Applications of Gegenbauer Polynomials to a Certain Subclass of

p-Valent Functions

WALEED AL-RAWASHDEH

Department of Mathematics

University of Zarqa

Zarqa 13110

JORDAN

Abstract:

The paper presents a subclass of p-valent functions deﬁned by the means of Gegenbauer Polynomials in the

open unit disk D. We investigate the properties of this new class and provide estimations for the modulus of

the coeﬃcients ap+1 and ap+2, where p∈N, for functions belong to this subclass. Moreover, we examine the

classical Fekete-Szeg¨o inequality for functions fbelong to the presenting subclass.

Key-Words: Analytic Functions; holomorphic Functions; Univalent Functions; p-Valent Functions; Principle of Subordination;

Gegenbauer Polynomials; Chebyshev polynomials; Coeﬃcient estimates; Fekete-Szeg¨o Inequality.

1 Introduction

Let Hbe the class of all functions f(z) that are holo-

morphic in the open unit disk D={z∈C:|z|<1}. An

analytic function fin a domain D⊂Cis called p-valent,

if for each w∈Cthe equation f(z) = whas at most p

roots in D. Therefore, there exists w0∈Csuch that the

equation f(z) = w0has exactly proots in D. Let Ap

be the class of all holomorphic functions f∈Hthat are

given by

f(z) = zp+

∞

n=p+1

anzn,where z∈D.(1)

Let Sdenote the class of all functions fin the class

A=A1that are univalent in D. Let S∗

pbe the class of

p-valent starlike functions such that we say f(z)∈Ap

in the class S∗

pif the following condition satisﬁes for all

z∈D:

Rzf′(z)

f(z)>0.

Also, let Sc

pbe the class of p-valent functions such that

wesayf(z)∈Apin the class Sc

pif the following condition

satisﬁes for all z∈D:

R1 + zf′′(z)

f′(z)>0.

It is well known (see, for details [1]) and [2]) that if

fis analytic in a convex domain D⊂Cand

Reiθf′(z)>0 for some real θand for all z∈D,

then f(z)is univalent in D. In [3] the auther extended

the previous result, in fact he showed that if f(z) of

the form (1) is analytic in a convex domain D⊂Cand

Rneiθf(p)(z)o>0 for some real θand for all z∈D,

then f(z)is at most p-valent in D. Moreover, it can be

shown that if f∈Apand Rnf(p)(z)o>0 for all z∈D,

then f(z) is at most p-valent in D. According to [4] we

have if f∈Ap,p≥2, and arg nf(p)(z)o<3π

4for all

z∈D, then f(z) is at most p-valent in D. For more

information about p-valent we refer the readers to the

articles [5], [6], [7], [8] and the references therein.

Let the functions fand gbe analytic in D, we say the

function fis subordinate by the function gin D, denoted

by f(z)≺g(z) for all z∈D, if there exists a Schwarz

function w, with w(0) = 0 and |w(z)|<1 for all z∈D,

such that f(z) = g(w(z)) for all z∈D. In particular,

if the function gis univalent over Dthen f(z)≺g(z)

equivalent to f(0) = g(0) and f(D)⊂g(D. For more

information about the Subordination Principle we refer

the readers to the monographs [9], [10], [11] and [12].

The research in the geometric function theory has

been very active in recent years, the typical problem in

this ﬁeld is studying a functional made up of combina-

tions of the initial coeﬃcients of the functions f∈A.

For a function in the class S, it is well-known that |an|is

Received: April 19, 2023. Revised: December 5, 2023. Accepted: December 17, 2023. Published: December 31, 2023.

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bounded by n. Moreover, the coeﬃcient bounds give in-

formation about the geometric properties of those func-

tions. For instance, the bound for the second coeﬃcients

of functions in the class Sgives the growth and distor-

tion bounds for the class.

