On Certain Multivalent Starlike or Convex Functions with Negative Coefficients

Journal of Inequalities and Applications Volume 2010, Article ID 751721,11pages doi:10.1155/2010/751721

Research Article

On Certain Multivalent Starlike or

Convex Functions with Negative Coefficients

Neslihan Uyanik,

_{Erhan Deniz,}

_{Ekrem Kadio ˇglu,}

and Shigeyoshi Owa

1_{Department of Mathematics, Kazim Karabekir Faculty of Education, Atat ¨urk University,}

Erzurum 25240, Turkey

2_{Department of Mathematics, Science and Art Faculty, Atat ¨urk University, Erzurum 25240, Turkey} 3_{Department of Mathematics, Kinki University, Higashi-Osaka, Osaka 577-8502, Japan}

Correspondence should be addressed to Shigeyoshi Owa,[email protected]

Received 2 April 2010; Accepted 3 June 2010

Academic Editor: N. Govil

Copyrightq2010 Neslihan Uyanik et al. 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.

By means of a differential operator, we introduce and investigate some new subclasses of p -valently analytic functions with negative coefficients, which are starlike or convex of complex order. Relevant connections of the definitions and results presented in this paper with those obtained in several earlier works on the subject are also pointed out.

1. Introduction

LetAmpdenote the class of functions of the following form:

fz zp₋ ∞ kpm

akzk

ak≥0; m, p∈N:{1,2,3, . . .}

, 1.1

which are analytic and multivalent in the open unit dicsU{z:z∈Cand|z|<1}.

Letfq_{denote theqth-order ordinary differential operator for a functionf ∈ Amp,} that is,

fqz p! p−q!z

p−q₋ ∞ kpm

k!

k−q!akz

k−q_{, 1.2}

Next, we define the differential operatorDn_fq_as

Dnfqz p! p−q!z

p−q₋ ∞ kpm

k!

k−q!

k−q p−q

n

akzk−q m∈N; z∈ U. 1.3

In view of1.3, it is clear that

D0fqz fqz, D1fqz Dfqz 1 p−qz

fqz,

Dnfqz Dn−1Dfqz.

1.4

If we takep1 andq0 forDn_fq_{, thenD}n_fq_{become the differential operator defined by} S˘al˘agean1 .

Finally, in terms of a differential operator Dn_fq_{defined by1.3above, let E}n m,pq denote the subclass of Ampconsisting of functions f which satisfy the following inequality:

Em,pn

q

⎧ ⎨ ⎩f∈ Am

p: D

n_fq_z

zp−q /0, z∈C− {0}, fz z p₋ ∞

kpm

akzk, ak≥0 ⎫ ⎬ ⎭,

1.5

wherek∈N,n∈N0,k > n,m∈N;k−q/p−q≥p−q−n−1≥0;z∈ U.

Form∈N,n∈N0, andγ∈C− {0}, we define the next subclasses ofEnm,pq.

Enm,p

q, γ

f ∈Enm,p

q: Re

11

γ

Dn1_fq_z

Dn_fq_z −pqn

>0, z∈ U

,

Nm,pn

q, γ

⎧ ⎨

⎩f∈Enm,p

q: ∞

kpm

k!

k−q!

k−q p−q

n_k−q

p−q −pqn

_Re

γ

_γ γ

ak

≤ p!

p−q!

_γ

−pqn1Re γ

γ

,

Knm,p

q, γ

⎧ ⎨

⎩f∈Enm,p

q: ∞

kpm

k!

k−q!

k−q p−q

n

k−q

p−q−pqnγ

ak

≤ p!

p−q!γ−pqn1

,

1.6

Remark 1.1. 1E0_m,10, γ S∗bwas studied by Nasr and Aouf2 also see Bulboac˘a et al. 3 .

2E0

m,10,1−α TαmandE1m,10,1−α Cαm,α∈ 0,1were introduced by

Srivastava et al.4 .

