ON CERTAIN PROPERTIES OF NEIGHBORHOODS OF MULTIVALENT FUNCTIONS INVOLVING THE GENERALIZED SAITOH OPERATOR

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Generalized Saitoh Operator Hesam Mahzoon and S. Latha vol. 10, iss. 4, art. 112, 2009

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ON CERTAIN PROPERTIES OF NEIGHBORHOODS OF MULTIVALENT FUNCTIONS INVOLVING THE

GENERALIZED SAITOH OPERATOR

HESAM MAHZOON S. LATHA

Department of Studies in Mathematics Department of Mathematics

Manasagangotri University of Mysore Yuvaraja’s College University of Mysore

India India

EMail:mahzoon_hesam@yahoo.com EMail:drlatha@gmail.com

Received: 10 June, 2009

Accepted: 03 November, 2009

Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 30C45.

Key words: Coefficient bounds,(n, δ)- neighborhood and generalized Saitoh operator.

Abstract: In this paper, we introduce the generalized Saitoh operator Lp(a, c, η) and using this operator, the new subclasses H^p,bn,m(a, c, η), L^p,bn,m(a, c, η;µ), H^p,b,α_n,m (a, c, η)and L^p,b,α_n,m (a, c, η;µ)of the class of multivalent functions de- noted byAp(n)are defined. Further for functions belonging to these classes, certain properties of neighborhoods are studied.

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1. Introduction

LetA_p(n)be the class of normalized functionsf of the form

(1.1) f(z) =z^p +

∞

X

k=n+p

a_kz^k, (n, p∈N),

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

LetT_p(n)be the subclass ofA_p(n),consisting of functionsf of the form

(1.2) f(z) = z^p−

∞

X

k=n+p

a_kz^k, (a_k≥0, n, p∈N),

which arep-valent inU.

The Hadamard product of two power series f(z) =z^p+

∞

X

k=n+p

a_kz^k and g(z) =z^p+

∞

X

k=n+p

b_kz^k

is defined as

(f ∗g)(z) = z^p+

∞

X

k=n+p

a_kb_kz^k.

Definition 1.1. For a ∈ R, c ∈ R \Z⁻0, where Z⁻0 = {...,−2,−1,0} and η ∈ R(η≥0),the operatorL_p(a, c, η) :A_p(n)→ A_p(n), is defined as

(1.3) L_p(a, c, η)f(z) =φ_p(a, c, z)∗D_ηf(z), where

D_ηf(z) = (1−η)f(z) + η

pzf⁰(z), (η≥0, z∈ U)

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and

φ_p(a, c, z) =z^p+

∞

X

k=n+p

(a)k−p

(c)k−p

z^k, z ∈ U

and(x)_kdenotes the Pochammer symbol given by (x)_k =

( 1 if k = 0,

x(x+ 1)· · ·(x+k−1) if k ∈N={1,2,3, ...}.

In particular, we have,L₁(a, c, η)≡L(a, c, η).

Further, iff(z) = z^p +P∞

k=n+pa_kz^k,then L_p(a, c, η)f(z) =z^p+

∞

X

k=n+p

1 +

k p −1

η

(a)k−p

(c)k−p

a_kz^k.

Remark 1. Forη= 0andn = 1,we obtain the Saitoh operator [7] which yields the Carlson - Shaffer operator [1] forη= 0 andn=p= 1.

For any function f ∈ T_p(n)andδ ≥ 0,the(n, δ)-neighborhood of f is defined as,

(1.4) N_n,δ(f) = (

g∈ T_p(n) :g(z) =z^p−

∞

X

k=n+p

b_kz^kand

∞

X

k=n+p

k|a_k−b_k| ≤δ )

.

For the function h(z) = z^p, (p∈N)we have, (1.5) N_n,δ(h) =

(

g∈ T_p(n) :g(z) =z^p−

∞

X

k=n+p

b_kz^k and

∞

X

k=n+p

k|b_k| ≤δ )

.

The concept of neighborhoods was first introduced by Goodman [2] and then generalized by Ruscheweyh [6] .

