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

ISSN: 2168-9679

Journal of Applied & Computational Mathematics

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

Darwish H, Lashin Abd AM and Sowileh S*
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt
*Corresponding Author: Sowileh S, Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt, Tel: +20502383781, Email: [email protected]

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-ŨC(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]:

image (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:

image(1.2)

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

image (1.3)

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

image

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

image(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 imageconsisting of functions starlike with respect to conjugate points and image 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

image(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:

image(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:

image

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

image

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

image(1.7)

And a function f(z)εA is in the class image if and only if zf’(z)ε image The classes image of starlike functions with respect to β-symmetric points of order [23] and imageof 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:

image

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.

image

for 0 ≤ γ<1; 0 ≤ γ ≤ 1 and k ≥ 0:

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

image(1.9)

For convenience, we write k-UCβ(λ;γ)∩T simple as k-ŨCβ(λ;γ); and k- UQCβ(λ;γ)∩T simple as k- ŨQCβ(λ;γ).

Remark 1.1

(i) For k=0; the classes k-ŨC(β)(λ;γ) and k-ŨQCβ(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:

image (2.1)

Where,

image(2.2)

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

image

Proof:

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

image

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

image

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

image

This completes the proof of Theorem 2.1.

Corollary 2.1:

If f(z)ε k-ŨC(β)(λ;γ); then,

image

where bn 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

image(2.3)

where bn 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:

image

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 bn is dened by (2.2). We define the α-neighbourhood of a function f εT and denote by Nα(f) consisting of all functions image satisfying:

image

Theorem 3.1

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

image(3.1)

then Nα(f)⊂ k-ŨC(β)(λ;γ)

Proof:

It is easy to see that

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

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

image

the above inequality may be rewritten as follows:

image

or equivalently,

image

Now, by setting

image (3.7)

And

image (3.5)

the inequality (3.3) becomes equivalent to

image

Then

image

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

image

which is equivalent to

image(3.6)

Where

image(3.7)

image

It follows from (3.7) that

image

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

image(3.8)

Now, if we suppose that

image(3.9)

We easily see that

image

Then,

image

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

image

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

image

Theorem 4.1:

If f(z) ε k- ŨC(β)(λ,γ) then

image(4.1)

And

image(4.2)

Where

image

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 image . It is not difficult to verify thatimage [29].

image(4.4)

We may write

image(4.5)

image

Then, from (4.5) we can obtain,

image

And

image

Now |w(z)| ≤ 1 if and only if

image

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

image (4.6)

gives sharp results, we observe that for z=γeiπ/m

image

image

Where

image

Now |w(z)| ≤ 1 if and only if,

image

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,

image (4.7)

image (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.

References

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.

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