Certain Results on Subclasses of Analytic and Bi-Univalent Functions Associated with Coefficients Estimates and Quasi-Subordination

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Certain Results on Subclasses of Analytic and Bi-Univalent Functions Associated with Coefficients Estimates and Quasi-Subordination

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A peer-reviewed article of this preprint also exists.

Elaf Ibrahim Badiwi,

Waggas Galib Atshan^*

,Ameera N. Alkiffai,

Alina Alb Lupas

Elaf Ibrahim Badiwi,

Waggas Galib Atshan^*

,Ameera N. Alkiffai,

Alina Alb Lupas

This version is not peer-reviewed

Submitted:

06 November 2023

Posted:

07 November 2023

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Abstract

The purpose of the present paper is to introduce and investigate new subclasses of analytic function class of bi univalent function defined in open unit disk connected with a linear q-convolution operator, which are associated with the quasi-subordination. We find coefficients estimate |h_2 |,|h_3 | for functions in these subclasses. Several known and new consequences of these results are also pointed out.

Keywords:

Subject: Computer Science and Mathematics - Analysis

MSC: 30C45; 30C50

1. Introduction

The theor

y

of q-calculus pla

y

s an important rôle in man

y

areas of mathematical ph

y

sical and engineering scienc

e

s. Jackson (see [11] and [10]) was th

e

first to have s

o

me applicati

o

ns of the q-calculus and introduc

e

d the q-analogue of the classical derivati

v

e and integral

o

perators (see also [30]).

Let

A

be the class of analytic functions

T

in an open unit disk

U = {ε \in C

| ε | < 1}

of the form:

T (ε) = ε + \sum_{j = 2}^{\infty} a_{j} ε^{j}, (ε \in U) . (1.1)

and satisfying the normalization conditions (see

[1]

T (0) = T^{'} (0) - 1 = 0

Assume that

\sum_{U}

denotes the class of all functions in

A

defined by (1.1), which are univalent in

U

e

well-known Koebe-One Quarter Theorem

[5]

states that the imag

e

of the op

e

n unit disk

U

under each univalent functi

o

n in a disk with the radius

\frac{1}{4}

. Thus, every univalent functi

o

T

has an inv

e

rse

T^{- 1}

, such that

T^{- 1} (T (ε)) = ε, (z \in U),

and

{T (T}^{- 1} (ς)) = ς (|ς| < r_{0} (T); r_{0} (T) \geq \frac{1}{4}) .

In fact, the inverse function

{ξ = T}^{- 1}

is given by

ξ (ς) = ς - a_{2} ς^{2} + {(2 a}_{2}^{2} - a_{3}) ς^{3} - {(5 a}_{2}^{2} - {5 a}_{2} a_{3} + a_{4}) ς^{4} + \dots . (1.3)

= ς + \sum_{n = 2}^{\infty} A_{n} ς^{n}

The function

T \in A

is said to be bi-univalent in

U

b o t h T

and

i t s i n v e r s e T^{- 1}

are univalent functions in

U

given by (1.1).

The class of bi-univalent functions was introduced by Lewin

[14]

and proved that

|a_{2}| \leq 1.51

for the function of the form (1.1). Subsequently, Brannan and Clunie

[3]

conjectured that

|a_{2}| \leq \sqrt{2}

. Later Netanyahu

i n [17]

proved that

\max_{T \in \sum} |a_{2}| = \frac{4}{3}

. Also sev

e

ral authors studi

e

d class

e

s of bi-unival

e

nt analytic functi

o

ns a

n

d f

o

und estimat

e

s of th

e

coefficie

n

|a_{2}|

and

|a_{3}|

for functi

o

ns in th

e

se classes [ For two analytic functi

o

T

n

ξ, T

is quasi-sub

o

rdinate to

ξ

w

ritten as follo

w

T (ε) ≺_{q} ξ (ε) (ε \in U) (1.3)

if the

r

e e

x

ist analytic functions

h (ε) a n d k (ε)

w

ith

|h (z)| \leq 1

k (0) = 0 a n d |k (ε)| < 1

(ε \in U),

c

h that

T (ε) = h (ε) ξ (k (ε)), (ε \in U) .

Note that if (

h (ε) = 1)

, then

T (ε) = ξ (k (ε)),

hance

T (ε) ≺ ξ (ε)

(z \in U) .

ξ b e

univalent in

U, t h e n

T ≺ ξ

if and only if

T (0) = ξ (0)

and

T (U) \subset ξ (U)

For the functions

T, ρ

∈

\sum_{U}

defined by

T (ε) = \sum_{j = 1}^{\infty} a_{j} ε^{j} and ρ (ε) = \sum_{j = 1}^{\infty} h_{j} ε^{j} (ε \in U),

the convoluti

o

n of

T

and

ρ

denot

e

d by

T * ρ

(T * ρ) (ε) = \sum_{j = 1}^{\infty} a_{j} {h_{j} ε}^{j} = (ρ * T) (ε) (ε \in U) .

To start with, we recall the follo

w

ing differential and integral operators. For

0 < q < 1

, El-Deeb et al.

