Research Article -Bazilevic Functionsdownloads.hindawi.com/journals/aaa/2012/383592.pdfST γ CV γ K γ, γ

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2012, Article ID 383592, 10 pagesdoi:10.1155/2012/383592

Research Articlen-Bazilevic Functions

F. M. Al-Oboudi

Department of Mathematical Sciences, College of Science, Princess Nora bint Abdul Rahman University,P.O. Box 4384, Riyadh 11491, Saudi Arabia

Correspondence should be addressed to F. M. Al-Oboudi, [email protected]

Received 31 December 2011; Revised 2 March 2012; Accepted 3 March 2012

Academic Editor: Khalida Inayat Noor

Copyright q 2012 F. M. Al-Oboudi. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

The aim of this paper is to define and study a class of Bazilevic functions using the generalizedSalagean operator. Some properties of this class are investigated: inclusion relation, someconvolution properties, coefficient bounds, and other interesting results.

1. Introduction

Let H be the set of analytic functions in the open unit disc E = {z : |z| < 1}. Let A be theset of functions f ∈ H, with f(z) = z +

∑∞k=2 akz

k, and let A0 be the set of functions f ∈ H,with f(0) = 1. Let S be the class of functions f ∈ A, which are univalent in E. Denote byST(γ)(CV (γ))(K(γ)), γ < 1, the class of starlike (convex)(close-to-convex) functions of orderγ . Note that when 0 ≤ γ < 1, then ST(γ)(CV (γ))(K(γ)) ⊂ S, let ST(0)(CV (0))(K(0)) ≡ST(CV )(K). A function F ∈ A0, where F /= 0 belongs to the Kaplan class K(α, β), α ≥ 0, β ≥ 0,[1] if

−απ +12(α − β

)(θ2 − θ1) ≤ argF

(reiθ2

)− argF

(reiθ1

)≤ 1

2(α − β

)(θ2 − θ1) + βπ, (1.1)

for θ1 < θ2 < θ1 + 2π and 0 < r < 1.The Dual of ν ⊂ A0 is defined as

ν� ={g ∈ A0 : f ∗ g /= 0 in Δ, f ∈ ν

}, (1.2)

where ∗ denotes Hadamard product (convolution).

2 Abstract and Applied Analysis

A set ν ⊂ A0 is called a test set for u ⊂ A (denoted by ν � u) if ν ⊂ u ⊂ ν��. Note thatif ν � u, then ν� ⊂ u�.

Denote by P the class of functions p ∈ A0, such that Re p > 0, in E, and let Pα = {fα ∈A0, f ∈ P}. Note that for 0 ≤ α ≤ β,

K(α, β)= Pα ·K(0, β − α

), (1.3)

and that f ∈ ST(α), α < 1, if, and only if, f/z ∈ K(0, 2 − 2α).For α ≥ 0 and β ≥ 0, define the class T(α, β) as

T(α, β)=

{(1 + xz)[α]

(1 + yz

)α−[α]

(1 + uz)β: |x| = ∣∣y∣∣ = |u| = 1

}

. (1.4)

Note that T(α, β) � K(α, β), α ≥ 1, β ≥ 1.A function f ∈ A0, is called prestarlike of order α, α ≤ 1, (denoted by R(α)) if and only

if f/z ∈ T(0, 3 − 2α)�, or f ∈ R(α) if and only if

f ∗ z

(1 − z)2−2α∈ ST(α) α < 1,

Ref

z>

12, z ∈ E α = 1.

(1.5)

Let B(α, β), α > 0, β ∈ R, denote the class of Bazilevic functions in E, introduced byBazilevic [2], f ∈ B(α, β), α > 0, β ∈ R, if and only if there exists a g ∈ ST(1 − α), such that forz ∈ E

zf ′

g

(f

z

)α+iβ−1∈ P, (1.6)

where (f/z)α+iβ = 1 at z = 0. We denote B(α, 0) by B(α). Bazilevic shows that B(α, β) ⊂ S, forα > 0, β ∈ R. Note that

CV ⊂ ST ⊂ K ⊂ B(α, β). (1.7)

For further information, see [3–7].The generalized Salagean operator Dn

λf : A −→ A, n ∈ N0 = N ∪ {0}, λ ≥ 0, is defined

[8] as

Dnλf(z) = hn,λ(z) ∗ f(z), (1.8)

Abstract and Applied Analysis 3

where

hn,λ(z) = hλ(z) ∗ · · · ∗ hλ(z)︸︷︷︸

(n-times)

,

hλ(z) =z − (1 − λ)z2

(1 − z)2= z +

∞∑

k=2

[1 + λ(k − 1)]zk.

