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volume 6, issue 5, article 132, 2005.

Received 15 March, 2005;

accepted 04 October, 2005.

Communicated by:H.M. Srivastava

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Journal of Inequalities in Pure and Applied Mathematics

A UNIFIED TREATMENT OF CERTAIN SUBCLASSES OF PRESTARLIKE FUNCTIONS

MASLINA DARUS

School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia Bangi 43600 Selangor D.E., Malaysia.

EMail:maslina@pkrisc.cc.ukm.my

URL:http://www.webspawner.com/users/maslinadarus/

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2000Victoria University ISSN (electronic): 1443-5756 083-05

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A Unified Treatment of Certain Subclasses of Prestarlike

Functions Maslina Darus

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J. Ineq. Pure and Appl. Math. 6(5) Art. 132, 2005

http://jipam.vu.edu.au

Abstract

In this paper we introduce and study some properties of a unified class U[µ, α, β, γ, λ, η, A, B]of prestarlike functions with negative coefficients in a unit diskU. These properties include growth and distortion, radii of convexity, radii of starlikeness and radii of close-to-convexity.

2000 Mathematics Subject Classification:30C45.

Key words: Analytic functions, Prestarlike functions, radii of starlikeness, convexity and close-to-convexity, Cauchy-Schwarz inequality.

This paper is based on the talk given by the author within the “International Conference of Mathematical Inequalities and their Applications, I”, December 06- 08, 2004, Victoria University, Melbourne, Australia [http://rgmia.vu.edu.au/

conference]

The work presented here is supported by IRPA 09-02-02-10029 EAR.

1. Introduction

LetAdenote the class of normalized analytic functions of the form:

(1.1) f(z) =z+

∞

X

n=2

a_nzⁿ,

in the unit disk U = {z : |z| < 1}. Further let S denote the subclass of A consisting of analytic and univalent functionsf in the unit diskU. A function f inSis said to be starlike of orderαif and only if

(1.2) Re

zf⁰(z) f(z)

> α

for someα(0≤α <1).We denote byS^∗(α)the class of all starlike functions of orderα. It is well-known thatS^∗(α)⊆S^∗(0) ≡S^∗.

Let the function

(1.3) S_α(z) = z

(1−z)^2(1−α), (z ∈U; 0≤α <1)

which is the extremal function for the classS^∗(α).We also note thatS_α(z)can be written in the form:

(1.4) S_α(z) = z+

∞

X

n=2

|c_n(α)|zⁿ,

where

(1.5) c_n(α) = Πⁿ_j=2(j−2α)

(n−1)! (n ∈N{1}, N:={1,2,3, . . .}).

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We note thatc_n(α)is decreasing inαand satisfies

(1.6) lim

n→∞c_n(α) =











∞ ifα < ¹₂ , 1 ifα= ¹₂ , 0 ifα > ¹₂.

Also a functionf inS is said to be convex of orderαif and only if

(1.7) Re

1 + zf⁰⁰(z) f⁰(z)

> α

for someα(0≤ α <1).We denote byK(α)the class of all convex functions of orderα. It is a fact thatf ∈K(α)if and only ifzf⁰(z)∈S^∗(α).

The well-known Hadamard product (or convolution) of two functionsf(z) given by (1.1) andg(z)given byg(z) =z+P∞

n=2b_nzⁿis defined by

(1.8) (f∗g)(z) =z+

∞

X

n=2

a_nb_nzⁿ, (z ∈U).

LetR[µ, α, β, γ, λ, A, B]denote the class of prestarlike functions satisfying the following condition

(1.9)

zH_λ⁰(z) H_λ(z) −1 2γ(B−A)_zH0

λ(z) Hλ(z) −µ

−B_zH0 λ(z) Hλ(z) −1

< β,

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where H_λ(z) = (1 −λ)h(z) +λzh⁰(z), λ ≥ 0, h = f ∗ S_α, 0 < β ≤ 1, 0≤µ <1, and

B

2(B−A) < γ ≤







B

2(B−A)µ, µ6= 0,

1, µ= 0

for fixed−1≤A≤B ≤1and0< B ≤1.

We also note that a function f is a so-called α-prestarlike (0 ≤ α < 1) function if, and only if, h = f ∗ Sα ∈ S^∗(α) which was first introduced by Ruscheweyh [3], and was rigorously studied by Silverman and Silvia [4], Owa and Ahuja [5] and Uralegaddi and Sarangi [6]. Further, a functionf ∈ A is in the classC[µ, α, β, γ, λ, A, B]if and only if,zf⁰(z)∈ R[µ, α, β, γ, λ, A, B].