[8,24]

defined the q-c

o

nvolution operator (see also

[10])

for

T * ρ

Q_{q} (T * ρ) (ε) = Q_{q} (ε + \sum_{j = 2}^{\infty} a_{j} h_{j} ε^{j})

\frac{(T * ρ) (ε) - (T * ρ) (q ε)}{ε (1 - q)} = 1 + \sum_{j = 2}^{\infty} {[j]}_{q} a_{j} h_{j} ε^{j - 1}, ε \in U,

where

{[j]}_{q} = \frac{1 - q^{j}}{1 - q} = 1 + \sum_{j = 1}^{j - 1} q^{j}, {[0]}_{q} = 0 . (1.4)

We used the linear operator

Y_{ρ}^{ζ, q}

A

→

A

acco

r

ding to El-Deeb et al. [

8] (s e e a l s o [25])

for and

ζ > - 1, 0 < q < 1

. If

Y_{ρ}^{ζ, q} T (ε) * I_{q}^{ζ + 1} (ε) = ε Q_{q} (T * ρ) (ε), ε \in U,

where

I_{q}^{ζ + 1}

is given by

I_{q}^{ζ + 1} (ε) = ε + \sum_{j = 2}^{\infty} \frac{{[ζ + 1]}_{q, ε - 1}}{{[ε - 1]}_{q}!} ε^{j}, ε \in U,

then

Y_{ρ}^{ζ, q} T (ε) = ε + \sum_{j = 2}^{\infty} \frac{{[j]}_{q}!}{{[ζ]}_{q, ε - 1}} a_{j} h_{j} ε^{j} (ζ > - 1, 0 < q < 1, ε \in U) . (1.5)

Using the operator

Y_{ρ}^{ζ, q}

, we define a new operator as follows:

Q_{ρ, σ, ϑ}^{ζ, q, 0} T (ε) = Y_{ρ}^{ζ, q} T (ε)

Q_{ρ, σ, ϑ}^{ζ, q, 1} T (ε) = (σ - ϑ) ε^{3} {(Y_{ρ}^{ζ, q} T (ε))}^{'''} + (1 + 2 (σ - ϑ)) ε^{2} {(Y_{ρ}^{ζ, q} T (ε))}^{''} + ε {(Y_{ρ}^{ζ, q} T (ε))}^{'} (1.6)

Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε) =

{(σ - ϑ) ε}^{3} {(Y_{ρ}^{ζ, q, n - 1} T (ε))}^{'''} + (1 + 2 (σ - ϑ)) ε^{2} {(Y_{ρ}^{ζ, q, n - 1} T (ε))}^{''} + ε {(Y_{ρ}^{ζ, q, n - 1} T (ε))}^{'}

= ε + \sum_{j = 2}^{\infty} j^{2 n} {((σ - ϑ) (j - 1) + 1)}^{n} \frac{{[j]}_{q}!}{{[ζ]}_{q, ε - 1}} a_{j} h_{j} ε^{j}

Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε) = ε + \sum_{j = 2}^{\infty} ψ_{j} h_{j} ε^{j} (\begin{matrix} ζ > - 1, 0 < q < 1, ϑ \geq 0, σ > 0, σ \neq ϑ, \\ n \in N_{O} \cup \{0\} a n d ε \in U \end{matrix}), (1.7)

where

{ψ_{j} = j}^{2 n} {((σ - ϑ) (j - 1) + 1)}^{n} \frac{{[j]}_{q}!}{{[ζ]}_{q, ε - 1}} a_{j},

and by

[10]

,let 0 < q < 1 and

{[j]}_{q}

is defined by

{[j]}_{q} = \frac{1 - q^{j}}{1 - q} = 1 + \sum_{j = 1}^{j - 1} q^{j}, {[0]}_{q} = 0 .

The

q

- number shift factorial is giv

e

n b

y

{[j]}_{q}! = \{\begin{matrix} {[j]}_{q} {[j - 1]}_{q} . . . {[2]}_{q} {[1]}_{q}, i f j = 1, 2,3, \dots, \\ 1, i f j = 0, \end{matrix}

From the d

e

finition r

e

lation

(1.5),

e

get

(i) {[ζ + 1]}_{q} Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε) = {[ζ]}_{q} Q_{ρ, σ, ϑ}^{ζ + 1, q, n} T (ε) + q^{ζ} ε Q_{q} (Q_{ρ, σ, ϑ}^{ζ + 1, q, n} T (ε)), ε \in U; (1.8)

(i i) R_{ρ, σ, ϑ}^{ζ, n} T (ε) = \lim_{q \to 1^{-}} Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε) = ε + \sum_{j = 2}^{\infty} j^{2 n} {((σ - ϑ) (j - 1) + 1)}^{n} \frac{{[j]}_{q}!}{{[ζ]}_{q, ϵ - 1}} a_{j} h_{j} ε^{j} (1.9)

The

q -

generalized Pochhammersymbol is defined by

{[ζ]}_{q, ϵ - 1}

\frac{⎾_{q} (ζ + ϵ - 1)}{⎾_{q} (ζ)}

ϵ - 1 \in N,

ζ \in N .

For,

q \to 1 -,

then

{[ζ]}_{q, ϵ - 1}

reduces to

(ζ)_{ϵ - 1}

\frac{⎾ (ζ + ϵ - 1)}{⎾ (ζ)}

Remark(1.1):

e

find the foll

o

wing special cases for the

o

perator

Q_{ρ, σ, ϑ}^{ζ, q, n}

by c

o

nsidering sev

e

ral particular cases f

o

r the coefficients

a_{j}

and n :

(i): $P u t t i n g a_{j} = 1, ϑ = 0 a n d n = 0$ into this operator, we obtain the operator $Q T R c a l B_{q}^{α}$ defined by Srivastava et al. $[23];$
(ii): Putting $a_{j} = \frac{{(- 1)}^{j} Γ (ρ + 1)}{4^{j - 1} (j - 1)! Γ (r + ρ)}$ $(ρ > 0), ϑ = 0$ and n = 0 in this operator, we obtain the operator $N_{p, q}^{σ}$ defined by El-Deeb and Bulboac˘a $[9]$ and El-Deeb $[8];$
(iii): Putting $a_{j} = {(\frac{τ + 1}{τ + j})}^{r} (r > 0, τ \geq 0$ ) $, ϑ = 0$ and n = 0 in this operator, we obtain the operator $M_{τ, q}^{σ, r}$ defined by El-Deeb and Bulboac˘a $[24]$ and Srivastava and El-Deeb $[25];$
(iv): P $u$ tting $a_{j} = \frac{ς^{j - 1}}{(j - 1)!} ϱ^{- ς}$ ( $ς$ > 0) and n = 0 in this operator, we obtain the q-analogu $e$ of Poisson $o$ perator $Ι_{q}^{ϑ, ς}$ defin $e$ d by El-Deeb et al. $[8];$
(v): Putting $a_{j} = 1, ϑ = 0$ in this operator, w $e$ obtain the operat $o$ r $Q T R c a l B_{ϑ, σ}^{δ, q, n}$ defined as follows:

$B_{ϑ, σ}^{δ, q, n} Ϝ (ℇ) = ε + \sum_{j = 2}^{\infty} j^{2 n} {((σ - ϑ) (j - 1) + 1)}^{n} \frac{{[j]}_{q}!}{{[ζ]}_{q, ε - 1}} h_{j} ε^{j}; (1.10)$
(vi): Putting $a_{j} = \frac{{(- 1)}^{j} Γ (ρ + 1)}{4^{j - 1} (j - 1)! Γ (r + ρ)}$ $(ρ > 0)$ in this operat $o$ r, we obtain the op $e$ rator $N_{ς, p, q}^{σ, n}$ defined as follows:

$N_{ς, p, q}^{σ, n} Ϝ (ℇ) = ε + \sum_{j = 2}^{\infty} j^{2 n} {((σ - ϑ) (j - 1) + 1)}^{n} \frac{{[j]}_{q}!}{{[ζ + 1]}_{q, ε - 1}} \frac{{(- 1)}^{j} Γ (ρ + 1)}{4^{j - 1} (j - 1)! Γ (r + ρ)} h_{j} ε^{j} = ε + \sum_{j = 2}^{\infty} φ_{j} h_{j} ε^{j}, (1.11)$

where

${φ_{j} = j}^{2 n} {((σ - ϑ) (j - 1) + 1)}^{n} \frac{{[j]}_{q}!}{{[ζ]}_{q, ε - 1}} \frac{{(- 1)}^{j} Γ (ρ + 1)}{4^{j - 1} (j - 1)! Γ (r + ρ)}, (1.12)$
(vii): Putting $a_{j} = {(\frac{τ + 1}{τ + j})}^{r} (r > 0, τ \geq 0$ ) in this operator, we $o$ btain the operator $M_{τ, θ, q}^{σ, n, r}$ defin $e$ d as follows:

$M_{τ, θ, q}^{σ, n, r} Ϝ (ℇ) = ε + \sum_{j = 2}^{\infty} j^{2 n} {((σ - ϑ) (j - 1) + 1)}^{n} {(\frac{τ + 1}{τ + j})}^{r} \frac{{[j]}_{q}!}{{[ζ + 1]}_{q, ε - 1}} h_{j} ε^{j} .$

Ma and Minda have giv

e

n a unifi

e

d treatment of various subclass c

o

nsisting of starlike and convex functi

o

ns for either on

e

of the quantities

\frac{ε T^{'} (ε)}{T (ε)}

1 + \frac{ε T^{″} (ε)}{T (ε)}

is sub

o

rdinate to a m

o

re general super

o

rdinate functi

o

n. The

S^{*} (ϕ)

introduc

e

d by Ma and Minda

[15]

consists of function

T \in A s a t i s f y i n g \frac{ε T^{'} (ε)}{T (ε)} ≺ ϕ (z), z \in U

and corresp

o

nding class

k (ϕ)

of conv

e

x functions

T \in A s a t i s f y i n g 1 + \frac{ε T^{″} (ε)}{T (ε)} ≺ ϕ (z), z \in U

, Ma and Minda

[15],

where

ϕ

is ana

l y

tic and univalent functi

o

n with positive real part in the unit disc U, satisfying

ϕ (0) = 1

ϕ^{'} (0) > 0

and

ϕ (U)

is a starlike region with the respect to 1 and s

y

mmetric with the re

s

pect to the real axis. The functi

o

ns in the classes

S^{*} (ϕ)

and

K (ϕ),

are called starlike of Ma-Minda type or convex of Ma-Minda type respectivel

y

. By

S_{\sum_{U}}^{*} (ϕ)

and

K_{\sum_{U}} (ϕ)

, we den

o

te to bi-starlike of Ma-Minda type and bi-convex of Ma-Minda type respectively.

[15] .

In this investigation,

w

e assume that

ϕ (ε) = 1 + {B_{1} ε + B_{2} ε}^{2} {+ B_{3} ε}^{3} + \dots, B_{1} > 0 . (1.13)

and

h (ε) = h_{0} + h ε + h_{2} ε^{2} {+ h_{3} ε}^{3} + \dots . (1.14)

The aim of this pap

e

r is to introduce n

e

w subclasses of the class

\sum_{U}

and d

e

termine estimates of bounds on the c

o

efficient

|h_{2}| a n d |h_{3}|

and for the functi

o

ns in above subclass

e

In [3] (see also [35,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38]), c

e

rtain subclasses of the bi-unival

e

nt analytic functi

o

ns class B

w

ere introduced and non-sharp estimat

e

s on the first two c

o

efficients

|h_{2}| a n d |h_{3}|

were found. The object of the present pap

e

r is to introduce two ne

w

subclasses as in Definiti

o

2.1

and 3.1 of the functi

o

n class B using the lin

e

ar q-convolution operat

o

r and determine estimat

e

s of the coefficients

|h_{2}| a n d |h_{3}|

for the functi

o

ns in these new subclasses of the functi

o

n class.