3 E0_1,10,1 −α T∗αand E₁1_,10,1−α Cα,α ∈ 0,1 were introduced by Silverman5 .

4 K_m,0 ₁0, γ O0

mγand K1m,10, γ O1mγ were introduced by Parvathan and Ponnusanny6, pages 163-164 .

5For p 1 andq 0, the classesEn

m,pq, γ,Nm,pn q, γ, andKnm,pq, γare closely related withTn,mγ,On,mγ, andPn,mγwhich are defined by Owa and S˘al˘agean in7 .

In this paper we give relationships between the classes ofEnm,pq, γ,Nm,pn q, γ, and

Kn

m,pq, γ.In the particular case whenm∈Nandn0,p1, andq0, we obtain the same results as in8 .

2. Main Results

Our main results are contained in

Theorem 2.1. Letm∈N,n∈N0 and letγ∈C− {0}; then

1Kn

m,pq, γ⊆Enm,pq, γ; 2En

m,pq, γ⊆Nnm,pq, γ; 3ifγ∈0,∞, then

Knm,p

q, γEnm,p

q, γNm,pn

q, γ; 2.1

4ifγ∈−∞,0, thenNn

m,pq, γ/⊆Enm,pq, γ; 5ifγ∈−∞,0, thenEn

m,pq, γ/⊆Km,pn q, γ. Proof. 1Letf∈Kn

m,pq, γ.We prove that

Dn1_fq_z

Dn_fq_z −pqn

<γ, z∈ U. 2.2

Iff has the series expansion

fz zp−

∞

kpm

then

Dn1_fq_z

Dn_fq_z −pqn

−γ

≤ −

p!/p−q!1−pqn

p!/p−q!−∞_kpmk!/k−q!k−q/p−qnak|z|k−p

_∞

kpm

k!/k−q!k−q/p−qnak

k−q/p−q−pqn|z|k−p p!/p−q!−_k∞_pmk!/k−q!k−q/p−qnak|z|k−p −

_γ.

2.4

We use the fact thatDn_fq_z/zp−q

/

0 forz∈ U − {0}and limz→oDnfqz/zp−q

p!/p−q!; these imply

p!

p−q!− ∞

kpm

k!

k−q!

k−q p−q

n

ak|z|k−p>0 2.5

forz∈ U.

From2.4and2.5, we deduce

D

n1_fq_z

Dn_fq_z −pqn −γ

< −

p!/p−q!1−pqnγ

p!/p−q!−∞_kpmk!/k−q!k−q/p−qnak

.

_∞

kpm

k!/k−q!k−q/p−qnak

k−q/p−q−pqnγ p!/p−q!−∞_kpmk!/k−q!k−q/p−qnak

.

2.6

By using the definition ofKn

m,pq, γfrom this last inequality we, obtain2.2which implies

Re

1

γ

Dn1_fq_z

Dn_fq_z −pqn

>−1 z∈ U, 2.7

2Letf be inEn

m,pq, γ.Then2.7holds and, by using2.3, this is equivalent to

Re ⎧ ⎨ ⎩ 1 γ ⎛ ⎝

p!/p−q!zp−q₋∞ kpm

k!/k−q!k−q/p−qn1akzk−q

p!/p−q!zp−q₋∞ kpm

k!/k−q!k−q/p−qnakzk−q

−pqn

⎞ ⎠

⎫ ⎬ ⎭

>−1 z∈ U.

2.8

Forzt∈0,1ift → 1−, from2.8we obtain ⎧ ⎨ ⎩ ⎛ ⎝−

p!/p−q!∞_kpmk!/k−q!k−q/p−qn1ak

p!/p−q!−∞_kpmk!/k−q!k−q/p−qnak ⎞ ⎠

⎫ ⎬ ⎭≤

_γ2

Reγ −pqn

2.9

which is equivalent to

∞

kpm

k!

k−q!

k−q p−q

n

k−q p−q

_γ2

Reγ −pqn

ak≤

p!

p−q!

