On univalent functions defined by a generalized Sălăgean operator

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PII. S0161171204108090 http://ijmms.hindawi.com © Hindawi Publishing Corp.

ON UNIVALENT FUNCTIONS DEFINED BY A GENERALIZED

S˘

AL˘

AGEAN OPERATOR

F. M. AL-OBOUDI

Received 17 August 2001 and in revised form 18 March 2002

We introduce a class of univalent functionsRn_(λ,α)_{deﬁned by a new diﬀerential} opera-torDn_{f (z)}_,_n_∈_N₀_{= {}₀_,₁_,₂_,..._}_{, where}_D0_{f (z)}₌_{f (z)}_,_D1_{f (z)}₌₍₁₋_{λ)f (z)}₊_λzf_(z)₌ D_λf (z),λ≥0, andDn_{f (z)}₌_D

λ(Dn−1f (z)). Inclusion relations, extreme points ofRn(λ,α), some convolution properties of functions belonging toRn_(λ,α)_{, and other results are given.}

2000 Mathematics Subject Classiﬁcation: 30C45.

1. Introduction. LetAdenote the class of functions of the form

f (z)=z+∞

k=2

akzk, (1.1)

analytic in the unit disc∆= {z:|z|<1}.

We denote byR(α)the subclass ofAfor which Ref_{(z) > α}_in_∆_{. For a function}_f_in

A, we deﬁne the following diﬀerential operator:

D0_{f (z)}₌_{f (z),} _(1.2)

D1_{f (z)}₌₍₁₋_{λ)f (z)}₊_λzf_(z)₌_D_λ_{f (z), λ}₀_, _(1.3) Dn_{f (z)}₌_D

λDn−1f (z). (1.4)

Iff is given by (1.1), then from (1.3) and (1.4) we see that

Dn_{f (z)}₌_z₊∞ k=2

1+(k−1)λnakzk. (1.5)

Whenλ=1, we get S˘al˘agean’s diﬀerential operator [8].

LetRn_(λ,α)_{denote the class of functions}_f_∈_A_{which satisfy the condition}

ReDn_{f (z)}_{> α, z}_∈_∆_, _(1.6)

for some 0≤α≤1,λ 0, andn∈N0= {0,1,2,...}. It is clear thatR0(λ,α)≡R(α)≡ Rn₍₀_,α)_{and that}_R1_(λ,α)_≡_R(λ,α)_{, the class of functions}_f_∈_A_satisfying

Ref(z)+λzf(z)> α, z∈∆, (1.7)

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The Hadamard product or convolution of two power seriesf (z)=∞k=0akzk and g(z)=∞k=0bkzkis deﬁned as the power series(f∗g)(z)=

_∞

k=0akbkzk,z∈∆.

The object of this paper is to derive several interesting properties of the classRn_(λ,α)

such as inclusion relations, extreme points, some convolution properties, and other results.

2. Inclusion relations. Theorem 2.3shows that the functions inRn_(λ,α)_{belong to} R(α)and hence are univalent. We need the following lemmas.

Lemma2.1. Ifp(z)is analytic in∆,p(0)=1andRep(z) >1/2,z∈∆, then for any functionF analytic in∆, the functionp∗Ftakes its values in the convex hull ofF(∆).

The assertion ofLemma 2.1follows by using the Herglotz representation forp. The next lemma is due to Fejér [3].

A sequencea0,a1,...,an,...of nonnegative numbers is called aconvex null sequence

ifan→0 asn→ ∞and

a0−a1 a1−a2··· an−an+1···0. (2.1)

Lemma2.2. Let{ck}_k=∞0be a convex null sequence. Then the functionp(z)=c0/2+ ∞

k=1ckzk,z∈∆, is analytic andRep(z) >0in∆.

Now we prove the following theorem.

Theorem2.3.