Lemma (1.1)[8]:

Let

p (ε) \in P

,then

| p i | \leq 2

for each

i

where

P

is the family of all fun

c

tions

p

, anal

y

tic in 𝔘, for

w

hich

R e (p (ε)) > 0, (ε \in U), w

here

p (z) = 1 + {p_{1} ε + p_{2} ε}^{2} {+ p_{3} ε}^{3} + \dots .

2. Coefficients Estimates for the Class $f_{q, \sum}^{μ} (ζ, n, ρ, σ, ϑ, γ, δ, φ) .$

Definition (2.1):

For a function

T \in \sum_{U}

defined by

(1.1)

is said to be in the class

f_{q, \sum}^{μ} (ζ, n, ρ, σ, ϑ, γ, δ, φ)

if the follo

w

ing quasi-subordinati

o

n conditi

o

ns are satisfi

e

[\frac{{ε [{(Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε))}^{'} + γ {ε (Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε))}^{''}]}^{μ}}{γ ε ({(Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε))}^{'} + δ {(Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε))}^{''}) + (1 - γ) ((1 - δ) Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε) + δ {ε (Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε))}^{'}}] - {1 ≺}_{q} (φ (ε) - 1), (2.1)

and

[\frac{ς {[{(Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς))}^{'} + γ ς {(Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς))}^{''}]}^{μ}}{γ ς ({(Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς))}^{'} + δ {(Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς))}^{''}) + (1 - γ) ((1 - δ) Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς) + δ ς {(Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς))}^{'}}] - 1 ≺_{q} (φ (ς) - 1), (2.2)

where

γ, δ, μ \in [0, 1]

and

Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε)

is defined in (1.7) and

(ε, ς \in U) .

For special values to parameters

μ, δ, γ, ζ, n, ρ, σ, ϑ a n d φ (ε),

leads to get Known and new classes.

Remark (3.1):

For

δ = 0,

a function

T \in \sum_{U}

define by

(1.7)

is said to be in the class

f_{q, \sum}^{μ} (ζ, n, ρ, σ, ϑ, γ, δ, φ)

if the follo

w

ing quasi-subordinati

o

n conditions are satisfi

e

[\frac{ε {[{(Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε))}^{'} + γ {ε (Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε))}^{''}]}^{μ}}{γ {(Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε))}^{'} + + (1 - γ) Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε)}] - {1 ≺}_{q} (φ (ε) - 1),

and

[\frac{{ς [{(Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς))}^{'} + γ ς {(Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς))}^{''}]}^{μ}}{γ ς {(Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς))}^{'} + (1 - γ) Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς)}] - {1 ≺}_{q} (φ (ς) - 1),

where

ξ

is the inverse function of

T

and

(ε, ς \in U) .

Remark (3.3):

For

δ = 1,

a functi

o

T \in \sum_{U}

define by

(1.7)

is said to be in the class

f_{q, \sum}^{μ} (ζ, n, ρ, σ, ϑ, γ, δ, φ)

if the follo

w

ing quasi-subordination c

o

nditions are

s

atisfied:

[\frac{{ε [{(Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε))}^{'} + γ {ε (Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε))}^{''}]}^{μ}}{γ {ε (Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε))}^{''} + (1 - γ) ε {(Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε))}^{'}}] - {1 ≺}_{q} (φ (ε) - 1),

and

[\frac{ς {[{(Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς))}^{'} + γ ς {(Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς))}^{''}]}^{μ}}{γ ς {(Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς))}^{''} + (1 - γ) ς {(Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς))}^{'}}] - {1 ≺}_{q} (φ (ς) - 1),

where

ξ

is the inverse function of

T

and

(ε, ς \in U) .

Theorem (2.1):

If the function

T

belongs to the class

f_{q, \sum}^{μ} (ζ, n, ρ, σ, ϑ, γ, δ, φ),

then, we have

{| h}_{2} | \leq \frac{|A_{0}| B_{1} \sqrt{B_{1}}}{\sqrt{(1 + 2 γ) (3 μ - 2 δ - 1) {A_{0} B_{1}^{2} ψ}_{3} - {(1 + γ)}^{2} [{(2 μ - δ - 1)}^{2} (B_{2} - B_{1}) - [2 μ (μ - 1) - (2 μ - δ - 1) (1 + δ)] {ψ_{2}^{2} A}_{0} B_{1}^{2}]}} (3.3)

and

{| h}_{3} | \leq \frac{B_{1} (|A_{0}| + |A_{1}|)}{(1 + 2 γ) (3 μ - 2 δ - 1) ψ_{3}} + \frac{A_{0}^{2} B_{1}^{2}}{4 {{(1 + γ)}^{2} (2 μ - δ - 1)}^{2} ψ_{2}^{2}}, B_{1} > 1, (2.4)

Proof:

Let

T \in f_{q, \sum}^{μ} (ζ, n, ρ, σ, ϑ, γ, δ, φ),

there e

x

ist two anal

y

tic functions

u, v

and

u, v : U ⟶ U

with

u (0) = v (0) = 0, |u (ε)| < 1 a n d |v (ς)| \leq 1

, for all

ε,

ς \in U

satisf

y

ing the follo

w

ing c

o

nditions.

[\frac{{ε [{(Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε))}^{'} + γ {ε (Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε))}^{''}]}^{μ}}{γ ε ({(Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε))}^{'} + δ {(Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε))}^{''}) + (1 - γ) ((1 - δ) Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε) + δ {ε (Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε))}^{'}}] - {1 ≺}_{q} h (ε) (φ (u (ε) - 1)), ε \in U (2.5)

and

[\frac{ς {[{(Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς))}^{'} + γ ς {(Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς))}^{''}]}^{μ}}{γ ς ({(Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς))}^{'} + δ {(Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς))}^{''}) + (1 - γ) ((1 - δ) Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς) + δ ς {(Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς))}^{'}}] - {1 ≺}_{q} h (ς) (φ (v (ς) - 1)), ς \in U, (2.6)

where

ξ

is the inverse function of

T

and

(ε, ς \in U) .