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Definition 1.2. A functionf ∈ T_p(n)is said to be in the class H^p,b_n,m(a, c, η) if

(1.6)

1 b

z(L_p(a, c, η)f(z))^(m+1)

(L_p(a, c, η)f(z))^(m) −(p−m)

!

<1,

where p∈N, m∈N0, a >0, η≥0, p > m, b∈C\ {0} and z ∈ U. Definition 1.3. A functionf ∈ T_p(n)is said to be in the class L^p,b_n,m(a, c, η;µ) if

(1.7) 1 b

"

p(1−µ)

L_p(a, c, η)f(z) z

(m)

+µ(L_p(a, c, η)f(z))^(m+1)−(p−m)

#

< p−m, where p∈N, m∈N0, a >0, η≥0, p > m, µ≥0, b∈C\ {0}and z∈ U.

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2. Coefficient Bounds

In this section, we determine the coefficient inequalities for functions to be in the subclassesH^p,b_n,m(a, c, η)andL^p,b_n,m(a, c, η;µ).

Theorem 2.1. Letf ∈ T_p(n). Then, f ∈ H_n,m^p,b (a, c, η) if and only if

(2.1)

∞

X

k=n+p

1 +

k p −1

η

(a)_k−p (c)k−p

k m

(k+|b| −p)a_k ≤ |b|

p m

.

Proof. Let f ∈ H^p,b_n,m(a, c, η).Then, by(1.6)and(1.7)we can write,

(2.2) <











∞

P

k=n+p

h 1 +

k p −1

ηi _(a)

k−p

(c)k−p

k m

(p−k)a_kz^k−p

p m

−

∞

P

k=n+p

h 1 +

k p −1

ηi_(a)

k−p

(c)k−p

k m

a_kz^k−p











>−|b|, (z ∈ U).

Takingz = r, (0≤ r < 1)in(2.2),we see that the expression in the denominator on the left hand side of(2.2),is positive forr = 0and for allr, 0≤r < 1.Hence, by lettingr 7→1⁻ through real values, expression(2.2)yields the desired assertion (2.1).

Conversely, by applying the hypothesis(2.1)and letting|z|= 1,we obtain,

z(L_p(a, c, η)f(z))^(m+1)

(L_p(a, c, η)f(z))^(m) −(p−m)

=

P∞ k=n+p

h 1 +

k p −1

ηi_(a)

k−p

(c)k−p

k m

(p−k)a_kz^k−m

p m

z^p−m−P∞ k=n+p

h 1 +

k p −1

ηi _(a)

k−p

(c)k−p

k m

a_kz^k−m

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≤

|b|h

p m

−P∞ k=n+p

h 1 +

k p −1

ηi _(a)

k−p

(c)k−p

k m

a_ki

p m

−P∞ k=n+p

h 1 +

k p −1

ηi_(a)

k−p

(c)k−p

k m

a_k

=|b|.

Hence, by the maximum modulus theorem, we have f ∈ H^p,b_n,m(a, c, η).

On similar lines, we can prove the following theorem.

Theorem 2.2. A function f ∈ L^p,b_n,m(a, c, η;µ) if and only if

(2.3)

∞

X

k=n+p

1 +

k p −1

η

(a)k−p

(c)k−p

k−1 m

[µ(k−1) + 1]a_k

≤(p−m)

|b| −1

m! +

p m

.

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

In this section, we prove certain inclusion relationships for functions belonging to the classes H^p,b_n,m(a, c, η)andL^p,b_n,m(a, c, η;µ).

Theorem 3.1. If

(3.1) δ = (n+p)|b| _m^p

(n+|b|)

1 + ⁿ_pη_(a)

n

(c)n

n+p m

, (p >|b|),

then H^p,b_n,m(a, c, η)⊂ N_n,δ(h).

