View of Application Of Horadam Polynomial On Sakaguchi Type Bi-Univalent Functions Satisfying Certain Subordination Constraints

Turkish Journal of Computer and Mathematics Education

Vol.12 No.10 (2021), 5615-5620

Research Article

Application Of Horadam Polynomial On Sakaguchi Type Bi-Univalent Functions

Satisfying Certain Subordination Constraints

Balakrishnan Senthil

_{and B. Srutha Keerthi}

1_{Department of Mathematics}

Jerusalem College of Engineering, Chennai, India *Research Scholar, School of Advanced Sciences VIT Chennai Campus, India

E-mail: senkrishh@gmail.com

2_{School of Advanced Sciences}

VIT Chennai Campus, India

Article History: Received: 10 January 2021; Revised: 12 February 2021; Accepted: 27 March 2021; Published

online: 28 April 2021

Abstract” In this current investigation, we apply Horadam polynomial to establish sharp upper bound for the

second and third coefficient of functions from new subclass of sakaguchi type bi-univalent functions defined in the open unit disk 𝕌. Also, we discuss Fekete-Szego inequality for functions belongs to this subclass.

Keywords: Holomorphic function, Univalent functions, Bi-univalent functions, Horadam Polynomial, Starlike

functions, Convex functions, Sakaguchi-type functions, Coefficient bounds, Fekete-Szego inequality.

Mathematics Subject Classification: 30C45. 1 Introduction

Let 𝕌 = {𝜉 ∶ |𝜉 | < 1} denote the open unit disk on the complex plane. The class of all holomorphic functions of the form

𝑢(𝜉) = 𝜉 + 𝑎2 𝜉2 + 𝑎3 𝜉3 + ⋯ (1)

defined in the open unit disk 𝕌 with Montel normalization 𝑢(0) = 0 = 𝑢′(0) − 1 is denoted by 𝒜 and the class 𝒮 ⊂ 𝒜 is the class which consists of univalent functions in 𝕌.

The Koebe one quarter theorem [1], states that the image of 𝕌 under every univalent function 𝑢 ∈ 𝒜 contains a disk of radius 1

4 Thus Koebe one quarter theorem guarantees that for every univalent function 𝑢 ∈ 𝒜, there exists

inverse function 𝑢−1 = 𝑣 satisfying

𝑢−1{(𝑢(𝜉)} = 𝜉, 𝜉 ∈ 𝕌 𝑎𝑛𝑑 𝑢{𝑢−1(𝜁)} = 𝜁, 𝑤ℎ𝑒𝑟𝑒 |𝜁| < 𝑟𝑢, 𝑟𝑢≥

1 4

A function 𝑢 ∈ 𝒜 is said to be bi-univalent in 𝕌 if both 𝑢 and 𝑢−1 _{are univalent in 𝕌. Let Σ denote the class of}

all function 𝑢 ∈ 𝒜 which are bi-univalent functions defined in the unit disk 𝕌 and whose Taylor series expansion

is given by (1). A simple computation shows that its inverse 𝑣 = 𝑢−1 _{also has the expansion.}

𝑣 (𝜁) = 𝑢−1_{(𝜁) = 𝜁 − 𝑎}

2𝜁2+ (2𝑎22− 𝑎3)𝜁3− (5𝑎23− 5𝑎2𝑎3+ 𝑎4)𝜁4+ ⋯ (2)

Many authors have established and examined subclasses of bi-univalent function and attained sharp bounds for the initial coefficients. (see [2,3,4,5,6])

A holomorphic function 𝑢 is subordinate to an holomorphic function 𝐺 in 𝕌 denoted as 𝑢 ≺ 𝐺, (𝜉 ∈ 𝕌). If 𝑢(𝜉) = 𝐺(𝜔(𝜉)), |𝜉| < 1 for some holomorphic schwarz function 𝜔(𝜉) with 𝜔(0) = 0 and |𝜔(𝜉)| < 1. It follows from schwarz lemma that

𝑢(𝜉) ≺ 𝐺(𝜉) ⇔ 𝑢(0) = 𝐺(0) 𝑎𝑛𝑑 𝑢(𝕌) ⊂ 𝐺(𝕌), 𝜉 ∈ 𝕌 One can refer [1,7] for details of subordination.

