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volume 7, issue 1, article 5, 2006.

Received 08 November, 2005;

accepted 15 November, 2005.

Communicated by:Th.M. Rassias

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Journal of Inequalities in Pure and Applied Mathematics

INCLUSION AND NEIGHBORHOOD PROPERTIES OF SOME ANALYTIC AND MULTIVALENT FUNCTIONS

R.K. RAINA AND H.M. SRIVASTAVA

Department of Mathematics

College of Technology and Engineering

Maharana Pratap University of Agriculture and Technology Udaipur 313001, Rajasthan, India.

EMail:rainark_7@hotmail.com

Department of Mathematics and Statistics University of Victoria

Victoria, British Columbia V8W 3P4, Canada.

EMail:harimsri@math.uvic.ca

c

2000Victoria University ISSN (electronic): 1443-5756 333-05

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Inclusion and Neighborhood Properties of Some Analytic and Multivalent Functions R.K. Raina and H.M. Srivastava

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J. Ineq. Pure and Appl. Math. 7(1) Art. 5, 2006

http://jipam.vu.edu.au

Abstract

By means of a certain extended derivative operator of Ruscheweyh type, the authors introduce and investigate two new subclasses of p-valently analytic functions of complex order. The various results obtained here for each of these function classes include coefficient inequalities and the consequent inclusion relationships involving the neighborhoods of thep-valently analytic functions.

2000 Mathematics Subject Classification:Primary 30C45.

Key words: Analytic functions, p-valent functions, Hadamard product (or convolution), Coefficient bounds, Ruscheweyh derivative operator, Neighbor- hood of analytic functions.

The present investigation was supported, in part, by AICTE-New Delhi (Government of India) and, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353.

1. Introduction, Definitions and Preliminaries

LetA_p(n)denote the class of functionsf(z)normalized by (1.1) f(z) =z^p−

∞

X

k=n+p

akz^k (ak=0; n, p∈N:={1,2,3, ...}),

which are analytic andp-valent in the open unit disk U={z :z ∈C and |z|<1}.

The Hadamard product (or convolution) of the functionf ∈ A_p(n)given by (1.1) and the functiong ∈ A_p(n)given by

(1.2) g(z) =z^p−

∞

X

k=n+p

b_kz^k (b_k =0; n, p∈N)

is defined (as usual) by

(1.3) (f∗g)(z) :=z^p+

∞

X

k=n+p

a_kb_kz^k =: (g∗f)(z).

We introduce here an extended linear derivative operator of Ruscheweyh type:

D^λ,p :Ap → Ap Ap :=Ap(1) , which is defined by the following convolution:

(1.4) D^λ,pf(z) = z^p

(1−z)^λ+p ∗f(z) (λ >−p; f ∈ A_p).

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In terms of the binomial coefficients, we can rewrite (1.4) as follows:

(1.5) D^λ,pf(z) =z^p−

∞

X

k=1+p

λ+k−1 k−p

a_kz^k (λ >−p; f ∈ A_p).

In particular, when λ = n (n ∈ N), it is easily observed from (1.4) and (1.5) that

(1.6) D^n,pf(z) = z^p z^n−pf(z)(n)

n! (n ∈N⁰ :=N∪ {0}; p∈N), so that

(1.7) D^1,pf(z) = (1−p)f(z) +zf⁰(z),

(1.8) D^2,pf(z) = (1−p)(2−p)

2! f(z) + (2−p)zf⁰(z) + z² 2!f⁰⁰(z), and so on.

By using the operator

D^λ,pf(z) (λ >−p; p∈N)

given by (1.5), we now introduce a new subclass H^p_n,m(λ, b)of thep-valently analytic function classAp(n), which includes functionsf(z)satisfying the following inequality:

(1.9)

1 b

z D^λ,pf(z)(m+1)

D^λ,pf(z)(m) −(p−m)

!

<1

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(z ∈U; p∈N; m∈N0; λ∈R; p > max(m,−λ); b ∈C\ {0}). Next, following the earlier investigations by Goodman [3], Ruscheweyh [5]

and Altinta¸s et al. [2] (see also [1], [4] and [6]), we define the(n, δ)-neighborhood of a functionf(z)∈ A_n(p)by (see, for details, [2, p. 1668])

(1.10) N_n,δ(f) :=

(

g ∈ A_p(n) :g(z) =z^p−

∞

X

k=n+p

b_kz^k and

∞

X

k=n+p

k|ak−bk|5δ )

.

It follows from (1.10) that, if

(1.11) h(z) = z^p (p∈N),

then

(1.12) N_n,δ(h) = (

g ∈ A_p(n) :g(z) =z^p−

∞

X

k=n+p

b_k z^k and

∞

X

k=n+p

k|b_k|5δ )

.

