EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 17, No. 1, 2024, 158-170
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

A family of Analytic Functions Subordinate to
Horadam Polynomials

Waleed Al-Rawashdeh

Department of Mathematics, Zarqa University, 2000 Zarqa, 13110, Jordan

Abstract. In this paper, we introduce and investigate a family of analytic functions, denoted
by F(Π, α, β, λ, δ, µ), defined by means of Horadam polynomials. For functions in this family, we
derive the estimations for the initial Taylor-Maclaurin coefficients |a2| and |a3|. Moreover, we
obtain the classical Fekete-Szegö inequality of functions belonging to this family.

2020 Mathematics Subject Classifications: 30C45, 30C50, 33C45, 33C05, 11B39

Key Words and Phrases: Analytic Functions, Taylor-Maclaurin Series, Principle of Subordina-
tion, Horadam Polynomials, Chebyshev polynomials, Coefficient estimates, Fekete-Szegö Inequality

1. Introduction

Let A be the family of all analytic functions f that are defined on the open unit disk
D = {z ∈ C : |z| < 1} and normalized by the conditions f(0) = 1 − f ′(0) = 0. Any
function f ∈ A has the following Taylor-Maclarin series expansion:

f(z) = z +
∞∑
n=2

anz
n, where z ∈ D. (1)

Let S denote the class of all functions f ∈ A that are univalent in D. Let the functions
f and g be analytic in D, we say the function f is subordinate by the function g in 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 g is univalent over D then f(z) ≺ g(z) equivalent to f(0) = g(0) and f(D) ⊂ g(D.
For more information about the subordination principle and univalent functions we refer
the readers to to the monographs [11], [10], [14], [23], [25] and the references therein.

In the year 1965, for a, b, p, q ∈ R, Horadam [17] introduced the sequence Wn =
Wn(a, b; p, q) that is defined by the following recurrence relation

Wn+2 = pWn+1 + qWn, for n ≥ 2,

DOI: https://doi.org/10.29020/nybg.ejpam.v17i1.5022

Email address: walrawashdeh@zu.edu.jo (W. Al-Rawashdeh)

https://www.ejpam.com 158 © 2024 EJPAM All rights reserved.



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (1) (2024), 158-170 159

with the initial values W0 = a and W1 = b. The characteristic equation of this sequence
is given by

t2 − pt− q = 0.

In addition, the generating function of Horadam sequence is

f(t) =
a+ t(b− ap)

1− pt− qt2
.

The Horadam sequences generalize many famous sequences such as Fibonacci, Lucas,
Pell, Pell-Lucas and Jacobsthal sequences. These sequences have been studied for a long
time. For more information about these sequences, we refer the readers to the articles
[16], [15], the monograph [21] and the references therein.

In the year 1985, Horadam and Mahon [16] defined the Horadam polynomials hn(x) =
hn(a, b; p, q) by the following recurrence relation:

hn(x) = pxhn−1(x) + qhn−2(x), for n ∈ N \ {1, 2}, (2)

with initial values,

h1(x) = a, h2(x) = bx, and h3(x) = pbx2 + qa. (3)

Moreover, the generating function of Horadam ploynomials is given by

Π(x, z) =
∞∑
n=1

hn(x)z
n−1 =

a+ (b− ap)xz

1− pxz − qz2
.

In this paper, the argument of x ∈ R is independent of the argument z ∈ C; that is
x ̸= R(z). For particular values of a, b, p and q the Horadam polynomials leads to many
known polynomials. Below, we list some particular cases of Horadam Polynomials.

• If a = b = p = q = 1, we get Fibonacci polynomials Fn(x) whose recurrence relation
is

Fn(x) = xFn−1(x) + Fn−2(x); with F1(x) = 1, F2(x) = x.

• If a = 2 and b = p = q = 1, we get Lucas Polynomials Ln(x) whose recurrence
relation is

Ln−1(x) = xLn−2(x) + Ln−3(x); with L0(x) = 2, L1(x) = x.

• If a = q = 1 and b = p = 2, we get Pell Polynomials Pn(x) whose recurrence relation
is

Pn(x) = 2xPn−1(x) + Pn−2(x); with P1(x) = 1, P2(x) = 2x.