The Horadam Polynomial ℎ𝑛(𝜎) are defined by the following repetition relation (see [9,10]):

ℎ𝑛(𝜎) = 𝑥𝜎 ℎ𝑛−1 (𝜎) + 𝑦 ℎ𝑛−2(𝜎), (𝜎 ∈ ℝ, 𝑛 ∈ ℕ − {1,2} )

with

ℎ1(𝜎) = 𝑥 𝑎𝑛𝑑 ℎ2(𝜎) = 𝑦𝜎 (3)

for some real constants 𝑎, 𝑏, 𝑥 and 𝑦.

The generating function of the Horadam polynomials ℎ𝑛(𝜎) (see [9,10]) is given by

Π(𝜎, 𝜉) = ∑ ℎ𝑛(𝜎)𝜉𝑛−1

∞

𝑛=1

=𝑎 + (𝑏 − 𝑎𝑥)𝜎𝜉

1 − 𝑥𝜎𝜉 − 𝑦𝜉2 (4) 2 Bi-Univalent Function Class 𝓗𝓑𝝆,𝝁,𝒕{𝚷(𝝈, 𝝃)}

In this section, we introduce a new subclass of Sakaguchi type bi-univalent functions with the application

of Horadam polynomial by subordination technique and obtain bound for initial Taylor coefficient |𝑎2| and |𝑎3|

Turkish Journal of Computer and Mathematics Education

Vol.12 No.10 (2021), 5615-5620

Research Article

For 0 ≤ 𝜌 ≤ 1, 0 ≤ 𝜇 < 1 and |𝑡| ≤ 1, but 𝑡 ≠ 1, a function 𝑢 ∈ Σ of the form (1) is said to be in the class ℋℬ𝜌,𝜇,𝑡{Π(𝜎, 𝜉)}, if the following subordination hold:

(1 − 𝑡)[𝜌𝜇 𝜉3 _𝑢′′′_{(𝜉) + (2𝜌𝜇 + 𝜌 − 𝜇)𝜉}2 _𝑢′′_{(𝜉) + 𝜉 𝑢}′_(𝜉)] 𝜌𝜇 𝜉2_[𝑢′′_{(𝜉) − 𝑡}2_𝑢′′_{(𝑡𝜉)] + (𝜌 − 𝜇)𝜉[𝑢}′_{(𝜉) − 𝑡𝑢}′_{(𝑡𝜉)] + (1 − 𝜌 + 𝜇)[𝑢(𝜉) − 𝑢(𝑡𝜉)]} ≺ Π(𝜎, 𝜉) + 1 − 𝑎 (5) and (1 − 𝑡)[𝜌𝜇𝜁3 _{𝑣′′′(𝜁) + (2𝜌𝜇 + 𝜌 − 𝜇)𝜁}2 _𝑣′′_{(𝜁) + 𝜁 𝑣}′_(𝜁)] 𝜌𝜇𝜁2_[𝑣′′_{(𝜁) − 𝑡}2_𝑣′′_{(𝑡𝜁)] + (𝜌 − 𝜇)𝜁[𝑣}′_{(𝜁) − 𝑡𝑣}′_{(𝑡𝜁)] + (1 − 𝜌 + 𝜇)[𝑣(𝜁) − 𝑣(𝑡𝜁)]} ≺ Π(𝜎, 𝜁) + 1 − 𝑎 (6) where 𝑣 is given by (2).

Specializing the parameter 𝜌 = 0, 𝜇 = 0, 𝑡 = 0 and 𝜌 = 1, 𝜇 = 0, 𝑡 = 0, we have the following respectively.

Definition 2.

A function 𝑢 ∈ Σ of the form (1) is said to be in the class 𝒮ℋℬΣ {Π(𝜎, 𝜉)}, if the following subordination hold:

𝜉 𝑢′_(𝜉) 𝑢(𝜉) ≺ Π(𝜎, 𝜉) + 1 − 𝑎 and 𝜁 𝑣′_(𝜁) 𝑣(𝜁) ≺ Π(𝜎, 𝜁) + 1 − 𝑎 where 𝑣 is given by (2). Definition 3.

A function 𝑢 ∈ Σ of the form (1) is said to be in the class 𝒦ℋℬΣ{Π(𝜎, 𝜉)}, if the following subordination hold:

1 +𝜉 𝑢′′(𝜉) 𝑢′_(𝜉) ≺ Π(𝜎, 𝜉) + 1 − 𝑎 and 1 +𝜁 𝑣′′(𝜁) 𝑣′_(𝜁) ≺ Π(𝜎, 𝜁) + 1 − 𝑎 where 𝑣 is given by (2).

