Journal of Indian Acad. Math.  
ISSN: 0970-5120  
Vol. 47, No. 1 (2025) pp. 113-131  
Girish D. Shelake1  
BI-CONCAVE AND OZAKI-TYPE  
BI-CLOSE-TO-CONVEX FUNCTIONS  
ASSOCIATED WITH MILLER-ROSS  
TYPE POISSON DISTRIBUTION  
SUBORDINATE TO INVOLUTION  
NUMBERS  
Sarika K. Nilapgol2  
and  
Santosh B. Joshi3  
Abstract: The purpose of this article is to study new subclasses of bi-  
univalent functions related to the Miller-Ross type Poisson distribution,  
which is subordinate to the generalized telephone numbers. Here, we  
introduce two new subclasses of Ozaki-type bi-close-to-convex functions  
and bi-concave functions. For the functions, In these new classes, we  
estimate the first two Taylor-Maclaurin coefficients and Fekete-Szegö  
problem.  
Keywords: Univalent Functions, Bi-Univalent Functions, Bi-Convex  
Function,  
Miller-Ross  
Type  
Poisson  
Distribution,  
Subordination, Fekete-Szegö Problem.  
Mathematics Subject Classification (2020) No.: 30C45.  
1. Introduction  
We begin by considering that  
defined as  
B
represents the class of analytic functions  
(z) z   
drzr  
z O  
,
(1.1)  
r2  
those are analytic in open unit disk O {z : z , z  1}. Let us denote the  
as the family of all analytic and univalent functions in  
S
O
.
114  
G.D. SHELAKE, S.K. NILAPGOL AND S.B. JOSHI  
Koebe one-quarter Theorem [11] states that, the image of  
O
under any  
univalent function  
S contains the disk of radius 1/4. As a result, every function  
has an inverse 1 given by  
1(z) (z) w d2w2 (2d22 d3)w3 (5d23 5d2d3 d4)w4 · · ·.  
A function  
. Let us denote  
S is bi-univalent in  
O
if both  
and 1 are univalent in  
O
as the class of bi-univalent functions.  
Assume that g1 and g2 are analytic functions that are defined in  
that g1 is subordinate to g2 i.e.g1(z) g2(z), when we can identify a function  
with analytic properties in domain , as follows:  
O
. We say  
w
O
w(0) 0, w(z)  1 and g1(z) g2(w(z))  
.
In particular, g2 is univalent in  
O
then the below equivalence is obtained.  
g1 g2 g1(0) g2(0) and g1(O) g2(O)  
.
A function  
satisfies conditions listed below:  
: S belongs to the class of concave functions if  
•
is analytic in  
O
and satisfying normalization conditions  
(0)  
'(0) 1 0  
.
•
•
maps  
convex.  
O
conformally onto a set whose complement with respect to  
is  
The opening angle (O) at  
is less than or equal to , (1, 2].  
The class  () represents the class of concave analytic and univalent  
functions (for details, see [5; 3; 4; 25; 24]) and the functions of this class satisfy  
below inequality:  
BI-CONCAVE AND OZAKI-TYPE BI-CLOSE-TO-CONVEX FUNCTIONS  
115  
z
''(z)   
R 1   
0  
z O  
'(z)  
Bhowmik B., Ponnusamy S., Wirths K. [7] established that a function maps  
O
onto  
an angled concave domain if and only if  
1
(1)(1 z)  
z
''(z)  
R
1   
0  
.
1  
2(1 z)  
'(z)  
Numerous studies on bi-univalent function subclasses may be found in the  
varied publications [9; 8; 16; 21]. Motivated by works [27; 30; 28; 31; 20; 2], we  
analyze the novel subclasses of concave and bi-close-to-convex functions.  
Let us consider the Miller-Ross function [17] (also see [15; 29]) and is denoted as  
(z)r  
,(z) z  
,
, , z .  
(r 1)  
r 0  
The two parameter Mittag-Leffler function [32], ,µ(z) is given by  
zr  
,(z)   
,
, , z , R{, } 0.  
(r )  
r 0  
For µ 1, we have the Mittag-Leffler function [18],  
zr  
(z)   
,
, z , R{} 0.  
(r 1)  
r 0  
The Miller-Ross function [12] can be represented as  
G
,(z) z1,1(z).  
Recently, Şeker et al. [26] represented a power series with the corresponding  
coefficients are the Miller-Ross type Poisson distribution, which is as shown below:  
116  
G.D. SHELAKE, S.K. NILAPGOL AND S.B. JOSHI  
()r 1  
,(z) z  
zr,  
z O, 0.  
(1.2)  
(r )  
,()  
r 2  
Now, consider the convolution of functions (1.1) and (1.2), an operator  
  