Determine the definition of the functions

p (ε)

and

q (ς)

p (ε) = \frac{1 + u (ε)}{1 - u (ε)} = 1 + c_{1} ε^{2} + c_{2} ε^{2} + \dots (2.7)

and

q (ς) = \frac{1 + v (ς)}{1 - v (ς)} = 1 + d_{1} ς^{2} + d_{2} ς^{2} + \dots . (2.8)

Equivalently,

u (ε) : = \frac{p (ε) - 1}{p (ε) + 1} = \frac{1}{2} \{c_{1} ε + (c_{2} - \frac{c_{1}^{2}}{2}) ε^{2} + \dots\}, (2.9)

and

v (ς) : = \frac{q (ς) - 1}{q (ς) + 1} = \frac{1}{2} \{b_{1} ς + (b_{2} - \frac{b_{1}^{2}}{2}) ς^{2} + \dots\} . (2.10)

Applying (3.9) ,(3.10) in (3.5) and (3.6),r

e

spectively, we h

a

[\frac{{ε [{(Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε))}^{'} + γ {ε (Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε))}^{''}]}^{μ}}{γ ε ({(Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε))}^{'} + δ {(Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε))}^{''}) + (1 - γ) ((1 - δ) Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε) + δ {ε (Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε))}^{'}}] - 1 = h (ε) (φ (\frac{p (ε) - 1}{p (ε) + 1}) - 1), (2.11)

and

[\frac{ς {[{(Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς))}^{'} + γ ς {(Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς))}^{''}]}^{μ}}{γ ς ({(Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς))}^{'} + δ {(Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς))}^{''}) + (1 - γ) ((1 - δ) Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς) + δ ς {(Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς))}^{'}}] - 1 = h (ς) (φ (\frac{q (ς) - 1}{q (ς) + 1}) - 1) . (2.12)

Utilize

(2.8)

and

(2.9)

in the right hand

s

RH of the relati

o

ns (3.1( and (3.13),

w

e obtain

h (ε) (φ (\frac{p (ε) - 1}{p (ε) + 1}) - 1) = \frac{1}{2} A_{0} B_{1} c_{1} ε + \{\frac{1}{2} A_{1} B_{1} c_{1} + \frac{1}{2} A_{0} B_{1} (c_{2} - \frac{c_{1}^{2}}{2}) + \frac{A_{0} B_{2}}{4} c_{1}^{2}\} ε^{2} + \dots . (2.13)

and

h (ς) (φ (\frac{q (ς) - 1}{q (ς) + 1}) - 1) = \frac{1}{2} A_{0} B_{1} d_{1} ς + \{\frac{1}{2} A_{1} B_{1} d_{1} + \frac{1}{2} A_{0} B_{1} (d_{2} - \frac{d_{1}^{2}}{2}) + \frac{A_{0} B_{2}}{4} d_{1}^{2}\} ς^{2} + \dots . (2.14)

y

equalizing

(2.11)

(2.12)

,(3.13) and (3.14),r

e

spectively ,we g

e

(1 + γ) {(2 μ - δ - 1) h}_{2} ψ_{2} = \frac{1}{2} A_{0} B_{1} c_{1}, (2.15)

[(1 + 2 γ) {(3 μ - 2 δ - 1) h}_{3} ψ_{3} + {(1 + γ)}^{2} [2 μ (μ - 1) - (1 + δ) (2 μ - δ - 1)] h_{2}^{2} ψ_{2}^{2}]

= \frac{1}{2} A_{1} B_{1} c_{1} + \frac{1}{2} A_{0} B_{1} (c_{2} - \frac{c_{1}^{2}}{2}) + \frac{A_{0} B_{2}}{4} c_{1}^{2} . (2.16)

and

- (1 + γ) {(2 μ - δ - 1) h}_{2} ψ_{2} = \frac{1}{2} A_{0} B_{1} b_{1} (2.17)

[\begin{matrix} [{(1 + γ)}^{2} [2 μ (μ - 1) - (2 μ - δ - 1) (1 + δ)] + 2 (1 + 2 γ) (3 μ - 2 δ - 1)] h_{2}^{2} ψ_{2}^{2} \\ - (1 + 2 γ) {(3 μ - 2 δ - 1) h}_{3} ψ_{3} \end{matrix}]

= \frac{1}{2} A_{1} B_{1} d_{1} + \frac{1}{2} A_{0} B_{1} + (d_{2} - \frac{d_{1}^{2}}{2}) + \frac{A_{0} B_{2}}{4} d_{1}^{2} . (2.18)

From

(2.15)

and

(2.17)

, we have

h_{2} = \frac{A_{0} B_{1} c_{1}}{2 (1 + γ) (2 μ - δ - 1) ψ_{2}} = - \frac{A_{0} B_{1} d_{1}}{2 (1 + γ) (2 μ - δ - 1) ψ_{2}} (2.19)

It follows that

c_{1} = - d_{1}, (2.20)

and

8 {{(1 + γ)}^{2} (2 μ - δ - 1)}^{2} h_{2}^{2} ψ_{2}^{2} = A_{0}^{2} B_{1}^{2} {(d}_{1}^{2} + c_{1}^{2}) . (2.21)

Now , by summing (3.19) and (3.31), in light

o

(2.19) a n d (2.20)

,we obtain

8 [{(1 + γ)}^{2} [2 μ (μ - 1) - (2 μ - δ - 1) (1 + δ)] A_{0} B_{1}^{2} ψ_{2}^{2} + (1 + 2 γ) (3 μ - 2 δ - 1) ψ_{3} A_{0} B_{1}^{2}] h_{2}^{2}

