Initial seven coefficient estimates for a subclass of bi-starlike functions
UDC 517.5 In the present article, a subclass of bi-starlike functions is studied and initial seven Taylor–Maclaurin coefficient estimates $|a_{2}|, |a_{3}|, \ldots , |a_{7}|$ for functions in the subclass of the function class $ \Sigma $ are obtained for the first time in the literature. Few new or...
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2021
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| author | Janani, T. Yalçın, S. Janani, T. Yalçın, S. Yalçın, Sibel |
| author_facet | Janani, T. Yalçın, S. Janani, T. Yalçın, S. Yalçın, Sibel |
| author_sort | Janani, T. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-03-31T08:46:33Z |
| description | UDC 517.5
In the present article, a subclass of bi-starlike functions is studied and initial seven Taylor–Maclaurin coefficient estimates $|a_{2}|, |a_{3}|, \ldots , |a_{7}|$ for functions in the subclass of the function class $ \Sigma $ are obtained for the first time in the literature. Few new or known consequences of the results are also pointed out. |
| doi_str_mv | 10.37863/umzh.v73i11.1046 |
| first_indexed | 2026-03-24T02:04:27Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v73i11.1046
UDC 517.5
T. Janani (School Comput. Sci. and Eng., Vellore Inst. Technology, India),
S. Yalçın (Bursa Uludag Univ., Turkey)
INITIAL SEVEN COEFFICIENT ESTIMATES
FOR A SUBCLASS OF BI-STARLIKE FUNCTIONS
ПОЧАТКОВI ОЦIНКИ СЕМИ КОЕФIЦIЄНТIВ
ДЛЯ ПIДКЛАСУ БIЗIРКОВИХ ФУНКЦIЙ
In the present article, a subclass of bi-starlike functions is studied and initial seven Taylor – Maclaurin coefficient estimates
| a2| , | a3| , . . . , | a7| for functions in the subclass of the function class \Sigma are obtained for the first time in the literature. Few
new or known consequences of the results are also pointed out.
Дослiджується пiдклас бiзiркових функцiй та вперше отримано оцiнки сiмох початкових коефiцiєнтiв Тейлора –
Маклорена | a2| , | a3| , . . . , | a7| для функцiй у пiдкласi функцiонального класу \Sigma . Також вказано деякi наслiдки
результатiв, як новi, так i вiдомi ранiше.
1. Introduction. Let \scrA denote the class of functions of the form
f(z) = z +
\infty \sum
n=2
anz
n, (1.1)
which are analytic in the open unit disc \Delta = \{ z : | z| < 1\} and normalized by the conditions f(0) = 0
and f \prime (0) = 1. Further, let \scrS denote the class of all functions in \scrA which are univalent in \Delta . The
important and well analyzed subclasses of the univalent function class \scrS includes, the class \scrS \ast (\alpha )
of starlike functions of order \alpha in \Delta and the class \scrK (\alpha ) of convex functions of order \alpha , 0 \leq \alpha < 1,
in \Delta . It is well-known that every function f \in \scrS has an inverse f - 1, defined by
f - 1(f(z)) = z, z \in \Delta ,
and
f(f - 1(w)) = w, | w| < r0(f), r0(f) \geq 1/4,
where
f - 1(w) = g(w) = w - a2w
2 + (2a22 - a3)w
3 - (5a32 - 5a2a3 + a4)w
4 + (14a42 -
- 21a22a3 + 6a4a2 + 3a23 - a5)w
5 + ( - 42a52 + 84a32a3 - 28a4a
2
2 -
- 28a2a
2
3 + 7a5a2 + 7a4a3 - a6)w
6 + (132a62 - 330a42a3 + 120a32a4 +
+ 180a22a
2
3 - 36a5a
2
2 - 72a2a3a4 + 8a6a2 - 12a33 + 8a5a3 + 4a24 - a7)w
7 + . . . . (1.2)
A function f(z) \in \scrA is said to be bi-univalent in \Delta if both f(z) and f - 1(z) are univalent in \Delta .
Let \Sigma denote the class of bi-univalent functions in \Delta given by (1.1). Earlier, Brannan and Taha [4]
introduced certain subclasses of bi-univalent function class \Sigma , namely bi-starlike functions of order
c\bigcirc T. JANANI, S. YALÇIN, 2021
1576 ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
INITIAL SEVEN COEFFICIENT ESTIMATES . . . 1577
\alpha denoted by \scrS \ast
\Sigma (\alpha ) and bi-convex function of order \alpha denoted by \scrK \Sigma (\alpha ) corresponding to the
function classes \scrS \ast (\alpha ) and \scrK (\alpha ), respectively.