:
B B written as:  
(z) (z) (z)  
()r 1  
z   
drzr  
(r ),()  
r 2  
z   
rdrzr,  
z O, 0.  
r 2  
()r 1  
where, r   
.
(r ),()  
In particular,  
()  
()2  
2   
, and 3   
.
(1.3)  
(2 ),()  
(3 ),()  
1.1 Involution Numbers: Considering the involution numbers (that are also  
referred to as telephone numbers (TN)), the recurrence relation is  
(r) (r 1) (r 1)(r 2)  
with (0) (1) 1  
,
r 2  
,
.
New generalized telephone numbers (GTNs) were recently identified in 2019  
by Bednarz and Wolowiec-Musial [6] and are represented as  
(r) (r 1) (r 1)(r 2)  
,
r 2, 1  
,
with (0) (1) 1  
.
BI-CONCAVE AND OZAKI-TYPE BI-CLOSE-TO-CONVEX FUNCTIONS  
117  
GTNs are presented in exponential series form by  
2
x
ex   
(r) xr  
,
1  
.
2
r !  
r 0  
Thus, for 1 , we have TNs (r) and for specific values of r, (r) is  
provided as  
1.  
(0) (1) 1  
(2) 1   
(3) 1 3  
,
2.  
3.  
4.  
5.  
,
,
(4) 1 632 ,  
(5) 1 10152 .  
Let us consider the function  
2
x
(x) ex   
2
(r) xr  
r!  
r 0  
1 x (1 ) x2 (1 3) x3 (1 632) x4 .  
2
6
24  
For x O. (see also [19; 10])  
We introduce two novel subclasses of bi-univalent functions connected with  
the Poisson distribution of Miller-Ross type that are subordinate to GTNs in our  
current paper. In addition, we estimate the Fekete-Szego inequality and the Taylor-  
Maclaurin coefficients d2  
,
d3 , for the newly defined classes.  
2. Ozaki-type Bi-Close-to-Convex Functions  
In 1952, Kaplan [13] introduced the class  
K
of close-to-convex functions. In  
1935, Ozaki [22] had already identified these functions, satisfying the following  
inequality:  
118  
G.D. SHELAKE, S.K. NILAPGOL AND S.B. JOSHI  
z
'(z)   
1
2
R 1   
   
,
z O  
.
(2.1)  
(z)  
Kaplan [8] states that the function which satisfy inequality (2.1) are close-to-  
convex functions and which are categorized under class . The Ozaki inequality  
was further generalized by Kargar and Gebadian [14]. (For details see [22; 1]) A  
S
function  
B is locally univalent and is said to be Ozaki close-to-convex  
function if it satisfy the condition:  
z
'(z)   
1
2
1
2
R 1   
,  
z O,  12 , ]  
.
(z)  
Definition 1: The class OBCV(, , ,) includes all functions  
B if it satisfies the following subordination conditions:  
((z,(z))')'  
21  
2
(z)  
,
,
(2.2)  
(2.3)  
(,  
(z))'   
21  
21  
and  
((w,(w))')'  
21  
2
(z)  
(,(w))'  
21  
21  
1
2
where  
1(w) (w) and  
1 .  
1
2
Remark 1: For  
  