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (1) (2024), 158-170 160

• If a = b = p = 2 and q = 1, we get Pell-Lucas Polynomials Qn(x) whose recurrence
relation is

Qn−1(x) = 2xQn−2(x) +Qn−3(x); with Q0(x) = 2, Q1(x) = 2x.

• If a = b = p = x = 1 and q = 2y, we get Jacobsthal Polynomials Jn(y) whose
recurrence relation is

Jn(y) = Jn−1(y) + 2yJn−2(y); with J1(y) = 1, J2(y) = 1.

• If a = 2, b = p = x = 1 and q = 2y, we get Jacobsthal-Lucas Polynomials Jn(y)
whose recurrence relation is

Jn−1(y) = Jn−2(y) + 2yJn−3(y); with J0(y) = 2, J1(y) = 1.

• If a = 1 and b = p = 2, and q = −1, we get Chebyshev Polynomials Hn(x) of the
second kind whose recurrence relation is

Hn−1(x) = 2xHn−2(x)−Hn−3(x); with H0(x) = 1, H1(x) = 2x.

• If a = b = 1 and p = 2, and q = −1, we get Chebyshev Polynomials Tn(x) of the
first kind whose recurrence relation is

Tn−1(x) = 2xTn−2(x)− Tn−3(x); with T0(x) = 1, T1(x) = x.

For more information about Horadam polynomials and its special interesting cases, we
refer the readers to the articles [1], [3], [2], [16], [18], [24], [26], [27], [30], the monograph
[21], [29] and the references therein.

Recently, many researchers have been studying the geometric function theory, the typ-
ical problem in this field is studying a functional made up of combinations of the initial
coefficients of the functions f ∈ A. For a function in the class S, it is well-known that |an|
is bounded by n. Moreover, the coefficient bounds give information about the geometric
properties of those functions. For instance, the bound for the second coefficients of the
class S gives the growth and distortion bounds for the class. In addition, the Fekete-Szegö
functional arises naturally in the investigation of univalency of analytic functions. In the
year 1933, Fekete and Szegö [13] found the maximum value of |a3 − λa22|, as a function
of the real parameter 0 ≤ λ ≤ 1 for a univalent function f . Since then, the problem of
dealing with the Fekete-Szegö functional for f ∈ A with any complex λ is known as the
classical Fekete-Szegö problem. There are many researchers investigated the Fekete-Szegö
functional and the other coefficient estimates problems, for example see the articles [1],
[5], [4], [6], [9], [13], [20], [22], [27], [30] and the references therein.

Motivated by the aforementioned research and the papers [3], [2] and [28] we introduce a
novel class of analytic functions defined using Horadam polynomials. For functions belong
to this function class, we derive estimations for the Taylor-Maclaurin initial coefficients
and Fekete-Szegö functional problem. We also present corollaries for subclasses of our
class defined by the means of special cases of Horadam polynomials.



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (1) (2024), 158-170 161

2. Preliminaries

In this section we present some information that are curial for the main results of
this paper. First, we define our family of analytic functions subordinated by Horadam
polynomials, which we denote by F(Π, α, β, λ, δ, µ).

Definition 1. We say that a function f ∈ A in the class F(Π, α, β, λ, δ, µ) if it fulfills the
subordination conditions, associated with the Horadam Polynomials, for all z ∈ D:

β

(
zG′(z)

G(z)

)α

+ (1− β)

(
zG′(z)

G(z)

)λ(
1 +

zG′′(z)

G′(z)

)1−λ

≺ Π(x, z) + 1− a, (4)

where
G(z) = δµz2f ′′(z) + (µ− δ)zf ′(z) + (1− µ+ δ)f(z),

and
1 ≤ α ≤ 2, 0 ≤ β ≤ 1, 0 ≤ λ ≤ 1 and 0 ≤ δ ≤ µ ≤ 1.

The following are interesting cases related to our presenting class.

(a) If we replace Horadam polynomials by Gegenbaure polynomials H
(γ)
n (z, t), we obtain

the class F(H
(γ)
n (z, t), α, β, λ, δ, µ) which was investigated by Sirvastava et al. [28].