A function f(z) \in \scrA is in the class of strongly bi-starlike functions \scrS \ast
\Sigma [\alpha ] [4, 15] of order \alpha ,
0 < \alpha \leq 1, if each of the following conditions is satisfied:\bigm| \bigm| \bigm| \bigm| \mathrm{a}\mathrm{r}\mathrm{g}\biggl( zf \prime (z)
f(z)
\biggr) \bigm| \bigm| \bigm| \bigm| < \alpha \pi
2
,
\bigm| \bigm| \bigm| \bigm| \mathrm{a}\mathrm{r}\mathrm{g}\biggl( wg\prime (w)
g(w)
\biggr) \bigm| \bigm| \bigm| \bigm| < \alpha \pi
2
and strongly bi-convex functions \scrK \ast
\Sigma [\alpha ] [4, 15] of order \alpha , 0 < \alpha \leq 1,\bigm| \bigm| \bigm| \bigm| \mathrm{a}\mathrm{r}\mathrm{g}\biggl( 1 + zf \prime \prime (z)
f \prime (z)
\biggr) \bigm| \bigm| \bigm| \bigm| < \alpha \pi
2
,
\bigm| \bigm| \bigm| \bigm| \mathrm{a}\mathrm{r}\mathrm{g}\biggl( 1 + wg\prime \prime (w)
g\prime (w)
\biggr) \bigm| \bigm| \bigm| \bigm| < \alpha \pi
2
,
where g is given by (1.2). For each of the function classes \scrS \ast
\Sigma [\alpha ] and \scrK \Sigma [\alpha ], non-sharp estimates
on the first two Taylor – Maclaurin coefficients | a2| and | a3| were found [4, 15]. Though intensive
research is happening to settle the coefficient problem of obtaining each of the following Taylor –
Maclaurin coefficients:
| an| , n \in \BbbN \setminus \{ 1, 2\} , \BbbN := \{ 1, 2, 3, . . . \} ,
it is still an open problem (see [3, 4, 10, 12, 15]). Many researchers (see [14, 16, 17]) have introduced
and investigated several interesting subclasses of the bi-univalent function class \Sigma and obtained non-
sharp estimates on the first few Taylor – Maclaurin coefficients | a2| , | a3| and | a4| .
An analytic function f is subordinate to an analytic function g, written f(z) \prec g(z), provided
there is an analytic function w defined on \Delta with w(0) = 0 and | w(z)| < 1 satisfying f(z) =
= g(w(z)). Ma and Minda [19] unified various subclasses of starlike and convex functions for which
either of the quantity
z f \prime (z)
f(z)
or 1+
z f \prime \prime (z)
f \prime (z)
is subordinate to a more general superordinate function.
For this analysis, an analytic function \phi with positive real part in the unit disk \Delta was considered,
with \phi (0) = 1, \phi \prime (0) > 0, and \phi maps \Delta onto a region starlike with respect to 1 and symmetric
with respect to the real axis. The class of Ma – Minda starlike functions consists of functions f \in \scrA
satisfying the subordination
z f \prime (z)
f(z)
\prec \phi (z). Similarly, the class of Ma – Minda convex functions of
functions f \in \scrA satisfying the subordination 1 +
z f \prime \prime (z)
f \prime (z)
\prec \phi (z).
A function f is bi-starlike of Ma – Minda type or bi-convex of Ma – Minda type if both f and
f - 1 are, respectively, Ma – Minda starlike or convex. These classes are denoted, respectively, by
\scrS \ast
\Sigma (\phi ) and \scrK \Sigma (\phi ).
Motivated by the earlier work of coefficient estimate analysis study of various subclasses of bi-
univalent function class [1 – 3, 5 – 11, 13, 14, 16, 17], in the present article, a bi-starlike function sub-
class of \Sigma is considered and initial seven Taylor – Maclaurin coefficient estimates | a2| , | a3| , . . . , | a7|
for functions in the subclass of the function class \Sigma are obtained. It can be easily observed in the
literature that only | a2| , | a3| , | a4| coefficient estimates are obtained for many subclasses of function
class \Sigma so far, where as | a5| , | a6| , | a7| are estimated for the first time in the literature without any
assumptions.
In the sequel, it is assumed that \phi is an analytic function with positive real part in the unit disk \Delta ,
satisfying \phi (0) = 1, \phi \prime (0) > 0, and \phi (\Delta ) is symmetric with respect to the real axis. Such a function
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
1578 T. JANANI, S. YALÇIN
has a series expansion of the form
\phi (z) = 1 +B1z +B2z
2 +B3z
3 + . . . , B1 > 0.
The following bi-starlike class definition is considered from the literature for the study.
