The Extension of Chebyshev Polynomial Bounds Involving Bazilevic Function

Abstract

The Chebychev’s polynomial has vast applications in GFT. The powerful tool called convolution (Or Hadamard product), subordination techniques are used in designing the new class.
In establishing the core results, derivative tests, triangle inequality and appropriate results that are existing are used.
Findings:The trigonometric polynomials are applied and a class of Bi-univalent functions Pa;b;cS (l ;t ;q;q ) involving Bazilevic Sakaguchi function is derived. More over, the maximum bounds for initial coefficients and Fekete-Szego functional for the underlying class are computed. This finding opens the door to young researchers to move further with successive coefficient estimates and related research.
Keywords: Bistarlike functions; Bi-Starlike Functions; Bi-Univalent Functions; Sakaguchi Type Functions; Subordination; Trigonometric Polynomials

Full text

INDIAN JOURNAL OF SCIENCE AND TECHNOLOGY
RESEARCH ARTICLE
OPEN ACCESS
Received: 26-05-2023
Accepted: 01-06-2023
Published: 18-07-2023
Citation: Chinthamani S, Lokesh P
(2023) The Extension of Chebyshev
Polynomial Bounds Involving
Bazilevic Function. Indian Journal of
Science and Technology 16(27):
2040-2046. https://doi.org/
10.17485/IJST/v16i27.icrms-207
∗Corresponding author.
chinthamani@stellamariscollege.edu.in
Funding: None
Competing Interests: None
Copyright: © 2023 Chinthamani &
Lokesh. This is an open access
article distributed under the terms
of the Creative Commons
Attribution License, which permits
unrestricted use, distribution, and
reproduction in any medium,
provided the original author and
source are credited.
Published By Indian Society for
Education and Environment (iSee)
ISSN
Print: 0974-6846
Electronic: 0974-5645
The Extension of Chebyshev Polynomial
Bounds Involving Bazilevic Function
S Chinthamani1∗, P Lokesh2
1Department of Mathematics, Stella Maris College, Cathedral Road, Tamil Nadu, India
2Department of Mathematics, Adhiparasakthi College of Engineering, Kalavai, Tamil Nadu,
India
Abstract
Objectives: To propose a new class of bi-univalent function based on
Bazilevic Sakaguchi function using the trigonometric polynomials Tnq,ei
θ
and
to find the Taylor – Maclaurin coefficient inequalities and Fekete – Szego
inequality for upper bounds. Methods: The Chebychev’s polynomial has vast
applications in GFT. The powerful tool called convolution (Or Hadamard
product), subordination techniques are used in designing the new class.
In establishing the core results, derivative tests, triangle inequality and
appropriate results that are existing are used. Findings:The trigonometric
polynomials are applied and a class of Bi-univalent functions Pa,b,c
Σ(
λ
,
τ
,q,
θ
)
involving Bazilevic Sakaguchi function is derived. More over, the maximum
bounds for initial coefficients and Fekete-Szego functional for the underlying
class are computed. This finding opens the door to young researchers to move
further with successive coefficient estimates and related research. Novelty:In
recent days, several studies on Chebyshev’s polynomial are revolving around
univalent function classes among researchers. But in this article a significant
amount of interplay between Chebyshev’s polynomial and Bazilevic Sakaguchi
function associated with Bi-univalent functions is clearly established.
Keywords: Bistarlike functions; Bi-Starlike Functions; Bi-Univalent Functions;
Sakaguchi Type Functions; Subordination; Trigonometric Polynomials
1 Introduction
Let A represent the family of functions f that are analytic in the open unit disk ∆=
(z∈C:(z|<1}of the form:
f(z) = z+∑∞
k=2
ρ
kzk(1)
For h(z)∈A, given by
h(z) = z+∑∞
k=2hkzk
Let S mean the subclass of A consisting of univalent functions in∆. It is well known
(refer(1,2)) that every function of f∈Svirtually possesses an inverse of f, dened by
f−1[f(z)] = z,(z∈∆)and f[f−1(w)] = w,((w|<r0(f);r0(f)≥1
4), where
f−1(w) = w−
ρ
2w2+ (2
ρ
2
2−
ρ
3)w3−(5
ρ
3
2−5
ρ
2
ρ
3+
ρ
4)w4+....... (2)
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When the function f∈Ais bi-univalent, both fand f−1are univalent in∆. Let ∑be the class of bi-univalent functions in ∆