For z ∈ D, t ∈ (12 , 1], and γ ≥ 0 the generating function of Gegenbauer polynomials
is given by

H(γ)
n (z, t) = (z2 − 2tz + 1)−γ .

(b) If we replace Horadam polynomials by Chebyshev polynomials Hn(z, t) of the second
kind, where γ = 1, we obtain the class F(Hn(z, t), α, β, λ, δ, µ) which was introduced
and studied by Kamali et al. [19].

Now, we present some particular special subclasses which obtained by taking specific
values of the parameters involved in our class F(Π, α, β, λ, δ, µ).

• If λ = δ = µ = 0, α = 1, and β = 1 − η where 0 ≤ η ≤ 1, then we get the class
F(Π, 1, 1 − η, 0, 0, 0). We say f ∈ A belong to this class if it satisfies the following
subordination:

(1− η)

(
zf ′(z)

f(z)

)
+ η

(
1 +

zf ′′(z)

f ′(z)

)
≺ Π(x, z) + 1− a. (5)

This class investigated by Abrami et al. [1].

• If β = 0 and δ = µ = 0, we get the class F(Π, α, 0, λ, 0, 0). We say f ∈ A belong to
this class if it satisfies the following subordination:(

zf ′(z)

f(z)

)λ(
1 +

zf ′′(z)

f ′(z)

)1−λ

≺ Π(x, z) + 1− a. (6)

This class investigated by Abrami et al. [1].



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (1) (2024), 158-170 162

• If α = β = 1, we get the class F(Π, 1, 1, λ, δ, µ). We say f ∈ A belong to this class if
it satisfies the following subordination:

µδz3f ′′′(z) + (2µδ + µ− δ)z2f ′′(z) + zf ′(z)

µδz2f ′′(z) + (µ− δ)zf ′(z) + (1− µ+ δ)f(z)
≺ Π(x, z) + 1− a. (7)

If we replace Π(x, z) + 1 − a by the Chebyshev polynomials Hn(z, t) of the second
kind, we get the class F(Hn(z, t), 1, 1, λ, δ, µ) which introduced and studied by Çağlar
et al. [8].

• If µ = δ = 0, we get the class F(Π, α, β, λ, 0, 0). We say f ∈ A belong to this class
if it satisfies the following subordination:

β

(
zf ′(z)

f(z)

)α

+ (1− β)

(
zf ′(z)

f(z)

)λ(
1 +

zf ′′(z)

f ′(z)

)1−λ

≺ Π(x, z) + 1− a. (8)

If we replace Π(x, z)+1−a by the Chebyshev polynomialsHn(z, t) of the second kind,
we get the class F(Hn(z, t), α, β, λ, 0, 0) which introduced and studied by Szatmari
et al. [31].

• If δ = λ = 0, α = 1 and β = 1 − η, we get the class F(Π, 1, 1 − η, 0, 0, µ).We say
f ∈ A belong to this class if it satisfies the following subordination:

(1− η)

(
zG′(z)

G(z)

)
+ η

(
1 +

zG′′(z)

G′(z)

)
≺ Π(x, z) + 1− a, (9)

where G(z) = µzf ′(z) + (1−µ)f(z). If we replace Π(x, z) + 1− a by the Chebyshev
polynomials Hn(z, t) of the second kind, we get the class F(Hn(z, t), α, 1− η, λ, 0, 0)
which introduced and studied by Bulut et al. [7].

• If δ = µ = β = λ = 0, we get the class F(Π, α, 0, 0, 0, 0). We say f ∈ A belong to
this class if it satisfies the following subordination:

1 +
zf ′′(z)

f ′(z)
≺ Π(x, z) + 1− a. (10)

If we replace Π(x, z) + 1 − a by the Chebyshev polynomials Hn(z, t) of the second
kind, we get the class F(Hn(z, t), α, 0, 0, 0, 0) which introduced and studied by Dziok
et al. [12].

The following lemma (see, for details [20]) is a well-known fact, but it is crucial for the
main results of this paper.