Definition 1.1. A function f(z) \in \Sigma given by (1.1) is said to be in the class \scrS \ast
\Sigma (\phi ) if the
following conditions are satisfied:
z f \prime (z)
f(z)
\prec \phi (z) (1.3)
and
w g\prime (w)
g(w)
\prec \phi (w) (1.4)
where z, w \in \Delta and the function g is given by (1.2).
2. Coefficient estimates for the function class \bfscrS \ast
\Sigma (\bfitphi ). The following lemma is used to derive
our main result.
Lemma 2.1 [18]. If h \in \scrP , then | ck| \leq 2 for each k, where \scrP is the family of all functions h
analytic in \Delta for which real part of h(z) > 0 and
h(z) = 1 + c1z + c2z
2 + . . . for z \in \Delta .
Theorem 2.1. Let f(z) given by (1.1) be in the class \scrS \ast
\Sigma (\phi ). Then the initial seven Taylor –
Maclaurin coefficient estimates are
| a2| \leq B1,
| a3| \leq B2
1 +B1/2,
| a4| \leq 2B3
1/3 + 5B2
1/4 + 4B1/3 + 4| B2| /3 + | B3| /3,
| a5| \leq 7B1/4 + | B3| (B1 + 3/4) + 35B2
1/8 + 5B3
1/4 + | B2| (4B1 + 9/4),
| a6| \leq 16B1/5 + 8| B4| /5 + | B5| /5 + | B3| (B2
1 + 77B1/24 + 24/5) + 63B2
1/8 +
+ 19B3
1/4 + 7B4
1/12 +B5
1/5 + | B2| (4B2
1 + 77B1/8 + 32/5),
| a7| \leq 31B1/6 + | B3| (10B3
1/9 + 115B2
1/24 + 2023B1/90 + 16| B2| /9 + 35/3) +
+ 1787B2
1/90 + 231B3
1/16 + 32| B2| 2/9 + 5909B4
1/36 + 2| B3| 2/9 +
+ 2899B5
1/12 + 49B6
1/45 + | B5| (4B1/5 + 5/6) + | B4| (32B1/5 + 5) +
+ | B2| (1124B3
1/9 + 385B2
1/24 + 3349B1/90 + 25/2).
Proof. It follows from (1.3) and (1.4) that
z f \prime (z)
f(z)
= \phi (u(z)) (2.1)
and
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
INITIAL SEVEN COEFFICIENT ESTIMATES . . . 1579
w g\prime (w)
g(w)
= \phi (v(w)). (2.2)
Define the functions p(z) and q(z) by
p(z) =
1 + u(z)
1 - u(z)
= 1 + p1z + p2z
2 + p3z
3 + p4z
4 + p5z
5 + p6z
6 + p7z
7 + . . .
and
q(z) =
1 + v(z)
1 - v(z)
= 1 + q1z + q2z
2 + q3z
3 + q4z
4 + q5z
5 + q6z
6 + q7z
7 + . . . ,
or, equivalently,
u(z) =
p(z) - 1
p(z) + 1
= (p1z)/2 + (p2/2 - p21/4)z
2 + (p3/2 - p1(p2/4 - p21/8) - (p1p2)/4)z
3 +
+ (p4/2 - p2(p2/4 - p21/8) - (p1p3)/4 + p1((p1(p2/4 - p21/8))/2 - p3/4 +
+ (p1p2)/8))z
4 + (p5/2 - p3(p2/4 - p21/8) - (p1p4)/4 + p2((p1(p2/4 -
- p21/8))/2 - p3/4 + (p1p2)/8) - p1(p4/4 - (p2(p2/4 - p21/8))/2 -
- (p1p3)/8 + (p1((p1(p2/4 - p21/8))/2 - p3/4 + (p1p2)/8))/2))z
5 + . . .
and
v(z) =
q(z) - 1
q(z) + 1
= (q1z)/2 + (q2/2 - q21/4)z
2 + (q3/2 - q1(q2/4 - q21/8) - (q1q2)/4)z
3 +
+ (q4/2 - q2(q2/4 - q21/8) - (q1q3)/4 + q1((q1(q2/4 - q21/8))/2 - q3/4 +
+ (q1q2)/8))z
4 + (q5/2 - q3(q2/4 - q21/8) - (q1q4)/4 + q2((q1(q2/4 -
- q21/8))/2 - q3/4 + (q1q2)/8) - q1(q4/4 - (q2(q2/4 - q21/8))/2 -
- (q1q3)/8 + (q1((q1(q2/4 - q21/8))/2 - q3/4 + (q1q2)/8))/2))z
5 + . . . .