, the class OBCV(, , , ) CV()  
includes all functions  
if  
((z,(z))')'  
(,(z))'  
((w,(w))')'  
(z)  
,
and  
(z)  
(,(w))'  
where  
1(w) (w)  
.
The following lemma [23] will be necessary for proving the main findings.  
BI-CONCAVE AND OZAKI-TYPE BI-CLOSE-TO-CONVEX FUNCTIONS  
119  
Lemma 1: If h , then ck   2 for each k, where  
is the family  
of all functions h, analytic in , for which [h(z)] 0 (z O) , where  
O
h(z) 1 c1z c2z2 · · · (z O).  
Theorem  
1: A function  
B  
form  
(1.1) is in class  
OBCV(, , ,), then  
21  
21  
d   min  
,
,
2
2 (21)(33 222) 2(1 )22  
42  
(2.4)  
and  
21  
(21)2 21  
(21)2  
d   min  
,
2
1222  
4[(21)(33 22) 2(1 )2  
123  
123  
2   
2
(2.5)  
where 2 and 3 are as given in (1.3).  
Proof: Let us consider s(z) and t(z) as  
1 l(z)  
s(z)   
1 s1z s2z2 ,  
(2.6)  
(2.7)  
1 l(z)  
1 m(w)  
t(w)   
1 t1w t2w2 ,  
1 m(w)  
or, equivalently,  
l(z)   
s12  
s(z) 1  
1
2
s1z s2   
z ,  
(2.8)  
(2.9)  
s(z) 1  
2
2
and  
t12  
t(w) 1  
1
m(w)   
t w t   
w2 ,  
1
2
t(w) 1  
2
2
120  
G.D. SHELAKE, S.K. NILAPGOL AND S.B. JOSHI  
Then s(0) t(0) 1 , and s(z) and t(z)are analytic in  
real part in  
O
with a positive  
O
.
Now consider,  
2
[l(z)]  
(l(z)) el(z)  
2
s12  
s1  
s2  
2
(l(z)) 1   
z   
(1 )  
z
2
2
8
s13  
s3  
s1s2  
3
(1)  
(1 3)  
z
.  
(2.10)  
(2.11)  
2
4
48  
Similarly,  
t12  
t1  
t2  
2
(m(w)) 1   
w   
(1 )  
w
2
2
8
t13  
t3  
t1t2  
3
w .  
(1)  
(1 3)  
2
4
48  
From (2.2) and (2.3), we have  
((z,(z))')'  
21  
2
l(z)),  
(2.12)  
(2.13)  
(,  
(z))'   
21  
21  
and  
((w,(w))')'  
21  
2
(m(w)).  
(,(w))'  
21  
21  
Using (2.10), (2.11) in (2.12), (2.13) and comparing the coefficients, we the  
following relations  
4
s1  
d22  
,
(2.14)  
21  
2
BI-CONCAVE AND OZAKI-TYPE BI-CLOSE-TO-CONVEX FUNCTIONS  
121  
s12  
4
s2  
(3d33 2d22)   
(1)  
,
(2.15)  
2
2
21  
2
8
4
t1  
d22   
,
(2.16)  
(2.17)  
21  
2
s12  
4
s2  
(3[2d2d3 ]3 2d22)   
(1)  
.
2
2
2
21  
2
8
From (2.14) and (2.16), it follows that  
s1   t1  
.
(2.18)  
(2.19)  
Squaring and adding (2.14) and (2.16), we obtain that  
128  
d2222 s12t12  
.
(21)2  
Now, applying Lemma 1 to (2.19), we get  
21  
422  
d2    
.
(2.20)  