Lemma 1. Let the Schwarz function w(z) be given by:

w(z) = w1z + w2z
2 + w3z

3 + · · · where z ∈ D,

then |w1| ≤ 1 and for t ∈ C

|w2 − tw2
1| ≤ 1 + (|t| − 1)|w1|2 ≤ max{1, |t|}.

The result is sharp for the functions w(z) = z and w(z) = z2.



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (1) (2024), 158-170 163

In this presenting paper, we investigate a family of analytic functions on the open unit
disk D, which we denoted by F(Π, α, β, λ, δ, µ) subordinate to Horadam polynomials. For
functions in this family, we derive upper bounds for the initial Taylor-Maclarin coefficients
|a2| and |a3|. Furthermore, we examine the corresponding Fekete-Szegö functional problem
for functions belong to this family.

3. Coefficient estimates for the function class F(Π, α, β, λ, δ, µ)

In this section, we provide bounds for the initial Taylor-Maclaurin coefficients for the
functions belong to the class F(Π, α, β, λ, δ, µ) which are given by equation (1).

Theorem 1. Let the function f(z) given by (1) be in the class F(Π, α, β, λ, δ, µ). Then

|a2| ≤
|bx|

A(1 + 2µδ + µ− δ)
, (11)

|a3| ≤
|bx|
2B

max

{
1,

∣∣∣∣bxK2A2
− pbx2 + qa

bx

∣∣∣∣} , (12)

where
A = αβ + (1− β)(2− λ),

B = (αβ + (1− β)(3− 2λ))(1 + 2(3µδ + µ− δ)),

and
K = αβ(α− 3) + (1− β)(λ2 + 5λ− 8).

Proof. Let f belong to the class F(Π, α, β, λ, δ, µ). Then, using Definition 1, we can
find an analytic function u such that

β

(
zG′(z)

G(z)

)α

+ (1− β)

(
zG′(z)

G(z)

)λ(
1 +

zG′′(z)

G′(z)

)1−λ

≺ Π(x, u(z)) + 1− a (13)

where u : D → D is given by

u(z) =

∞∑
n=1

unz
n for z ∈ D,

such that u(0) = 0 and |u(z)| < 1 for all z ∈ D. Moreover, it is well-known that (see, for
details [11]), |uj | ≤ for all j ∈ N.

Now, upon comparing the coefficients in both sides of equation (13), we obtain the
following equations, where γ = 2µδ + µ− δ,

(αβ + (1− β)(2− λ))(γ + 1)a2 = h2(x)u1. (14)

[αβ(α− 3) + (1− β)(λ2 + 5λ− 8)](γ + 1)2a22+ (15)



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (1) (2024), 158-170 164

+4(αβ + (1− β)(3− 2λ))(2(γ + µδ) + 1)a3 = 2h2(x)u2 + 2h3(x)u
2
1.

Using equation (14), |u1| ≤ 1 and h2(x) = bx, we get the desired bound of |a2|:

|a2| ≤
|bx|

[αβ + (1− β)(2− λ)](γ + 1).

In view of equation (14), we can write equation (15) as:

a3 =
h2(x)

2B

(
u2 −

1

h2(x)

(
K[h2(x)]

2

2A2
− h3(x)

)
u21

)
.

Using Lemma 1, we obtain

|a3| ≤
|h2(x)|
2B

max

{
1,

∣∣∣∣K[h2(x)]
2 − 2A2h3(x)

2A2h2(x)

∣∣∣∣} .

Using the initial values (3), we get the desired estimate of |a3|. This complete the
proof.

The following corollaries are just consequences of Theorem 1.

Corollary 1. If the function f ∈ A satisfies the subordination (5), then

|a2| ≤
|bx|
1 + η

,

and

|a3| ≤
|bx|

2(1 + 2η)
max

{
1,

∣∣∣∣bx(1 + 3η)

(1 + η)2
+

pbx2 + qa

bx

∣∣∣∣} .

Corollary 2. If the function f ∈ A satisfies the subordination (6), then

|a2| ≤
|bx|
2− λ

,

and

|a3| ≤
|bx|

2(3− 2λ)
max

{
1,

∣∣∣∣bx(λ2 + 5λ− 8)

2(2− λ)2
− pbx2 + qa

bx

∣∣∣∣} .

