Coefficient estimates for Ruscheweyh derivatives

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PII. S0161171204309051 http://ijmms.hindawi.com © Hindawi Publishing Corp.

COEFFICIENT ESTIMATES FOR RUSCHEWEYH DERIVATIVES

MASLINA DARUS and AJAB AKBARALLY

Received 5 September 2003

We consider functions f, analytic in the unit disc and of the normalized formf (z)= z+∞_n=2anzn. For functionsf∈R¯δ(β), the class of functions involving the Ruscheweyh derivatives operator, we give sharp upper bounds for the Fekete-Szegö functional|a3−µa22|.

2000 Mathematics Subject Classiﬁcation: 30C45.

1. Introduction. LetSdenote the class of normalized analytic univalent functionsf

deﬁned by

f (z)=z+

∞

n=2

anzn (1.1)

in the unit discD= {z:|z|<1}. Suppose that

S∗₌_f_∈_S_{: Re}zf(z)

f (z)

>0, z∈D

,

S∗_(β)₌_f_∈_S_:_argzf(z)

f (z)

<βπ₂ , z∈D

(1.2)

are classes of starlike and strongly starlike functions of orderβ (0< β≤1), respec-tively. Note thatS∗_(β)_⊂_S∗_{for 0}_{< β <}_{1 and}_S∗₍₁₎₌_S∗_[₅_{]. Kanas [}₂_{] introduced the} subclass ¯Rδ(β)of functionf∈S as the following.

Definition_1.1. _Forδ≥_0,β∈(₀,₁]_{, a function}f _{normalized by (}_1.1_{) belongs to}

¯

Rδ(β)if, forz∈D−{0}andDδf (z)≠0, the following holds:

argz

Dδ_{f (z)}

≤βπ₂ , (1.3)

whereDδ_f_{denotes the generalized Ruscheweyh derivative which was originally deﬁned} as the following.

Definition_1.2_[₆_]. _LetDnf_andf_{be deﬁned by (}_1.1_{). Then for}n∈N∪{₀}_,

Dn_{f (z)}₌ z

(1−z)n+1∗f (z), (1.4)

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Later in [1], Al-Amiri generalized the Ruscheweyh derivative Dδ _{for real numbers}

δ≥ −1 as a Hadamard product of the functionsfandz/(1−z)δ+1_.

Note that ¯R0(β)=S∗(β)for eachβ∈(0,1]and ¯R0(1)=S∗. In this note, we obtain

sharp estimates for|a2|,|a3|and the Fekete-Szegö functional for the class ¯Rδ(β). For the subclassS∗_{, sharp upper bounds for the functional}_|_a₃₋_µa2_2|_{have been obtained}

for all realµ[3,4].

2. Preliminary results. In proving our results, we will need the following lemmas. However, we ﬁrst denotePto be the class of analytic functions with positive real part inD.

Lemma_2.1. _Letp∈P_{and let it be of the form}p(z)=₁+∞_n=₁c_nzn_with_Rep(z) >_0.

Then

(i) |cn| ≤2forn≥1, (ii) |c2−c12/2| ≤2−|c1|2/2.

Lemma_2.2. _Letδ≥₀_andβ∈(₀,₁]_{. If}f∈R¯_δ(β)_{and is given by (}_1.1_{), then}

_a₂_≤ 2β δ+1,

_a₃_≤         

2β

(δ+2)(δ+1) ifβ≤ 1 3, 6β2

(δ+2)(δ+1) ifβ≥

1 3.

(2.1)

Proof. _LetF(z)=Dδf (z)=z+A₂z2+A₃z3+ ···_{. Since}f∈R¯_δ(β)_andDδf (z)∈

S∗_(β)_{, then}

zF_(z)

F(z) =pβ(z) (2.2)

and so

z1+2A2z+3A3z2+···

z+A2z2+A3z3+··· =

1+c1z+c2z2+··· β, (2.3)

which implies that

z+2A2z2+3A3z3+··· =z+βc1+A2 z2+

βc2+β(β−1)

2 c

2

1+βA2c1+A3

z3_+···_.