Then p(z) and q(z) are analytic in \Delta with p(0) = 1 = q(0). Since u, v : \Delta \rightarrow \Delta , the functions
p(z) and q(z) have a positive real part in \Delta , and, for each i,
| pi| \leq 2 \mathrm{a}\mathrm{n}\mathrm{d} | qi| \leq 2.
Since p(z) and q(w) in \scrP we have the following forms:
\phi (u(z)) = \phi
\biggl(
1
2
\biggl[
p1z +
\biggl(
p2 -
p21
2
\biggr)
z2 + . . .
\biggr] \biggr)
=
= ((B1p1)/2)z + ((B2p
2
1)/4 +B1(p2/2 - p21/4))z
2 + ( - B1(p1(p2/4 - p21/8) -
- p3/2 + (p1p2)/4) + (B3p
3
1)/8 +B2p1(p2/2 - p21/4))z
3 + ((B4p
4
1)/16 -
- B2(p1(p1(p2/4 - p21/8) - p3/2 + (p1p2)/4) - (p2/2 - p21/4)
2) +B1(p4/2 -
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
1580 T. JANANI, S. YALÇIN
- p2(p2/4 - p21/8) - (p1p3)/4 + p1((p1(p2/4 - p21/8))/2 - p3/4 + (p1p2)/8)) +
+ (3B3p
2
1(p2/2 - p21/4))/4)z
4 + ( - B3( - p1(p2/2 - p21/4)
2 + (p1(p1(p1(p2/4 -
- p21/8) - p3/2 + (p1p2)/4) - (p2/2 - p21/4)
2))/2 + (p21(p1(p2/4 - p21/8) - p3/2 +
+ (p1p2)/4))/4) - B2(2(p2/2 - p21/4)(p1(p2/4 - p21/8) - p3/2 + (p1p2)/4) -
- p1(p4/2 - p2(p2/4 - p21/8) - (p1p3)/4 + p1((p1(p2/4 - p21/8))/2 - p3/4 +
+ (p1p2)/8))) + (B5p
5
1)/32 - B1( - p5/2 + p3(p2/4 - p21/8) + (p1p4)/4 -
- p2((p1(p2/4 - p21/8))/2 - p3/4 + (p1p2)/8) + p1(p4/4 - (p2(p2/4 -
- p21/8))/2 - (p1p3)/8 + (p1((p1(p2/4 - p21/8))/2 - p3/4 + (p1p2)/8))/2)) +
+ (B4p
3
1(p2/2 - p21/4))/2)z
5 + . . .
and
\phi (v(w)) = \phi
\biggl(
1
2
\biggl[
q1w +
\biggl(
q2 -
q21
2
\biggr)
w2 + . . .
\biggr] \biggr)
=
= ((B1q1)/2)w + ((B2q
2
1)/4 +B1(q2/2 - q21/4))w
2 + ( - B1(q1(q2/4 - q21/8) -
- q3/2 + (q1q2)/4) + (B3q
3
1)/8 +B2q1(q2/2 - q21/4))w
3 + ((B4q
4
1)/16 -
- B2(q1(q1(q2/4 - q21/8) - q3/2 + (q1q2)/4) - (q2/2 - q21/4)
2) +B1(q4/2 -
- q2(q2/4 - q21/8) - (q1q3)/4 + q1((q1(q2/4 - q21/8))/2 - q3/4 + (q1q2)/8)) +
+ (3B3q
2
1(q2/2 - q21/4))/4)w
4 + ( - B3( - q1(q2/2 - q21/4)
2 + (q1(q1(q1(q2/4 -
- q21/8) - q3/2 + (q1q2)/4) - (q2/2 - q21/4)
2))/2 + (q21(q1(q2/4 - q21/8) - q3/2 +
+ (q1q2)/4))/4) - B2(2(q2/2 - q21/4)(q1(q2/4 - q21/8) - q3/2 + (q1q2)/4) -
- q1(q4/2 - q2(q2/4 - q21/8) - (q1q3)/4 + q1((q1(q2/4 - q21/8))/2 - q3/4 +
+ (q1q2)/8))) + (B5q
5
1)/32 - B1( - q5/2 + q3(q2/4 - q21/8) + (q1q4)/4 -
- q2((q1(q2/4 - q21/8))/2 - q3/4 + (q1q2)/8) + q1(q4/4 - (q2(q2/4 - q21/8))/2 -
- (q1q3)/8 + (q1((q1(q2/4 - q21/8))/2 - q3/4 + (q1q2)/8))/2)) +
+ (B4q
3
1(q2/2 - q21/4))/2)w
5 + . . . .