In the following theorem, we determine the bound for initial Taylor coefficient |𝑎2| and |𝑎3| for the function class

ℋℬ𝜌,𝜇,𝑡{Π(𝜎, 𝜉)}. Later we will reduce these bounds to other classes for special cases. Theorem 1.

Let 𝑢 given by (1) be in the class ℋℬ𝜌,𝜇,𝑡{Π(𝜎, 𝜉)}. Then

|𝑎2| ≤ |𝑏𝜎|√|𝑏𝜎| √|[2(3𝜌𝜇 + 𝜌 − 𝜇) + 1] { (3 − 𝑇3)[𝑏 2_𝜎2_] −(2 − 𝑇2)[(2𝜌𝜇 + 𝜌 − 𝜇) + 1]{𝑏2𝜎2𝑇2+ (2 − 𝑇2)[𝑥𝑏𝜎2+ 𝑦𝑎]} }| and |𝑎3| ≤ |𝑏𝜎| |3 − 𝑇3|[2(3𝜌𝜇 + 𝜌 − 𝜇) + 1] + |𝑏 2_𝜎2_| (2 − 𝑇2)2[(2𝜌𝜇 + 𝜌 − 𝜇) + 1]2 where 𝑇𝑛 = 1 − 𝑡𝑛 1 − 𝑡 = 1 + 𝑡 + 𝑡 2 _{+ ⋯ + 𝑡}𝑛−1 _(7). Proof.

Let 𝑢 ∈ ℋℬ𝜌,𝜇,𝑡{Π(𝜎, 𝜉)}. Then there are two holomorphic schwarz functions 𝑓, 𝑔 ∶ 𝕌 → 𝕌 given by

𝑓(𝜉) = 𝛼1𝜉 + 𝛼2𝜉2+ 𝛼3𝜉3+ ⋯ (𝜉 ∈ 𝕌) (8)

𝑔(𝜁) = 𝛽1𝜁 + 𝛽2𝜁2+ 𝛽3𝜁3+ ⋯ (𝜁 ∈ 𝕌) (9)

with 𝑓(0) = 𝑔(0) = 0 and |𝑓(𝜉)| < 1, |𝑔(𝜁)| < 1 (𝜉 , 𝜁 ∈ 𝕌) Hence, we have

|𝛼 𝑖| < 1 𝑎𝑛𝑑 |𝛽𝑖| < 1, ∀ 𝑖 ∈ ℕ (10)

Now using (8) and (9) in (5) and (6), we have

(1 − 𝑡)[𝜌𝜇 𝜉3 _𝑢′′′_{(𝜉) + (2𝜌𝜇 + 𝜌 − 𝜇)𝜉}2 _𝑢′′_{(𝜉) + 𝜉 𝑢}′_(𝜉)]

𝜌𝜇 𝜉2_[𝑢′′_{(𝜉) − 𝑡}2_𝑢′′_{(𝑡𝜉)] + (𝜌 − 𝜇)𝜉[𝑢}′_{(𝜉) − 𝑡𝑢}′_{(𝑡𝜉)] + (1 − 𝜌 + 𝜇)[𝑢(𝜉) − 𝑢(𝑡𝜉)]}

= Π(𝜎, 𝑓(𝜉)) + 1 − 𝑎

(11) and

Turkish Journal of Computer and Mathematics Education

Vol.12 No.10 (2021), 5615-5620

Research Article

(1 − 𝑡)[𝜌𝜇 𝜁3 _𝑣′′′_{(𝜁) + (2𝜌𝜇 + 𝜌 − 𝜇)𝜁}2 _{𝑣′′(𝜁) + 𝜁 𝑣}′_(𝜁)]

𝜌𝜇 𝜁2_{[𝑣′′(𝜁) − 𝑡}2_{𝑣′′(𝑡𝜁)] + (𝜌 − 𝜇)𝜁[𝑣}′_{(𝜁) − 𝑡𝑣}′_{(𝑡𝜁)] + (1 − 𝜌 + 𝜇)[𝑣(𝜁) − 𝑣(𝑡𝜁)]} = Π(𝜎, 𝑔(𝜁)) + 1 − 𝑎

(12) where 𝜉, 𝜁 ∈ 𝕌 and 𝑣 is given by (2).