given by (1). In fact, Feras Yousef et al.(2)have revived the study of analytic and bi-univalent functions in recent years. Many
researchers investigated and propounded various subclasses of bi-univalent functions and xed the initial coecients (
ρ
2|and
(
ρ
3|(3–6)
For analytic functions f and g, f is said to be subordinate to g, denotedf(z)≺g(z), if there is an analytic function w such
that w(0) = 0,(w(z)|<1and f(z) = g(w(z)).
A function f∈Sis said to be Bazilevic function if it satises (see(7)):
ℜz1−
λ
f′(z)
(f(z))1−
λ
>0,(z∈∆,
λ
≥0)
is class of the function was denoted by B
λ
. Consequently when
λ
=0, the class of starlike function is obtained.
In recently, P.Lokesh et al(8)investigated the inequalities of coecient for certain classes of Sakaguchi type functions that
satisfy geometrical condition as
R(s−t)z(f′(z)
f(sz)−f(tz)>
α
(3)
for complex numbers s, t but s =t and
α
(0 ≤
α
< 1) .
e convolution or Hadamard product of two functions f,g∈Ais dened by f∗gand is dened by
(f∗g)(z) = z+∑∞
n=2
ρ
n
δ
nzn.
where fis given by (1) and g(z) = z+∑∞
n=2
δ
nzn.
Let R= (−∞,∞)be the set of real numbers. Cbe the complex numbers and
N:={1,2,3,...}=N0/(0}
be the set of positive integers. Let ∆= (z∈C:(z|<1}be open unit disc in C. A well known, the trigonometric polynomials
Tnq,ei
θ
are expressed by the generating function
ξ
qei
θ
,z=1
(1−zei
θ
)(1−qze−i
θ
)
=∑∞
n=0Tnq,ei
θ
zn,(q∈(−1,1),
θ
∈(−
π
,
π
],z∈∆).
where
Tnq,ei
θ
=ei(n+1)
θ
−qn+1e−i(n+1)
θ
ei
θ
−qe−i
θ
(n≥2)
with
T0q,ei
θ
=1,T1q,ei
θ
=ei
θ
+qe−i
θ
,T2q,ei
θ
=e2i
θ
+q2e−2i
θ
+q....
e obtained results for q=1give the corresponding ones for Chebyshev polynomials of the second kind. e classical
Chebyshev polynomials which are used in this paper, have been in the late eighteenth century, when was dened using de
Moivre’s formula by Chebyshev(refer(9)). Such polynomials as (for example) the Fibonacci polynomials, the Lucas polynomials,
the Pell polynomials and the families of orthogonal polynomials and other special polynomials as well as their generalizations
are potentially important in the elds of probability, statistics, mechanics and number theory(10–14)
2 Methodology
In the present work, the convolution operator Ia,b,cdue to Hohlov (refer(15,16), which is special case of the Rajavadivelu
emangani et al (refer(17)) is recalled.
For the complex parameters a,b&c(c=0,−1,−2,....)the Gaussian hyper geometric function 2F1(a,b,c:z)is dened as
2F1(a,b,c:z) = ∑∞
n=0
(a)n(b)n
(c)n
zn
n!=1+∑∞
n=2
(a)n−1(b)n−1
(c)n−1
zn−1
(n−1)!(z∈∆).
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Chinthamani & Lokesh / Indian Journal of Science and Technology 2023;16(27):2040–2046
where (a)nis the Pochhammer symbol (or the shied factorial) given by
(a)n:=Γ(a+k)
Γ(a)=1,n=0
a(a+1)(a+2)... (a+n−1),n∈N:={1,2, ...}
Now, let us consider a linear operator introduced by Isra Al-shbeil et al (refer(18)) and
Ia,b,c:A→A.
dened by the Hadamard productIa,b,cf(z)=(z2F1(a,b,c:z)) ∗f(z).
It is observed that, for a function fof the form (1),
Ia,b,cf(z) = z+∑∞
n=2Ψn
ρ
nzn,(z∈∆).
where Ψn=(a)n−1(b)n−1
(c)n−1(n−1)!.
In this paper, a new class of bi-univalent function based on Bazilevic Sakaguchi function using the trigonometric polynomials
Tnq,ei
θ
is established. Furthermore, the coecient bounds and Fekete – Szego inequalities are also derived for this class.
Denition 1 : For 0≤
λ
<1,(
τ
|≤1,
τ
=1,q∈(−1,1),
θ
∈(−
π
,
π
], a function f∈Σgiven by (1) is said to be in the class
Pa,b,c
Σ(
λ
,
τ
,q,
θ
)if it satises the following conditions,
((1−
τ
)z)1−
λ
Ia,b,cf(z)′
Ia,b,cf(z)−Ia,b,cf(
τ
z)1−
λ
≺
φ
qei
θ
,z,(z∈∆)(4)
((1−
τ
)z)1−
λ
Ia,b,cg(w)′
Ia,b,cg(w)−Ia,b,cg(
τ
w)1−
λ
≺
φ
qei
θ
,w,(w∈∆)(5)
where the function g=f−1.
By taking the parameters
λ
=0and
τ
=0, which was introduced by Sahsene Altinkaya et al (refer(19)).
3 Result and Discussion
In this section, we obtain the extension of Chebyshev polynomial bounds(
ρ
2|and (
ρ
3|for the function of the
class Pa,b,c
Σ(
λ
,
τ
,q,
θ
).
eorem 1 Let the function fgiven by (1) be in the class Pa,b,c
Σ(
λ
,
τ
,q,
θ
).
en
|
ρ
2|≤ei
θ
+qe−i
θ
2|ei
θ
+qe−i
θ
|