Corollary 3. If the function f ∈ A satisfies the subordination (7), then

|a2| ≤
|bx|

1 + 2µδ + µ− δ
,

and

|a3| ≤
|bx|

2(1 + 2(3µδ + µ− δ))
max

{
1,

∣∣∣∣b2x2 + pbx2 + qa

bx

∣∣∣∣} .



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (1) (2024), 158-170 165

Corollary 4. If the function f ∈ A satisfies the subordination (8), then

|a2| ≤
|bx|

αβ + (1− β)(2− λ)
,

and

|a3| ≤
|bx|

2(αβ + (1− β)(3− 2λ))
max

{
1,

∣∣∣∣bxK2A2
− pbx2 + qa

bx

∣∣∣∣} ,

Corollary 5. If the function f ∈ A satisfies the subordination (9), then

|a2| ≤
|bx|

(1 + η)(1 + µ)
,

and

|a3| ≤
|bx|

2(1 + 2η)(1 + 2µ)
max

{
1,

∣∣∣∣bx(1 + 3η)

(1 + η)2
+

pbx2 + qa

bx

∣∣∣∣} .

4. Fekete-Szegö functional of the class F(Π, α, β, λ, δ, µ)

In this section, we consider the classical Fekete-Szegö problem for our presenting class
F(Π, α, β, λ, δ, µ).

Theorem 2. Let the function f given by (1) be in the class F(Π, α, β, λ, δ, µ). Then for
bx > 0 and for some ζ ∈ R,

|a3 − ζa22| ≤

{
bx
2B , if ζ ∈ [ζ1, ζ2]
|2(pbx2+qa)BA2(γ+1)2−b2x2(K(γ+1)2+4ζB)|

4BA2(γ+1)2
, if ζ /∈ [ζ1, ζ2],

(16)

where

ζ1 =
(γ + 1)2(2A2(pbx2 − bx+ qa)− b2x2K)

4Bb2x2
,

and

ζ2 =
(γ + 1)2(2A2(pbx2 + bx+ qa)− b2x2K)

4Bb2x2
.

Proof. In view of equations (14) and (15), we get the following

a3 =
h2(x)u2

2B
+

h3(x)u
2
1

2B
− K[h2(x)]

2u21
4BA2

.

For some real number ζ, using equation (15), we have

a3 − ζa22 =
h2(x)u2

2B
+

h3(x)u
2
1

2B
− K[h2(x)]

2u21
4BA2

− ζ[h2(x)]
2u21

(γ + 1)2A2

=
h2(x)

2B

{
u2 +

(
h3(x)

h2(x)
− Kh2(x)

2A2
− 2ζBh2(x)

(γ + 1)2A2

)
u21

}



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (1) (2024), 158-170 166

Using Lemma 1 and the initial values (3), we obtain

|a3 − ζa22| ≤
|bx|
2B

max

{
1,

∣∣∣∣pbx2 + qa

bx
− bxK

2A2
− 2ζBbx

(γ + 1)2A2

∣∣∣∣} (17)

Since bx > 0, we have ∣∣∣∣pbx2 + qa

bx
− bxK

2A2
− 2ζBbx

(γ + 1)2A2

∣∣∣∣ ≤ 1.

Solving for ζ we get:

(γ + 1)2(2A2(pbx2 − bx+ qa)− b2x2K)

4Bb2x2
≤ ζ ≤ (γ + 1)2(2A2(pbx2 + bx+ qa)− b2x2K)

4Bb2x2

Hence, inequality (17) becomes

|a3 − ζa22| ≤


bx
2B , if ζ ∈ [ζ1, ζ2]∣∣∣∣pbx2+qa

2B − b2x2K
4BA2 − ζ

(
bx

A(γ+1)

)2
∣∣∣∣ , if ζ /∈ [ζ1, ζ2].

Simplifying the last inequality, we get the desired inequality (16), this completes the proof
of Theorem 2.

The following corollaries are just consequences of Theorem 2.