Proof. Let f ∈ H^p,b_n,m(a, c, η). By Theorem2.1, we have, (n+|b|)

1 + n

pη (a)_n

(c)_n

n+p m

^∞ X

k=n+p

a_k ≤ |b|

p m

,

which implies (3.2)

∞

X

k=n+p

ak≤ |b| _m^p

(n+|b|)

1 + ⁿ_pη

(a)n

(c)n

n+p m

. Using(2.1)and(3.2), we have,

1 + n

pη (a)_n

(c)n

n+p m

^∞ X

k=n+p

ka_k

≤ |b|

p m

+ (p− |b|)

1 + n pη

(a)_n (c)_n

n+p m

^∞ X

k=n+p

a_k

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≤ |b|

p m

+ (p− |b|)

1 + n pη

(a)_n (c)_n

n+p m

|b| _m^p (n+|b|)

1 + ⁿ_pη (a)n

(c)n

n+p m

=|b|

p m

n+p n+|b|. That is,

∞

X

k=n+p

ka_k ≤ |b|(n+p) _m^p (n+|b|)

1 + ⁿ_pη

(a)n

(c)n

n+p m

=δ, (p > |b|).

Thus, by the definition given by(1.5), f ∈ N_n,δ(h).

Similarly, we prove the following theorem.

Theorem 3.2. If

(3.3) δ=

(p−m)(n+p)h_|b|−1

m! + _m^pi [µ(n+p−1) + 1]

1 + ⁿ_pη

(a)n

(c)n

n+p m

, (µ >1)

thenL^p,b_n,m(a, c, η;µ)⊂ N_n,δ(h).

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4. Further Neighborhood Properties

In this section, we determine the neighborhood properties of functions belonging to the subclassesH^p,b,α_n,m(a, c, η) andL^p,b,α_n,m (a, c, η;µ).

For 0≤ α < p and z ∈ U, a function f is said to be in the classH^p,b,α_n,m (a, c, η) if there exists a function g ∈ H_n,m^p,b (a, c, η) such that

(4.1)

f(z) g(z) −1

< p−α.

For 0≤α < p and z ∈ U, a function fis said to be in the classL^p,b,α_n,m (a, c, η;µ) if there exists a function g ∈ L^p,b_n,m(a, c, η;µ) such that the inequality(4.1)holds true.

Theorem 4.1. If g ∈ H^p,b_n,m(a, c, η) and

(4.2) α =p−

δ(n+|b|)

1 + ⁿ_pη

(a)n

(c)n

n+p m

(n+p)h

(n+|b|)

1 + ⁿ_pη

(a)n

(c)n

n+p m

− |b| _m^pi, then N_n,δ(g)⊂ H^p,b,α_n,m (a, c, η).

Proof. Let f ∈ N_n,δ(g). Then, (4.3)

∞

X

k=n+p

k|ak−bk| ≤δ,

which yields the coefficient inequality, (4.4)

∞

X

k=n+p

|a_k−b_k| ≤ δ

n+p, (n ∈N).

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Since g ∈ H^p,b_n,m(a, c, η), by(3.2)we have, (4.5)

∞

X

k=n+p

b_k ≤ |b| _m^p

(n+|b|)

1 + ⁿ_pη

(a)n

(c)n

n+p m

so that,

f(z) g(z) −1

<

P∞

k=n+p|a_k−b_k| 1−P∞

k=n+pb_k

≤ δ n+p

(n+|b|)

1 + ⁿ_pη

(a)n

(c)n

n+p m

h

(n+|b|)

1 + ⁿ_pη (a)n

(c)n

n+p m

− |b| _m^pi

=p−α.

Thus, by definition, f ∈ H^p,b,α_n,m (a, c, η) forαgiven by(4.2).

On similar lines, we prove the following theorem.

Theorem 4.2. If g ∈ L^p,b_n,m(a, c, η;µ) and

(4.6) α=p− 1 (n+p)

×

δ[µ(n+p−1) + 1]

1 + ⁿ_pη (a)n

(c)n

n+p−1 m

h{µ(n+p−1) + 1}

1 + ⁿ_pη_(a)

n

(c)n

n+p−1 m

−(p−m)_|b|−1

m! + _m^pi, thenN_n,δ(g)⊂ L^p,b,α_n,m (a, c, η;µ).

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References

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