2

e2i
θ
+q2e−2i
θ
+2q3−(1−
λ
)1+
τ
+
τ
2Ψ3
−e2i
θ
+q2e−2i
θ
+2q(1−
λ
)(1+
τ
)(2(1−
τ
) +
λ
(1+
τ
))
+2e2i
θ
+q2e−2i
θ
+q(2−(1−
λ
)(1+
τ
))2Ψ2
2
|(6)
and
(
ρ
3|≤ei
θ
+qe−i
θ

(3−(1−
λ
)(1+
τ
+
τ
2))Ψ3
+e2i
θ
+q2e−2i
θ
+2q
(2−(1−
λ
)(1+
τ
))2Ψ2
2
,(7)
Proof. Since f∈Pa,b,c
Σ(
λ
,
τ
,q,
θ
), there is two analytic functions
ϕ
,
χ
such that
ϕ
(0) = 0,(
ϕ
(z)|=r1z+r2z2+r3z3+...<1,(z∈∆)
χ
(0) = 0,(
χ
(w)|=s1w+s2w2+s3w3+...<1,(w∈∆)
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We can express as
((1−
τ
)z)1−
λ
Ia,b,cf(z)′
Ia,b,cf(z)−Ia,b,cf(
τ
z)1−
λ
=1+T1q,ei
θ

ϕ
(z) + T2q,ei
θ

ϕ
2(z) + ...
and
((1−
τ
)w)1−
λ
Ia,b,cg(w)′
Ia,b,cg(w)−Ia,b,cg(
τ
w)1−
λ
=1+T1q,ei
θ

χ
(w) + T2q,ei
θ

χ
2(w) + ...
or, equivalently,
((1−
τ
)z)1−
λ
ℑa,b,cf(z)′
ℑa,b,cf(z)−Ia,b,cf(
τ
z)1−
λ
=1+T1q,ei
θ
r1z+T1q,ei
θ
r2+T2q,ei
θ
r2
1z2+.. . (8)
and
((1−
τ
)w)1−
λ
Ia,b,cg(w)′
Ia,b,cg(w)−Ia,b,cg(
τ
w)1−
λ
=1+T1q,ei
θ
s1w+T1q,ei
θ
s2+T2q,ei
θ
s2
1w2+... (9)
It is well known that
|ri|≤1and |si|≤1,(∀i∈N)(10)
From the equations (8) and (9), we obtain
(2−(1−
λ
)(1+
τ
))Ψ2
ρ
2=T1q,ei
θ
r1(11)
3−(1−
λ
)1+
τ
+
τ
2Ψ3
ρ
3−(1−
λ
)(1+
τ
)
2(2(1−
τ
) +
λ
(1+
τ
))Ψ2
2
ρ
2
2
=T1q,ei
θ
r2+T2q,ei
θ
r2
1
(12)
−(2−(1−
λ
)(1+
τ
))Ψ2
ρ
2=T1q,ei
θ
s1(13)