Coeﬃcient related investigations of functions belong

to the class Σ began around the 1970. It is worth men-

tioning that, in [13] the author studied the class of bi-

univalent functions and derived the bound for |a2|. Ac-

cording to [14] we know that the maximum value of |a2|

is 4

3for functions belong to the class Σ. Moreover, in

[15] the authors proved that |a2| ≤ √2 for functions

in the class Σ. Since then, many researchers investi-

gated the coeﬃcient bounds for various subclasses of

the bi-univalent function class Σ. However, not much

is known about the bounds of the general coeﬃcients

|a2|for n≥4. In fact, the coeﬃcient estimate problem

for the general coeﬃcient |an|is still an open problem.

The Fekete-Szeg¨o functional is well known for its rich

history in the geometric function theory. Its origin was in

[16] when they found the maximum value of |a3−λa2

2|, as

a function of the real parameter 0 ≤λ≤1 for a univalent

function f. Since then, maximizing the modulus of the

functional Ψλ(f) = a3−λa2

2for f∈Awith any complex

λis called the Fekete-Szeg¨o problem. There are many

researchers investigated the Fekete-Szeg¨o functional and

the other coeﬃcient estimates problems, for example see

the articles [17], [18], [19], [20], [21], [22], [23], [24] and

the references therein.

2 Preliminaries

In this section we present some information that are cru-

cial for the main results of this paper. In [25] the author

introduced and studied a subclass F(γ) of the class A

consisting of functions of the form

f(z) = Z1

−1

K(z, x)dσ(x),(2)

where K(z, x) = z

(z2−2tz + 1)γ,γ≥0, −1≤x≤1,

and σis the probability measure on [−1,1]. Moreover,

the function K(z, x) has the following Taylor-Maclaurin

series expansion

K(z, x) = z+Cγ

1(x)z2+Cγ

2(x)z3+Cγ

3(x)z4+···,

where Cγ

n(t) denotes the Gegenbauer polynomials of or-

der α. Furthermore, for any real numbers γ, x ∈R,with

γ≥0 and −1≤x≤1, the generating function of Gegen-

bauer polynomials is given by

Hγ(z, x) = (z2−2xz + 1)−γ,where z∈D.

Thus, for any ﬁxed xthe function Hγ(z, x) is analytic

on the unit disk Dand its Taylor-Maclaurin series is

given by

Hγ(z, x) =

∞

n=0

Cγ

n(x)zn.

Moreover, Gegenbauer polynomials can be deﬁned in

terms of the following recurrence relation:

Cγ

n(x) = 2x(n+γ−1)Cγ

n−1(x)−(n+ 2γ−2)Cγ

n−1(x)

(3)

with initial values

Cγ

0(x)=1, Cγ

1(x)=2γx, and

Cγ

2(x)=2γ(γ+ 1)x2−γ. (4)

It is well-known that the Gegenbauer polynomials

and their special cases such as Legendre polynomials

Ln(x) and the Chebyshev polynomials of the second kind

Tn(x) are orthogonal polynomials, where the values of γ

are γ= 1/2 and γ= 1 respectively, more precisely

Ln(x) = C1/2

n(x),and Tn(x) = C1

n(x).

For more information about the Gegenbauer polynomials

and their special cases, we refer the readers to the arti-

cles [26], [27], [28], the monographs [29], [30], and the

references therein. next, we deﬁne our class of p-valent

functions which we denote by Ap(α, β, γ).

Deﬁnition 2.1. A function f(z)in the family Apis said

to be in the class Ap(α, β, γ)if, for all z∈D, it satisﬁes

the following subordination

zf′(z)−pf(z)

βpf(z)+αf′(z)−pzp−1

βpzp−1≺Hγ(x, z),

where γ > 0,x∈[−1,1],0≤α≤1, and βis a non-zero

complex number.

The following lemma (see, for details [22]) is a well-

known fact, but it is crucial for our presented work.

Lemma 2.2. Let the Schwarz function w(z)be given by:

w(z) = w1z+w2z2+w3z3+··· where z∈D,

then |w1| ≤ 1and for t∈C

|w2−tw2

1| ≤ 1+(|t| − 1)|w1|2≤max{1,|t|}.