The analytic conditions given in the equation (2.1) and (2.2) are estimated using MatLab functions
and by comparing the coefficients on both sides of the conditions given, we can easily get the
following:
p1 = - q1,
a2 = (p1/2)B1,
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
INITIAL SEVEN COEFFICIENT ESTIMATES . . . 1581
a3 = (B2
1p
2
1)/4 + (p2/8 - q2/8)B1,
a4 = ( - q31/24)B3 + (p31/12)B
3
1 + (p3/12 - q3/12 + (p2q1)/12 + (q1q2)/12 -
- q31/24)B1 + ((5p1p2)/32 - (5p1q2)/32)B
2
1 + (q31/12 - (q1q2)/12 - (p2q1)/12)B2,
a5 = ((3p22)/128 - (3p2q2)/64 + (p1q1p2)/8 + (3q22)/128 + (p1q1q2)/8 +
+ (p1p3)/8 - (p1q3)/8 - (p1q
3
1)/16)B
2
1 + ((5p21p2)/64 - (5p21q2)/64)B
3
1 +
+ B1(p4/16 - q4/16 + (p3q1)/16 + (q1q3)/16 + (3p2q
2
1)/64 - (3q21q2)/64 -
- p22/32 + q22/32) + ((3p2q
2
1)/64 - (3q21q2)/64 - (B1p1q
3
1)/16)B3 +
+ ((3q21q2)/32 - (p3q1)/16 - (q1q3)/16 - (3p2q
2
1)/32 - B1((p1p2q1)/8 -
- (p1q
3
1)/8 + (p1q1q2)/8) + p22/32 - q22/32)B2,
a6 = ( - q51/160)B5 +B2
1((7p1p4)/64 + (7p2p3)/192 - (7p1q4)/64 - (7p2q3)/192 -
- (7p3q2)/192 + (7q2q3)/192 - (7p1p
2
2)/128 + (7p1q
2
2)/128 + (7p22q1)/192 -
- ( 7p2q
3
1)/384 - (7q1q
2
2)/192 + (7q31q2)/384 + (21p1p2q
2
1)/256 -
- (21p1q
2
1q2)/256 + (7p1p3q1)/64 + (7p1q1q3)/64) +B1(p5/20 - q5/20 -
- (p2p3)/20 + (p4q1)/20 + (q1q4)/20 + (q2q3)/20 - (3p22q1)/80 + (p2q
3
1)/40 +
+ (3p3q
2
1)/80 - (3q1q
2
2)/80 - (3q21q3)/80 + (q31q2)/40 - q51/160) + ((7p31q2)/384 -
- ( 7p31p2)/384)B
4
1 + ((3p1p
2
2)/128 + (p21p3)/16 + (3p1q
2
2)/128 - (p21q3)/16 -
- (p21q
3
1)/32 + (p21p2q1)/16 + (p21q1q2)/16 - (3p1p2q2)/64)B
3
1 + ( - p51/160)B
5
1 +
+ (q51/40 - (q31q2)/40 - (p2q
3
1)/40)B4 + ((p2p3)/20 - (p4q1)/20 - (q1q4)/20 -
- (q2q3)/20 - B1((7p1q
2
2)/128 - (7p1p
2
2)/128 + (7p22q1)/192 - (7p2q
3
1)/192 -
- (7q1q
2
2)/192 + (7q31q2)/192 + (21p1p2q
2
1)/128 - (21p1q
2
1q2)/128 + (7p1p3q1)/64 +
+ (7p1q1q3)/64) - B2
1((p
2
1p2q1)/16 - (p21q
3
1)/16 + (p21q1q2)/16) + (3p22q1)/40 -
- (3p2q
3
1)/40 - (3p3q
2
1)/40 + (3q1q
2
2)/40 + (3q21q3)/40 - (3q31q2)/40 + q51/40)B2 +
+ ((3p2q
3
1)/40 - (3p22q1)/80 - B1((7p2q
3
1)/384 - (7q31q2)/384 - (21p1p2q
2
1)/256 +
+ (21p1q
2
1q2)/256) + (3p3q
2
1)/80 - (3q1q
2
2)/80 - (3q21q3)/80 + (3q31q2)/40 -
- (3q51)/80 - (B2
1p
2
1q
3
1)/32)B3,
a7 = ((3q21q
2
2)/32 - B2
1((q
3
1(15p1p2 - 15p1q2))/576 - (p1p2q
3
1)/128 + (p1q
3
1q2)/128 -
- (21p21p2q
2
1)/512 + (21p21q
2
1q2)/512) + (5p2q
4
1)/64 + (p3q
3
1)/16 + (p4q
2
1)/32 -
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
1582 T. JANANI, S. YALÇIN