Rn+1_(λ,α)_⊂_Rn_(λ,α). _(2.2)

Proof. Letf belong toRn+1(λ,α)and let it be given by (1.1). Then from (1.5), we have

Re

1+₂₍₁1₋_α)

∞

k=2

k1+(k−1)λn+1akzk−1

>1

2. (2.3)

Now

_Dn_{f (z)}₌₁₊∞

k=2

k1+(k−1)λnakzk−1

=

1+₂₍₁1₋_α)

∞

k=2

k1+(k−1)λn+1akzk−1

∗

1+2(1−α) ∞

k=2

zk−1

1+(k−1)λ

.

(2.4)

ApplyingLemma 2.2, withc0=1 andck=1/(1+kλ),k=1,2,..., we get

Re

1+2(1−α) ∞

k=2

zk−1

1+(k−1)λ

> α. (2.5)

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ON UNIVALENT FUNCTIONS DEFINED BY A GENERALIZED 1431

We also have a better result thanTheorem 2.3.

Theorem2.4. Letf∈Rn+1_(λ,α)_{. Then}_f_∈_Rn_(λ,β)_{, where}

β=2₍λ₁2₊+_λ)((1₁+₊3λ)α₂_λ) α. (2.6)

Proof. Letf∈Rn+1_(λ,α)_{. It is shown in [}₉_{], as an example, that if}_λ_{0 and}

g(z)=z+

∞

k=2 zk

1+(k−1)λ, (2.7)

then

Reg(z) z >

4λ2₊₃_λ₊₁

2(1+λ)(1+2λ). (2.8)

Hence

Re

1+2(1−α) ∞

k=2

zk−1

1+(k−1)λ

>2λ2+(1+3λ)α

(1+λ)(1+2λ). (2.9)

Now an application ofLemma 2.1to(Dn_{f (z))}_{in the previous theorem completes}

the proof.

Remark2.5. If we putn=1 inTheorem 2.4, then we have

Ref_(z)₊_λzf_(z)_{> α} _⇒_Re_f_{(z) >}2λ2+(1+3λ)α

(1+λ)(1+2λ), (2.10)

which is an improvement of the result of Saitoh [7] forλ≥1, where he shows that, for λ >0,

Ref(z)+λzf(z)> α⇒Ref(z) >2α+λ

2+λ . (2.11)

UsingTheorem 2.4 ((n−m)times ) we get, after some calculations, the following theorem.

Theorem2.6. Letf∈Rn_(λ,α)_{and let}_{n > m}_{0. Then}_f_∈_Rm_(λ,β)_if

β= ₍ 1+3λ

1+λ)(1+2λ) n−m

α+₍ 2λ2

1+λ)(1+2λ) n−m−1

k=0

₁₊₃_λ (1+λ)(1+2λ)

k α.

(2.12)

If we putm=0 inTheorem 2.6, we obtain the following interesting result.

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Remark2.8. SinceDλ(given by (1.3)) is a linear function ofλ, it is clear that

Rn_(λ,α)_⊂_Rn_(λ/_,α), _(2.13)

whereλ > λ/_.

The following theorem deals with the partial sum of the functions inRn_(λ,α)_{. For}

the proof we need the following result, due to Ahuja and Jahangiri [2].

Lemma2.9. Let−1< t S=4.567802. Then

Re m

k=2 zk−1 k+t−1

>−₁1₊_t, z∈∆. (2.14)

Theorem2.10. LetSm(z,f )denote themth partial sum of a functionfinRn(λ,α).

Iff∈Rn_(λ,α)_and_λ_≥₁_/s₌₀_._{21892, then}_S

m(z,f )∈Rn−1(λ,β), where

β=2α_λ+₊λ₁−1. (2.15)

Proof. Letf∈Rn_(λ,α)_{and let it be given by (}_1.1_{). Then from (}_1.5_{) we have}

Re

1+

∞

k=2

> α (2.16)

or

Re

1+_λ₊2₁

∞

k=2

>2α+λ−1

λ+1 . (2.17)

Now

Dn−1_S

m(z,f )=1+ m

k=2

k1+(k−1)λn−1akzk−1

=

1+ 2

λ+1

∞

k=2

∗

1+λ+1

2λ m

k=2

zk−1

1/λ+(k−1)

, λ >0.

