Some New Inclusion and Neighborhood Properties for Certain Multivalent Function Classes Associated with the Convolution Structure

Volume 2008, Article ID 318582,9pages doi:10.1155/2008/318582

Research Article

Some New Inclusion and Neighborhood Properties

for Certain Multivalent Function Classes Associated

with the Convolution Structure

J. K. Prajapat1_{and R. K. Raina}2

1_{Department of Mathematics, Sobhasaria Engineering College, NH-11, Gokulpura, Sikar,} Rajasthan 332001, India

2_{University of Agriculture and Technology, Udaipur 313001, India}

Correspondence should be addressed to R. K. Raina,rainark [email protected]

Received 13 August 2007; Accepted 7 January 2008

Recommended by Brigitte Forster-Heinlein

We use the familiar convolution structure of analytic functions to introduce two new subclasses of multivalently analytic functions of complex order, and prove several inclusion relationships asso-ciated with then, δ-neighborhoods for these subclasses. Some interesting consequences of these results are also pointed out.

Copyrightq2008 J. K. Prajapat and R. K. Raina. 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.

1. Introduction and preliminaries

LetApndenote the class of functions of the form

fz zp∞ kn

akzk p < n;n, p∈N{1,2, . . .}, 1.1

which are analytic andp-valent in the open unit disk

Uz;z∈C:|z|<1. 1.2

Iff ∈ Apnis given by1.1andg∈ Apnis given by

gz zp∞ kn

then the Hadamard productor convolutionf∗goffandgis definedas usualby

f∗gz:zp

∞

kn

akbkzk: g∗fz. 1.4

We denote byTpnthe subclass ofApnconsisting of functions of the form

fz zp₋∞ kn

akzk p < n;ak≥0k≥n;n, p∈N, 1.5

which arep-valent inU.

For a fixed functiongz∈ Apndefined by

gz zp∞ kn

bkzk p < n;bk≥0k≥n;n, p∈N, 1.6

we introduce a new classSλ_pg;n, b, mof functions belonging to the subclass ofTpn,which

consists of functionsfzof the form1.5, satisfying the following inequality:

_b1_λzz_ff_∗∗g_gmm11_zz λz₁₋2f_λ∗g_f∗gm2mz_z−p−m <1

z∈U;p∈N;m∈N0;p > m; 0≤λ≤1;b∈C\ {0}

.

1.7

We note that there exist several interesting newor knownsubclasses of our function classSλ_pg;n, b, m. For example, ifλ0 in1.7, we obtain the classSpg;n, b, mstudied very

recently by Prajapat et al.1. On the other hand, if the coefficients bk in1.6are chosen as

follows:

bk _kμ

pμ

r

μ≥0;k≥n;r, p, n∈N, 1.8

andnis replaced bynpin1.4and1.5, then we obtain the classSn,mp μ, r, λ, bofp-valently

analytic functionsinvolving the multiplier transformation operatorIpr, μdefined in 2

which was studied recently by Srivastava et al.3. Also, if we setλ 0 in 1.7and if the arbitrary sequencebkin1.6is selected as follows:

bk

μk−1

k−p

μ >−p;k≥n;p, n∈N, 1.9

also ifnis replaced bynpin1.4and1.5, then we obtain the classHpn,mμ, bofp-valently

analytic functionsinvolving the familiar Ruscheweyh derivative operatorinvestigated by Raina and Srivastava4. Further, when

in1.7, then Sλ_pg;n, b, mreduces to the class studied recently by Ali et al. 5. Moreover, when

gz zp∞ kn

α1k−p· · ·αqk−p β_1k−p· · ·β_sk−pk−p!z

k_,

αj∈Cj1,2, . . . , q, β_j∈C\ {0,−1,−2, . . .}j1,2, . . . , s

1.11

with the parameters

α1, . . . , αq, β₁, . . . , β_s 1.12

being so chosen that the coefficientsbkin1.6satisfy the following condition:

bk

α1

k−p· · ·

αq_k−p

β_1k−p· · ·β_sk−pk−p! ≥0, 1.13

then the classS_pλg;n, b, mtransforms into apresumablynew classSλ_pn, b, mdefined by

Sλ pn, b, m

f∈ Tpn:_b1

_zHq sα1

fm1_{z λz2H}q sα1

fm2

λzHsqα1

fm1_z

1−λHsqα1

fm −p−m <1

z∈U;q≤s1;m, q, s∈N0; 0≤λ≤1, p∈N;b∈C\ {0}

.

1.14

The operator

Hsqα1fz:Hsqα1, . . . , αq;β₁, . . . , β_sfz, 1.15

involved in1.14, is the Dziok-Srivastava linear operatorsee for details6; see also7,8 which contains such well-known operators as the Hohlov linear operator, Saitoh generalized linear operator, Carlson-Shaffer linear operator, Ruscheweyh derivative operator as well as its generalized version, the Bernardi-Libera-Livingston operator, and the Srivastava-Owa frac-tional derivative operator. One may refer to7or6for further details and references for these operators. The Dziok-Srivastava linear operator defined in6has further been generalized by Dziok and Raina7 see also8,9.

