On Subclass of Analytic Univalent Functions Associated with Negative Coefficients

Volume 2010, Article ID 343580,11pages doi:10.1155/2010/343580

Research Article

On Subclass of Analytic Univalent Functions

Associated with Negative Coefficients

Ma’moun Harayzeh Al-Abbadi and Maslina Darus

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

Correspondence should be addressed to Maslina Darus,[email protected]

Received 18 November 2009; Accepted 9 January 2010 Academic Editor: Stanisława R. Kanas

Copyrightq2010 M. H. Al-Abbadi and M. Darus. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

M. H. Al-Abbadi and M. Darus2009recently introduced a new generalized derivative operator

μn,m

λ1,λ2, which generalized many well-known operators studied earlier by many different authors. In this present paper, we shall investigate a new subclass of analytic functions in the open unit disk

U{z∈C:|z|<1}which is defined by new generalized derivative operator. Some results on coefficient inequalities, growth and distortion theorems, closure theorems, and extreme points of analytic functions belonging to the subclass are obtained.

1. Introduction and Definitions

LetAxdenote the class of functions of the form

fz z ∞ kx1

akzk, ak is complex number, 1.1

the form

fz z− ∞ kx1

|ak|zk_{, x∈N{1,2,3, . . .}, 1.2}

defined on the open unit diskU{z ∈C: |z|< 1}. A functionf∈ Txis called a function with negative coefficient and the classT1was introduced and studied by Silverman1 .

In1 Silverman investigated the subclasses ofT1 denoted by S∗_Tα and CTαfor 0 ≤

α <1. That are, respectively, starlike of orderαand convex of orderα. Nowx_kdenotes the Pochhammer symbolor the shifted factorialdefined by

x_k Γxk Γx

⎧ ⎨ ⎩

1 fork0, x∈C\ {0},

xx1x2. . .xk−1 fork∈N{1,2, . . .}, x∈C. 1.3

The authors in2 have recently introduced a new generalized derivative operator μn,m_λ1,λ2as follows.

Definition 1.1. Forf ∈ A A1,the generalized derivative operator μn,m_λ1,λ2 : A → A is defined by

μn,m_λ1,λ2fz z∞ k2

1λ1k−1m−1 1λ2k−1m

cn, kakzk, z∈U, 1.4

wheren, m∈N0{0,1,2, . . .}, λ2≥λ1≥0, and cn, k _nk−1

n

n1_k−1/1k−1.

1Special cases of this operator include the Ruscheweyh derivative operator in the cases μn,1_λ1,0 ≡ μn,m_0,0 ≡ μn,0_0,λ2 ≡ Rn 3 , the Salagean derivative operator μ0,m_1,01 ≡ Sn

4 , the generalized Ruscheweyh derivative operatorμn,2_λ1,0≡Rn_λ5 , the generalized

Salagean derivative operator introduced by Al-Oboudi μ0,m_λ1,01 ≡ Sn_β 6 , and the

generalized Al-Shaqsi and Darus derivative operator μλ,n_β,01 ≡ Dn_λ,β where n λ, m n 1, λ1 β, and λ2 0 can be found in 7 . It is easily seen that μ0,1_λ1,0fz μ0,m_0,0fz μ0,0_0,λ2fz μ1,1_λ1,1fz fz, μ1,1_λ1,0fz μ1,m_0,0fz μ1,0_0,λ2fz μ0,2_1,0fz zf_{z, and alsoμ}n−1,0

λ1,0 fz μn0,0−1,mfzwheren1,2,3, . . . .

By making use of the generalized derivative operatorμn,m_λ1,λ2,the authors introduce a new subclass as follows.

Definition 1.2. For 0≤α <1, n, m∈N0{0,1,2, . . .}and λ2≥λ1≥0, letHn,m_λ1,λ2x, αbe the subclass ofSxconsisting of functionsfsatisfying

Re ⎛ ⎜ ⎝z

μn,m_λ1,λ2fz μn,m_λ1,λ2fz

⎞ ⎟

where

μn,m_λ1,λ2fz z ∞ kx1

1λ1k−1m−1 1λ2k−1m

cn, kakzk, 1.6

{x1,2,3, . . . ,z∈U}, andμn,m_λ1,λ2fz/0.