(2.18)

FromLemma 2.9, we see that, forλ≥1/s=0.21892,

Re

m

k=2

zk−1

1/λ+(k−1)>− λ

λ+1, (2.19)

hence

Re

1+λ₂+_λ1

m

k=2

zk−1

1/λ+(k−1)

>1

2, (2.20)

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Now we prove the following theorem.

Theorem2.11. The setRn_(λ,α)_{is convex.} Proof. Let the functions

fi(z)=z+ ∞

k=2

akizk (i=1,2) (2.21)

be in the classRn_(λ,α)_{. It is suﬃcient to show that the function} _h(z)₌_µ

1f1(z)+ µ2f2(z), withµ1andµ2nonnegative andµ1+µ2=1, is in the classRn(λ,α).

Since

h(z)=z+

∞

k=2

µ1ak1+µ2ak2zk, (2.22)

then from (2.4) we have

Dn_h(z)₌₁₊∞ k=2

kµ1ak1+µ2ak21+(k−1)λnzk−1, (2.23)

hence

ReDn_h(z)₌_Re₁₊_µ 1

∞

k=2

k1+(k−1)λnak1zk−1

+Re

1+µ2 ∞

k=2

k1+(k−1)λnak2zk−1

.

(2.24)

Sincef1,f2∈Rn(λ,α), this implies that

Re

1+µi ∞

k=2

k1+(k−1)λnakizk−1

>1+µi(α−1) (i=1,2). (2.25)

Using (2.25) in (2.24), we obtain

ReDn_h(z)_>₁₊_α_µ 1+µ2

−µ1+µ2

, (2.26)

and sinceµ1+µ2=1, the theorem is proved.

Hallenbeck [4] showed that

Ref_{(z) > α} _⇒_Ref (z)

z > (2α−1)+2(1−α)log 2. (2.27)

UsingTheorem 2.3and (2.27), we obtain the following theorem.

Theorem2.12. Letf∈Rn_(λ,α)_{. Then}

ReD

n_{f (z)}

z > (2α−1)+2(1−α)log 2. (2.28)

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3. Extreme points. The extreme points of the closed convex hull ofR(α)were de-termined by Hallenbeck [4]. We denote the closed convex hull of a familyF by clcoF, and we make use of some results in [4] to determine the extreme points ofRn_(λ,α)_.

Theorem3.1. The extreme points ofRn_(λ,α)_are

fx(z)=z+2(1−α) ∞

k=2

xk−1_zk

k1+(k−1)λn, |x| =1, z∈∆. (3.1)

Proof. SinceDn_:_f_→_Dn_f _{is an isomorphism from}_Rn_(λ,α)_to_R(α)_{, it preserves}

the extreme points and, in [4], it is shown that the extreme points ofR(α)are

z+2(1−α) ∞

k=2

1 kxk−

1_zk_, _|_x_{| =}₁_{, z}_∈_∆_. _(3.2)

Hence from (1.5), we see that the extreme points of clcoRn_(λ,α)_{are given by (}_3.1_).

Since the familyRn_(λ,α)_{is convex (}_{Theorem 2.6}_{) and therefore equal to its convex hull,}

we get the required result.

As consequences ofTheorem 3.1, we have the following corollary.

Corollary3.2. Letf belong toRn_(λ,α)_{and let it be given by (}_1.1_{). Then}

ak_k 2(1−α)

1+(k−1)λn, k 2. (3.3)

This result is sharp as shown by the functionfx(z)given by (3.1).

Corollary3.3. Iff∈Rn(λ,α), then

f (z) r+

∞

k=2

2(1−α) k1+(k−1)λnr

k_, _|_z_{| =}_{r ,}

_f_(z)₁₊∞ k=2

2(1−α)

1+(k−1)λnrk−

1_, _|_z_{| =}_{r .}

(3.4)

This result is sharp as shown by the functionfx(z)given by (3.1) atz=xr.

