Properties of Certain Subclass of Multivalent Functions with Negative Coefficients

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International Journal of Mathematics and Mathematical Sciences Volume 2012, Article ID 362312,21pages

doi:10.1155/2012/362312

Research Article

Properties of Certain Subclass of Multivalent

Functions with Negative Coefficients

Jingyu Yang

1, 2

_{and Shuhai Li}

1

1_{Department of Mathematics, Chifeng University, Inner Mongolia, Chifeng 024000, China} 2_{School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, China}

Correspondence should be addressed to Shuhai Li,[email protected]

Received 26 December 2011; Revised 26 March 2012; Accepted 27 March 2012

Academic Editor: Marianna Shubov

Copyrightq2012 J. Yang and S. Li. 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.

Making use of a linear operator, which is defined by the Hadamard product, we introduce and

study a subclass Ya,cA, B;p, λ, α of the class Ap. In this paper, we obtain the coeﬃcient

inequality, distortion theorem, radius of convexity and starlikeness, neighborhood property, modified convolution properties of this class. Furthermore, an application of fractional calculus operator is given. The results are presented here would provide extensions of some earlier works. Several new results are also obtained.

1. Introduction

LetApdenote the class of functions

fz zp−

∞

k1

_a_k_p_zkp _p_∈_N_{₁_,₂_{, . . .}_} _1.1

which are analytic andp-valent in the open unit discU{z:|z|<1}on the complex plane

C.

We denote byS∗_pA, BandKpA, B −1≤B < A≤1the subclass ofp-valent starlike functions and the subclass ofp-valent convex functions, respectively, that is,see for details

1,2

S∗_pA, B

fz∈Ap:zf

_z

pfz ≺

1Az

1Bz z∈U; −1≤B < A≤1

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KpA, B

fz∈Ap: 1

p zfz pfz ≺

1Az

1Bz z∈U; −1≤B < A≤1

.

1.2

A functionf ∈Apis said to be analytic starlike of orderξif it satisfies

Re

zfz

pfz > ξ 1.3

for someξ 0≤ξ <1and for allz∈U.

Further, a functionf ∈Apis said to be analytic convex of orderξif it satisfies

Re

1

p zfz

pfz > ξ 1.4

for someξ0≤ξ <1and for allz∈U. Forfz zp∞

k1akpzkpandgz zp

_∞

k1bkpzkp, the Hadamard product or convolutionis defined by

f∗gz zp

∞

k1

akpbkpzkp

g∗fz. 1.5

The linear operatorLpa, cis defined as followssee Saitoh3:

Lpa, cfz ϕpa, c;z∗fz fz∈Ap, 1.6

andϕpa, c;zis defined by

ϕpa, c;z zp

∞

k1

a_k

c_kz

kp _a_∈_R_; _c_\_z−

0; z−0 :{0,−1,−2, . . .}

, 1.7

wherev_kis the pochharmmer symbol definedin terms of the Gamma functionby

ν_n Γνn

Γν

1, n0,

νν1· · ·νn−1, n∈N. 1.8

The operator Lpa, c was studied recently by Srivastava and Patel 4. It is easily verified from1.6and1.7that

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Moreover, forfz∈Ap,

Lpa, afz fz, Lp

p1, pfz zf

_z

p ,

Lp

np,1fz Dnp−1fz n >−p,

1.10

where Dnp−1_f_z _{denotes the Ruscheweyh derivative of a function}_f_z _∈ _A_z _{of order}

np−1see5.

Aouf, Silverman and Srivastava 6 introduced the class of P_a,c A, B;p, λ by use of linear operator Lpa, c, further investigated the properties of this class. In 7, Sok óŁinvestigated several properties of the linear Aouf-Silverman-Srivastava operator and furthermore obtained the corresponding characterizations of multivalent analytic functions which were studied by Aouf et al. in paper 6. The properties of multivalent functions with negative coefficients were studied in8–10. References11,12gave the results of the univalent function with negative coefficients.

In this paper, we will use operatorLpa, cto define a new subclass ofApas follows. Forp∈N, a >0, c >0, α≥0 and for the parametersλ, A, andBsuch that

−1≤B < A≤1, −1≤B <0, 0≤λ < p, 1.11

we say that a functionfz∈Apis in the classY_a,c A, B;p, λ, αif it satisfies the following subordination:

1

p−λ

Lpa, cfz zp−1 −α

Lpa, cfz pzp−1 −1

−λ

≺ 1Az

1Bz z∈U, 1.12

or equivalently, if the following inequality holds true:

Lpa, cfz/zp−1−αLpa, cfz/pzp−1−1−p

BLpa, cfz/zp−1−αLpa, cfz/pzp−1−1−pB A−Bp−λ

<1.