LetT denote the subclass ofAconsisting of functions of the form

(1.10) f(z) = z−

∞

X

n=2

a_nzⁿ, (a_n ≥0).

Let us write

R_T[µ, α, β, γ, λ, A, B] =R[µ, α, β, γ, λ, A, B]∩T and

CT[µ, α, β, γ, λ, A, B] =C[µ, α, β, γ, λ, A, B]∩T

where T is the class of functions of the form (1.10) that are analytic and univalent in U. The idea of unifying the study of classes RT[µ, α, β, γ, λ, A, B]

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andCT[µ, α, β, γ, λ, A, B]thus, forming a new classU[µ, α, β, γ, λ, η, A, B]is somewhat or rather motivated from the work of [1] and [2].

In this paper, we will study the unified presentation of prestarlike functions belonging toU[µ, α, β, γ, λ, η, A, B]which include growth and distortion theorem, radii of convexity, radii of starlikeness and radii of close-to-convexity.

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2. Coefficient Inequality

Our main tool in this paper is the following result, which can be easily proven, and the details are omitted.

Lemma 2.1. Let the function f be defined by (1.10). Then f ∈ RT[µ, α, β, γ, λ, A, B]if and only if

(2.1)

∞

X

n=2

Λ(n, λ)D[n, β, γ, A, B]|a_n|c_n(α)≤E[β, γ, µ, A, B]

where

Λ(n, λ) = (1 + (n−1)λ),

D[n, β, γ, A, B] =n−1 + 2βγ(n−µ)(B−A)−Bβ(n−1), E[β, γ, µ, A, B] = 2βγ(1−µ)(B−A).

The result is sharp.

Next, by observing that

(2.2) f ∈ CT[µ, α, β, γ, λ, A, B]⇔zf⁰(z)∈ RT[µ, α, β, γ, λ, A, B], we gain the following Lemma2.2.

Lemma 2.2. Let the function f be defined by (1.10). Then f ∈ C_T[µ, α, β, γ, λ, A, B]if and only if

(2.3)

∞

X

n=2

nΛ(n, λ)D[n, β, γ, A, B]|an|cn(α)≤E[β, γ, µ, A, B]

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where

Λ(n, λ) = (1 + (n−1)λ),

D[n, β, γ, A, B] =n−1 + 2βγ(n−µ)(B−A)−Bβ(n−1), E[β, γ, µ, A, B] = 2βγ(1−µ)(B−A)

andc_n(α)given by (1.5).

In view of Lemma 2.1 and Lemma 2.2, we unified the classes RT[µ, α, β, γ, λ, A, B]andCT[µ, α, β, γ, λ, A, B]and so a new classU[µ, α, β, γ, λ, η, A, B]

is formed. Thus we say that a function f defined by (1.10) belongs to U[µ, α, β, γ, λ, η, A, B]if and only if,

(2.4)

∞

X

n=2

(1−η+nη)Λ(n, λ)D[n, β, γ, A, B]|an|cn(α)≤E[β, γ, µ, A, B],

(0≤α <1; 0< β ≤1; η ≥0; λ≥0; −1≤A≤B ≤1and0< B ≤1), whereΛ(n, λ),D[n, β, γ, A, B],E[β, γ, µ, A, B]andcn(α)are given in (Lemma 2.1and Lemma2.2) and given by (1.5), respectively.

Clearly, we obtain

U[µ, α, β, γ, λ, η, A, B] = (1−η)RT[µ, α, β, γ, A, B] +ηCT[µ, α, β, γ, A, B], so that

U[µ, α, β, γ, λ,0, A, B] =RT[µ, α, β, γ, A, B], and

U[µ, α, β, γ, λ,1, A, B] =C_T[µ, α, β, γ, A, B].

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3. Growth and Distortion Theorem

A distortion property for functionf in the classU[µ, α, β, γ, λ, η, A, B]is given as follows:

Theorem 3.1. Let the function f defined by (1.10) be in the class U[µ, α, β, γ, λ, η, A, B], then

(3.1) r− E[β, γ, µ, A, B]

2(1 +η)Λ(2, λ)D[2, β, γ, A, B](1−α)r²

≤ |f(z)| ≤r+ E[β, γ, µ, A, B]

2(1 +η)Λ(2, λ)D[2, β, γ, A, B](1−α)r², (η≥0; 0≤α <1; 0< β ≤1; z ∈U)

and

(3.2) 1− E[β, γ, µ, A, B]

(1 +η)Λ(2, λ)D[2, β, γ, A, B](1−α)r

≤ |f⁰(z)| ≤1 + E[β, γ, µ, A, B]

(1 +η)Λ(2, λ)D[2, β, γ, A, B](1−α)r, (η ≥0; 0≤α <1; 0< β≤1; z ∈U).