4. Convolution properties. Ruscheweyh and Sheil-Small [6] veriﬁed the Polya-Schoenberg conjecture and its analogous results, namely,C∗C⊂C,C∗S_⊂_S_{, and} C∗K⊂K, whereC, S_{, and} _K_{denote the classes of convex, starlike, and}

close-to-convex univalent functions, respectively. In the following, we prove the analogue of the Polya-Schoenberg conjecture for the classRn_(λ,α)_.

Theorem4.1. Letf∈Rn_(λ,α)_and_g_∈_C_{. Then}_f_∗_g_∈_Rn_(λ,α)_. Proof. It is known that ifgis convex univalent in∆, then

Reg(z) z >

1

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Using convolution properties, we have

ReDn_(f_∗_g)(z)₌_Re_Dn_{f (z)}_∗g(z) z

, (4.2)

and the result follows by application ofLemma 2.1.

Theorem4.2. Letf andgbelong toRn_(λ,α)_{. Then}_f_∗_g_∈_Rn_(λ,β)_{, where}

β=λ(2α₂+_(λ1)₊+₁4₎α−1 α. (4.3)

Proof. Letg(z)=z+∞k=2bkzk∈Rn(λ,α), then

Re

1+

∞

k=2

k1+(k−1)λnbkzk−1

> α. (4.4)

Letc0=1 and

ck=_(k λ+1

+1)[1+kλ]n, k 1. (4.5)

Then{ck}∞_k=0is a convex null sequence. Hence, byLemma 2.2, we have

Re

1+

∞

k=2

λ+1

k1+(k−1)λnzk−1

>1

2. (4.6)

Now we take the convolution of (4.4) and (4.6) and applyLemma 2.1to obtain

Re

1+λ+1

∞

k=2 bkzk−1

> α (4.7)

or

Reg(z) z =Re

1+

∞

k=2 bkzk−1

>λ+α

λ+1. (4.8)

Hence

Re _g(z)

z −

2α+λ−1 2(λ+1)

>1

2. (4.9)

Sincef∈Rn_(λ,α)_{, by applying}_{Lemma 2.1}_{, we obtain}

ReDn_{f (z)}_∗g(z)

z −

2α+λ−1 2(λ+1)

> α (4.10)

or

ReDn_{f (z)}_∗g(z) z

>λ(2α+1)+4α−1

2(λ+1) =β, (4.11)

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Remark4.3. If we putλ=0 inTheorem 4.2, we get the corresponding result for functions inR(α), given by Ahuja [1].

References

[1] O. P. Ahuja,Univalent functions whose derivatives have a positive real part, Rend. Mat. (7)2 (1982), no. 1, 173–187.

[2] O. P. Ahuja and M. Jahangiri,On the derivatives of a family of analytic functions, Math. Japon. 47(1998), no. 1, 67–72.

[3] L. Fejér,Über die positivität von summen, die nach trigonometrischen oder Legendreschen funktionen fortschreiten. I, Acta Szeged2(1925), 75–86 (German).

[4] D. J. Hallenbeck,Convex hulls and extreme points of some families of univalent functions, Trans. Amer. Math. Soc.192(1974), 285–292.

[5] S. Ponnusamy,Diﬀerential subordination and starlike functions, Complex Variables Theory Appl.19(1992), no. 3, 185–194.

[6] St. Ruscheweyh and T. Sheil-Small,Hadamard products of Schlicht functions and the Pólya-Schoenberg conjecture, Comment. Math. Helv.48(1973), 119–135.

[7] H. Saitoh,Properties of certain analytic functions, Proc. Japan Acad. Ser. A Math. Sci. 65 (1989), no. 5, 131–134.

[8] G. ¸S. S˘al˘agean,Subclasses of univalent functions, Complex Analysis—Fifth Romanian-Finnish Seminar, Part 1 (Bucharest, 1981), Lecture Notes in Math., vol. 1013, Springer, Berlin, 1983, pp. 362–372.

[9] Z. Zhongzhu and S. Owa,Convolution properties of a class of bounded analytic functions, Bull. Austral. Math. Soc.45(1992), no. 1, 9–23.

F. M. Al-Oboudi: Mathematics Department, Science Sections, Girls College of Education, Sitteen Street, Malaz, Riyadh 11417, Saudi Arabia