(1.9)

The Operator Dnλf satisfies the following identity:

Dn+1λ f(z) = (1 − λ)Dn

λf(z) + λz(Dn

λf(z))′. (1.10)

Not that for λ = 1, Dn1f(z) ≡ Dnf(z), Salagean differential operator [9].

Let

kn,λ(z) = h(−1)n,λ (z) = z +

∞∑

k=2

zk

[1 + λ(k − 1)]n, λ > 0, (1.11)

we mean by f (−1), the solution of f ∗ f (−1) = z/(1 − z). Hence

kn,λ(z) = kλ(z) ∗ · · · ∗ kλ(z)︸︷︷︸

(n-times)

, (1.12)

where kλ(z) = h(−1)λ

(z). It is known [10] that kλ/z ∈ T(1, 1+ 1/λ)∗, hence kλ ∈ R(1− 1/λ), andthat

kλ(z) ∗ z

(1 − z)(1/λ)+1=

z

(1 − z)1/λ. (1.13)

The class STn(γ), γ ≤ 1 is defined as f ∈ STn(γ) if and only if Dnλf ∈ ST(γ). For λ = 1,

we get Salagean-type n-starlike functions [9].The operator Dn

λf is now called “Al-Oboudi Operator” and has been extensivelystudied latly, [5, 11, 12].

In this paper we define and study a class of Bazilevic functions using the operatorDnλf

and study some of its basic properties, inclusion relation, convolution properties coefficientbounds, and other interesting results.

2. Definition and Preliminaries

In this section, the class of n-Bazilevic functions Bnλ , λ > 0, where B0

λ ≡ B(1/λ, 0), is definedand some preliminary lemmas are given.


2.1. Definition

Let f ∈ A. Then f ∈ Bnλ, n ∈ N0, λ > 0, if and only if there exists a g ∈ STn(1 − 1/λ), such that

Dn+1λ z(f/z)1/λ

Dnλg(z)

∈ P, z ∈ E, (2.1)

where the power (f/z)1/λ is chosen as a principal one.Denote by Bn

1λ the class of functions f ∈ Bnλ, where g ≡ z.

Using (1.10), we see that Dn+1λ z(f/z)1/λ = f ′(z)Dn

λz(f/z)1/λ−1, from which the

following special cases are clear.

2.1.1. Special Cases

(1) For n = 0, B01/α ≡ B(α), α > 0, Bazilevic [2].

(2) For λ = 1, Bn1 ≡ Kn(0), Salagean type close to convex functions, Blezu [13].

(3) For n = 0, λ = 1, B01 ≡ K, Kaplan [14].

(4) For λ = 1, Bn11 ≡ Bn+1(1), Abdul Halim [15] and Bn

11 ≡ T1n+1(0), Opoola [16].

2.2. Lemmas

The following lemmas are needed to prove our results.

Lemma 2.1 (see [10]). Let α, β ≥ 1 and f ∈ T(α, β)∗, g ∈ K(α − 1, β − 1). Then for F ∈ A

(ϕ ∗ g F

)

(ϕ ∗ g) (E) ⊂ CO (F(E)). (2.2)

Lemma 2.2. If Dn+1λ f ∈ ST(1 − 1/λ), λ > 0, then Dn

λf ∈ ST(1 − 1/λ).

Proof. Since Dnλf = kλ ∗Dn+1

λf , we will show that (kλ ∗Dn+1

λf) ∈ ST(1 − 1/λ). Now Dn+1

λf ∈

ST(1 − 1/λ) ⊂ ST(1/2 − 1/2λ), implies

(z

(1 − z)1/λ+1

)(−1)∗Dn+1

λ f(z) ∈ R

(12− 12λ

)

⊂ R

(

1 − 12λ

)

. (2.3)

Hence

(z

(1 − z)1/λ

)(z

(1 − z)1/λ+1

)(−1)∗Dn+1

λ f(z) ∈ ST

(

1 − 12λ

)

⊂ ST

(

1 − 1λ

)

. (2.4)

From (1.13), we get the required result.