1.13

From the above definition, we can imply that the function class P_a,c A, B;p, λ in

6 is the special case ofY_a,c A, B;p, λ, αin our present paper because Y_a,c A, B;p, λ,0

P_a,c A, B;p, λ.

Since Y_a,c A, B;p, λ,0 P_a,c A, B;p, λ, then, like 6, we have the following subclasses which were studied in many earlier works:

1Y_n_p,1−1,1;p, λ,0 Qnp−1λ 0 ≤ λ < 1; n > −p; p ∈ N Aouf and Darwish

8,

2Y_a,a A, B;p,0,0 P∗p, A, B Shukla and Dashrath9,

3Y_a,a 1,−1;p, λ,0 Fp1, λ 0≤λ < p; p∈N Lee et al.10,

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5Y_n_p,1−1,1; 1, λ,0 Qnλ n ∈ N0 : N ∪ {0}; 0 ≤ λ < 1 Uralegaddi and

Sarangi12.

The purpose of this paper is to give various properties of classY_a,c A, B;p, λ, α. We extend the results of basic paper6.

2. Necessary and Sufficient Condition for

f

z

∈

Y

a,c

A, B;

p, λ, α

Theorem 2.1. Let the functionfz ∈ Apbe given by1.1, thenfz ∈ Y_a,c A, B;p, λ, αif and only if

∞

k1

φk

p, α;B, a, cakp≤A−B, 2.1

where

φk

p, α;B, a, c

pα1−Bkpa_k

pp−λc_k . 2.2

Proof. For the suﬃcient condition, letfz zp−∞_k₁|akp|zkp, we have

BLpa, cfz/zp−1−αLpa, cfz/pzp−1−pB A−Bp−λ

pLpa, cfz−αeiθLpa, cfz−pzp−1−p2zp−1

BpLpa, cfz−αeiθ_L_p_{a, c}_f_z₋_pzp−1₋_p_pB _A₋_B_p₋_λ_zp−1

,

2.3

then

−BpLpa, cfz−αeiθLpa, cfz−pzp−1−ppB A−Bp−λzp−1

−∞

k1

pkpakp

a_k

c_kz

kp−1₋_αeiθ

−

∞

k1

kpakp

a_k

c_kz

kp−1

−−∞

k1

Bpkpakp

a_k

c_kz

kp−1₋_αBeiθ

−

∞

k1

kpakp

a_k

c_kz

kp−1

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≤∞

k1

pkpakp

a_k

c_k|z|

kp−1_α_eiθ∞

k1

kpakp

a_k

c_k|z|

kp−1

|B|

∞

k1

pkpakp

a_k

c_k|z|

kp−1_α_|_B_|∞

k1

kpakp

a_k

c_k|z|

kp−1

−pA−Bp−λ|z|p−1

pα

∞

k1

kp1−Bakp

a_k

c_k −pA−B

p−λ z∈∂U. 2.4

By the maximum modulus theorem, for anyz∈U, we have

−BpLpa, cfz−αeiθLpa, cfz−pzp−1−ppB A−Bp−λzp−1

≤pα

∞

k1

kp1−Bakp

a_k

c_k −pA−B

p−λ

∞

k1

φk

p, α;B, a, cakp−A−B

≤0,

2.5

this impliesfz∈Y_a,c A, B;p, λ, α.

For necessary condition, letfz∈Y_a,c A, B;p, λ, αbe given by1.1, then

Lpa, cfz zp−1 p−

∞

k1

kpakp

a_k

c_kz

k_, _2.6

from1.13and2.6, we find that

BLpa, cfz/zp−1−αLpa, cfz/pzp−1−1−pB A−Bp−λ

−

_∞

k1

kpakpak/ckzk−α−

_∞

k1

kp/pakpak/ckzk

−A−Bp−λ−B∞_k₁kpakpak/ckzk−αB−

_∞

k1

kp/pakpak/ckzk

<1.