- (q21q4)/32 + (q31q3)/16 - (5q41q2)/64 + p32/96 - q32/96 - B1((3p1q
5
1)/40 -
- (3q21q
2
2)/128 + (q31(8p3 - 8q3 + 8p2q1 + 8q1q2 - 4q31))/576 - (3p22q
2
1)/128 +
+ (3p1p
2
2q1)/40 - (3p1p2q
3
1)/20 - (3p1p3q
2
1)/40 + (3p1q1q
2
2)/40 + (3p1q
2
1q3)/40 -
- (3p1q
3
1q2)/20 + (3p2q
2
1q2)/64) - (3p22q
2
1)/32 + (B2q
3
1(8p2q1 + 8q1q2 - 8q31))/576 +
+ (5B3
1p
3
1q
3
1)/288 - (p2p3q1)/16 - (q1q2q3)/16)B3 +B3
1((7p
2
1p4)/128 + (7p2q
2
2)/1024 -
- (7p22q2)/1024 - (7p21q4)/128 + (7p32)/3072 - (7q32)/3072 + ((15p1p2 - 15p1q2)
(8p3 - 8q3 + 8p2q1 + 8q1q2 - 4q31))/2304 - (7p21p
2
2)/256 + (7p21q
2
2)/256 - (p1p
2
2q1)/64 +
+ (p1p2q
3
1)/128 + (7p21p3q1)/128 + (p1q1q
2
2)/64 - (p1q
3
1q2)/128 + (7p21q1q3)/128 +
+ (21p21p2q
2
1)/512 - (21p21q
2
1q2)/512 - (p1p2p3)/64 + (p1p2q3)/64 + (p1p3q2)/64 -
- (p1q2q3)/64) + ((49p61)/2880)B
6
1 + (q61/288)B
2
3 - B2((3q
2
1q
2
2)/32 +B3
1((p
3
1(8p2q1 +
+ 8q1q2 - 8q31))/288 + (85p31((p2q1)/12 + (q1q2)/12 - q31/12))/16 + (97p31q
3
1)/192 -
- (97p31p2q1)/192 - (97p31q1q2)/192) - (p2p4)/24 + (p5q1)/24 + (q1q5)/24 + (q2q4)/24 +
+ B1((3q
2
1q
2
2)/64 + ((8p2q1 + 8q1q2 - 8q31)(8p3 - 8q3 + 8p2q1 + 8q1q2 - 4q31))/2304 +
+ (p2q
2
2)/64 + (p22q2)/64 - (p1q
5
1)/20 - p32/64 - q32/64 + (3p22q
2
1)/64 - (3p1p
2
2q1)/20 +
+ (3p1p2q
3
1)/20 + (3p1p3q
2
1)/20 - (3p1q1q
2
2)/20 - (3p1q
2
1q3)/20 + (3p1q
3
1q2)/20 -
- (3p2q
2
1q2)/32 - (p1p2p3)/10 + (p1p4q1)/10 + (p2p3q1)/32 + (p1q1q4)/10 + (p1q2q3)/10 + . . .
. . .+ (p2q1q3)/32 - (p3q1q2)/32 - (q1q2q3)/32) + (5p2q
4
1)/96 + (p3q
3
1)/16 + (p4q
2
1)/16 -
- (q21q4)/16 + (q31q3)/16 - (5q41q2)/96 + p32/48 - p23/48 - q32/48 + q23/48 +
+ B2
1(((15p1p2 - 15p1q2)(8p2q1 + 8q1q2 - 8q31))/2304 - (7p21p
2
2)/256 + (7p21q
2
2)/256 -
- (p1p
2
2q1)/64 + (p1p2q
3
1)/64 + (7p21p3q1)/128 + (p1q1q
2
2)/64 - (p1q
3
1q2)/64 + (7p21q1q3)/128 +
+ (21p21p2q
2
1)/256 - (21p21q
2
1q2)/256) - (3p22q
2
1)/32 - (p2p3q1)/8 - (q1q2q3)/8) +
+ ((255p21(p2 - q2)
2)/1024 + (15p1p2 - 15p1q2)
2/4608 - (97p31p3)/192 + (97p31q3)/192 +
+ (p31(8p3 - 8q3 + 8p2q1 + 8q1q2 - 4q31))/288 + (85p31(p3/12 - q3/12 + (p2q1)/12 + (q1q2)/12 -
- q31/24))/16 - (323p21p
2
2)/1024 - (323p21q
2
2)/1024 + (97p31q
3
1)/384 + (323p21p2q2)/512 -
- (97p31p2q1)/192 - (97p31q1q2)/192)B
4
1 +B1(p6/24 - q6/24 + (q21q
2
2)/32 - (p2p4)/24 +
+ (p5q1)/24 + (q1q5)/24 + (q2q4)/24 + (5p2q
4
1)/384 + (p3q
3
1)/48 + (p4q
2
1)/32 - (q21q4)/32 +
+ (q31q3)/48 - (5q41q2)/384 + p32/96 - p23/48 - q32/96 + q23/48 - (p22q
2
1)/32 - (p2p3q1)/16 -