Lemma 2.3 (see [1]). Let α, β ≥ 1 and f ∈ T(α, β)∗, g ∈ K(α, β). Then f ∗ g ∈ K(α, β).

For X ⊂ A, let rc(X) denote the largest positive number so that every f ∈ X is convexin |z| < rc(X). The following result is due to Al-Amiri [17].

Lemma 2.4. One has

rc(hλ) = rc =1

1 + |c| +√

1 + |c| + |c|2, c = 2λ − 1, 0 < λ ≤ 1. (2.5)

Lemma 2.5 (see [18]). Let f, g ∈ H, with f(0) = g(0) = 0 and f ′(0)g ′(0)/= 0. Let ϕ ∈ ν� in|z| < r < 1, where

ν ={1 + xz

1 + yzg(z) : |x| = ∣∣y∣∣ = 1

}

. (2.6)

Then for each F ∈ H,

(ϕ ∗ Fg)(ϕ ∗ g) (|z| < r) ⊂ CO (F(E)), (2.7)

where CO stands for closed convex hull.

Remark 2.6. In [1], it was shown that condition (2.6) is satisfied for all z in E whenever ϕ is inCV and g is in ST .

Lemma 2.7 (see [10]). Let α, β, γ, δ, μ, ν ∈ R be such that

0 ≤ γ ≤ α − 1, 0 ≤ δ ≤ β − 1, 0 ≤ μ ≤ α − γ, 0 ≤ γ ≤ β − δ, (2.8)

and let f ∈ T(α, β)∗, g ∈ K(γ, δ), F ∈ K(μ, ν). Then

(f ∗ gF)(f ∗ g) ∈ Pmax{μ,ν}. (2.9)

From (1.12) and (1.13), we immediately have;

Lemma 2.8. One has

kn+1,λ =z

(1 − z)(1/λ)−n∗(

z

(1 − z)(1/λ)+1

)(−1), (2.10)


Lemma 2.9 (see [10]). Let f ∈ K(α, β), α, β ≥ 1. Then

f � (1 + z)α

(1 − z)β, (2.11)

where� stands for coefficient majorization.

3. Main Results

Theorem 3.1. One has

Bn+1λ ⊂ Bn

λ, λ > 0. (3.1)

Proof. Let f ∈ Bn+1λ . Then there exists g ∈ STn+1(1 − 1/λ), such that

Dn+2λ z

(f

z

)1/λ

= Dn+1λ g(z) · p(z), p ∈ K(1, 1). (3.2)

Hence

(f

z

)1/λ

=kn+1,λz

∗ kλz

∗ Dn+1λ g

z· p, p ∈ K(1, 1). (3.3)

Since Dn+1λ

g/z ∈ K(0, 2/λ), and kλ/z ∈ T(1, 1 + 1/λ)∗, application of Lemma 2.3 gives

kλz

∗ Dn+1λ

g

z· p =

(kλz

∗ Dn+1λ

g

z

)

p0, p0 ∈ K(1, 1) (3.4)

hence

(f

z

)1/λ

=kn+1,λz

∗ Dnλg

zp0, p0 ∈ K(1, 1) (3.5)

Using Lemma 2.2 we deduce that f ∈ Bnλ .

As a consequence of (3.1) we immediately have the following.


Corollary 3.2. One has

Bnλ ⊂ S. (3.6)

Corollary 3.3. If f ∈ Bnλ, n ∈ N0, λ > 0, then, for z ∈ E

(f

z

)1/λ

∈ K

(

1, 1 +2λ

)

. (3.7)

Proof. Since f ∈ Bnλ , there exists a g ∈ STn(1 − 1/λ) or Dn

λg/z ∈ K(0, 2/λ) such that

(f

z

)1/λ

=kn+1,λz

∗ Dnλg

z· p, p ∈ K(1, 1), (3.8)

=kn+1,λz

∗ F, F ∈ K

(

1, 1 +2λ

)

, (3.9)

using (1.3). From (3.6), we conclude that

(f

z

)1/λ

=kn+1,λz

∗ F /= 0, 0 < |z| < 1, (3.10)

which implies that

kn+1,λz

∈ K

(

1, 1 +2λ

)∗≡ T

(

1, 1 +2λ

)∗, (3.11)

Applying Lemma 2.3 to (3.9), we get the result.

