# On Neighbourhood and Partial Sums of a Certain Subclasses of Analytic Functions with Respect to β-Symmetric Points

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**Corresponding Author:**Sowileh S, Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt, Tel: +20502383781, Email: [email protected]

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Received Date: Apr 10, 2018 /
Accepted Date: Apr 11, 2018 /
Published Date: Apr 15, 2018 *

### Abstract

In the present investigation, we introduce a new class k-Ã Â¨C(β)(λ;γ)of analytic functions in the open unit disc U with negative coefficients. The object of the present paper is to determine coefficients estimates, neighbourhoods and partial sums for functions f(z) belonging to this class.

**Keywords:**
Analytic function; Uniformly starlike function; Quasiconvex functions; Coefficient estimate; Neighbourhood; Partial sums; β-Symmetric points; 2010 Mathematics subject classification; 30C45

#### Introduction

Let A denote the family of functions f of the form [1]:

(1.1)

that are analytic in the open unit disc U={z:|z|<1}.

For fεA given by (1.1) and g(z) given [2] by:

(1.2)

their convolution (or Hadamard product) denoted by(f*g) is defined [3] as:

(1.3)

Let be the usual subclasses [4-7] of A consisting of starlike, convex, close-to-convex and quasi-convex in , respectively. Thus, by definition, we have [8-15],

Let be the subclass of S consisting of functions of the [16-28] form eqn. (1.1) satisfying:

(1.4)

These functions are called starlike with respect to symmetric points and were introduced by Sakaguchi [19] (see also Robertson [17]; Stankiewicez [25]; Wu [28] and Owa [14]. In El-Ashwah and Thomas [9], introduced and studied two other classes namely the class consisting of functions starlike with respect to conjugate points and consisting of functions [18] starlike with respect to symmetric conjugate points. Recently several subclasses of analytic functions with respect to k-symmetric points were introduced and studied by various authors [5,16].

A function fεA is said to be in k- US(γ), the class of k-uniformly starlike functions of order γ; 0 ≤ γ<1; if it satisfies the condition

(1.5)

and a function fεA is said to be k- UC(γ), the class [19] of k-uniformly convex functions of order γ; 0 ≤ γ<1; if it satisfies the condition:

(1.6)

Uniformly starlike and uniformly convex functions were [20] first introduced by Goodman [11] and then studied by various authors. It follows from (1.5) and (1.6) that:

Let denote the class of functions in A [21] satisfying the following inequality:

where 0 ≤ γ<1; β ≥ 2 is a fixed positive integer and f_{(β)}(z) is [22] defined by the following equality:

(1.7)

And a function f(z)εA is in the class if and only if zf’(z)ε The classes of starlike functions with respect to β-symmetric points of order [23] and of convex functions with respect to β- symmetric points of order were considered recently by Singh [4] and Wang et al. [26] respectively. We now introduce and investigate the following subclasses of A with respect [24-28] to β-symmetric points and obtain some interesting results.

**Definition 1.1**

We say that fεA belongs to the class k-UC(β)(λ;γ) if for all zεU:

for 0 ≤ γ<1, 0 ≤ λ ≤ 1 and k ≥ 0.

**Definition 1.2**

We say that fεA belongs to the class k-UQC^{(β)}(λ;γ) if for all zεU.

for 0 ≤ γ<1; 0 ≤ γ ≤ 1 and k ≥ 0:

Denote by T the subclass of A consisting of functions of the form:

(1.9)

For convenience, we write k-UC^{β}(λ;γ)∩T simple as k-ŨC^{β}(λ;γ); and k- UQC^{β}(λ;γ)∩T simple as k- ŨQC^{β}(λ;γ).

**Remark 1.1**

(i) For k=0; the classes k-ŨC^{(β)}(λ;γ) and k-ŨQC^{β}(λ;γ) reduce to the classes C^{(β)}(λ;γ) and QC^{(β)}(λ;γ) respectively introduced by Wang et al. [27].

The aim of the present paper to study the coefficient bounds, partial sums and certain neighbourhood results of the classes k-ŨC^{(β)}(λ,γ) and k-ŨQC^{(β)}(λ, γ).

(ii) Coefficient bounds of the function classes k-ŨC^{(β)}(λ,γ) if and only if:

(2.1)

Where,

(2.2)

The result is sharp for the function f(z) given by:

**Proof:**

Assume that the inequality (2.1) holds true. Then we have:

Conversely, if f(z)ε k-ŨC^{(β)}(λ;γ) and z is real, then

Letting z→1 - along the real axis, we obtain:

This completes the proof of Theorem 2.1.

