑐(𝛼, 𝛽). For the function (5.1) becomes 

(1 + 𝛼)𝐸(𝑐, 𝜏, 𝑛)

2(1 − 𝛽) + (1 + 𝛼)𝐸(𝑐, 𝜏, 𝑛)
𝐹(𝑧) ≺

𝑧

1 − 𝑧
. 

It is easily verified that 

min𝑅𝑒 {
(1 + 𝛼)𝐸(𝑐, 𝜏, 𝑛)

2(1 − 𝛽) + (1 + 𝛼)𝐸(𝑐, 𝜏, 𝑛)
𝐹(𝑧)} =

−1

2
, 𝑧 ∈ 𝑈. 

This shows that the constant 
(1+𝛼)𝐸(𝑐,𝜏,𝑛)

2(1−𝛽)+(1+𝛼)𝐸(𝑐,𝜏,𝑛)
𝐹(𝑧) ≺

𝑧

1−𝑧
 is best possible.               

Refrences 

[1] A. Baricz, Geometric properties of generalized Bessel function, Publ. Math. Debrecan, 73 (2008), 155- 178.   

[2] A. Baricz, Generalized Bessel functions of the first kind, Ph.D thesis, Babes-Bolyai University, 

ClujNapoca, 2008. 

[3] A. Baricz, Generalized Bessel functions of the first kind, Lecture Notes in Math., 1994, Springer, Berlin, 

2010 . 

[4] A. Baricz and B. A. Frasin, Univalence of integral operators involving Bessel functions, Appl. Math. Lett., 

23 (2010), 371-376. 

[5] E. Deniz, H. Orhan and H. M. Srivastava, Some sufficient conditions for univalence of certain families of 

integral operators involving generalized Bessel functions, Taiwan. J. Math., 15(2) (2011), 883-917. 

[6] J.E. Littlewood, On inequalities in theory of functions, Proc. London Math. Soc., 23 ( 1925), 481-519 .  

[7] F., Rønning, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. 

Math. Soc., 118 (1993), 189-196. ‘ 

[8] H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc., 51 (1975) 109– 116.  

[9] P. Thirupathi Reddy and B. Venkateswarlu, On a certain subclass of uniformly convex functions defined by 

Bessel functions, Transylvanian J. of Math. and Mech., 10(1) (2018), 43-49. 

[10] P. Thirupathi Reddy and B. Venkateswarlu, A certain subclass of uni- formly convex fucntions defined by 

Bessel functions, Proyecciones J. of Math., 38 (4) (2019) 719- 731. 

[11] H. S. Wilf, Subordinating factor sequence for convex maps of the unit circle, Proc. Amer. Math. Soc., 12 ( 

1961), 689-693.  

[12] J. A. Baker, J. Lawrence, and F. Zorzitto, The stability of the equation f(x + y) = f(x)f(y), Proc. Amer. 

Math. Soc. 74(1979), 242-246.  

[13] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27(1941), 222-

224. 

[14] L. Szekelyhidi, On a theorem of Baker, Lawrence and Zorzitto, Proc. Amer. Math. Soc. 84(1982), 95-96.  

[15] S. M. Ulam, Problems in Modern Mathematics, Science Editions, Wiley, New York, 1960.