Further, we define the classTHn,m_λ1,λ2x, αby

THn,m_λ1,λ2x, α Hn,m_λ1,λ2x, α∩ Tx, {x1,2,3, . . .}, 1.7

for 0≤α <1, n, m∈N0{0,1,2, . . .}, andλ2≥λ1≥0.

Also note that various subclasses ofHn,m_λ1,λ2x, αandTHn,m_λ1,λ2x, αhave been studied by many authors by suitable choices ofn, m, λ1, λ2, andx. For example,

TH0,0_0,λ21, α≡ TH0,1_λ1,01, α≡ TH0,m_0,01, α≡ TH1,1_λ1,11, α≡S∗

Tα, 1.8

starlike of orderαwith negative coefficients. And

TH1,0_0,λ21, α≡ TH1,1_λ1,01, α≡ TH_0,01,m1, α≡ TH0,2_1,01, α≡CTα, 1.9

class of convex function of orderαwith negative coefficients. Also

TH0,0_0,λ2x, α≡ TH0,1_λ1,0x, α≡ TH0,m_0,0x, α≡ TH1,1_λ1,1x, α≡S∗ Tx, α,

TH1,0_0,λ2x, α≡ TH1,1_λ1,0x, α≡ TH_0,01,mx, α≡ TH0,2_1,0x, α≡CTx, α,

TH0,2_λ1,0x, α≡Px, λ1, α 0≤λ1 <1,

TH1,2_λ1,0x, α≡Cx, λ1, α 0≤λ1<1.

1.10

The classesS∗_Tx, αandCTx, αwere studied by Chatterjea8 see also Srivastava et al.

9 , whereas the classes Px, λ1, αand Cx, λ1, α were, respectively, studied by Altintas¸ 10 and Kamali and Akbulut11 . Whenλ1λ2 0 orm1, λ2 0, orm0, λ1 0 in the classHn,m_λ1,λ2x, α, we have the classRnαintroduced and studied by Ahuja12 . Finally we note that whenm 2, λ2 0 in the classHn,m_λ1,λ2x, αwe have the classKn_λx, αintroduced and studied by Al-Shaqsi and Darus13 .

2. Coefficient Inequalities

Theorem 2.1. For 0≤α <1 andλ2 ≥λ1≥0, letf∈ Sxbe defined by1.1. If

∞

kx1

k−α1λ1k−1m−1 1λ2k−1m

cn, k|ak| ≤1−α, x1,2, . . . , 2.1

thenf ∈ Hn,m_λ1,λ2x, α, wheren∈N{1,2, . . .}andm∈N0{0,1,2, . . .}.

Proof. Assume that2.1holds true. Then we shall prove condition1.5. It is sufficient to

show that

zμn,m_λ1,λ2fz μn,m_λ1,λ2fz −1

≤1−α, z∈U. 2.2

So, we have that

zμn,m_λ1,λ2fz μn,m_λ1,λ2fz −1

zμn,m_λ1,λ2fz−μn,m_λ1,λ2fz μn,m_λ1,λ2fz

_∞ kx1

k−11λ1k−1m−1/1λ2k−1m

cn, kakzk z∞

kx1

1λ1k−1m−1/1λ2k−1m

cn, kakzk

|z|<1,

≤ _∞

kx1

k−11λ1k−1m−1/1λ2k−1m

cn, k|ak|

1−∞_kx11λ1k−1m−1/1λ2k−1m

cn, k|ak| ,

2.3

and expression2.3is bounded by1−α. Hence2.2holds if

∞

kx1

k−11λ1k−1m−1 1λ2k−1m

cn, k|ak|

≤1−α

1−

∞

kx1

1λ1k−1m−1 1λ2k−1m

cn, k|ak|

,

2.4

which is equivalent to

∞

kx1

k−α1λ1k−1m−1 1λ2k−1m

by2.1. Thusf ∈ Hn,m_λ1,λ2x, α. Note that the denominator in2.3is positive provided that 2.1holds.