The results is sharp for the functions w(z) = zand

w(z) = z2.

The primary goal of this article is to investigate a

class of p-valent functions in the open unit disk D, which

we denote by Ap(α, β, γ). For functions in this class,

we obtain the estimates for the initial coeﬃcients |ap+1|

and |ap+2|. Furthermore, we examine the correspond-

ing Fekete-Szeg¨o functional problem for functions in this

class.

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3 Coeﬃcient bounds of functions

in the class Ap(α, β, γ)

In this section, we provide bounds for the initial coef-

ﬁcients for the functions belong to the class Ap(α, β, γ)

which are given by equation (1).

Theorem 3.1. If a function f∈Apis given by (1)

belong to the class Ap(α, β, γ), then

|ap+1| ≤ 2γp|x||β|

αp +α+ 1,(5)

and

|ap+2| ≤ γp|x||β|

αp + 2α+ 1×

max (1,

2(γ+ 1)x2−1

2x+2xpγβ

(αp + 1α+ 1)2).

(6)

Proof. Let fbe in the class Ap(α, β, γ). Then, using

Deﬁnition 2.1, there exists a holomorphic function ψon

the unit disk Dsuch that

zf′(z)−pf(z)

βpf(z)+αf′(z)−pzp−1

βpzp−1

≺Hγ(x, ψ(z)),

(7)

where the holomorphic function ψis given by ψ(z) =

∞

n=1

bnznsuch that ψ(0) = 0 and |ψ(z)|<1 for all

z∈D. Moreover, it is well-known that (see, for de-

tails [9]), |bj| ≤ 1 for all j∈N.

Now, upon comparing the coeﬃcients of both-sides

of equation (7), we obtain the equations

α(p+ 1) + 1

pβ ap+1 =Cγ

1(x)b1,(8)

and

2α(p+ 2) + 2

pβ ap+2 −1

pβ a2

p+1

=Cγ

1(x)b2+Cγ

2(x)b2

(9)

Hence, using equation (8), we get

ap+1 =pβCγ

1(x)b1

α(p+ 1) + 1.(10)

In view of the initial values (4) and the fact |b1| ≤ 1,

we get

|ap+1| ≤ 2pγ|x||β|

α(p+ 1) + 1,

which is the desired estimate of |ap+1|.

Secondly, we look for the coeﬃcient estimate of ap+2.

Using equation (9), we obtain

2(α(p+ 2) + 1)ap+2 −a2

p+1

=pβ[Cγ

1(x)b2+Cγ

2(x)b2

1].(11)

Using equation (10), the last equation becomes

ap+2 =pβ

2(α(p+ 2) + 1)×

Cγ

1(x)b2+Cγ

2(x)b2

1+pβ[Cγ

1(x)]2b2

(α(p+ 1) + 1)2

=pβCγ

1(x)

2(α(p+ 2) + 1)×

b2+Cγ

2(x)

Cγ

1(x)b2

1+pβCγ

1(x)b2

(α(p+ 1) + 1)2.

Hence, using Lemma 2.2, we obtain

|ap+2| ≤ p|β||Cγ

1(x)|

2(α(p+ 2) + 1)×

max 1,

Cγ

2(x)

Cγ

1(x)+pβCγ

1(x)

(α(p+ 1) + 1)2.

Finally, using the initial values (4), we get the desired

estimate of |ap+2|. Therefore, this completes the proof

of Theorem 3.1.

The following corollary is just a consequence of The-

orem 3.1 when taking γ= 1. These initial coeﬃcient

estimates are related to Chebyshev polynomials of the

second kind. The proof is similar to the previous theo-

rem’s proof, so we omit the proof’s details.

Corollary 3.2. If a function f∈Apis given by (1)

belong to the class Ap(α, β, 1), then

|ap+1| ≤ 2p|x||β|

αp +α+ 1,

and

|ap+2| ≤ p|x||β|

αp + 2α+ 1 max (1,

4x2−1

2x+2xpγβ

(αp +α+ 1)2).