(2.4)

Equating the coeﬃcients, we have

A2=βc1, (2.5)

A3=β

2

c2−c 2 1

2

+3₄β2_c2

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Now, forδ≥ −1,Dδ_f_{has the Taylor expansion}

Dδ_{f (z)}₌_z₊∞ n=2

Γ(n+δ)

(n−1)!Γ(1+δ)anzn, z∈D, (2.7)

whereΓ(n+δ)denotes Euler’s functions with

Γ(n+δ)=δ(δ+1)···(δ+n−1)Γ(δ). (2.8)

Then

z+A2z2+A3z3+··· =z+Γ(

2+δ)

Γ(1+δ)a2z2+ Γ(

3+δ)

2Γ(1+δ)a3z3+···. (2.9)

Equating the coeﬃcients in (2.9), we have

a2Γ(

2+δ)

Γ(1+δ)=a2(δ+1)=A2. (2.10)

Then, from (2.5), we obtain

a2= βc1

δ+1. (2.11)

It follows that fromLemma 2.1(i)

_a₂_≤ 2β

δ+1, (2.12)

whereas the coeﬃcient ofz3_{in (}_2.9_{) is}

a3 Γ(

3+δ)

2Γ(1+δ)=a3

(δ+1)(δ+2)

2 =A3. (2.13)

From (2.6), we obtain

a3=

2

(δ+1)(δ+2)

β

2

c2−c 2 1

2

+3

4β

2_c2 1

. (2.14)

It follows fromLemma 2.1(ii) that

_a₃_≤ 2

(δ+1)(δ+2)

β

2

2−c1

2

+3

4β

2_c 12

, (2.15)

that is,

_a₃_≤         

2β

(δ+2)(δ+1) ifβ≤

1 3, 6β2

(δ+2)(δ+1) ifβ≥

1 3.

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3. Results. We ﬁrst consider the functional|a3−µa2_2|_{for complex}_µ_.

Theorem3.1. Letf∈R¯_δ(β)andβ∈(0,1]. Then forµcomplex,

_a₃₋_µa2 2≤

2β

(δ+1)(δ+2)max

1,β

3(δ+1)−2µ(δ+2) (δ+1)

. (3.1)

For eachµthere is a function in_R¯_δ_(β)such that equality holds.

Proof. From (2.11) and (2.14), we write

a3−µa22=

2

(δ+1)(δ+2)

β

2

c2−c 2 1

2

+3

4β

2_c2 1

−µ

_βc

1

δ+1 2

,

= 1

(δ+1)(δ+2)

β

c2−c 2 1

2

+β2

3(δ+1)−2µ(δ+2) 2(δ+1)2_(δ₊₂₎ c

2 1.

(3.2)

It follows from (3.2) andLemma 2.1(ii) that

_a₃₋_µa2 2≤

β (δ+1)(δ+2)

2−c12

2

+β2

3(δ+1)−2µ(δ+2)

2(δ+1)2_(δ₊₂₎

c12,

=_(δ₊₁2_)(δβ ₊₂₎+β2

3(δ+1)−2µ(δ+2) −β(δ+1)

2(δ+1)2_(δ₊₂₎ c1 2

,

(3.3)

which on usingLemma 2.1(i), that is,|c1| ≤2, gives

_a₃₋_µa2 2≤

        

2β

(δ+1)(δ+2) ifκ(δ)≤β(δ+1), _β2₆_(δ₊₁₎₋₄_µ(δ₊₂₎

(δ+1)2_(δ₊₂₎ ifκ(δ)≥β(δ+1),

(3.4)

whereκ(δ)= |β2₍₃_(δ₊₁₎₋₂_µ(δ₊₂₎₎_|_.

Equality is attained for functions in ¯Rδ(β)given by

zDδ_{f (z)}

Dδ_{f (z)} = ₁₊_z2

1−z2

β

, z

Dδ_{f (z)}

Dδ_{f (z)} = ₁₊_z

1−z

β

, (3.5)

respectively.

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Theorem3.2. Letf∈R¯_δ(β)andβ∈(0,1]. Then forµreal,

_a₃₋_µa2 2≤

                    

β2₆_(δ₊₁₎₋₄_µ(δ₊₂₎

(δ+1)2_(δ₊₂₎ ifµ≤

(6β−2)(δ+1)

4β(δ+2) , 2β

(δ+1)(δ+2) if

(6β−2)(δ+1)

4β(δ+2) ≤µ≤

(2+6β)(δ+1)

4β(δ+2) ,

β2₄_µ(δ₊₂₎₋₆_(δ₊₁₎

(δ+1)2_(δ₊₂₎ ifµ≥

(2+6β)(δ+1)

4β(δ+2) .

(3.6)

For eachµ, there is a function inR¯δ(β)such that equality holds.

Proof. _{Here we consider two cases.}

Case (i): µ≤3(δ+1)/2(δ+2).