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
INITIAL SEVEN COEFFICIENT ESTIMATES . . . 1583
- (q1q2q3)/16) + ((p31(15p1p2 - 15p1q2))/288 + (85p31((5p1p2)/32 - (5p1q2)/32))/16 -
- (971p41p2)/512 + (971p41q2)/512 + (255p41(p2 - q2))/256)B
5
1 +B2
1((3q
2
1q
2
2)/128 +
+ (8p3 - 8q3 + 8p2q1 + 8q1q2 - 4q31)
2/4608 + (p1p5)/10 + (p2p4)/32 - (p1q5)/10 -
- (p2q4)/32 - (p4q2)/32 + (q2q4)/32 + (p2q
2
2)/64 + (p22q2)/64 - (p1q
5
1)/80 - p32/64 -
- q32/64 + (3p22q
2
1)/128 - (3p1p
2
2q1)/40 + (p1p2q
3
1)/20 + (3p1p3q
2
1)/40 - (3p1q1q
2
2)/40 -
- (3p1q
2
1q3)/40 + (p1q
3
1q2)/20 - (3p2q
2
1q2)/64 - (p1p2p3)/10 + (p1p4q1)/10 + (p2p3q1)/32 +
+ (p1q1q4)/10 + (p1q2q3)/10 + (p2q1q3)/32 - (p3q1q2)/32 - (q1q2q3)/32) + ((8p2q1 + 8q1q2 -
- 8q31)
2/4608)B2
2 + ((5p2q
4
1)/384 - (5q41q2)/384 - (B1p1q
5
1)/80)B5 + ((5q41q2)/96 -
- (5p2q
4
1)/96 - (p3q
3
1)/48 - (q31q3)/48 - (q21q
2
2)/32 - B1((p1p2q
3
1)/20 - (p1q
5
1)/20 +
+ (p1q
3
1q2)/20) + (p22q
2
1)/32)B4.
After applying modulus on both sides of the equations and applying the Lemma 2.1, we obtain
| a2| \leq B1,
| a3| \leq B2
1 +B1/2,
| a4| \leq 2B3
1/3 + 5B2
1/4 + 4B1/3 + 4| B2| /3 + | B3| /3,
| a5| \leq 7B1/4 + | B3| (B1 + 3/4) + 35B2
1/8 + 5B3
1/4 + | B2| (4B1 + 9/4),
| a6| \leq 16B1/5 + 8| B4| /5 + | B5| /5 + | B3| (B2
1 + 77B1/24 + 24/5) + 63B2
1/8 + 19B3
1/4 +
+ 7B4
1/12 +B5
1/5 + | B2| (4B2
1 + 77B1/8 + 32/5),
| a7| \leq 31B1/6 + | B3| (10B3
1/9 + 115B2
1/24 + 2023B1/90 + 16| B2| /9 + 35/3) +
+ 1787B2
1/90 + 231B3
1/16 + 32| B2| 2/9 + 5909B4
1/36 + 2| B3| 2/9 + 2899B5
1/12 +
+ 49B6
1/45 + | B5| (4B1/5 + 5/6) + | B4| (32B1/5 + 5) + | B2| (1124B3
1/9 +
+ 385B2
1/24 + 3349B1/90 + 25/2).
The theorem is proved.
Considering the function \phi to be
\phi (z) =
\biggl(
1 + z
1 - z
\biggr) \alpha
= 1 + 2\alpha z + 2\alpha 2z2 + . . . , 0 < \alpha \leq 1, (2.3)
which gives B1 = 2\alpha and B2 = 2\alpha 2, for the class of strongly starlike functions.
Also, if we consider
\phi (z) =
1 + (1 - 2\beta )z
1 - z
= 1 + 2(1 - \beta )z + 2(1 - \beta )z2 + . . . , 0 \leq \beta < 1, (2.4)
then we have B1 = B2 = 2(1 - \beta ).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
1584 T. JANANI, S. YALÇIN
Remark 2.1. By choosing \phi (z) of the form (2.3) and (2.4), we can easily obtain the initial seven
Taylor coefficient estimates | a2| , | a3| , . . . , | a7| based on the result discussed in the Theorem 2.1,
of which the higher estimates like | a5| , | a6| and | a7| are obtained for the first time without any
assumption in the estimation process.