Remark 1. The following are special cases of our class

of p-valent functions Ap(α, β, γ).

•Putting, α= 0 in Deﬁnition 2.1, we get a subclass

that satisﬁes the following subordination:

βzf′(z)

pf(z)−1≺Hγ(x, z) (12)

•Putting, β= 1 in Deﬁnition 2.1, we get a subclass

that satisﬁes the following subordination:

zf′(z)

pf(z)−1+αf′(z)

pzp−1−1≺Hγ(x, z),(13)

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Corollary 3.3. If a function f∈Apis given by (1)

satisﬁes the subordination (12), then

|ap+1| ≤ 2γp|x||β|,

and

|ap+2| ≤ γp|x||β|max 1,

(4pγβ + 2γ+ 2)x2−1

2x.

Corollary 3.4. If a function f∈Apis given by (1)

satisﬁes the subordination (13), then

|ap+1| ≤ 2γp|x|

αp +α+ 1,

and

|ap+2| ≤ γp|x|

αp + 2α+ 1×

max (1,

2(γ+ 1)x2−1

2x+2xpγ

(αp +α+ 1)2).

4 Fekete-Szeg¨o inequality of the

class Ap(α, β, γ)

In this section, we maximize the modulus of the func-

tional Ψλ(f) = ap+2 −λa2

p+1 for real numbers λand for

functions fbelong to the class Ap(α, β, γ).

Theorem 4.1. If a function f∈Apis given by (1)

belong to the class Ap(α, β, γ), then for some λ∈Rand

for x∈(0,1],

|a3−λa2

2| ≤ (pγx|β|

αp+2α+1 ,if λ∈[λ1, λ2]

pγx|β||A|

αp+2α+1 ,if λ /∈[λ1, λ2],(14)

where

λ1=(2(γ+ 1)x2−2x−1)(αp +α+ 1)2+ 4γβpx2

8γβpx2(αp + 2α+ 1) ,

λ2=(2(γ+ 1)x2+ 2x−1)(αp +α+ 1)2+ 4γβpx2

8γβpx2(αp + 2α+ 1) ,

and

A=2(γ+ 1)x2−1

2x+2xγβ(1 −2λ(αp + 2α+ 1))

(αp +α+ 1)2.

Proof. For any real number λ, using equations (11) and

(10), we get

ap+2 −λa2

p+1 =pβ[Cγ

1(x)b2+Cγ

2(x)b2

2(αp + 2α+ 1)

+1

2(αp + 2α+ 1) −λa2

p+1

=βp[Cγ

1(x)b2+Cµ

2(t)b2

2(αp + 2α+ 1)

+βp(1 −2λ(αp + 2α+ 1))[Cµ

1(t)]2b2

2(αp + 2α+ 1)(αp +α+ 1)2

=γβpx

αp + 2α+ 1 b2+Ab2

1.

Hence, using Lemma 2.2, we get

|ap+2 −λa2

p+1| ≤ γp|β||x|

αp + 2α+ 1 max{1,|A|}.

For x > 0, if |A| ≤ 1, then



Cγ

2(x)

Cγ

1(t)+βp(1 −2λ(αp + 2α+ 1)[Cγ

1(t)]

(αp +α+ 1)2≤1.

Therefore, solving for λwe get

−[Cγ

2(x) + Cγ

1(x)](αp +α+ 1)2

βp[Cγ

1(x)]2

≤1−2λ(αp + 2α+ 1)

≤[Cγ

1(x)−Cγ

2(x)](αp +α+ 1)2

βp[Cγ

1(x)]2.

Hence, simple calculations give us the following inequal-

ity

(Cγ

2(x)−Cγ

1(x)) (αp +α+ 1)2+βp[Cγ

1(x)]2

2βp(αp + 2α+ 1)[Cγ

1(x)]2≤λ

≤(Cγ

2(x) + Cγ

1(x)) (αp +α+ 1)2+βp[Cγ

1(x)]2

2βp(αp + 2α+ 1)[Cγ

1(x)]2.