Corollary 6. If the function f ∈ A satisfies the subordination (5), then for bx > 0 and
for some ζ ∈ R

|a3 − ζa22| ≤


bx

2(1+2η) , if ζ ∈ [ζ1, ζ2]∣∣∣pbx2+qa
2(1+2η) +

b2x2(1+3η−2ζ(1+2η))
2(1+2η)(1+η)2

∣∣∣ , if ζ /∈ [ζ1, ζ2],

where

ζ1 =
(1 + η)2(pbx2 − bx+ qa) + b2x2(1 + 3η)

2(1 + 2η)b2x2
,

and

ζ2 =
(1 + η)2(pbx2 + bx+ qa) + b2x2(1 + 3η)

2(1 + 2η)b2x2
.

Corollary 7. If the function f ∈ A satisfies the subordination (6), then for bx > 0 and
for some ζ ∈ R

|a3 − ζa22| ≤

{
bx

2(3−2λ) , if ζ ∈ [ζ1, ζ2]
|2(pbx2+qa)(3−2λ)(2−λ)2−b2x2(λ2+5λ+4ζ(3−2λ)−8)|

4(3−2λ)(2−λ)2
, if ζ /∈ [ζ1, ζ2],



W. Al-Rawashdeh / Eur. J. Pure Appl. Math, 17 (1) (2024), 158-170 167

where

ζ1 =
2(2− λ)2(pbx2 − bx+ qa)− b2x2(λ2 + 5λ− 8))

4(3− 2λ)b2x2
,

and

ζ2 =
2(2− λ)2(pbx2 + bx+ qa)− b2x2(λ2 + 5λ− 8))

4(3− 2λ)b2x2
.

Corollary 8. If the function f ∈ A satisfies the subordination (7), then for bx > 0 and
for some ζ ∈ R

|a3 − ζa22| ≤


bx

2(1+2(3µδ+µ−δ)) , if ζ ∈ [ζ1, ζ2]∣∣∣∣ pbx2+qa+b2x2

2(1+2(3µδ+µ−δ)) − ζ
(

bx
1+2µδ+µ−δ

)2
∣∣∣∣ , if ζ /∈ [ζ1, ζ2],

where

ζ1 =
(1 + 2µδ + µ− δ)2(pbx2 − bx+ qa+ b2x2)

2(1 + 2(3µδ + µ− δ))b2x2
,

and

ζ2 =
(1 + 2µδ + µ− δ)2(pbx2 + bx+ qa+ b2x2)

2(1 + 2(3µδ + µ− δ))b2x2
.

Corollary 9. If the function f ∈ A satisfies the subordination (8), then for bx > 0 and
for some ζ ∈ R

|a3 − ζa22| ≤


bx

2(αβ+(1−β)(3−2λ)) , if ζ ∈ [ζ1, ζ2]∣∣∣ pbx2+qa
2(αβ+(1−β)(3−2λ)) −

b2x2K
4(αβ+(1−β)(3−2λ))A2 − ζ

(
bx
A

)2∣∣∣ , if ζ /∈ [ζ1, ζ2].

where

ζ1 =
2A2(pbx2 − bx+ qa)− b2x2K

4(αβ + (1− β)(3− 2λ)b2x2
,

and

ζ2 =
2A2(pbx2 + bx+ qa)− b2x2K

4(αβ + (1− β)(3− 2λ)b2x2
.

Corollary 10. If the function f ∈ A satisfies the subordination (9), then for bx > 0 and
for some ζ ∈ R

|a3 − ζa22| ≤


bx

2(1+2η)(1+2µ) , if ζ ∈ [ζ1, ζ2]∣∣∣ (pbx2+qa)(1+η)2+b2x2(1+3η)
2(1+2η)(1+2µ)(1+η)2

− ζb2x2

(1+η)2(1+µ)2

∣∣∣ , if ζ /∈ [ζ1, ζ2].

where

ζ1 =
(1 + µ)2[(1 + η)2(pbx2 − bx+ qa) + b2x2(1 + 3η)]

2(1 + 2η)(1 + 2µ)b2x2
,

and

ζ2 =
(1 + µ)2[(1 + η)2(pbx2 + bx+ qa) + b2x2(1 + 3η)]

2(1 + 2η)(1 + 2µ)b2x2
.



REFERENCES 168

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