Remark 2.2. Based on the coefficient estimation procedure discussed in the above remark, one
can easily notice that the initial estimates | a2| and | a3| leads to the well-known results given earlier
by Brannan and Taha [4].
References
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№ 1, 35 – 40 (2019).
3. D. A. Brannan, J. G. Clunie (Eds), Aspects of contemporary complex analysis, Proc. NATO Adv. Study Inst., Univ.
Durham, Durham, July, 1979, Acad. Press, New York, London (1980), p. 1 – 20.
4. D. A. Brannan, T. S. Taha, On some classes of bi-univalent functions, Stud. Univ. Babeş-Bolyai Math., Some Math. J.,
31, № 2, 70 – 77 (1986).
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49 – 60 (2013).
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of complex order, Novi Sad J. Math., 48, № 1, 93 – 102 (2018).
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11. G. Murugusundaramoorthy, Subclasses of bi-univalent functions of complex order based on subordination conditions
involving wright hypergeometric functions, J. Math. and Fundam. Sci., 47, 60 – 75 (2015).
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function in | z| < 1, Arch. Ration. Mech. and Anal., 32, 100 – 112 (1969).
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associated with the Hohlov operator, J. Complex Anal., 2014, Article 693908, (2014), p. 1 – 6.
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Received 21.08.19,
after revision — 26.06.20
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 11
|
| id | umjimathkievua-article-1046 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:04:27Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/d1/e8352e75f8e8a757310406a0e858abd1.pdf |
| spelling | umjimathkievua-article-10462025-03-31T08:46:33Z Initial seven coefficient estimates for a subclass of bi-starlike functions Initial seven coefficient estimates for a subclass of bi-starlike functions Janani, T. Yalçın, S. Janani, T. Yalçın, S. Yalçın, Sibel Analytic functions, Univalent functions, Bi-univalent functions, Bi-starlike, Analytic functions, Univalent functions, Bi-univalent functions, Bi-starlike, UDC 517.5 In the present article, a subclass of bi-starlike functions is studied and initial seven Taylor–Maclaurin coefficient estimates $|a_{2}|, |a_{3}|, \ldots , |a_{7}|$ for functions in the subclass of the function class $ \Sigma $ are obtained for the first time in the literature. Few new or known consequences of the results are also pointed out. УДК 517.5 Початковi оцiнки семи коефiцiєнтiв для пiдкласу бiзiркових функцiй Досліджується підклас бізіркових функцій та вперше отримано оцінки сімох початкових коефіцієнтів Тейлора–Маклорена $|a_{2}|, |a_{3}|, \ldots , |a_{7}|$ для функцій у підкласі функціонального класу $\Sigma.$ Також вказано деякі наслідки результатів, як нові, так і відомі раніше. Institute of Mathematics, NAS of Ukraine 2021-11-23 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1046 10.37863/umzh.v73i11.1046 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 11 (2021); 1576 - 1584 Український математичний журнал; Том 73 № 11 (2021); 1576 - 1584 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1046/9154 Copyright (c) 2021 Thambidurai Janani, Sibel Yalçın |
| spellingShingle | Janani, T. Yalçın, S. Janani, T. Yalçın, S. Yalçın, Sibel Initial seven coefficient estimates for a subclass of bi-starlike functions |
| title | Initial seven coefficient estimates for a subclass of bi-starlike functions |
| title_alt | Initial seven coefficient estimates for a subclass of bi-starlike functions |
| title_full | Initial seven coefficient estimates for a subclass of bi-starlike functions |
| title_fullStr | Initial seven coefficient estimates for a subclass of bi-starlike functions |
| title_full_unstemmed | Initial seven coefficient estimates for a subclass of bi-starlike functions |
| title_short | Initial seven coefficient estimates for a subclass of bi-starlike functions |
| title_sort | initial seven coefficient estimates for a subclass of bi-starlike functions |
| topic_facet | Analytic functions Univalent functions Bi-univalent functions Bi-starlike, Analytic functions Univalent functions Bi-univalent functions Bi-starlike, |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1046 |
| work_keys_str_mv | AT jananit initialsevencoefficientestimatesforasubclassofbistarlikefunctions AT yalcıns initialsevencoefficientestimatesforasubclassofbistarlikefunctions AT jananit initialsevencoefficientestimatesforasubclassofbistarlikefunctions AT yalcıns initialsevencoefficientestimatesforasubclassofbistarlikefunctions AT yalcınsibel initialsevencoefficientestimatesforasubclassofbistarlikefunctions |