In this case, (3.2) andLemma 2.1(ii) give

_a₃₋_µa2 2≤

β (δ+1)(δ+2)

2−c1 2

2

+β2

6(δ+1)−4µ(δ+2) 4(δ+1)2_(δ₊₂₎ c1

2

,

= 2β

(δ+1)(δ+2)+

β2₆_(δ₊₁₎₋₄_µ(δ₊₂₎ ₋₂_β(δ₊₁₎

4(δ+1)2_(δ₊₂₎ c1 2

,

(3.7)

and so, using the fact that|c1| ≤2, we obtain

_a₃₋_µa2 2≤

          

β2₆_(δ₊₁₎₋₄_µ(δ₊₂₎

(δ+1)2_(δ₊₂₎ ifµ≤

(6β−2)(δ+1)

4β(δ+2) ,

2β

(δ+1)(δ+2) if

(6β−2)(δ+1)

4β(δ+2) ≤µ≤

3(δ+1)

2(δ+2).

(3.8)

Equality is attained on choosingc1=c2=2 andc1=0,c2=2, respectively, in (3.2).

Case (ii): µ≥3(δ+1)/2(δ+2).

It follows from (3.2) andLemma 2.1(ii) that

_a₃₋_µa2

2≤_(δ₊₁_)(δβ ₊₂₎

2−c1 2

2

+β2

4µ(δ+2)−6(δ+1)

4(δ+1)2_(δ₊₂₎ c1 2

,

=_(δ₊₁2_)(δβ ₊₂₎+β2

4µ(δ+2)−6(δ+1) −2β(δ+1)

4(δ+1)2_(δ₊₂₎ c1 2

,

(3.9)

and so, using the fact that|c1| ≤2, we obtain

_a₃₋_µa2 2≤

         2β

(δ+1)(δ+2) if

3(δ+1)

2(δ+2)≤µ≤

(6β+2)(δ+1)

4β(δ+2) ,

β2₄_µ(δ₊₂₎₋₆_(δ₊₁₎

(δ+1)2_(δ₊₂₎ ifµ≤

(6β+2)(δ+1)

4β(δ+2) .

(3.10)

Equality is attained on choosing c1=0, c2=2 andc1=2i, c2= −2, respectively, in

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Theorem3.3. Letf∈R¯_δ(β)and let it be given by (1.1). Then

_a₃₋_a₂_≤ 2β

(δ+1)(δ+2) ifβ≤

3(δ+1)

5δ+1 . (3.11)

Proof. _Write

_a₃₋_a₂_≤_a₃₋2 3a

2 2

+2₃a2 2

−a2. (3.12)

Then since(6β−2)(δ+1)/4β(δ+2)≤2/3 forβ≤3(δ+1)/(5δ+1), it follows from Theorem 3.2that

_a₃₋_a₂_≤ 2β

(δ+1)(δ+2)+

2 3a2

2

−a2=λ(x), (3.13)

wherex= |a2| ∈[0,2β/(δ+1)]. Sinceλ(x)attains its maximum value atx=0, the theorem follows. This is sharp.

Acknowledgments. The work presented here was supported by IRPA Grant

09-02-02-0080-EA208. The author would like to thank all the referees for their suggestions regarding the contents of the note.

References

[1] H. S. Al-Amiri,Certain generalizations of prestarlike functions, J. Austral. Math. Soc. Ser. A 28(1979), no. 3, 325–334.

[2] S. Kanas,Class of functions deﬁned by Ruscheweyh derivatives, Bull. Malaysian Math. Soc. 18(1995), no. 1, 1–8.

[3] F. R. Keogh and E. P. Merkes,A coeﬃcient inequality for certain classes of analytic functions, Proc. Amer. Math. Soc.20(1969), 8–12.

[4] W. Ma and D. Minda,A uniﬁed treatment of some special classes of univalent functions, Proceedings of the Conference on Complex Analysis (Tianjin, 1992) (Z. Li, F. Y. Ren, L. Yang, and S. Y. Zhang, eds.), Conf. Proc. Lecture Notes Anal., vol. 1, International Press, Massachusetts, 1994, pp. 157–169.

[5] M. Obradovi´c and B. J. Santosh,On certain classes of strongly starlike functions, Taiwanese J. Math.2(1998), no. 3, 297–302.

[6] S. Ruscheweyh,New criteria for univalent functions, Proc. Amer. Math. Soc.49(1975), 109– 115.

Maslina Darus: School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor Darul Ehsan, Malaysia

E-mail address:[email protected]