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Now, since|Rez| ≤ |z|for allz, we have

Re

−∞

k1

_∞

k1

kp/pakpak/ckzk

_∞

k1

kp/pakpak/ckzk

<1.

2.8

We choose value ofzon the real axis so that the expressionLpa, cfz/zp−1is real. Then, we have

Re −

_∞

k1

_∞

k1

kp/pakpak/ckzk

_∞

k1

kp/pakpak/ckzk

−

_∞

k1

_∞

k1

kp/pakpak/ckzk

_∞

k1

kp/pakpak/ckzk

,

2.9

−∞_k₁kpakpak/ckzk−α−

_∞

k1

kp/pakpak/ckzk

_∞

k1

kp/pakpak/ckzk

−

_∞

k1

kpakpak/ckzk−α

_∞

k1

kp/pakpak/ck|z|k

−A−Bp−λ−B∞_k₁kpakpak/ckzk−αB

_∞

k1

kp/pakpak/ck|z|k

<1.

2.10

Lettingz → 1−throughout real values in2.10, we get

−∞

k1

kpakpak/ck−α

_∞

k1

kp/pakpak/ck

−A−Bp−λ−B∞_k₁kpakpak/ck−αB

_∞

k1

kp/pakpak/ck

<1.

2.11

So we have

_∞

k1p

kpakpak/ck α

_∞

k1

kpakpak/ck

A−Bp−λB∞_k₁pkpakpak/ck αB

_∞

k1

kpakpak/ck

<1,

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it is

∞

k1

pα1−Bkpakp

a_k

c_k ≤pA−B

p−λ. 2.13

This completes the proof of the theorem.

Corollary 2.2. Let the functionfz∈Apbe given by1.1. Iffz∈Y_a,c A, B;p, λ, α,then

akp≤

pA−Bp−λc_k

pα1−Bkpa_k

k, p∈N. 2.14

The result is sharp for the function given by

fz zp−

∞

k1

pA−Bp−λc_k

pα1−Bkpa_kz

kp_. _2.15

3. Distortion Inequality of Class

Y

_a,c

A, B;

p, λ, α

Theorem 3.1. If a functionfzdefined by1.1is in the classY_a,c A, B;p, λ, α, then

p!

p−m! −

cpA−Bp−λp!

apα1−Bp1−m!|z|

|z|p−m≤fmz

≤

p!

p−m!

cpA−Bp−λp!

apα1−Bp1−m!|z|

|z|p−m z∈U; a≥c >0; m∈N0; p > m

.

3.1

The result is sharp for the functionfzgiven by

fz zp−cA−B

p−λ a1−Bpαz

1p _p_∈_N_. _3.2

Proof. Sincefz zp₋∞

k1|akp|zkp, then

fmz p! p−m!z

p−m₋∞

k1

kp!

kp−m!akpz

kp−m_. _3.3

Fromfz∈Y_a,c A, B;p, λ, α, usingTheorem 2.1, we have

apα1−B1p cpA−Bp−λp1!

∞

k1

kp!akp≤ ∞

k1

pαkp1−Ba_k

pA−Bp−λc_k akp≤1

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which readily yields

∞

k1

kp!akp≤

cpA−Bp−λp!

apα1−B . 3.5

By3.3and3.5, we can imply3.1.

4. Radius of Starlikeness and Convexity

Theorem 4.1. Let the functionfzdefined by1.1be in the classY_a,c A, B;p, λ, α. Then,fzis starlike ofη 0≤η <1in|z|< rηp, k, α, A, B, a, cwhere

rη

p, k, α, A, B, a, cinf

p1−ηφk

p, α;B, a, c

A−Bkp1−η

1/k

4.1

andφkp, α;B, a, cis defined by2.1.

Proof. We must show that

zf_pf_zz−1<1−η for|z|< rη

p, k, α, A, B, a, c. 4.2

Since

zf_pf_zz−1

−∞k1kakpzk

p−∞_k₁pakpzk

≤

_∞

k1kakp|z|k

p−∞_k₁pakp|z|k

, 4.3

to prove4.1, it is suﬃcient to prove

_∞

k1kakp|z|k

p−∞_k₁pakp|z|k

<1−η. 4.4

It is equivalent to

∞

k1

kp1−ηakp

p1−η |z|

k_≤₁_. _4.5

ByTheorem 2.1, we have

∞

k1

φk

p, α;B, a, c

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hence4.5will be true if

kp1−ηakp

p1−η |z|

k_≤ φk

p, α;B, a, c

A−B akp. 4.7

It is equivalent to

|z| ≤

p1−ηφk

p, α;B, a, c

A−Bkp1−η

1/k

. 4.8

This completes the proof.