Theorem 3.4. Let f ∈ Bnλ . Then

⎛

⎝Dn+1

λz(f/z)1/λ

z

⎞

⎠

λ

∈ K(λ,λ + 2). (3.12)

Proof. From (2.1), we see that

Dn+1λ z

(f

z

)1/λ

= Dnλg(z) · p(z), p ∈ K(1, 1). (3.13)


Since (Dnλg/z)

λ ∈ K(0, 2), and p(z)λ ∈ K(λ, λ), then

⎛

⎝Dn+1

λz(f/z)1/λ

z

⎞

⎠

λ

=

(Dn

λg

z

)λ

p(z)λ ∈ K(λ,λ + 2), (3.14)

which is the required result.

In the following we prove the converse of Theorem 3.1, for 0 < λ ≤ 1.

Theorem 3.5. Let f ∈ Bnλ , 0 < λ ≤ 1. Then f ∈ Bn+1

λ in |z| < rc, where rc is given by (2.5)

Proof. f ∈ Bnλimplies (2.1), where Dn

λg ∈ ST(1 − 1/λ) ⊂ ST .

Now

Dn+2λ

z(f/z)1/λ

Dn+1λ

g=

hλ ∗Dn+1λ

z(f/z)1/λ

hλ ∗Dnλg

. (3.15)

Using Lemma 2.4, we see that hλ ∈ CV in |z| < rc, for 0 < λ ≤ 1, where rc is given by (2.5).From Remark 2.6, we conclude

hλ ∗{1 + xz

1 + yzDn

λg : |x| = ∣∣y∣∣ = 1}

/= 0. (3.16)

Applying Lemma 2.5, we deduce

⎛

⎜⎝

hλ ∗(Dn+1

λ z(f/z)1/λ

/Dnλg)Dn

λg

hλ ∗Dnλg

⎞

⎟⎠(|z| < rc) ⊂ CO

⎛

⎝Dn+1

λz(f/z)1/λ

Dnλg

⎞

⎠(E), (3.17)

hence Dn+2λ

z(f/z)1/λ/Dn+1λ

g ∈ P in |z| < rc, as required.

Corollary 3.3 can be improved for 0 < λ ≤ 1, as follows.

Theorem 3.6. Let f ∈ Bnλ , 0 < λ ≤ 1. Then

f

z∈ K(1, 2). (3.18)

Proof. Wewill use Ruscheweyh’s.method of proof [10]. f ∈ Bnλimplies (3.8), whereDn+1

λg/z ∈

K(0, 2/λ), kn+1,λ/z ∈ T(1, 1 + 2/λ)∗.Let Dn

λg/z = l ·m, where l = (Dn

λg/z)(1+λ)/2 and m = (Dn

λg/z)(1−λ)/2.

Then l ∈ K(0, 1/λ + 1) ·m ∈ K(0, 1/λ − 1) andm · p = F ∈ K(1, 1/λ). This implies

(f

z

)1/λ

=kn+1,λz

∗ lF. (3.19)


Using Lemma 2.7, we get

(kn+1,λ/z) ∗ lF(kn+1,λ/z) ∗ l = F0 ∈ K

(1λ,1λ

)

, 0 < λ ≤ 1. (3.20)

Hence

(f

z

)1/λ

=(kn+1,λz

∗ l)

F0, F0 ∈ K

(1λ,1λ

). (3.21)

To prove that (f/z)1/λ ∈ K(1/λ, 2/λ), we have to show that ((kn+1,λ/z) ∗ l) ∈ K(0, 1/λ), orequivalently kn+1,λ ∗ z l ∈ ST(1 − 1/2λ).

Since z l ∈ ST((λ − 1)/2λ), then from (1.5)

(z

(1 − z)1/λ+1

)(−1)∗ z l ∈ R

(λ − 12λ

)

⊂ R

((n + 2)λ − 1

2λ

)

,

z

(1 − z)1/λ−n∗(

z

(1 − z)1/λ+1

)(−1)∗ z l ∈ ST

((n + 2)λ − 1

2λ

)

⊂ ST

(

1 − 12λ

)

.