Finally, we denote byL^p_n,m(λ, b;µ)the subclass ofA_p(n)consisting of func-

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tionsf(z)which satisfy the inequality (1.13) below:

(1.13) 1

b p(1−µ)

D^λ,pf(z) z

(m)

+µ D^λ,pf(z)(m+1)

−(p−m)

< p−m

(z ∈U; p∈N; m∈N0; λ∈R; p >max(m,−λ); µ=0; b∈C\ {0}). The object of the present paper is to investigate the various properties and characteristics of analyticp-valent functions belonging to the subclasses

H^p_n,m(λ, b) and L^p_n,m(λ, b;µ),

which we have introduced here. Apart from deriving a set of coefficient bounds for each of these function classes, we establish several inclusion relationships involving the(n, δ)-neighborhoods of analyticp-valent functions (with negative and missing coefficients) belonging to these subclasses.

Our definitions of the function classes

H^p_n,m(λ, b) and L^p_n,m(λ, b;µ)

are motivated essentially by two earlier investigations [1] and [4], in each of which further details and references to other closely-related subclasses can be found. In particular, in our definition of the function classL^p_n,m(λ, b;µ)involving the inequality (1.13), we have relaxed the parametric constraint05µ51, which was imposed earlier by Murugusundaramoorthy and Srivastava [4, p. 3, Equation (1.14)] (see also Remark3below).

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2. A Set of Coefficient Bounds

In this section, we prove the following results which yield the coefficient inequalities for functions in the subclasses

H^p_n,m(λ, b) and L^p_n,m(λ, b;µ).

Theorem 1. Let f(z) ∈ A_p(n) be given by(1.1). Then f(z) ∈ H^p_n,m(λ, b)if and only if

(2.1)

∞

X

k=n+p

λ+k−1 k−p

k m

(k+|b| −p)a_k 5|b|

p m

.

Proof. Let a function f(z) of the form (1.1) belong to the class H_n,m^p (λ, b).

Then, in view of (1.5), (1.9) yields the following inequality:

(2.2) <

P∞ k=n+p

λ+k−1 k−p

_k

m

(p−k)z^k−p

p m

−P∞ k=n+p

λ+k−1 k−p

_k

m

z^k−p

!

>− |b| (z ∈U).

Putting z = r (0 5 r < 1) in (2.2), we observe that the expression in the denominator on the left-hand side of (2.2) is positive forr = 0and also for all r (0< r < 1). Thus, by lettingr → 1−through real values, (2.2) leads us to the desired assertion (2.1) of Theorem1.

Conversely, by applying (2.1) and setting|z|= 1, we find by using (1.5) that

z D^λ,pf(z)(m+1)

(D^λ,pf(z))^(m) −(p−m)

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=

P∞ k=n+p

λ+k−1 k−p

_k

m

(p−k)z^k−m

p m

z^p−m−P∞ k=n+p

λ+k−1 k−p

_k

m

z^k−m

5

|b|h

p m

−P∞ k=n+p

λ+k−1 k−p

_k

m

a_ki

p m

−P∞ k=n+p

λ+k−1 k−p

_k

m

a_k =|b|.

Hence, by the maximum modulus principle, we infer that f(z) ∈ H_n,m^p (λ, b), which completes the proof of Theorem1.

Remark 1. In the special case when

(2.3) m= 0, p= 1, and b =βγ (0< β51; γ ∈C\ {0}), Theorem1corresponds to a result given earlier by Murugusundaramoorthy and Srivastava [4, p. 3, Lemma 1].

By using the same arguments as in the proof of Theorem1, we can establish Theorem2below.

Theorem 2. Letf(z)∈ A_p(n)be given by(1.1).Thenf(z)∈ L^p_n,m(λ, b;µ)if and only if

(2.4)

∞

X

k=n+p

λ+k−1 k−p

k−1 m

[µ(k−1) + 1]ak

5(p−m)

|b| −1

m! +

p m

.

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Remark 2. Making use of the same parametric substitutions as mentioned above in (2.3), Theorem 2 yields another known result due to Murugusun- daramoorthy and Srivastava [4, p. 4, Lemma 2].

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3. Inclusion Relationships Involving (n, δ) -Neighborhoods

In this section, we establish several inclusion relationships for the function classes

H^p_n,m(λ, b) and L^p_n,m(λ, b;µ) involving the(n, δ)-neighborhood defined by (1.12).

Theorem 3. If

(3.1) δ= (n+p)|b| _m^p

(n+|b|) ^λ+n+p−1_n _n+p

m

(p >|b|), then

(3.2) H^p_n,m(λ, b)⊂ N_n,δ(h).

Proof. Letf(z)∈ H^p_n,m(λ, b). Then, in view of the assertion (2.1) of Theorem 1, we have

(3.3) (n+|b|)

λ+n+p−1 n

n+p m

^∞ X

k=n+p

ak 5|b|

p m

.

This yields (3.4)

∞

X

k=n+p

a_k 5 |b| _m^p

(n+|b|) ^λ+n+p−1_n _n+p

m

.