γ2

Reγ −pqn1

.

2.10

Then multiplying the relation last inequality with Reγ/|γ|,we obtainf∈Nn m,pq, γ. 3ifγis a real positive number, then the definitions ofNn

m,pq, γandKnm,pq, γare equivalent, henceNn

m,pq, γ Knm,pq, γ.By using1and2from this theorem, we obtain 3.

4We have the following two cases.

Case 1. γ∈p−q−n−1−m/p−q,0.

Letfm,αp, q, n;zbe defined by

fm,α

p, q, n;zzp−α

pm−q p−q

−n _p!

pm!

pm−q!

p−q! z

pm _2.11

and letα >0.We have

∞

kpm

k!

k−q!

k−q p−q

n_k−q

p−q −pqn

_Re

γ

_γ γ

ak

≤

pm!

pm−q!

pm−q p−q

n

pm−q

p−q −pqn

Reγ

_γ γ

×α

pm−q p−q

₋n _p!

pm!

pm−q!

p−q!

α p! p−q!

pm−q

p−q −pqn

γ

−γ −γ

or

∞

kpm

k!

k−q!

k−q p−q

n

k−q

p−q−pqn

_Re

γ

_γ γ

ak

≤ −α p! p−q!

−pqn1 m

p−qγ

≤0

< p! p−q!

−pqn1Re γ

γ γ

,

2.13

and then fm,αp, q, n;z∈Nm,pn q, γ see the definition ofNm,pn q, γ. Let now

Fz 11

γ

⎛

⎝Dn1fm,αq

p, q, n;z Dn_fq

m,α

p, q, n;z −pqn

⎞

⎠ _{z∈ U. 2.14}

Then, by a simple computation and by using the fact that

fm,αq

p, q, n;zfm,αqz

p!

p−q!z p−q_−α

pm!

pm−q!

pm−q p−q

−n _p!

pm!

pm−q!

p−q! z pm−q

p!

p−q!z

p−q_−α p!

p−q!

pm−q p−q

₋n

zpm−q_.

Dnfm,αqz

p!

p−q!z

p−q_−α p!

p−q!

pm−q p−q

−n_pm−q

p−q

n

zpm−q

p!

p−q!z

p−q_1−αzm_,

Dn1fm,αqz

p!

p−q!z p−q

1−αpm−q p−q z

m

,

we obtain

Fz 11

γ

Dn1_fq m,αz

Dn_fq m,αz

−pqn

11

γ

p!/p−q!zp−q_{1−αpm−q/p−qz}m

p!/p−q!zp−q_1−αzm −pqn

1−pqn1−αz

m_{−pqnpm−q/p−q}

γ1−αzm

1a−αbζ

γ1−αζ 1ϕζ,

2.16

whereζzm_{,a−pqn1, b−pqn1m/p−q, and}

ϕζ a−αbζ

γ1−αζ . 2.17

Forα >1 we, haveϕU C∞−Dc, d,whereDis the disc with the center

c α

2_b−a

γα2₋₁ 2.18

and the radius

d αb−a

γ1−α2. 2.19

We haveFU C∞−Dc1, dwhereDc, d {w:|w−c|< d}and we deduce that ReFz>0 for allz∈ Udoes not hold.

We have obtained that forα >1, fm,α∈Nm,pn q, γ,butfm,α/∈Enm,pq, γand in this case

Nn

m,pq, γ/⊆En_m,pq, γ.

Case 2. γ∈−∞, p−q−n−1−m/p−q.

We consider the functionfm,αdefined by2.11forα∈ 1,−pqn1γ/−p

qn1m/p−q. In this case, the inequality2.13 holds too and this implies that

fm,α∈Nm,pn q, γ.

We also obtain that f /∈En

5Letffm,αbe given by2.11, where α >|γ| −pqn1/|γ| −pqn1

m/p−qand|γ| −pqn1m/p−q>0.Then

∞

kpm

k!

k−q!

k−q p−q

n

k−q

p−q−pqnγ

ak

pm!

pm−q!

pm−q p−q

n

pm−q

p−q −pqnγ

×α

pm−q p−q

₋n _p!

pm!·

pm−q!

p−q!