= 2 {A_{0}^{2} B}_{1}^{3} (c_{2} + d_{2}) + (8 {{(1 + γ)}^{2} (2 μ - δ - 1)}^{2} {(B_{2} - B_{1}) h}_{2}^{2} ψ_{2}^{2}), (2.22)

which implies

h_{2}^{2} = \frac{2 {A_{0}^{2} B}_{1}^{3} (c_{2} + d_{2})}{8 \{(1 + 2 γ) (3 μ - 2 δ - 1) {A_{0} B_{1}^{2} ψ}_{3} - {(1 + γ)}^{2} [{(2 μ - δ - 1)}^{2} (B_{2} - B_{1}) - [2 μ (μ - 1) - (2 μ - δ - 1) (1 + δ)] A_{0} B_{1}^{2}]\}} . (2.23)

Applying lemma

(1.1) |c_{i}| \leq 2, |d_{i}| \leq 2,

in (3.33) ,we get the desired result (3.3).

Next, f

o

r the bound on

|a_{3}|,

by subt

r

acting (3.18) from ( 3.16), we obtain

{4 \{(1 + 2 γ) (3 μ - 2 δ - 1) ψ_{3} h_{3} - (1 + 2 γ) (3 μ - 2 δ - 1) ψ_{3} h_{2}^{2}\} = 2 A}_{1} B_{1} c_{1} + A_{0} B_{1} (c_{2} - d_{2}) (2.24)

By substituting (3.18) from (3.16), further computation using (3.30) and (3.31), we obtain

h_{3} = \frac{2 A_{1} B_{1} c_{1}}{4 (1 + 2 γ) (3 μ - 2 δ - 1) ψ_{3}} + \frac{A_{0} B_{1} (c_{2} - d_{2})}{4 (1 + 2 γ) (3 μ - 2 δ - 1) ψ_{3}} + \frac{A_{0}^{2} B_{1}^{2} {(c}_{1}^{2} {+ d}_{1}^{2})}{{{8 (1 + γ)}^{2} (2 μ - δ - 1)}^{2} ψ_{2}^{2}} (3.35)

Appl

y

ing Lemma

(1.1) |c_{i}| \leq 2, |d_{i}| \leq 2,

in (3.34), we get (3.4). This completes the proof of the Theor

e

(2.1)

y

putting

δ = 0

in Theor

e

m (3.1), we

o

btain the foll

o

wing Corollary:

Corollary (3.1):

If the functi

o

T (ε)

giv

e

n by

(1.1)

belong to th

e

class

f_{q, \sum}^{μ} (ζ, n, ρ, σ, ϑ, γ, 0, φ),

then

{| h}_{2} | \leq \frac{|A_{0}| B_{1} \sqrt{B_{1}}}{\sqrt{(1 + 2 γ) (3 μ - 1) {A_{0} B_{1}^{2} ψ}_{3} - {(1 + γ)}^{2} [{(2 μ - 1)}^{2} (B_{2} - B_{1}) - [2 μ (μ - 1) - (2 μ - 1)] {ψ_{2}^{2} A}_{0} B_{1}^{2}]}}

and

{| h}_{3} | \leq \frac{B_{1} (|A_{0}| + |A_{1}|)}{(1 + 2 γ) (3 μ - 1) ψ_{3}} + \frac{A_{0}^{2} B_{1}^{2}}{4 {{(1 + γ)}^{2} (2 μ - 1)}^{2} ψ_{2}^{2}} .

By putting

δ = 1

in Theorem (3.1), we obtain the following Corollary:

Corollary (3.3):

Let

T (ε)

giv

e

n by

(1.1)

belongs to th

e

class

f_{q, \sum}^{μ} (ζ, n, ρ, σ, ϑ, γ, 1, φ)

. Then

{| h}_{2} | \leq \frac{|A_{0}| B_{1} \sqrt{B_{1}}}{\sqrt{3 (1 + 2 γ) (μ - 1) {A_{0} B_{1}^{2} ψ}_{3} - {(1 + γ)}^{2} [{(2 μ - 2)}^{2} (B_{2} - B_{1}) - 2 [μ (μ - 1) - (2 μ - 2)] {ψ_{2}^{2} A}_{0} B_{1}^{2}]}}

and

{| h}_{3} | \leq \frac{B_{1} (|A_{0}| + |A_{1}|)}{3 (1 + 2 γ) (μ - 1) ψ_{3}} + \frac{A_{0}^{2} B_{1}^{2}}{8 {{(1 + γ)}^{2} (μ - 1)}^{2} ψ_{2}^{2}} .

By putting

γ = 1

in Theor

e

m (3.1), we have the follo

w

ing Corollar

y

Corollary (3.3)

Let

T (ε)

giv

e

n by

(1.1)

belongs to th

e

class

f_{q, \sum}^{μ} (ζ, n, ρ, σ, ϑ, 1, δ, φ)

. Then

{| h}_{2} | \leq \frac{|A_{0}| B_{1} \sqrt{B_{1}}}{\sqrt{3 (3 μ - 2 δ - 1) {A_{0} B_{1}^{2} ψ}_{3} - 4 [{(2 μ - δ - 1)}^{2} (B_{2} - B_{1}) - [2 μ (μ - 1) - (2 μ - δ - 1) (1 + δ)] {ψ_{2}^{2} A}_{0} B_{1}^{2}]}}

and

{| h}_{3} | \leq \frac{B_{1} (|A_{0}| + |A_{1}|)}{3 (3 μ - 2 δ - 1) ψ_{3}} + \frac{A_{0}^{2} B_{1}^{2}}{16 {(2 μ - δ - 1)}^{2} ψ_{2}^{2}}, B_{1} > 1 .