**Corollary 2.1:**

If f(z)ε k-ŨC^{(β)}(λ;γ); then,

where b_{n} is defined by (2.2) and 0 ≤ γ<1, 0 ≤ λ ≤ 1; β ≥ 2; and k ≥ 0.

**Theorem 2.2:**

A function f(z)εT is in the class k-ŨQC^{(β)}(λ;γ) if and only if

(2.3)

where b_{n} is defined by (2.2), and 0 ≤ γ<1, 0 ≤ λ<1; β ≥ 2, and k ≥ 0: The result is sharp for the function f(z) given by:

**Proof:**

By Denition 1.2, the proof is similar to that of Theorem 2.1.

**Neighbourhood of the Function Class k-ŨQC ^{(β)}(λ;γ)**

Following the earlier investigations (based upon the familiar concept of neighborhoods of analytic functions) by Goodman [10], Ruscheweyh [18], Altintas et al. [1,2] and others including Srivastava et al. [22,23], Orhan [13], Deniz et al. [7], Catas [3,6], we dene the neighborhood of a function f εT by:

**Definition 3.1**

Let 0 1; 0 ≤ λ ≤ 1, 0 ≤ γ<1, k ≥ 0, α ≥ 0 and b_{n} is dened by (2.2). We define the α-neighbourhood of a function f εT and denote by N_{α}(f) consisting of all functions satisfying:

**Theorem 3.1**

Let fε k-ŨC^{(β)}(λ;γ) and for all real θ we have γ(e^{iθ}-1)-2e^{iθ}≠0: For and complex number ε with |ε|<α(α ≥ 0); if it satisfies the following condition:

(3.1)

then N_{α}(f)⊂ k-ŨC^{(β)}(λ;γ)

**Proof:**

It is easy to see that

Rew>k|w-1|+γ⇔ Re{w(1+ke^{iθ})-ke^{iθ}}>γ; (-π<θ<π; 0 ≤ γ<1; k ≥ 0);

By definition 1.1, fεk-ŨC^{(β)}(γ,λ) if and only if

the above inequality may be rewritten as follows:

or equivalently,

Now, by setting

(3.7)

And

(3.5)

the inequality (3.3) becomes equivalent to

Then

(-π<θ<π); for any complex number s with |s|=1, we have

which is equivalent to

(3.6)

Where

(3.7)

It follows from (3.7) that

If f(z)εk-ŨC^{(β)}(λ,γ) given by (1.9), satisfies the inclusion property (3.1), then (3.6) yields

(3.8)

Now, if we suppose that

(3.9)

We easily see that

Then,

Thus, for any complex number s such that |s|=1, we have

which implies that g (z)εk-ŨC^{(β)}(λ,γ)so Nα(f)⊂k- ŨC^{(β)}(λ,γ).

#### Partial Sums of the Function Class K-Ũc(Β)(λ,γ)

In this section, applying methods used by Silverman [20] and Silvia [21], We investigate the ratio of the real part of f(z) which dened by (1.9) to its sequence of partial sums dened by f1(z)=z and

**Theorem 4.1:**

If f(z) ε k- ŨC^{(β)}(λ,γ) then

(4.1)

And

(4.2)

Where

The bounds in (4.1) and (4.2) are the best possible for each m 2N. Proof. We employ the same technique used by Silverman [20]: The function f(z)εk-ŨC(β)(λ,γ), If and only if . It is not difficult to verify that [29].

(4.4)

We may write

(4.5)

Then, from (4.5) we can obtain,

And

Now |w(z)| ≤ 1 if and only if

which is true by (4.4). This readily yields the assertion (4.1) of Theorem 4.1. To see that:

(4.6)

gives sharp results, we observe that for z=γe^{iπ/m}

Where

Now |w(z)| ≤ 1 if and only if,

which is true by (4.4). This immediately leads to the assertion (4.2) of Theorem 4.1. The estimate in (4.2) is sharp with the extremal function f(z) given by (4.6). This completes the proof of Theorem 4.1.

**Theorem 4.2:**

If f(z)εk-ŨC^{(β)}(λ,γ) then,

(4.7)

(4.8)

The estimates in (4.7) and (4.8) are sharp with the extremal function given by (4.6).

**Proof: **The proof is similar to that of Theorem 4.1.

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Citation: Darwish H, Lashin Abd AM, Sowileh S (2018) On Neighbourhood and Partial Sums of a Certain Subclasses of Analytic Functions with Respect to β-Symmetric Points. J Appl Computat Math 7: 397. DOI: 10.4172/2168-9679.1000397

Copyright: © 2018 Darwish H, et al. 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.