Theorem 4.2. Let the functionfzdefined by1.1be in the classY_a,c A, B;p, λ, α. Then,fzis convex ofξ0≤ξ <1in|z|< rξp, k, α, A, B, a, cwhere

rξ

p, k, α, A, B, a, cinf

p2₁₋_ξ_φ k

p, α;B, a, c

kpA−Bkp1−ξ

1/k

4.9

andφkp, α;B, a, cis defined by2.1.

Proof. It suﬃcient to show that

zf_pf_zz−

p−1

p

<1−ξ for|z|< rξ

p, k, α, A, B, a, c. 4.10

Since

zf_pf_zz−

p−1

p

−

_∞

k1k

kpakpzk

p2−∞

k1p

kpakpzk

≤

_∞

k1k

kpakp|z|k

p2₋∞

k1p

kpakp|z|k

, 4.11

to prove4.9, it is suﬃcient to prove

_∞

k1k

kpakp|z|k

p2₋∞

k1p

kpakp|z|k

<1−ξ. 4.12

It is equivalent to

∞

k1

kpkp1−ξakp

p2₁₋_ξ |z|

k_≤₁_. _4.13

ByTheorem 2.1, we have

∞

k1

φk

p, α;B, a, c

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hence4.13will be true if

kpkp1−ξakp

p2₁₋_ξ |z|

k_≤ φk

p, α;B, a, c

A−B akp. 4.15

It is equivalent to

|z| ≤

p2₁₋_ξ_φ k

p, α;B, a, c

kpA−Bkp1−ξ

1/k

. 4.16

This completes the proof.

5. δ

-Neighborhood of

Y

_a,c

A, B;

p, λ, α

Based on the earlier works by Aouf et al. 6, Altintas et al.13–15, and Aouf 16, we introduce theδ-neighborhood of a functionfz ∈ Apof the form1.1and present the relationship betweenδ-neighborhood and corresponding function class.

Definition 5.1. Forδ > 0, a > 0, c > 0, the δ-neighborhood of a functionfz ∈ Apis defined as follows:

N_δf

g :gz zp−

∞

k1

_b_k_p_zkp_∈_A_p_, ∞

k1

tkbkp−akp< δ , 5.1

where

tk

pα1−Bkpa_k

pA−Bp−λc_k . 5.2

Theorem 5.2. Let the functionfzdefined by1.1be in the classY_a_1,cA, B;p, λ, α. Then,

N_δf⊂Y_a,c A, B;p, λ, α 1 a1

. 5.3

This result is the best possible in the sense thatδcannot be increased.

Proof. Letfz∈Y_a_1,cA, B;p, λ, α, then byTheorem 2.1, we have

∞

k1

pα1−Bkpa1_k

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which is equivalent to

∞

k1

pα1−Bkpa_k

pA−Bp−λc_k akp≤ a

a1. 5.5

For any

gz zp−

∞

k1

bkpzkp∈Nδ

f, 5.6

we find from5.1that

∞

k1

pα1−Bkpa_k

pA−Bp−λc_k bkp−akp≤δ. 5.7

By5.5and5.7, we get

∞

k1

pα1−Bkpa_k pA−Bp−λc_k bkp

≤∞

k1

pα1−Bkpa_k

pA−Bp−λc_k akp

∞

k1

pα1−Bkpa_k

pA−Bp−λc_k bkp−akp

≤ a

a1 δ1.

5.8

ByTheorem 2.1, it implies thatgz∈Y_a,c A, B;p, λ, α.

To show the sharpness of the assertion ofTheorem 5.2, we consider the functionsfz

andgzgiven by

fz zp− cpA−B

p−λ

a1pα1−Bp1z

p1_∈_Y

a1,c

A, B;p, λ, α,

gz zp−

cpA−Bp−λ

a1pα1−Bp1

cpA−Bp−λδ apα1−B1p

zp1,

5.9

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Since

pαp11−Ba₁

pA−Bp−λc₁ b1p−a1p

pαp11−Ba₁ pA−Bp−λc₁ ·

δ,

5.10

sogz∈N_δf.