Theorem 2.2. Letfbe defined by1.2and 1−∞_kx1k1λ1k−1m−1/1λ2k−1mc(n, k)|ak| ≥0x1,2, . . .. Thenf∈ THn,m_λ1,λ2x, αif and only if 2.1is satisfied.

Proof. We only prove the right-hand side, since the other side can be justified using similar arguments in proof of Theorem2.1. Sincef ∈ THn,m_λ1,λ2x, αby condition1.5, we have that

Re ⎛ ⎜ ⎝z

μn,m_λ1,λ2fz μn,m_λ1,λ2fz

⎞ ⎟ ⎠

Re ⎧ ⎨ ⎩

z−∞_kx1

k1λ1k−1m−1/1λ2k−1m

cn, k|ak|zk z−∞

kx1

1λ1k−1m−1/1λ2k−1m

cn, k|ak|zk ⎫ ⎬ ⎭> α.

2.6

Choose values ofzon real axis so thatzμn,m_λ1,λ2fz/μn,m_λ1,λ2fzis real. Lettingz → 1−through real values, we have that

1−∞_kx1k1λ1k−1m−1/1λ2k−1m

cn, k|ak|zk

1−∞_kx11λ1k−1m−1/1λ2k−1m

cn, k|ak|zk > α. 2.7

Thus we obtain

∞

kx1

k−α1λ1k−1m−1

1λ2k−1m cn, k|ak| ≤

1−α, 2.8

which is2.1. Hence the proof is complete.

The result is sharp with the extremal functionfgiven by

fz z− 1−α1λ2xm

x1−α1λ1xm−1cn, x1

zx1_{, x1,2, . . .. 2.9}

Theorem 2.3. Let the functionfgiven by1.2be in the classTHn,m_λ1,λ2x, α. Then

|ak| ≤ 1−α1λ2k−1m k−α1λ1k−1m−1cn, k

, 2.10

Proof. Sincef ∈ THn,m_λ1,λ2x, α, then condition2.1gives

|ak| ≤ 1−α1λ2k−1m k−α1λ1k−1m−1cn, k

, 2.11

for eachkx1 wherex1,2,3, . . . .

Clearly the function given by2.9satisfies2.10, and therefore,fgiven by2.9is in THn,m_λ1,λ2x, αfor this function; the result is clearly sharp.

3. Growth and Distortion Theorems

In this section, growth and distortion theorems will be considered and covering property for function in the class will also be given.

Theorem 3.1. Let the functionfgiven by1.2be in the classTH_λ1,λ2n,m x, α. Then for 0<|z|r < 1,

r− 1−α1λ2xm

x1−α1λ1xm−1cn, x1 rx1

≤fz≤r 1−α1λ2xm

x1−α1λ1xm−1cn, x1

rx1_{, x1,2, . . . ,}

3.1

where 0≤α <1, m∈N0 {0,1,2. . .}, andn∈N{1,2, . . .}.

Proof. We only prove the right-hand side inequality in3.1, since the other inequality can be

justified using similar arguments. Sincef∈ THn,m_λ1,λ2x, αby Theorem2.2, we have that

∞

kx1

k−α1λ1k−1m−1 1λ2k−1m

cn, k|ak| ≤1−α. 3.2

Now

x1−α1λ1xm−1

1λ2xm cn, x1

_∞

kx1 |ak|

∞ kx1

x1−α1λ1xm−1

1λ2xm cn, x1|ak|,

≤ ∞ kx1

k−α1λ1k−1m−1 1λ2k−1m

cn, k|ak|,

≤1−α, x1,2,3, . . . .

And therefore,

∞

kx1

|ak| ≤ 1−α1λ2x m

x1−α1λ1xm−1cn, x1

, x1,2, . . . . 3.4

Since

fz z− ∞ kx1

|ak|zk_{, x1,2, . . . , 3.5}

then we have that

_fz z−

∞

kx1 |ak|zk

. 3.6

After that,

_{fz≤ |z||z|}x1 ∞ kx1

|ak||z|k−x1

≤rrx1 ∞ kx1

|ak|.