Now (11) ⟹ 1 + (2 − 𝑇2)[(2𝜌𝜇 + 𝜌 − 𝜇) + 1]𝑎2 𝜉 − { (2 − 𝑇2)[(2𝜌𝜇 + 𝜌 − 𝜇) + 1]2𝑎22 𝑇2 −(3 − 𝑇3)[2(3𝜌𝜇 + 𝜌 − 𝜇) + 1]𝑎3 } 𝜉2_{+ ⋯ = Π(𝜎, 𝑓(𝜉)) + 1 − 𝑎} (13) where Π(𝜎, 𝑓(𝜉)) + 1 − 𝑎 = 1 − 𝑎 + ℎ1(𝜎) + ℎ2(𝜎)𝑓(𝜉) + ℎ3(𝜎)𝑓2(𝜉) + ⋯ = 1 + ℎ2(𝜎)𝛼1 𝜉 + [ℎ2(𝜎)𝛼2+ ℎ3(𝜎)𝛼12]𝜉2+ ⋯ (14)

Equating coefficients of 𝜉 and 𝜉2 _{from (13) and (14), we get}

(2 − 𝑇2)[(2𝜌𝜇 + 𝜌 − 𝜇) + 1]𝑎2= ℎ2(𝜎)𝛼1 (15) { (3 − 𝑇3)[2(3𝜌𝜇 + 𝜌 − 𝜇) + 1]𝑎3 −(2 − 𝑇2)[(2𝜌𝜇 + 𝜌 − 𝜇) + 1]2𝑎22 𝑇2 } = ℎ2(𝜎)𝛼2+ ℎ3(𝜎)𝛼12 (16) Now (12) ⟹ 1 + (2 − 𝑇2)[(2𝜌𝜇 + 𝜌 − 𝜇) + 1]𝑎2 𝜁 − { (2 − 𝑇2)[(2𝜌𝜇 + 𝜌 − 𝜇) + 1]2𝑎22 𝑇2 −(3 − 𝑇3)[2(3𝜌𝜇 + 𝜌 − 𝜇) + 1](2𝑎22− 𝑎3) } 𝜁2_{+ ⋯} = Π(𝜎, 𝑔(𝜁)) + 1 − 𝑎 (17) where Π(𝜎, 𝑔(𝜁)) + 1 − 𝑎 = 1 − 𝑎 + ℎ1(𝜎) + ℎ2(𝜎)𝑔(𝜁) + ℎ3(𝜎)𝑔2(𝜁) + ⋯ = 1 + ℎ2(𝜎 )𝛽1 𝜁 + [ℎ2(𝜎)𝛽2+ ℎ3(𝜎)𝛽12]𝜁2+ ⋯ (18)

Equating coefficients of 𝜁 and 𝜁2 _{from (17) and (18), we get}

−(2 − 𝑇2)[(2𝜌𝜇 + 𝜌 − 𝜇) + 1]𝑎2= ℎ2(𝜎)𝛽1 (19) {(3 − 𝑇3)[2(3𝜌𝜇 + 𝜌 − 𝜇) + 1](2𝑎2 2_{− 𝑎} 3) −(2 − 𝑇2)[(2𝜌𝜇 + 𝜌 − 𝜇) + 1]2𝑎22 𝑇2 } = ℎ2(𝜎)𝛽2 + ℎ3(𝜎)𝛽12 (20)

From (15) and (19), we have

𝛼1= −𝛽1 (21) Now (15)2_{+ (19)}2 _⟹ 2𝑎22= (𝛼12+ 𝛽12) ℎ22(𝜎) (2 − 𝑇2)2[(2𝜌𝜇 + 𝜌 − 𝜇) + 1]2 using (21) in the above, we get

2𝑎22= (2𝛼₁2_{) ℎ} 22(𝜎) (2 − 𝑇2)2[(2𝜌𝜇 + 𝜌 − 𝜇) + 1]2 (22) ⟹ 𝛼12 = (2 − 𝑇2)2[(2𝜌𝜇 + 𝜌 − 𝜇) + 1]2 𝑎22 ℎ22(𝜎) (23) Now by summing (16) and (20)