23−(1−
λ
)1+
τ
+
τ
2Ψ3
−(1−
λ
)(1+
τ
)
2(2(1−
τ
) +
λ
(1+
τ
))Ψ2
2

ρ
2
2−3−(1−
λ
)1+
τ
+
τ
2Ψ3
ρ
3
=T1q,ei
θ
s2+T2q,ei
θ
s2
1
(14)
From the equations (11) and (13), we easily nd
r1=−s1(15)
2(2−(1−
λ
)(1+
τ
))2Ψ2
2
ρ
2
2=T2
1q,ei
θ
r2
1+s2
1,(16)
Summing the equations (12) and (14), we get
23−(1−
λ
)1+
τ
+
τ
2Ψ3−(1−
λ
)(1+
τ
)(2(1−
τ
) +
λ
(1+
τ
))Ψ2
2
ρ
2
2
=T1q,ei
θ
(r2+s2) + T2q,ei
θ
r2
1+s2
1(17)
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By substituting the values of r2
1+s2
1from (16) in the right side of (17), we get



23−(1−
λ
)1+
τ
+
τ
2Ψ3−(1−
λ
)(1+
τ
)(2(1−
τ
) +
λ
(1+
τ
))Ψ2
2
−2(2−(1−
λ
)(1+
τ
))2Ψ2
2T2(q,ei
θ
)
T2
1(q,ei
θ
)
=T1q,ei
θ
(r2+s2),



ρ
2
2
which yields
ρ
2
2=T3
1q,ei
θ
(r2+s2)


T2
1q,ei
θ
23−(1−
λ
)1+
τ
+
τ
2Ψ3
−(1−
λ
)(1+
τ
)(2(1−
τ
) +
λ
(1+
τ
))Ψ2
2
−2(2−(1−
λ
)(1+
τ
))2Ψ2
2T2q,ei
θ


(18)
By subtracting the equation (14) from equation (12), we nd
23−(1−
λ
)1+
τ
+
τ
2Ψ3
ρ
3−23−(1−
λ
)1+
τ
+
τ
2Ψ3
ρ
2
2
=T1q,ei
θ
(r2−s2)(19)
In view of equation (16), the equation (19) becomes
ρ
3=T1q,ei
θ
(r2−s2)
2(3−(1−
λ
)(1+
τ
+
τ
2))Ψ3
+T2
1q,ei
θ
r2
1+s2
1
2(2−(1−
λ
)(1+
τ
))2Ψ2
2
By applying equation (10), we can easily obtain the desired inequalities in eorem 1.
Taking the parameters
λ
=0and
τ
=0in the eorem 1, we get the following remark.
Remark 1 If f∈Pa,b,c
Σ(q,
θ
), then
(
ρ
2|≤(ei
θ
+qe−i
θ
|(ei
θ
+qe−i
θ
|
(2(e2i
θ
+q2e−2i
θ
+2q)Ψ3−(2e2i
θ
+2q2e−2i
θ
+3q)Ψ2
2|,
and
(
ρ
3|≤ei
θ
+qe−i
θ

2Ψ3
+e2i
θ
+q2e−2i
θ
+2q
Ψ2
2
.
which was investigated by Sahsene Altinkaya et al(19).
Fekete – Szego inequality for the function classPa,b,c
Σ(
λ
,
τ
,q,
θ
)
In this section, we provide Fekete – Szego inequalities for function in the class Pa,b,c
Σ(
λ
,
τ
,q,
θ
). is inequality is given in
the following theorem.
eorem 2 For
µ
∈Rthe function f∈Pa,b,c
Σ(
λ
,
τ
,
µ
,q,
θ
).

ρ
3−
µρ
2
2≤












|ei
θ
+qe−i
θ
|
(3−(1−
λ
)(1+
τ
+
τ
2))Ψ3,|
µ
−1| ≤
γ
2|1−
µ
||e2i
θ
+q2e−2i
θ
+2q∥ei
θ
+qe−i
θ
|
2(e2i
θ
+q2e−2i
θ
+2q)(3−(1−
λ
)(1+
τ
+
τ
2)Ψ3
−e2i
θ
+q2e−2i
θ
+2q(1−
λ
)(1+
τ
)(2(1−
τ
) +
λ
(1+
τ
))
+2e2i
θ
+q2e−2i
θ
+q(2−(1−
λ
)(1+
τ
))2Ψ2
2
|,|
µ
−1|≥
γ
where
γ
=