Following a recent investigation by Frasin and Darus10, letfz∈ Tpn, δ ≥0,then

aq, δ-neighborhood of the functionfzis defined by

Nq_n,δf

h:h∈ Tpn:hz zp− ∞

kn

ckzk, ∞

kn

kq1_a

k−ck≤δ

. 1.16

It follows from the definition1.16that if

then

Nq_n,δe h:h∈ Tpn:hz zp− ∞

kn

ckzk, ∞

kn

kq1_c k≤δ

. 1.18

We observe that

N0

2,δf Nδf,

N1

2,δf Mδf,

1.19

whereNδfandMδfdenote, respectively, theδ-neighborhoods of the function

gz z∞ k2

akzk ak≥0, z∈U, 1.20

defined by Ruscheweyh11and Silverman12. Finally, for a fixed function

gz zp∞ kn

bkzk∈ Apn p < n;bk>0k≥n;n, p∈N, 1.21

letP_pλg;n, b, m denote the subclass ofTpnconsisting of functionsfz of the form1.5

which satisfy the following inequality:

_b11−λp−m−1f∗gm1z λzf∗gm2z−p−m< p−m

z∈U, m∈N0;p∈N;p > m; 0≤λ≤1, b∈C\ {0}

.

1.22

The object of the present paper is to investigate the various properties and characteristics of functions belonging to the above-defined subclasses

Sλ

pg;n, b, m, Ppλg;n, b, m 1.23

of p-valently analytic functions in U. Apart from deriving coefficient inequalities for each of these function classes, we establish several inclusion relationships involving the n, δ -neighborhoods of functions belonging to these subclasses.

2. Coefficient bound inequalities

We begin by proving a necessary and sufficient condition for the functionfz∈ Tpnto be

in each of the classes

Sλ

pg;n, b, m, Ppλg;n, b, m. 2.1

Theorem 2.1. Letfz∈ Tpnbe given by1.5. Thenfzis in the classSλpg;n, b, mif and only

if

∞

kn

akbkλk−m−1 1k−p|b|

k m

≤ |b|λp−m−1 1

p m

Proof. Assume thatfz∈Sλ_pg;n, b, m.Then, in view of1.5–1.7, we get

R

λz2_f∗gm2_{z z1−λp−mf∗g}m1_{z−1−λp−mf∗g}m_z

λzf∗gm1_{z 1−λf∗g}m_z

>−|b|

2.3

which yields

R

∞

knakbkmkk−pλk−m−1 1zk−p p

mλp−m−1 1zp−m−∞knakbkmkλk−m−1 1zk−p

<|b| z∈U.

2.4

Puttingz r0 ≤ r < 1in2.4, the denominator expression on the left-hand side of2.4 remains positive for r 0,and also for allr ∈ 0,1. Hence, by lettingr→1−, through real values, inequality2.4leads to the desired assertion2.2ofTheorem 2.1.

Conversely, by applying the hypothesis2.2ofTheorem 2.1, and letting|z|1,we find that

_λzz_ff_∗∗_ggmm11_zz λz₁₋2_λf∗g_f∗m_g2mz_z−p−m

≤ |b|

mpλp−m−1 1−∞knakbkλk−m−1 1mk

p

mλp−m−1 1−∞knakbkλk−m−1 1mk

|b|.

2.5

Hence, by the maximum modulus principle, we infer thatfz∈ S_pλg;n, b, m,which completes the proof ofTheorem 2.1.

Remark 2.2. In the special case when

i bk _kμ

pμ

r

μ≥0;k≥n; r, p, n∈N;n→np. 2.6

Theorem 2.1corresponds to a result given recently by Srivastava et al.3, Theorem 1, page 3:

ii λ0; bk

μk−1

k−p

μ >−p;k≥n;n, p∈N; n→np. 2.7

Theorem 2.1yields the result given recently by Raina and Srivastava4, Theorem 1, page 3:

Theorem 2.1reduces to the result of Orhan and Kamali13, Lemma 1, page 57:

iv λ0, m0, bp1−α p∈N; 0≤α <1. 2.9

Theorem 2.1gives a recently established result due to Ali et al.5, Theorem 1, page 181.

The following results concerning the class of functionsP_pλg;n, b, mcan be proved on similar lines as given above forTheorem 2.1.

Theorem 2.3. Letfz∈ Tpnbe given by1.5. Thenfzis in the classPλpg;n, b, mif and only

if

∞

kn

λk−p 1k−m

k m

akbk≤p−m _{|b| −}

1

m!

p

m . 2.10

Remark 2.4. Making use of the same substitutions as mentioned above in2.6,Theorem 2.3 yields another known result due to Srivastava et al.3, Theorem 2, page 4. Also, using the same substitutions as mentioned above in2.8, we get the result of Orhan and Kamali13, Lemma 2, page 58.