α p! p−q!

pm−q

p−q −pqn

γ

> p!

p−q!γ

−pqn1

2.20

which implies that

fm,α/∈Knm,p

q, γ form∈N, n∈N0, γ∈−∞,0. 2.21

We have

Fz 1 1

γ

Dn1_fq m,αz

Dn_fq m,αz

−pqn

1 a−αbζ

γ1−αζ1ϕζ, 2.22

whereϕis given by2.17.

FromϕU Dc, dwherecanddare given by2.18and2.19, we obtain

ReFz≥1 αba

γα1. 2.23

If γ∈−∞, p−q−n−1−m/p−qand α∈|γ| −pqn1/|γ| −pqn1

m/p−q,1, then

αγbγa

and if

γ∈

p−q−n−1− m

p−q,0

,

α∈

γ−pqn1

_γ−pqn1m/p−q,

_γ−pqn1

_−pqn1m/p−q−γ

∩0,1,

2.25

then 2.24also holds. By combining2.24with 2.23and the definition ofEn

m,pq, γ, we obtain that

fm,α∈Enm,p

q, γ forα∈

γ−pqn1 _γ−pqn1m/

p−q,

_{γ−pqn1 −pqn1m/}

p−q−γ

∩0,1, γ∈−∞,0.

2.26

Appendix

In this paper, we discuss the classEn

m,pq, γof analytic functions with negative coefficients. Let us consider the functionsfgiven by

fz zp

∞

kp1

akzk A.1

which are analytic inU. For such a functionf, we say thatf∈Gn_1,pq, γif it satisfies

Re

11

γ

Dn1fqz

Dn_fq_z −pqn

>0 z∈ U A.2

for some complex numberγwith 0<Re1/γ<1/p−q−n−1. If we define the functionFforf∈Gn_1,pq, γby

Fz 1

1/γDn1_fq_z/Dn_fq_{z−pqn−i1−pqnIm1/γ}

then we know thatF is analytic in U,F0 1, and Refz > 0 z ∈ U. ThusF is the Carath´eodory function. Since the extremal function for the Carath´eodory functionFis given by

Fz 1z

1−z, A.4

we can write

11/γDn1_fq_z/Dn_fq_{z−pqn−i1−pqnIm1/γ}

11−pqnRe1/γ

1z

1−z. A.5

This shows us that

Dn1_fq_z

Dn_fq_z −pqnγ−iγ

1−pqnIm

1

γ

γ

11−pqnRe

1

γ

1z

1−z.

A.6

Noting that

Dn1fqz 1 p−qz

Dnfqz, A.7

we see that

1

p−q

Dn_fq_z

Dn_fq_z −

1

z

p−q−n−γiγ1−pqnIm

1

γ

γ

11−pqnRe

1

γ

2 1−z

1

z

,

A.8

that is,

1

p−q

Dnfqz Dn_fq_z −

1

z 2γ

11−pqnRe

1

γ

1

1−z. A.9

It follows from the above that

!_z

0

1

p−q

Dn_fq_t

Dn_fq_t − 1

t

dt2γ

11−pqnRe

1

γ

!z

0

1

1−tdt. A.10

Calculating the above integrations, we have that

1

p−qlogD

n_fq_{z−logz−2γ11−pqnRe}1

γ

Therefore, we obtain that

Dnfqz1/p−q z

1−z2γ11−pqnRe1/γ, A.12

that is,

Dn_fq_z

z

1−z2γ11−pqnRe1/γ

p−q

. A.13

Consequently, the function f defined by the above is the extremal function for the class

Gn_1,pq, γ. But our class En

m,pq, γ is defined with analytic functions f with negative coefficients. Thus we do not know how we can consider the extremal function for this class.

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