2e2i
θ
+q2e−2i
θ
+2q3−(1−
λ
)1+
τ
+
τ
2Ψ3
−e2i
θ
+q2e−2i
θ
+2q(1−
λ
)(1+
τ
)(2(1−
τ
) +
λ
(1+
τ
))
+2e2i
θ
+q2e−2i
θ
+q(2−(1−
λ
)(1+
τ
))2Ψ2
2


2(e2i
θ
+q2e−2i
θ
+2q)(3−(1−
λ
)(1+
τ
+
τ
2))Ψ3
.
Proof. From the equation (18) and the equation (19), we observe that
ρ
3−
µρ
2
2=T1q,ei
θ
(r2−s2)
2(3−(1−
λ
)(1+
τ
+
τ
2))Ψ3
+(1−
µ
)T3
1q,ei
θ
(r2+s2)


T2
1q,ei
θ
23−(1−
λ
)1+
τ
+
τ
2Ψ3
−(1−
λ
)(1+
τ
)(2(1−
τ
) +
λ
(1+
τ
))Ψ2
2
−2(2−(1−
λ
)(1+
τ
))2Ψ2
2T2q,ei
θ


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=T1q,ei
θ




ζ
(
µ
) + 1
2(3−(1−
λ
)(1+
τ
+
τ
2))Ψ3r2
+
ζ
(
µ
)−1
2(3−(1−
λ
)(1+
τ
+
τ
2))Ψ3s2




where
ζ
(
µ
) = (1−
µ
)T2
1q,ei
θ



T2
1q,ei
θ
23−(1−
λ
)1+
τ
+
τ
2Ψ3
−(1−
λ
)(1+
τ
)(2(1−
τ
) +
λ
(1+
τ
))Ψ2
2
−2(2−(1−
λ
)(1+
τ
))2Ψ2
2T2q,ei
θ


en, in view of equation (10), we get

ρ
3−
µρ
2
2≤


|T1(q,ei
θ
)|
(3−(1−
λ
)(1+
τ
+
τ
2))Ψ3,0≤ |
ζ
(
µ
)| ≤ 1
(3−(1−
λ
)(1+
τ
+
τ
2))Ψ3
2|
ζ
(
µ
)|T1q,ei
θ
,|
ζ
(
µ
)| ≥ 1
(3−(1−
λ
)(1+
τ
+
τ
2))Ψ3
is evidently completes the proof of eorem 2.
Taking the parameters
λ
=0and
τ
=0, in eorem 2,
Remark 2 For
µ
∈R, let the function f∈Pa,b,c
Σ(
µ
,q,
θ
). en

ρ
3−
µρ
2
2
≤










|ei
θ
+qe−i
θ
|
2Ψ3,|
µ
−1| ≤ 2(e2i
θ
+q2e−2i
θ
+2q)Ψ3−(2e2i
θ
+2q2e−2i
θ
+3q)Ψ2
2
2(e2i
θ
+q2e−2i
θ
+2q)Ψ3
|1−
µ
||e2i
θ
+q2e−2i
θ
+2q∥ei
θ
+qe−i
θ
|
|2(e2i
θ
+q2e−2i
θ
+2q)Ψ3−(2e2i
θ
+2q2e−2i
θ
+3q)Ψ2
2|,
|
µ
−1| ≥ 2(e2i
θ
+q2e−2i
θ
+2q)Ψ3−(2e2i
θ
+2q2e−2i
θ
+3q)Ψ2
2
2(e2i
θ
+q2e−2i
θ
+2q)Ψ3
4 Conclusion
In the present investigation, a new class of bi univalent function based on Bazilevic Sakaguchi function using the trigonometric
polynomials Tnq,ei
θ
is obtained in the open unit disc. Furthermore, belonging to this class, the Taylor – Maclaurin coecient
inequalities and the well knownFekete – Szego inequalities are also derived. ese ndings can further be improved by nding
sharpness. Moreover, Hankel Determinants and Toeplitz determinants for various integral orders can be computed in the future.
5 Declaration
is work has been presented in “International conference on Recent Strategies in Mathematics and Statistics (ICRSMS2022),
Organized by the Department of Mathematics of Stella Maris College and of IIT Madras during 19 to 21 May, 2022 at Chennai,
India. e Organizer claims the peer review responsibility.
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