3.7

By aid of inequality3.4, it yields the right-hand side inequality of3.1. Thus, this completes

the proof.

Theorem 3.2. Let the functionfgiven by1.2be in the classTH_λ1,λ2n,m x, α. Then for 0<|z|r < 1,

1− x11−α1λ2x m

x1−α1λ1xm−1cn, x1 rx

≤f_z≤1 x11−α1λ2xm x1−α1λ1xm−1cn, x1

rx_{, x1,2, . . . ,}

3.8

where 0≤α <1, λ2≥λ1≥0, m∈N0{0,1,2, . . .}, andn∈N{1,2, . . .}.

Proof. Sincef ∈ THn,m_λ1,λ2x, α, by Theorem2.2, we have that

∞

kx1

k−α1λ1k−1m−1 1λ2k−1m

Now

x1−α1λ1xm−1 1λ2xm

cn, x1

_∞

kx1 k|ak|

∞ kx1

x1−α1λ1xm−1

1λ2xm cn, x1k|ak|

≤x1∞ kx1

k−α1λ1k−1m−1 1λ2k−1m

cn, k|ak|

≤x11−α, k≥x1, x1,2,3, . . ..

3.10

Hence

∞

kx1

k|ak| ≤ x11−α1λ2x m

x1−α1λ1xm−1cn, x1

, x1,2, . . . . 3.11

Since

f_{z 1−} ∞

kx1

k|ak|zk−1, x1,2, . . . , 3.12

then we have that

1− |z|x

∞

kx1

k|ak||z|k−x1_≤fz≤1|z|x ∞ kx1

k|ak||z|k−x1_{, 3.13}

and therefore,

1−rx

∞

kx1

k|ak| ≤f_z≤1rx ∞ kx1

k|ak|, x1,2, . . . . 3.14

By using the inequality3.11in3.14, we get Theorem3.2. This completes the proof.

4. Extreme Points

The extreme points of the classTHn,m_λ1,λ2x, αare given by the following theorem.

Theorem 4.1. Letfxz zand

fkz z− 1−α1λ2k−1 m

k−α1λ1k−1m−1cn, k

zk_{, 4.1}

Thenf ∈ THn,m_λ1,λ2x, αif and only if it can be expressed in the form

fz ∞ kx

δkfkz, 4.2

whereδk≥0 and∞kxδk1.

Proof. Suppose thatfcan be expressed as in4.2. Our goal is to show thatf∈ THn,m_λ1,λ2x, α.

By4.2, we have that

fz ∞ kx

δkfkz

δxfxz

∞

kx1 δkfkz

δxfxz

∞

kx1 δk

z− 1−α1λ2k−1m k−α1λ1k−1m−1cn, k

zk

∞ kx

δkz−

∞

kx1

δk1−α1λ2k−1m k−α1λ1k−1m−1cn, k

zk

z− ∞ kx1

δk1−α1λ2k−1m k−α1λ1k−1m−1cn, k

zk_.

4.3

Now

fz z− ∞ kx1

|ak|zk_z− ∞ kx1

δk1−α1λ2k−1m k−α1λ1k−1m−1cn, k

zk_{, 4.4}

so that

|ak| δk1−α1λ2k−1m k−α1λ1k−1m−1cn, k

. 4.5

Now, we have that

∞

kx1

δk1−δx ≤1, x1,2,3, . . . . 4.6

Setting

∞

kx1 δk

∞

kx1

δk1−α1λ2k−1m k−α1λ1k−1m−1cn, k

k−α1λ1k−1m−1cn, k 1−α1λ2k−1m

we arrive to

∞

kx1

k−α1λ1k−1m−1

1−α1λ2k−1m cn, k|ak| ≤1. 4.8

And therefore,

∞

kx1

k−α1λ1k−1m−1 1λ2k−1m

cn, k|ak| ≤1−α, x1,2,3, . . . . 4.9

It follows from Theorem2.2thatf∈ THn,m_λ1,λ2x, α.