By putting

γ = 0

in Theorem (3.1), we have the following Corollary:

Corollary (3.4):

Let

T (ε)

given b

y (1.1)

belongs to th

e

class

f_{q, \sum}^{μ} (ζ, n, ρ, σ, ϑ, 0, δ, φ)

Then {| h}_{2} | \leq \frac{|A_{0}| B_{1} \sqrt{B_{1}}}{\sqrt{(3 μ - 2 δ - 1) {A_{0} B_{1}^{2} ψ}_{3} - [{(2 μ - δ - 1)}^{2} (B_{2} - B_{1}) - [2 μ (μ - 1) - (2 μ - δ - 1) (1 + δ)] {ψ_{2}^{2} A}_{0} B_{1}^{2}]}}

and

{| h}_{3} | \leq \frac{B_{1} (|A_{0}| + |A_{1}|)}{(3 μ - 2 δ - 1) ψ_{3}} + \frac{A_{0}^{2} B_{1}^{2}}{4 {(2 μ - δ - 1)}^{2} ψ_{2}^{2}}, B_{1} > 1 .

3. Coefficients Estimates for the Subclass $ℵ_{\sum}^{q, δ} (λ, ζ, n, ρ, σ, ϑ, φ) .$

Definition (3.1):

A function

T \in \sum_{U}

defined b

y (1.1)

is said to be in th

e

class

ℵ_{\sum}^{q, δ} (λ, ζ, n, ρ, σ, ϑ, φ)

if the follo

w

ing quasi-subordination c

o

nditions are satisfi

e

1 + \frac{1}{γ} [(1 - δ) \frac{ε {(Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε))}^{'}}{(1 - λ) ε + λ Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε)} + δ (\frac{ε {(Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε))}^{″} {+ (Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε))}^{'}}{λ {ε (Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε))}^{″} {+ (Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε))}^{'}}) - 1] ≺_{q} (φ (ε) - 1) (3.1)

and

1 + {\frac{1}{γ} [(1 - δ) \frac{ς {(Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς))}^{'}}{(1 - λ) ς + λ Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς)} + δ (\frac{ς {(Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς))}^{″} {+ (Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς))}^{'}}{λ ς {(Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς))}^{″} {+ (Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς))}^{'}}) - 1] ≺}_{q} (φ (ς) - 1), (3.2)

where (

0 \leq λ < 1, 0 \leq δ \leq 1, γ \in C ⧵ \{0\}

ε, \in U) .

o

r special values to param

e

ters

λ a n d δ

, we get n

e

w and well-kno

w

n classes.

Remark (3.1):

o

λ = 0,

a function

T \in \sum_{U}

define by

(1.1)

is said to be in the class

ℵ_{\sum}^{q, δ} (λ, ζ, n, ρ, σ, ϑ, φ)

if the follo

w

ing quasi-subordination conditions are satisfied:

1 + \frac{1}{γ} [(1 - δ) \frac{ε {(Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε))}^{'}}{ε} + δ (\frac{ε {{(Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε))}^{″} + (Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε))}^{'}}{{(Q_{ρ, σ, ϑ}^{ζ, q, n} T (ε))}^{'}}) - 1] ≺_{q} (φ (z) - 1)

and

{1 + \frac{1}{γ} [(1 - δ) \frac{ς {(Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς))}^{'}}{ς} + δ (\frac{ς {(Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς))}^{″} {+ (Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς))}^{'}}{{(Q_{ρ, σ, ϑ}^{ζ, q, n} ξ (ς))}^{'}}) - 1] ≺}_{q} (φ (w) - 1)

Theorem (3.1.):

If the functi

o

T b e l o n g

t o t h

c l a s s ℵ_{\sum}^{q, δ} (λ, ζ, n, ρ, σ, ϑ, φ)

, then we have

{| h}_{2} | \leq \frac{γ |A_{0}| B_{1} \sqrt{B_{1}}}{\sqrt{{2 (1 - λ) (1 + 2 δ) A_{0} B_{1}^{2} ψ}_{3} - {(1 - λ)}^{2} [(1 + 3 δ) A_{0} B_{1}^{2} - {(1 + δ)}^{2} (B_{2} - B_{1})] ψ_{2}^{2}}} (3.3)

and

{| h}_{3} | \leq \frac{{γ B}_{1} (|A_{0}| + |A_{1}|)}{{(1 - λ) (1 + 2 δ) ψ}_{3}} + \frac{A_{0}^{2} B_{1}^{2} γ^{2}}{{{(1 + δ)}^{2} {(1 - λ)}^{2} ψ}_{2}^{2}} {, B}_{1} > 1, (3.4)

where

0 \leq δ \leq 1, 0 \leq λ \leq 1, γ \in U - \{0\}

Proof:

Proceeding as in th

e

proof of Theorem

(2.1),

we can g

e

t the relations as follo

w

\frac{1}{γ} (1 + δ) (1 - λ) h_{2} ψ_{2} = \frac{1}{2} A_{0} B_{1} c_{1}, (3.5)

\frac{1}{γ} [2 (1 - λ) (1 + 2 δ) h_{3} ψ_{3} - (1 - λ) (1 + λ) ((1 + 3 δ)) h_{2}^{2} ψ_{2}^{2}]

= \frac{1}{2} A_{1} B_{1} c_{1} + \frac{1}{2} A_{0} B_{1} (c_{2} - \frac{c_{1}^{2}}{2}) + \frac{A_{0} B_{2}}{4} c_{1}^{2}\} (3.6)

and

- \frac{1}{γ} {(1 + δ) (1 - λ) h}_{2} ψ_{2} = \frac{1}{2} A_{0} B_{1} b_{1}, (3.7)

\frac{1}{γ} [[4 (1 - λ) (1 + 2 δ) ψ_{3} - (1 - λ) (1 + λ) (1 + 3 δ)] h_{2}^{2} ψ_{2}^{2} - 2 (1 - λ) {(1 + 2 δ) ψ_{3} h}_{3}]