But

pαp11−Ba₁ pA−Bp−λc₁

cpA−Bp−λ

a1pαp11−B

a

a1 δ

>1,

5.11

byTheorem 2.1gz∈Y_a,c A, B;p, λ, α.

6. Properties Associated with Modified Hadamard Product

Following early works by Aouf et al. in6, we provide the properties of modified Hadamard product ofY_a,c A, B;p, λ, α.

For the function

fjz zp−

∞

k1

_a_k_p,j_zkp _j₁_,_2; _p_∈_N_, _6.1

the modified Hadamard product of the functionsf1zandf2zwas denoted byf1·f2z

and defined as follows:

f1·f2

z zp−

∞

k1

akp,1akp,2zkp

f2·f1

z. 6.2

Theorem 6.1. Letfjz j1,2given by6.1be in the classY_a,c A, B;p, λ, α, then

f1·f2

z∈Y_a,c A, B;p, γ, α, 6.3

where

γ:p− cpA−B

p−λ2

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The result is sharp for the functionsfjz j1,2given by

fjz zp₋ cpA−B

p−λ apα1−B1pz

p1_. _6.5

Proof. ByTheorem 2.1, we need to find the largestγsuch that

∞

k1

pα1−Bkpa_k

pA−Bp−γc_k akp,1·akp,2≤1. 6.6

Sincefjz∈Y_a,c A, B;p, λ, α, j1,2, then we see that

∞

k1

pα1−Bkpa_k

pA−Bp−λc_k akp,j≤1

j1,2. 6.7

By Cauchy-Schwartz inequality, we obtain

∞

k1

pα1−Bkpa_k pA−Bp−λc_k

akp,1·akp,2≤1. 6.8

This implies that we only need to show that

akp,1·akp,2

p−γ ≤

akp,1·akp,2

p−λ k∈N 6.9

or equivalently that

akp,1·akp,2≤

p−γ

p−λ. 6.10

By making use of the inequality6.8, it is suﬃcient to prove that

pA−Bp−λc_k

pα1−Bkpa_k ≤ p−γ

p−λ k∈N. 6.11

From6.11, we have

γ≤p− pA−B

p−λ2c_k

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Define the functionΦkby

Φk:p− pA−B

p−λ2c_k

pα1−Bkpa_k k∈N. 6.13

We see thatΦkis an increasing function ofk. Therefore, we conclude that

γ≤Φ1 p− pA−B

p−λ2c₁

pα1−Bkpa₁. 6.14

By using arguments similar to those in the proof ofTheorem 6.1, we can derive the following result.

Theorem 6.2. Letf1zdefined by6.1be in the classY_a,c A, B;p, λ, α,f2zdefined by6.1be in the classY_a,c A, B;p, γ, α, then

f1·f2

z∈Y_a,c A, B;p, ξ, α, 6.15

where

ξ:p− cpA−B

p−λp−γ

apα1−B1p . 6.16

The result is sharp for the functionsfjz j1,2given by

f1z zp− cpA−B

p−λ apα1−B1pz

p1 _p_∈_N_,

f2z zp− cpA−B

p−γ apα1−B1pz

p1 _p_∈_N_.

6.17

Theorem 6.3. Letfjz j 1,2defined by6.1be in the classY_a,c A, B;p, λ, α, then the function

hzdefined by

hz zp−

∞

k1

akp,12akp,22

6.18

belongs to the classY_a,c A, B;p, χ, αwhere

χ:p− 2pcA−B

p−λ

apα1−B1p. 6.19

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Proof. ByTheorem 2.1, we want to find the largestχsuch that

∞

k1

pα1−Bkpa_k pA−Bp−χc_k

akp,12akp,22

≤1. 6.20

Sincefjz∈Y_a,c A, B;p, λ, α j1,2, we readily see that

∞

k1

pα1−Bkpa_k

pA−Bp−χc_k akp,j≤1

j1,2. 6.21

From6.21, we have

∞

k1

pα21−B2kp2 p2_A₋_B2_p₋_χ2

_a

k

c_k

2

akp,j2≤1

j1,2, 6.22

then we have

∞

k1

pα21−B2kp2 p2_A₋_B2_p₋_χ2

a_k

c_k

2

akp,12akp,22

≤2. 6.23

From6.23, if we want to prove6.20, it is suﬃcient to prove there exists the largest