(3.22)

From Lemma 2.8, (1.13), and (3.22), we see that ((kn+1,λ/z) ∗ l) ∈ K(0, 1/λ). Using (3.21), weobtain (f/z)1/λ ∈ K(1/λ, 2/λ). From (1.1) we get the required result.

Remark 3.7. For n = 0, λ = 1/α Theorem 3.6 and other stronger results depending on α, areproved by Sheil-Small [7].

For the coefficient bounds of f ∈ Bnλ, Theorem 3.6 is not strong enough to settle this

problem for 0 < λ < 1, In 1962, Zamorski [19] proved the Bieberbach conjecture for f ∈ B(α),when α = 1, 1/2, 1/3, . . ., in the following we prove this result for f ∈ Bn

λ , using the extremepoints of Kaplan class K(α, β).

Theorem 3.8. For f ∈ Bnλ, λ = m ∈ N,

f � z

(1 − z)2−mn. (3.23)

Proof. From (3.9) and Lemma 2.9, we get

(f

z

)1/λ

� kn+1,λz

∗ 1 + z

(1 − z)1+2/λ,

=1

(1 − z)2/λ−n,

(3.24)

using (2.10). Raising both sides of (3.24) to the mth power, where λ = m ∈ N, we get therequired result.


Remark 3.9. For n = 0, we get the result of Zamorski [19], and the result of Sheil-Small [7],from which we get the idea of proof.

References

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[2] I. E. Bazilevic, “On a case of integrability in quadratures of the Loewner-Kufarev equation,” Matema-ticheskii Sbornik, vol. 37, pp. 471–476, 1955.

[3] M. Arif, K. I. Noor, and M. Raza, “On a class of analytic functions related with generalized Bazilevictype functions,” Computers and Mathematics with Applications, vol. 61, no. 9, pp. 2456–2462, 2011.

[4] Y. C. Kim, “A note on growth theorem of Bazilevic functions,” Applied Mathematics and Computation,vol. 208, no. 2, pp. 542–546, 2009.

[5] A. T. Oladipo, “On a new subfamilies of Bazilevic functions,” Acta Universitatis Apulensis, no. 29, pp.165–185, 2012.

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[7] T. Sheil-Small, “Some remarks on Bazilevic functions,” Journal d’Analyse Mathematique, vol. 43, pp.1–11, 1983.

[8] F. M. Al-Oboudi, “On univalent functions defined by a generalized Salagean operator,” InternationalJournal of Mathematics and Mathematical Sciences, no. 25–28, pp. 1429–1436, 2004.

[9] G. S. Salagean, “Subclasses of univalent functions,” in Complex Analysis, Fifth Romanian-Finnish Semi-nar, Part 1 (Bucharest, 1981), vol. 1013 of Lecture Notes in Mathematics, pp. 362–372, Springer, Berlin,Germany, 1983.

[10] St. Ruscheweyh, Convolutions in Geometric Function Theory, vol. 83 of Seminaire de Mathematiques Super-ieures, Presses de l’Universite de Montreal, Montreal, Canada, 1982.

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[13] D. Blezu, “On the n-close-to-convex functions with respect to a convex set. I,” Mathematica Revued’Analyse Numerique et de Theorie de l’Approximation, vol. 28(51), no. 1, pp. 9–19, 1986.

[14] W. Kaplan, “Close-to-convex schlicht functions,” The Michigan Mathematical Journal, vol. 1, pp. 169–185, 1952.

[15] S. Abdul Halim, “On a class of analytic functions involving the Salagean differential operator,” Tam-kang Journal of Mathematics, vol. 23, no. 1, pp. 51–58, 1992.

[16] T. O. Opoola, “On a new subclass of univalent functions,” Mathematica, vol. 36, no. 2, pp. 195–200,1994.

[17] H. S. Al-Amiri, “On the Hadamard products of schlicht functions and applications,” InternationalJournal of Mathematics and Mathematical Sciences, vol. 8, no. 1, pp. 173–177, 1985.

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Research Article -Bazilevic Functionsdownloads.hindawi.com/journals/aaa/2012/383592.pdfST γ CV γ K γ, γ