The bounds in (3.1) and (3.2) are attained for the functionf given by f(z) = z− E[β, γ, µ, A, B]

2(1 +η)Λ(2, λ)D[2, β, γ, A, B](1−α)z².

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Proof. Observing thatc_n(α)defined by (1.5) is nondecreasing for(0≤α <1), we find from (2.4) that

(3.3)

∞

X

n=2

|a_n| ≤ E[β, γ, µ, A, B]

2(1 +η)Λ(2, λ)D[2, β, γ, A, B](1−α).

Using (1.10) and (3.3), we readily have(z ∈U)

|f(z)| ≥ |z| −

∞

X

n=2

|a_n|c_n(α)|zⁿ|

≥ |z| − |z²|

∞

X

n=2

|an|cn(α),

≥r− E[β, γ, µ, A, B]

2(1 +η)Λ(2, λ)D[2, β, γ, A, B](1−α)r², |z|=r <1 and

|f(z)| ≤ |z|+

∞

X

n=2

|an|cn(α)|zⁿ|

≤ |z|+|z²|

∞

X

n=2

|a_n|c_n(α),

≤r+ E[β, γ, µ, A, B]

2(1 +η)Λ(2, λ)D[2, β, γ, A, B](1−α)r², |z|=r <1, which proves the assertion (3.1) of Theorem3.1.

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Also, from (1.10), we find forz ∈U that

|f⁰(z)| ≥1−

∞

X

n=2

n|a_n|c_n(α)|zⁿ⁻¹|

≥1− |z|

∞

X

n=2

n|an|cn(α),

≥1− E[β, γ, µ, A, B]

(1 +η)Λ(2, λ)D[2, β, γ, A, B](1−α)r, |z|=r <1 and

|f⁰(z)| ≤1 +

∞

X

n=2

n|a_n|c_n(α)|zⁿ⁻¹|

≤1 +|z|

∞

X

n=2

n|a_n|c_n(α),

≤1 + E[β, γ, µ, A, B]

2(1 +η)Λ(2, λ)D[2, β, γ, A, B](1−α)r, |z|=r <1, which proves the assertion (3.2) of Theorem3.1.

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4. Radii Convexity and Starlikeness

The radii of convexity for classU[µ, α, β, γ, λ, η, A, B]is given by the following theorem.

Theorem 4.1. Let the functionf be in the classU[µ, α, β, γ, λ, η, A, B]. Then the function f is convex of order ρ(0 ≤ ρ < 1)in the disk |z| < r₁(µ, α, β, γ, λ, η, A, B) =r₁, where

(4.1) r₁ = inf

n

2(1−α)(1−ρ)Λ(n, λ)D[n, β, γ, A, B](1−η+nη) n(n−ρ)E[β, γ, µ, A, B]

_n−1¹ .

Proof. It sufficient to show that

zf⁰⁰(z) f⁰(z)

=

−P∞

n=2n(n−1)a_nzⁿ⁻¹ 1−P∞

n=2nanzⁿ⁻¹

≤ P∞

n=2n(n−1)a_n|z|ⁿ⁻¹ 1−P∞

n=2nan|z|¹⁻ⁿ (4.2)

which implies that (1−ρ)−

zf⁰⁰(z) f⁰(z)

≥(1−ρ)− P∞

n=2n(n−1)|a_n||z|ⁿ⁻¹ 1−P∞

n=2na_nzⁿ⁻¹

= (1−ρ)−P∞

n=2n(n−ρ)a_n|z|ⁿ⁻¹ 1−P∞

n=2na_n|z|ⁿ⁻¹ . (4.3)

Hence from (4.1), if (4.4) |z|ⁿ⁻¹ ≤ (1−ρ)

n(n−ρ)· 2(1−α)Λ(n, λ)D[n, β, γ, A, B](1−η+nη)

E[β, γ, µ, A, B] ,

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and according to (2.4) (4.5) 1−ρ−

∞

X

n=2

n(n−ρ)an|z|ⁿ⁻¹ >1−ρ−(1−ρ) = ρ.

Hence from (4.3), we obtain

zf⁰⁰(z) f⁰(z)

<1−ρ Therefore

Re

1 + zf⁰⁰(z) f⁰(z)

>0,

which shows thatf is convex in the disk|z|< r₁(µ, α, β, γ, λ, η, ρ, A, B).