2 { (3 − 𝑇3)[2(3𝜌𝜇 + 𝜌 − 𝜇) + 1] −(2 − 𝑇2)[(2𝜌𝜇 + 𝜌 − 𝜇) + 1]2 𝑇2 } 𝑎22 = ℎ2(𝜎)[𝛼2+ 𝛽2] + ℎ3(𝜎)[𝛼12+ 𝛽12] Since by (21), we have 2 { (3 − 𝑇3)[2(3𝜌𝜇 + 𝜌 − 𝜇) + 1] −(2 − 𝑇2)[(2𝜌𝜇 + 𝜌 − 𝜇) + 1]2 𝑇2 } 𝑎22 = ℎ2(𝜎)[𝛼2+ 𝛽2] + ℎ3(𝜎)[2𝛼12] (24) By substituting (23) in (24), we have 2 { (3 − 𝑇3)ℎ2 2_{(𝜎)[2(3𝜌𝜇 + 𝜌 − 𝜇) + 1]} −(2 − 𝑇2)[(2𝜌𝜇 + 𝜌 − 𝜇) + 1]2{ℎ22(𝜎)𝑇2+ (2 − 𝑇2)ℎ3(𝜎)} } 𝑎22= ℎ23(𝜎)[𝛼2+ 𝛽2] (25) Therefore, by using (10), we obtain

|𝑎2| ≤ |𝑏𝜎|√|𝑏𝜎| √|[2(3𝜌𝜇 + 𝜌 − 𝜇) + 1] { (3 − 𝑇3)[𝑏 2_𝜎2_] −(2 − 𝑇2)[(2𝜌𝜇 + 𝜌 − 𝜇) + 1]{𝑏2𝜎2𝑇2+ (2 − 𝑇2)[𝑥𝑏𝜎2+ 𝑦𝑎]} }|

Now we have to find bound for |𝑎3|, Lets subtract (19) from (15), then we get

2(3 − 𝑇3)[2(3𝜌𝜇 + 𝜌 − 𝜇) + 1]{𝑎3− 𝑎22} = ℎ2(𝜎)[𝛼2− 𝛽2] (26)

Turkish Journal of Computer and Mathematics Education

Vol.12 No.10 (2021), 5615-5620

Research Article

Now use (22) in (27), we obtain |𝑎3| ≤ |𝑏𝜎| |3 − 𝑇3|[2(3𝜌𝜇 + 𝜌 − 𝜇) + 1] + |𝑏 2_𝜎2_| (2 − 𝑇2)2[(2𝜌𝜇 + 𝜌 − 𝜇) + 1]2

where 𝑇2, 𝑇3 are given by (7).

If we take the parameters 𝜌 = 0, 𝜇 = 0, 𝑡 = 0 and 𝜌 = 1, 𝜇 = 0, 𝑡 = 0, in the above theorem, we have the following bounds of initial Taylor coefficients |𝑎2| and |𝑎3| for the function classes 𝒮ℋℬΣ{Π(𝜎, 𝜉)} and

𝒦ℋℬΣ{Π(𝜎, 𝜉)} respectively

Corollary 1.

Let 𝑢 given by (1) be in the class 𝒮ℋℬΣ{Π(𝜎, 𝜉)}, Then

|𝑎2| ≤ |𝑏𝜎|√|𝑏𝜎| √𝑏2_𝜎2_{− (𝑥𝑏𝜎}2_{+ 𝑦𝑎)} and |𝑎₃| ≤|𝑏𝜎| 2 + 𝑏 2_𝜎2 Corollary 2.

Let 𝑢 given by (1) be in the class 𝒦ℋℬΣ{Π(𝜎, 𝜉)}, Then

|𝑎2| ≤ |𝑏𝜎|√|𝑏𝜎| √2𝑏2_𝜎2_{− 4(𝑥𝑏𝜎}2_{+ 𝑦𝑎)} and |𝑎3| ≤ |𝑏𝜎| 6 + 𝑏2_𝜎2 4

3 Fekete-Szego Inequalities for the Function Class 𝓗𝓑𝝆,𝝁,𝒕{𝚷(𝝈, 𝝃)}

Fekete and Szego [12] introduced the generalized functional |𝑎3− 𝜆 𝑎22|, where 𝜆 is some real number. Due to

Zaprawa [13], in the following theorem we determine the Fekete-Szego functional for 𝑢 ∈ ℋℬ𝜌,𝜇,𝑡{Π(𝜎, 𝜉)}.

Theorem 2.

Let 𝑢 given by (1) be in the class ℋℬ𝜌,𝜇,𝑡{Π(𝜎, 𝜉)} and 𝜆 ∈ ℝ . Then we have

|a3− λ a22| ≤ { |bσ| |3−T3|[2(3ρμ+ρ−μ)+1], if |ϕ(λ, σ)| ≤ 1 2(3−T₃)[2(3ρμ+ρ−μ)+1] 2|bσ||ϕ(λ, σ)|, if |ϕ(λ, σ)| ≥ 1 2(3−T₃)[2(3ρμ+ρ−μ)+1] where 𝜙(𝜆, 𝜎) = (1−𝜆)ℎ22(𝜎) 2{ (3−𝑇₃)ℎ22(𝜎)[2(3𝜌𝜇+𝜌−𝜇)+1]−(2−𝑇2)[(2𝜌𝜇+𝜌−𝜇)+1]2{ℎ22(𝜎)𝑇2+(2−𝑇2)ℎ3(𝜎)}} (28) and 𝑇2, 𝑇3 are given by (7)

Proof.