χsuch that

1

p−χ ≤

pαkp1−Ba_k

2pA−Bp−λ2c_k k∈N, 6.24

that is

χ≤p− 2pA−B

p−λ2c_k

pαkp1−Ba_k k∈N. 6.25

Now we defineΨkby

Ψk p− 2pA−B

p−λ2c_k

pαkp1−Ba_k k∈N. 6.26

We see thatΨkis an increasing function ofk. Therefore, we conclude that

χ≤Ψ1 p− 2pcA−Bp−λ

2

apαkp1−B 6.27

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7. Application of Fractional Calculus Operator

References17–19have studied the fractional calculus operators extensively. In this part, we only investigate the application of fractional calculus operator which was defined by6in the class ofY_a,c A, B;p, λ, α.

Definition 7.1see6. The fractional integral of orderμis defined, for a functionfz, by

D−μz fz

1

Γμ

_z

0

fz

z−ξ1−μdξ

μ >0, 7.1

where the function fz is analytic in a simply connected domain of the complex z-plane containing the origin and the multiplicity ofz−ξμ−1is removed by requiring logz−ξto be real whenz−ξ >0.

Definition 7.2see6. The fractional integral of orderμis defined, for a functionfz, by

Dμzfz

1

Γ1−μ

_z

0

fz

z−ξμdξ

0≤μ <1, 7.2

where the function fz is constrained, the multiplicity of z−ξ−μ is removed as in

Definition 7.1.

In our investigation, we will use the operatorsJδ,pdefined bycf,20–22

Jδ,pf

z: δp

zp

_z

0

tδ−1ftdt f∈Ap; δ >−p; p∈N 7.3

as well asDzμfor which it is well known thatsee23

Dμzzρ

Γρ1

Γρ−μ1z

ρ−μ _{ρ >}₋_1; _μ_∈_R _7.4

in terms of Gamma.

Lemma 7.3see Chen et al.18. For a functionfz∈Ap,

Dμz

Jδ,pf

z Γ

p1

Γp−μ1z

p−μ∞

k1

δpΓkp1

δkpΓkp−μ1akpz

kp−μ_,

Jδ,p

Dμz

fz

δpΓp1

δp−μΓp−μ1z

p−μ

∞

k1

δpΓkp1

δkp−μΓkp−μ1akpz

kp−μ

7.5

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Remark 7.4. Throughout this section, we assume further thata≥c >0.

Theorem 7.5. Let function defined by1.1be in the classY_a,c A, B;p, λ, α, then

Dz−μ

Jδ,pf

z≥ Γ

p1

Γpμ1|z|

pμ

1− cp

δpA−Bp−λ

apαpδ1pμ11−B|z|

,

7.6

Dz−μ

Jδ,pf

z≤ Γ

p1

Γpμ1|z|

pμ

1 cp

δpA−Bp−λ

apαpδ1pμ11−B|z|

7.7

μ >0; δ >−p; p∈N; z∈U, each of the assertions is sharp.

Proof. Since

D−μz

Jδ,pf

z Γ

p1

Γp−μ1z

pμ₋∞

k1

δpΓkp1

δkpΓkpμ1akpz

kpμ_, _7.8

then

D−μz

Jδ,pf

z Γ

p1

Γp−μ1|z

μ_|

zp−

∞

k1

δpΓkp1Γpμ1

δkpΓkpμ1Γp1akpz

kp

.

7.9

LetGzis defined by

Gz zp−

∞

k1

δpΓkp1Γpμ1

δkpΓkpμ1Γp1akpz

kp_zp₋∞

k1

Qkakpzkp,

7.10

where

Qk

δpΓkp1Γpμ1

δkpΓkpμ1Γp1

k, p∈N; μ >0. 7.11

SinceQkis a decreasing function ofkwhenμ >0, we get

0< Qk≤Q1

δpp1

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sincefz∈Y_a,c A, B;p, λ, α, byTheorem 2.1, we get

apα1−Bp1

cpA−Bp−λ

∞

k1

_a_k_p_≤∞

k1

pαkp1−Ba_k

pA−Bp−λc_k akp≤1. 7.13

It is that

∞

k1

akp≤

cpA−Bp−λ

aαp1−Bp1, 7.14

then

|Gz|zp−

∞

k1

Qkakpzkp≥ |z|p−Q1|z|p1 ∞

k1

akp

≥ |z|p− cp

pδA−Bp−λ

apαδkpp1μ1−B|z|

p1_,

|Gz|zp−

∞

k1

Qkakpzkp≤ |z|pQ1|z|p1 ∞

k1

akp

≤ |z|p cp

pδA−Bp−λ

apαδkpp1μ1−B|z|

p1_.