3. Inclusion properties

We now obtain some inclusion relationships for the function classes

Sλ

pg;n, b, m, Ppλg;n, b, m, 3.1

involving then, δ-neighborhood defined by1.18. Theorem 3.1. Ifbk≥bnk≥nand

δ: n

λp−m−1 1|b|mp

n−p|b|λn−m−1 1mnbn

p >|b|, 3.2

then

Sλ

pg;n, b, m⊂ N0n,δe. 3.3

Proof. Letfz∈ S_pλg;n, b, m. Then, in view of assertion2.2ofTheorem 2.1, and the given conditionbk≥bnk≥n,we get

λn−m−1 1n−p|b|

n m

bn ∞

kn

ak

≤∞

kn

akbkλk−m−1 1k−p|b|

k m

≤ |b|λp−m−1 1

p m

,

which implies that

∞

kn

ak≤ |b|λp−m−1 1 p m

n−p|b|λn−m−1 1mnbn. 3.5

Applying the assertion2.2ofTheorem 2.1againin conjunction with3.5, we obtain

n m

λn−m−1 1bn ∞

kn

kak

≤ |b|λp−m−1 1

p m

p− |b|λn−m−1 1

n m

bn ∞

kn

ak

≤ |b|λp−m−1 1

p m

p− |b|λn−m−1 1

n m

bn

· |b|

λp−m−1 1mp

n−p|b|λn−m−1 1mnbn

n|b|

λp−m−1 1_mp

n−p|b| .

3.6

Hence,

∞

kn

kak≤ nλp−m−1 1|b| p m

n−p|b|λn−m−1 1mnbn :δ p >|b|, 3.7

which by virtue of1.18establishes the inclusion relation3.3ofTheorem 3.1.

In the analogous manner, by applying the assertion2.10ofTheorem 2.3instead of the assertion 2.2 of Theorem 2.1 to the functions in the classP_pλg;n, b, m, we can prove the following inclusion relationship.

Theorem 3.2. Ifbk≥bnk≥nand

δ: p−m

|b| −1/m! _mp

λn−p 1n−1_m bn , 3.8

then

Pλ

pg;n, b, m⊂Nn,δ0 e. 3.9

4. Neighborhood properties

This concluding section determines the neighborhood properties for each of the classes

Spλ,αg;n, b, m, Ppλ,αg;n, b, m 4.1

which are defined as follows.

A functionfz∈ Tpnis said to be in the classSpλ,αg;n, b, mif there exists a function

hz∈ Sλ

pg;n, b, msuch that

f_hz_z−1< p−α z∈U; 0≤α < p. 4.2

Analogously, a functionfz∈ Tpnis said to be in the classPpλ,αg;n, b, mif there exists a

functionhz∈ Ppλg;n, b, msuch that inequality4.2holds true.

Theorem 4.1. Ifhz∈ Sλ

pg;n, b, mand

αp− δ

nq1.

n−p|b|λn−m−1 1mnbn

n−p|b|λn−m−1 1mnbn− |b|λp−m−1 1mp

, 4.3

then

Nq_n,δh⊂ Spλ,αg;n, b, m. 4.4

Proof. Suppose thatfz∈ Nq_n,δh.We then find from1.16that

∞

kn

kq1_a

k−ck≤δ, 4.5

which readily implies that

∞

kn

|ak−ck| ≤ δ

nq1 n∈N. 4.6

Next, sincehz∈ Sλ_pg;n, b, m, we have in view of3.5that

∞

kn

ck≤ |b|λp−m−1 1 p m

n−p|b|λn−m−1 1mnbn, 4.7

so that

f_hzz−1≤

∞

kn|ak−ck|

1−∞_knck

≤ δ

nq1

1

1− |b|λp−m−1 1_mp/n−p|b|λn−m−1 1mnbn

≤ δ

nq1

n−p|b|λn−m−1 1mnbn

n−p|b|λn−m−1 1mnbn− |b|λp−m−1 1mp p−α,

4.8

provided thatαis given by4.3. Thus, by the above definition,f∈ Spλ,αg;n, b, mwhereαis

The proof ofTheorem 4.2 below is similar to that ofTheorem 4.1above, and its proof details are, therefore, omitted here.

Theorem 4.2. Ifhz∈ Pλ

pg;n, b, mand

αp− δ

nq1

λn−p 1n−mmnbn

λn−p 1n−mmnbn−p−m|b| −1/m! mp

, 4.9

then

Nq_n,δh⊂ Ppλ,αg;n, b, m. 4.10

Remark 4.3. Applying the parametric substitutions listed in2.6, Theorems4.1and4.2would yield the corresponding results of Srivastava et al.3, Theorems 5 and 6, page 6. Also using substitutions as mentioned above in2.8, we get the results due to Orhan and Kamali 13, Theorem 3, page 60; Theorem 4, page 61.

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