⇐⇒ λ1≤λ≤λ2.

Therefore, in view of the initial values (4), if |A| ≤ 1,

then λ∈[λ1, λ2] and hence we get

|ap+2 −λa2

p+1| ≤ γp|β||x|

αp + 2α+ 1.

Moreover, if |A|>1, then λ /∈[λ1, λ2] and hence we get

|ap+2 −λa2

p+1| ≤ γp|β||x||A|

αp + 2α+ 1.

This completes the Theorem’s proof.

The following corollary is just consequences of The-

orem 4.1. Taking γ= 1, we get the Fekete-Szeg¨o in-

equality that is related to Chebyshev polynomials of the

second kind.

Corollary 4.2. If a function f∈Apis given by (1)

belong to the class Ap(α, β, 1), then for some λ∈Rand

for x∈(0,1],

|a3−λa2

2| ≤ (px|β|

αp+2α+1 ,if λ∈[ζ1, ζ2]

px|β||B|

αp+2α+1 ,if λ /∈[ζ1, ζ2],

where

ζ1=(4x2−2x−1)(αp +α+ 1)2+ 4βpx2

8βpx2(αp + 2α+ 1) ,

ζ2=(4x2+ 2x−1)(αp +α+ 1)2+ 4βpx2

8βpx2(αp + 2α+ 1) ,

and

B=4x2−1

2x+2xβ(1 −2λ(αp + 2α+ 1))

(αp +α+ 1)2.

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Corollary 4.3. If a function f∈Apis given by (1)

satisﬁes the subordination (12), then for some λ∈R

and for x∈(0,1],

|a3−λa2

2| ≤ (pγx|β|,if λ∈[ζ3, ζ4]

pγx|β||K|,if λ /∈[ζ3, ζ4],

where

ζ3=(4γβp + 2γ+ 2)x2−2x−1

8γβpx2,

ζ4=(4γβp + 2γ+ 2)x2+ 2x−1

8γβpx2,

and

K=(4γβ(1 −2λ)+2γ+ 2)x2−1

2x.

Corollary 4.4. If a function f∈Apis given by (1)

satisﬁes the subordination (13), then for some λ∈R

and for x∈(0,1],

|a3−λa2

2| ≤ (pγx

αp+2α+1 ,if λ∈[ζ5, ζ6]

pγx|∆|

αp+2α+1 ,if λ /∈[ζ5, ζ6],

where

ζ5=(2(γ+ 1)x2−2x−1)(αp +α+ 1)2+ 4γpx2

8γpx2(αp + 2α+ 1) ,

ζ6=(2(γ+ 1)x2+ 2x−1)(αp +α+ 1)2+ 4γpx2

8γpx2(αp + 2α+ 1) ,

and

∆ = 2(γ+ 1)x2−1

2x+2xγ(1 −2λ(αp + 2α+ 1))

(αp +α+ 1)2.

5 Conclusion

This research paper has investigated a new class of p-

valent functions related to Gegenbauer polynomials. For

functions belong to this function class, the author has

derived estimates for the initial coeﬃcients and Fekete-

Szeg¨o functional problem. The work presented in this

paper will lead to many diﬀerent results for subclasses

deﬁned by the means of Horadam polynomials and their

special cases, such as: Fibonacci polynomials, Lucas Poly-

nomials, Pell Polynomials, and Pell-Lucas Polynomials.

Moreover, the presented work in this paper will inspire

researchers to extend its concepts to harmonic functions

and symmetric q-calculus such as q-Ruscheweyh and q-

Salagean diﬀerential operators.

Acknowledgment:

This research is partially funded by Zarqa University.

The author would like to express his sincerest thanks to

Zarqa University for the ﬁnancial support. The author

is very grateful to the unknown referees for their

valuable comments and suggestions which improved the

presentation in this paper.

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WSEAS TRANSACTIONS on MATHEMATICS

DOI: 10.37394/23206.2023.22.111

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