Adding (2.15) and (2.17), we can find out that  
2
2
s1 t1  
4
s2 t2  
[63d242d2]   
(1)  
.
(2.21)  
(2.22)  
2
2
2
21  
2
8
If we use (2.19) in (2.21), then we have  
(s2 t2)(21)2  
d22  
.
16[(21)(33 22) 2(1 )2]  
2
2
122  
G.D. SHELAKE, S.K. NILAPGOL AND S.B. JOSHI  
Employing Lemma 1, we obtain  
21  
d2    
.
2 (21)(33 22) 2(1 )2  
2
2
Subtracting (2.17) from (2.15) and using (2.18), it follows that  
24  
s2 t2  
(d3 d2)3   
,
2
21  
2
i.e.,  
(s2 t2)(21)  
d3  
d22  
.
(2.23)  
(2.24)  
483  
Substituting the value of d22 from (2.19) in (2.23), we obtain  
(s12t12)(21)2  
(s2 t2)(21)  
d3  
.
12822  
483  
Applying Lemma 1, we obtain  
21  
(21)2  
1222  
d3    
.
123  
Using (2.22) in (2.23), we obtain  
(s2 t2)(21)  
(s2 t2)(21)2  
d3  
. (2.25)  
16[(21)(33 22) 2(1 )2]  
483  
2
2
Using Lemma (1), it follows that  
21  
(21)2  
d3    
.
4[(21)(33 22) 2(1 )2]  
123  
2
2
BI-CONCAVE AND OZAKI-TYPE BI-CLOSE-TO-CONVEX FUNCTIONS  
123  
Lemma 2 [13]:  
c1 , c2   , then  
Let b1, b2  
and c1, c2 . Suppose that  
2b1 , b1   b2   
(b1 b2)c1 (b1 b2)c2    
2b2 , b1   b2   
Theorem 2:  
A function  
B  
form (1.1) is in class  
OBCV(, , ,) and , then  
21  
21  
,
0 (, , )   
12  
48  
d3 d2    
3
3
2
4(, , ),  
(, , ) 21   
48  
3
Proof: From (2.22) and (2.23), we have  
(s2 t2)(21)  
d3 d2 (1 )d22  
2
483  
(s2 t2)(1 )(21)2  
(s2 t2)(21)  
16[(21)(33 22) (1 )2]  
483  
2
2
21   
21   
  (, , )   
s2   (, , )   
t2  
,
483  
483  
where  
(1 )(21)2  
(, , )   
16[(21)(33 22) (1 )2]  
2
2
Applying Lemma 2, we deduce that  
124  
G.D. SHELAKE, S.K. NILAPGOL AND S.B. JOSHI  
21  
21  
,
0 (, , )   
12  
48  
d3 d2    
3
3
2
4(, , ),  
(, , ) 21   
48  
3
Corollary  
1: A function  
B  
form  
(1.1) is in class  
OBCV(, , ,), then  
21  
d3 d2    
.
2
123  
3. Bi-concave Functions  
Definition 2:  
A function  
B  
is belongs to the class  
BCV(, , , ) if it satisfies the following subordination conditions:  
z(,(z))''  
(,(z))'  
2
(1 )(1 z)  
1   
(z),  
(3.1)  
(3.2)  
1   
2(1 z)  
and  
w(,(w))''  
2
(1 )(1 w)  
1   
(w)  
(,(w))'  
1   
2(1 w)  
where  
then  
1(w) (w) and 1 2  
.
Theorem 3: A function  
B form (1.1) is in class BCV(, , ,)  
,
92 61  
d   min  
,
2
1622  
1   
2   
   