By setting η = 0and η = 1, we have the Corollary 4.2 and the Corollary 4.3, respectively.

Corollary 4.2. Let the function f be in the class RT(µ, α, β, γ, λ, ρ, A, B).

Then the function f is convex of order ρ (0 ≤ ρ < 1) in the disk |z| <

r₂(µ, α, β, γ, λ, ρ, A, B) =r₂, where

(4.6) r₂ = inf

n

2(1−α)(1−ρ)Λ(n, λ)D[n, β, γ, A, B] n(n−ρ)E[β, γ, µ, A, B]

_n−1¹ .

Corollary 4.3. Let the function f be in the class CT(µ, α, β, γ, λ, ρ, A, B).

Then the function f is convex of order ρ (0 ≤ ρ < 1) in the disk |z| <

r3(µ, α, β, γ, λ, ρ, A, B) =r3, where

(4.7) r3 = inf

n

2(1−α)(1−ρ)Λ(n, λ)D[n, β, γ, A, B] (n−ρ)E[β, γ, µ, A, B]

_n−1¹ .

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Theorem 4.4. Let the functionf be in the classU[µ, α, β, γ, λ, η, A, B]. Then the function f is starlike of orderρ(0≤ ρ <1)in the disk|z|< r4(µ, α, β, γ, λ, η, A, B) =r₄, where

(4.8) r₄ = inf

n

2(1−α)(1−ρ)Λ(n, λ)D[n, β, γ, A, B](1−η+nη) (n−ρ)E[β, γ, µ, A, B]

_n−1¹ .

zf⁰(z) f(z) −1

<1−ρ

Using a similar method to Theorem4.1 and making use of (2.4), we get (4.8).

Lettingη = 0andη = 1, we have the Corollary 4.5and the Corollary 4.6, respectively.

Corollary 4.5. Let the function f be in the class R_T(µ, α, β, γ, λ, ρ, A, B).

Then the function f is starlike of order ρ (0 ≤ ρ < 1) in the disk |z| <

r₅(µ, α, β, γ, λ, ρ, A, B) =r₅, where

(4.9) r₅ = inf

n

2(1−α)(1−ρ)Λ(n, λ)D[n, β, γ, A, B] (n−ρ)E[β, γ, µ, A, B]

_n−1¹ .

Corollary 4.6. Let the function f be in the class CT(µ, α, β, γ, λ, ρ, A, B).

Then the function f is starlike of order ρ (0 ≤ ρ < 1) in the disk |z| <

r₆(µ, α, β, γ, λ, ρ, A, B) =r₆, where

(4.10) r6 = inf

n

2n(1−α)(1−ρ)Λ(n, λ)D[n, β, γ, A, B]

(n−ρ)E[β, γ, µ, A, B]

_n−1¹ .

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Last, but not least we give the following result.

Theorem 4.7. Let the functionf be in the classU[µ, α, β, γ, λ, η, A, B]. Then the function f is close-to-convex of order ρ (0 ≤ ρ < 1) in the disk |z| <

r₇(µ, α, β, γ, λ, η, A, B) =r₇, where

(4.11) r₇ = inf

n

2(1−α)(1−ρ)Λ(n, λ)D[n, β, γ, A, B](1−η+nη) nE[β, γ, µ, A, B]

_n−1¹ .

|f⁰(z)−1|<1−ρ.

Using a similar technique to Theorem 4.1 and making use of (2.4), we get (4.11).

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References

[1] H.M. SRIVASTAVA, H.M. HOSSEN AND M.K. AOUF, A unified presen- tation of some classes meromorphically multivalent functions, Computers Math. Applic., 38 (1999), 63–70.

[2] R.K. RAINA AND H.M. SRIVASTAVA, A unified presentation of cer- tain subclasses of prestarlike functions with negative functions, Computers Math. Applic., 38 (1999), 71–78.

[3] S. RUSCHEWEYH, Linear operator between classes of prestarlike func- tions, Comm. Math. Helv., 52 (1977), 497–509.

[4] H. SILVERMAN AND E.M. SILVIA, Prestarlike functions with negative coefficients, Internat. J. Math. Math. Sci., 2 (1979), 427–439.

[5] S. OWAANDO.P. AHUJA, An application of the fractional calculus, Math.

Japon., 30 (1985), 947–955.

[6] B.A. URALEGADDI AND S.M. SARANGI, Certain generalization of prestarlike functions with negative coefficients, Ganita, 34 (1983), 99–105.