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Applying the assertion (2.1) of Theorem1again, in conjunction with (3.4), we obtain

λ+n+p−1 n

n+p m

^∞ X

k=n+p

kak

5|b|

p m

+ (p− |b|)

λ+n+p−1 n

n+p m

^∞ X

k=n+p

a_k

5|b|

p m

+ (p− |b|)

λ+n+p−1 n

n+p m

· |b| _m^p

(n+|b|) ^λ+n+p−1_n _n+p

m

=|b|

p m

n+p n+|b|

.

Hence (3.5)

∞

X

k=n+p

kak5 |b|(n+p) _m^p (n+|b|) ^λ+n+p−1_n _n+p

m

=:δ (p > |b|),

which, by virtue of (1.12), establishes the inclusion relation (3.2) of Theorem 3.

In an analogous manner, by applying the assertion (2.4) of Theorem2instead of the assertion (2.1) of Theorem 1to functions in the class L^p_n,m(λ, b;µ), we can prove the following inclusion relationship.

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Theorem 4. If

(3.6) δ =

(p−m)(n+p) h|b|−1

m! + _m^pi [µ(n+p−1) + 1] ^λ+n+p−1_n _n+p

m

(µ >1),

then

L^p_n,m(λ, b;µ)⊂ N_n,δ(h).

Remark 3. Applying the parametric substitutions listed in (2.3), Theorems 3 and4would yield the known results due to Murugusundaramoorthy and Srivas- tava [4, p. 4, Theorem 1; p. 5, Theorem 2]. Incidentally, just as we indicated in Section 2above, the conditionµ > 1is needed in the proof of one of these known results [4, p. 5, Theorem 2]. This implies that the constraint05µ5 1 in [4, p. 3, Equation (1.14)] should be replaced by the less stringent constraint µ=0.

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4. Further Neighborhood Properties

In this last section, we determine the neighborhood properties for each of the following (slightly modified) function classes:

H^p,α_n,m(λ, b) and L^p,α_n,m(λ, b;µ).

Here the classH^p,α_n,m(λ, b)consists of functions f(z) ∈ A_p(n)for which there exists another functiong(z)∈ H^p_n,m(λ, b)such that

(4.1)

f(z) g(z) −1

< p−α (z ∈U; 05α < p).

Analogously, the class L^p,α_n,m(λ, b;µ) consists of functions f(z) ∈ A_p(n) for which there exists another functiong(z)∈ L^p_n,m(λ, b;µ)satisfying the inequality (4.1).

The proofs of the following results involving the neighborhood properties for the classes

H^p,α_n,m(λ, b) and L^p,α_n,m(λ, b;µ)

are similar to those given in [1] and [4]. We, therefore, skip their proofs here.

Theorem 5. Letg(z)∈ H_n,m^p (λ, b).Suppose also that

(4.2) α=p− δ(n+|b|) ^λ+n+p−1_n _n+p

m

(n+p)h

(n+|b|)

λ+n+p−1 n+p

n+p m

− |b| _m^pi.

Then

(4.3) N_n,δ(g)⊂ H^p,α_n,m(λ, b).

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Theorem 6. Letg(z)∈ L^p_n,m(λ, b;µ).Suppose also that

(4.4) α=p− δ[µ(n+p−1)+1] ^λ⁺ⁿ⁺_n^p⁻¹ _n₊_p−1 m

(n+p)h

[µ(n+p−1)+1] ^λ⁺ⁿ⁺_n^p⁻¹ _n₊_p₋₁

m

−(p−m)n_|b|−

1

m! + _m^poi. Then

(4.5) N_n,δ(g)⊂ L^p,α_n,m(λ, b;µ).

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References

[1] O. ALTINTA ¸S, Ö. ÖZKAN AND H.M. SRIVASTAVA, Neighborhoods of a class of analytic functions with negative coefficients, Appl. Math. Lett., 13(3) (2000), 63–67.

[2] O. ALTINTA ¸S, Ö. ÖZKAN ANDH.M. SRIVASTAVA, Neighborhoods of a certain family of multivalent functions with negative coefficients, Comput.

Math. Appl., 47 (2004), 1667–1672.

[3] A.W. GOODMAN, Univalent functions and nonanalytic curves, Proc.

Amer. Math. Soc., 8 (1957), 598–601.

[4] G. MURUGUSUNDARAMOORTHY AND H.M. SRIVASTAVA, Neigh- borhoods of certain classes of analytic functions of complex order, J. In- equal. Pure Appl. Math., 5(2) (2004), Art. 24. 8 pp. [ONLINE: http:

//jipam.vu.edu.au/article.php?sid=374].

[5] S. RUSCHEWEYH, Neighborhoods of univalent functions, Proc. Amer.

Math. Soc., 81 (1981), 521–527.

[6] H.M. SRIVASTAVAANDS. OWA (Eds.), Current Topics in Analytic Func- tion Theory, World Scientific Publishing Company, Singapore, New Jersey, London and Hong Kong, 1992.