Conversely, let us suppose thatf ∈ THn,m_λ1,λ2x, α; our goal is, to get4.2. From4.2

and using similar last arguments, it is easily seen that

fz z− ∞ kx1

|ak|zk_z− ∞ kx1

δk1−α1λ2k−1m k−α1λ1k−1m−1cn, k

zk_{, 4.10}

which suffices to show that

|ak| δk1−α1λ2k−1 m

k−α1λ1k−1m−1cn, k

. 4.11

Now, we have thatf∈ THn,m_λ1,λ2x, α, then by previous Theorem2.3,

|ak| ≤ 1−α1λ2k−1m k−α1λ1k−1m−1cn, k

. 4.12

That is

k−α1λ1k−1m−1cn, k|ak| 1−α1λ2k−1m ≤1.

4.13

Since∞_kxδk1, we see δk≤1, for eachkx, x1, x2, . . . ,andx1,2,3, . . . . We can set that

δk k−α1λ1k−1

m−1_{cn, k|a} k| 1−α1λ2k−1m

. 4.14

Thus, the desired result is that

|ak| δk1−α1λ2k−1m k−α1λ1k−1m−1cn, k

. 4.15

Corollary 4.2. The extreme points of THn,m_λ1,λ2x, αare the functions

fxz z,

fkz z− 1−α1λ2k−1 m

k−α1λ1k−1m−1cn, k

zk_, 4.16

where 0≤α <1, n, m∈N0{0,1,2, . . .}, λ2≥λ1≥0, andkx1, x2, . . . ,x1,2,3, . . ..

Acknowledgments

This work is fully supported by UKM-GUP-TMK-07-02-107, Malaysia. The authors are also grateful to the referee for his/her suggestions which helped us to improve the contents of this article.

References

1 H. Silverman, “Univalent functions with negative coefficients,” Proceedings of the American Mathemat-ical Society, vol. 51, pp. 109–116, 1975.

2 M. H. Al-Abbadi and M. Darus, “Differential subordination for new generalized derivative operator,” Acta Universitatis Apulensis, no. 20, pp. 265–280, 2009.

3 St. Ruscheweyh, “New criteria for univalent functions,” Proceedings of the American Mathematical Society, vol. 49, pp. 109–115, 1975.

4 G. ˇS. S˘al˘agean, “Subclasses of univalent functions,” in Complex Analysis—Fifth Romanian-Finnish Seminar, Part 1 (Bucharest, 1981), vol. 1013 of Lecture Notes in Mathematics, pp. 362–372, Springer, Berlin, Germany, 1983.

5 K. Al-Shaqsi and M. Darus, “On univalent functions with respect tok-symmetric points defined by a generalized Ruscheweyh derivatives operator,” Journal of Analysis and Applications, vol. 7, no. 1, pp. 53–61, 2009.

6 F. M. Al-Oboudi, “On univalent functions defined by a generalized S˘al˘agean operator,” International Journal of Mathematics and Mathematical Sciences, vol. 2004, no. 27, pp. 1429–1436, 2004.

7 M. Darus and K. Al-Shaqsi, “Differential sandwich theorems with generalised derivative operator,” International Journal of Computational and Mathematical Sciences, vol. 2, no. 2, pp. 75–78, 2008.

8 S. K. Chatterjea, “On starlike functions,” Journal of Pure Mathematics, vol. 1, pp. 23–26, 1981.

9 H. M. Srivastava, S. Owa, and S. K. Chatterjea, “A note on certain classes of starlike functions,” Rendiconti del Seminario Matematico della Universit`a di Padova, vol. 77, pp. 115–124, 1987.

10 O. Altıntas¸, “On a subclass of certain starlike functions with negative coefficients,” Mathematica Japonica, vol. 36, no. 3, pp. 489–495, 1991.

11 M. Kamali and S. Akbulut, “On a subclass of certain convex functions with negative coefficients,” Applied Mathematics and Computation, vol. 145, no. 2-3, pp. 341–350, 2003.

12 O. P. Ahuja, “Integral operators of certain univalent functions,” International Journal of Mathematics and Mathematical Sciences, vol. 8, no. 4, pp. 653–662, 1985.