From (25) and (26), we obtain 𝑎3− 𝑎22 = ℎ2(𝜎)[𝛼2− 𝛽2] 2(3 − 𝑇3)[2(3𝜌𝜇 + 𝜌 − 𝜇) + 1] 𝑎3− 𝜆 𝑎22= ℎ2(𝜎)[𝛼2− 𝛽2] 2(3 − 𝑇3)[2(3𝜌𝜇 + 𝜌 − 𝜇) + 1] + (1 − 𝜆)𝑎22 = ℎ2(𝜎) [ 𝛼2− 𝛽2 2(3 − 𝑇3)[2(3𝜌𝜇 + 𝜌 − 𝜇) + 1] + (𝛼2+ 𝛽2)𝜙(𝜆, 𝜎)] = ℎ2(𝜎) [( 1 2(3 − 𝑇3)[2(3𝜌𝜇 + 𝜌 − 𝜇) + 1] + 𝜙(𝜆, 𝜎)) 𝛼2 + (𝜙(𝜆, 𝜎) − 1 2(3 − 𝑇3)[2(3𝜌𝜇 + 𝜌 − 𝜇) + 1] ) 𝛽2]

Turkish Journal of Computer and Mathematics Education

Vol.12 No.10 (2021), 5615-5620

Research Article

Then, by taking modulus, we conclude that

|a3− λ a22| ≤ { |bσ| |3 − T3|[2(3ρμ + ρ − μ) + 1] , if |ϕ(λ, σ)| ≤ 1 2(3 − T3)[2(3ρμ + ρ − μ) + 1] 2|bσ||ϕ(λ, σ)|, if |ϕ(λ, σ)| ≥ 1 2(3 − T3)[2(3ρμ + ρ − μ) + 1] where 𝜙(𝜆, 𝜎) is given by (28).

If we take the parameters 𝜌 = 0, 𝜇 = 0, 𝑡 = 0 and 𝜌 = 1, 𝜇 = 0, 𝑡 = 0, in the above theorem, we have the

following Fekete-Szego inequalities for the function classes 𝒮ℋℬΣ {Π(𝜎, 𝜉)} and 𝒦ℋℬΣ {Π(𝜎, 𝜉)}, respectively.

Corollary 3.

Let 𝑢 given by (1) be in the class 𝒮ℋℬΣ {Π(𝜎, 𝜉)} and 𝜆 ∈ ℝ, Then we have

|𝑎3− 𝜆 𝑎22| ≤ { |𝑏𝜎| 2 , 𝑖𝑓 |1 − 𝜆| ≤ |𝑏2_𝜎2_{− (𝑥𝑏𝜎}2_{+ 𝑦𝑎)|} 2|𝑏2_𝜎2_| |1 − 𝜆||𝑏3_𝜎3_| |𝑏2_𝜎2_{− (𝑥𝑏𝜎}2_{+ 𝑦𝑎)|}, 𝑖𝑓 |1 − 𝜆| ≥ |𝑏2_𝜎2_{− (𝑥𝑏𝜎}2_{+ 𝑦𝑎)|} 2|𝑏2_𝜎2_| Corollary 4.

Let 𝑢 given by (1) be in the class 𝒦ℋℬΣ {Π(𝜎, 𝜉)} and 𝜆 ∈ ℝ, Then we have

|𝑎3− 𝜆 𝑎22| ≤ { |𝑏𝜎| 6 , 𝑖𝑓 |1 − 𝜆| ≤ |𝑏2_𝜎2_{− 2(𝑥𝑏𝜎}2_{+ 𝑦𝑎)|} 3|𝑏2_𝜎2_| |1 − 𝜆||𝑏3_𝜎3_| 2|𝑏2_𝜎2_{− 2(𝑥𝑏𝜎}2_{+ 𝑦𝑎)|}, 𝑖𝑓 |1 − 𝜆| ≥ |𝑏2_𝜎2_{− 2(𝑥𝑏𝜎}2_{+ 𝑦𝑎)|} 3|𝑏2_𝜎2_| References

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