7.15

From7.15, we obtain7.6and7.7, respectively.

Equalities in7.6and7.7are attained for the functionfzgiven by

Dz−μ

Jδ,pf

z Γ

p1

Γp−μ1z

pμ

1− cp

δpA−Bp−λ

apαδkppμ11−Bz

.

7.16

Theorem 7.6. Let function defined by1.1be in the classY_a,c A, B;p, λ, α, then

Dzμ

Jδ,pf

z≥ Γ

p1

Γp−μ1|z|

p−μ

1− cp

δpA−Bp−λ

apαpδ1p−μ11−B|z|

,

7.17

Dzμ

Jδ,pf

z≤ Γ

p1

Γp−μ1|z|

p−μ

1 cp

δpA−Bp−λ

apαpδ1p−μ11−B|z|

,

7.18

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Proof. Since

Dμz

Jδ,pf

z Γ

p1

Γp−μ1z

p−μ₋∞

k1

δpΓkp1

δkpΓkp−μ1akpz

kp−μ_, _7.19

then

Dμz

Jδ,pf

z Γ

p1

Γp−μ1z

−μ

zp−

∞

k1

δpΓkp1Γp−μ1

δkpΓkp−μ1Γp1akpz

kp

.

7.20

LetHzbe defined by

Hz zp₋∞

k1

δpΓkp1Γp−μ1

δkpΓkp−μ1Γp1akpz

kp

zp−

∞

k1

Ekkpakpzkp,

7.21

where

Ek

δpΓkpΓp−μ1

δkpΓkp−μ1Γp1

k, p∈N; μ >0. 7.22

SinceEkis a decreasing function ofkwhen 0≤μ <1, then

0< Ek≤E1

δp

δ1p1p−μ, 7.23

sincefz∈Y_a,c A, B;p, λ, α, byTheorem 2.1, we get

apα1−B cpA−Bp−λ

∞

k1

kpakp≤ ∞

k1

pαkp1−Ba_k

pA−Bp−λc_k akp≤1. 7.24

It is that

∞

k1

kpakp≤

cpA−Bp−λ

(20)

then

|Hz|zp−

∞

k1

Ekkpakpzkp

≥ |z|p−E1|z|p1

∞

k1

kpakp

≥ |z|p− cp

pδA−Bp−λ

apαδkpp1−μ1−B|z|

p1_,

|Hz|zp−

∞

k1

Ekkpakpzkp

≤ |z|pE1|z|p1

∞

k1

kpakp

≤ |z|p cp

pδA−Bp−λ

apαδkpp1−μ1−B|z|

p1_.

7.26

From7.26, we obtain7.17and7.18, respectively.

Equalities in7.17and7.18are attained for the functionfzgiven by

Dμz

Jδ,pf

z Γ

p1

Γp−μ1z

p−μ

1− cp

δpA−Bp−λ

apαδkpp−μ11−Bz

7.27

or equivalently by

Jδ,pf

z zp₋ cp

δpA−Bp−λ

apαδkpp−μ11−Bz

p1_. _7.28

Acknowledgments

The author thanks the referee for numerous suggestions that helped make this paper more readable. The present investigation was partly supported by the Natural Science Foundation of Inner Mongolia under Grant 2009MS0113.

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Properties of Certain Subclass of Multivalent Functions with Negative Coefficients

Research Article

Properties of Certain Subclass of Multivalent

Functions with Negative Coefficients

Jingyu Yang

and Shuhai Li

1. Introduction

2. Necessary and Sufficient Condition for

f

z

∈

Y

A, B;

p, λ, α

3. Distortion Inequality of Class

Y

A, B;

p, λ, α

4. Radius of Starlikeness and Convexity

5.

δ

-Neighborhood of

Y

A, B;

p, λ, α

6. Properties Associated with Modified Hadamard Product

7. Application of Fractional Calculus Operator

Acknowledgments

References

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Mathematics

Complex Analysis

Optimization

Combinatorics

Function Spaces

The Scientific

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Discrete Mathematics

Stochastic Analysis

_{and Shuhai Li}