= \frac{1}{2} A_{1} B_{1} b_{1} + \frac{1}{2} A_{0} B_{1} (b_{2} - \frac{b_{1}^{2}}{2}) + \frac{A_{0} B_{2}}{4} b_{1}^{2}} . (3.8)

From

(3.5)

and

(3.7)

, we obtain

c_{1} = - d_{1} (3.9)

and

h_{2} = \frac{γ A_{0} B_{1} c_{1}}{2 (1 + δ) (1 - λ) ψ_{2}} = - \frac{γ A_{0} B_{1} b_{1}}{2 (1 + δ) (1 - λ) ψ_{2}} (3.10)

and

8 {(1 + δ)}^{2} {(1 - λ)}^{2} h_{2}^{2} ψ_{2}^{2} = A_{0}^{2} B_{1}^{2} {γ^{2} (d}_{1}^{2} + c_{1}^{2}) . (3.11)

Now, by summing (3.6) and (3.8 )and using

(3.11)

we obtain

\frac{8}{γ} {(2 (1 - λ) (1 + 2 δ {) ψ_{3} - (1 - λ) (1 + λ) (1 + 3 δ) ψ_{2}^{2}} h}_{2}^{2}

= 2 A_{0} B_{1} (c_{2} {+ d}_{2}) + {A_{0} (B_{2} {- B}_{1}) (c}_{1}^{2} {+ d}_{1}^{2}), (3.12)

which implies

h_{2}^{2} = \frac{2 {A_{0}^{2} B}_{1}^{3} (c_{2} + d_{2})}{8 \{{2 (1 - λ) (1 + 2 δ) A_{0} B_{1}^{2} ψ}_{3} - {(1 - λ)}^{2} [[(1 + 3 δ) A_{0} B_{1}^{2} - {(1 + δ)}^{2} (B_{2} - B_{1})] ψ_{2}^{2}]\}} . (3.13)

Applying lemma

(1.1)

in (3.13) ,we get the desir

e

d result (3.3).

e

xt ,for the b

o

und on

|h_{3}|,

by subtracting (3.6) from ( 3.8), we obtain

\frac{8}{γ} {\{{(1 - λ) (1 + 2 δ) ψ}_{3} h_{3} - {{(1 - λ) (1 + 2 δ) ψ}_{3} h}_{2}^{2}\} = 2 A}_{1} B_{1} c_{1} + A_{0} B_{1} (c_{2} - d_{2})

By substituting (3.18) from (3.16), further computation using (3.9) and (3.10), we obtain

h_{3} = \frac{2 γ A_{1} B_{1} c_{1}}{{4 (1 - λ) (1 + 2 δ) ψ}_{3}} + \frac{γ A_{0} B_{1} (c_{2} - d_{2})}{4 (1 - λ) (1 + 2 δ)} + \frac{A_{0}^{2} B_{1}^{2} γ^{2} {(c}_{1}^{2} {+ d}_{1}^{2})}{8 {{(1 + δ)}^{2} {(1 - λ)}^{2} ψ}_{2}^{2}} (3.14)

From (3.14) and (3.13), we get the desir

e

d result (3.4). The proof is c

o

mplete.

Corollary (3.1):

T (ε) \in ℵ_{\sum}^{q, δ} (1, ζ, n, ρ, σ, ϑ, φ)

defined in (1.1), then we have

{| h}_{2} | \leq \frac{γ |A_{0}| B_{1} \sqrt{B_{1}}}{\sqrt{{2 (1 + 2 δ) A_{0} B_{1}^{2} ψ}_{3} - [(1 + 3 δ) A_{0} B_{1}^{2} - {(1 + δ)}^{2} (B_{2} - B_{1})] ψ_{2}^{2}}}

and

{| h}_{3} | \leq \frac{{γ B}_{1} (|A_{0}| + |A_{1}|)}{{(1 + 2 δ) ψ}_{3}} + \frac{A_{0}^{2} B_{1}^{2} γ^{2}}{{{(1 + δ)}^{2} ψ}_{2}^{2}} {, B}_{1} > 1 .

Corollary (3.3):

T (ε) \in ℵ_{\sum}^{q, 1} (λ, ζ, n, ρ, σ, ϑ, φ)

defined in (1.1), then we have

{| h}_{2} | \leq \frac{γ |A_{0}| B_{1} \sqrt{B_{1}}}{\sqrt{{6 (1 - λ) A_{0} B_{1}^{2} ψ}_{3} - {(1 - λ)}^{2} [4 A_{0} B_{1}^{2} - 4 (B_{2} - B_{1})] ψ_{2}^{2}}}

and

{| h}_{3} | \leq \frac{{γ B}_{1} (|A_{0}| + |A_{1}|)}{3 {(1 - λ) ψ}_{3}} + \frac{A_{0}^{2} B_{1}^{2} γ^{2}}{{4 {(1 - λ)}^{2} ψ}_{2}^{2}} {, B}_{1} > 1 .

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Certain Results on Subclasses of Analytic and Bi-Univalent Functions Associated with Coefficients Estimates and Quasi-Subordination

Abstract

1. Introduction

2. Coefficients Estimates for the Class f q , ∑ μ ζ , n , ρ , σ , ϑ , γ , δ , φ .

3. Coefficients Estimates for the Subclass ℵ ∑ q , δ λ , ζ , n , ρ , σ , ϑ , φ .

References

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2. Coefficients Estimates for the Class $f_{q, \sum}^{μ} (ζ, n, ρ, σ, ϑ, γ, δ, φ) .$

3. Coefficients Estimates for the Subclass $ℵ_{\sum}^{q, δ} (λ, ζ, n, ρ, σ, ϑ, φ) .$