1
2
32 21  
2   
22  
(1)  
,
(3.3)  
4
(1)(22 33) 2(1 )   
   
BI-CONCAVE AND OZAKI-TYPE BI-CLOSE-TO-CONVEX FUNCTIONS  
125  
and  
92 61  
1  
d   min  
,
3
1622  
123  
1
32 21  
2   
22  
1  
(1)  
,
(3.4)  
(1)(222 33) 2(1 )   
4
123  
where 1 2 and 2 and 3 are as given in (1.3).  
Proof: From (3.1) and (3.2), we can write  
z(,(z))''  
(,(z))'  
2
(1 )(1 z)  
1   
l(z)),  
(3.5)  
(3.6)  
1   
2(1 z)  
and  
w(,(w))''  
2
(1 )(1 w)  
1   
m(w))  
.
(,(w))'  
1   
2(1 w)  
Using (2.10), (2.11) in (3.5), (3.6) respectively and equating the coefficients,  
we obtain  
2
s1  
[(1 ) 22d2 ]   
,
(3.7)  
(3.8)  
(3.9)  
1  
2
s12  
2
s2  
[(1 ) 42d263d3 ]   
(1)  
,
.
2
2
1  
2
8
2
t1  
[(1 ) 22d2 ]   
,
1  
2
t12  
2
t2  
[(1 ) 422d22 63(2d2d3)]   
(1)  
(3.10)  
(3.11)  
2
1  
2
8
From (3.7) and (3.9), it follows that  
s1 t1  
.
126  
G.D. SHELAKE, S.K. NILAPGOL AND S.B. JOSHI  
From (3.7) and (3.9), we can write  
1   
(1)s1  
d2   
,
(3.12)  
(3.13)  
22  
82  
1   
(1)t1  
d2   
.
22  
82  
Squaring and adding (3.12) and (3.13), we obtain  
(s12t12)(1)2  
(1 )2  
422  
(2 1)(s1 t1)  
d22  
,
(3.14)  
(3.15)  
12822  
1622  
i.e.,  
(s12t12)(1)2  
(1 )2  
222  
(2 1)(s1 t1)  
2d22  
.
6422  
822  
Applying Lemma 1 to (3.14), we have  
92 61  
d2   
.
1622  
Adding (3.8) and (3.10), we obtain  
2
2
s1t1  
2
(s2 t2)  
{2(1 ) 822d22 123d2}   
(1)  
.
(3.16)  
2
1  
2
8
Implies that,  
(1)(1)(s12t12)  
(1)(s2 t2)  
(1)  
(22 33)d2  
.
2
2
16  
64  
2
Multiplying both sides by (ϱ 1), thus, we get  
(1)(1)2(s12t12)  
(1)2(s2 t2)  
(2 1)  
(1)(222 33)d22  
. (3.17)  
16  
64  
2
BI-CONCAVE AND OZAKI-TYPE BI-CLOSE-TO-CONVEX FUNCTIONS  
127  
Using (3.15) in (3.17), we obtain  
(1)2(s2 t2)  
(1 )2  
222  
(1)(222 33)d22 2(1 )d22  
(1)  
16  
2
(2 1)  
822  
(1)  
. (3.18)  
(1)  
(s t )   
1
1
2
Thus, we have  
2
1
(1) (s2 t2)  
d22   
[(1)(22 3) 2(1 )]  
16  
3
2
2
(1 )2  
222  
(2 1)  
822  
(1)  
  
(1)  
(1)  
(s1 t1)   
.
(3.19)  
2
  
Applying Lemma 1 to (3.19), we obtain  
2
2
1
 21  
  
d22 2   
(1)  
.
(1)(22 3) 2(1 )  
22  
4
3
2
Now, subtracting (3.10) from (3.8) and using (3.11), we get  
(s2 t2)(1)  
d2 d3  
.
(3.20)  
2
483  
Using (3.14) in (3.20), we find that  
(s12 t12)(1)2  
(1 )2  
422  
(2 1)(s1 t1)  
(s2 t2)(1)  
d3  
. (3.21)  
12822  
1622  
483  
According to Lemma 1, we deduce that  
92 61  
1  
d3    
.
1622  
123  
128  
G.D. SHELAKE, S.K. NILAPGOL AND S.B. JOSHI  
Next we use the value of d22 form (3.19) in (3.20), we obtain  
2
1
(1) (s2 t2)  
d3   
[(1)(22 3) 2(1 )]  
16  
3
2
2
(1 )2  
222  
(2 1)  
822  
(1)  
(s2 t2)(1)  
  
(1)  
(1)  
(s1 t1)   
.
2
483  
  
(3.22)  
Using Lemma 1, we get  
2
2
1
321  
  
1  
d3   
(1)  
[(1)(22 3) 2(1 )]  
22  
4
123  
3
2
REFERENCES  
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1. Department of Mathematics,  
Willingdon College, Sangli,  
E-mail: shelakegd@gmail.com  
(Received, September 18, 2024)  
2. Department of Mathematics,  
Shivaji University, Kolhapur,  
E-mail: sarikanilapgol101@gmail.com  
3. Department of Mathematics,  
Walchand College of Engineering, Sangli,  
E-mail: joshisb@hotmail.com