(1)ON CERTAIN PROPERTIES OF NEIGHBORHOODS OF MULTIVALENT FUNCTIONS INVOLVING THE GENERALIZED SAITOH OPERATOR HESAM MAHZOON AND S

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

HESAM MAHZOON AND S. LATHA DEPARTMENT OFSTUDIES INMATHEMATICS

MANASAGANGOTRIUNIVERSITY OFMYSORE- INDIA.

mahzoon_hesam@yahoo.com DEPARTMENT OFMATHEMATICS

YUVARAJA’SCOLLEGEUNIVERSITY OFMYSORE-INDIA.

drlatha@gmail.com

Received 10 June, 2009; accepted 03 November, 2009 Communicated by S.S. Dragomir

ABSTRACT. In this paper, we introduce the generalized Saitoh operatorLp(a, c, η)and using this operator, the new subclasses H^p,b_n,m(a, c, η), L^p,b_n,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 denoted byAp(n)are defined. Further for functions belonging to these classes, certain properties of neighborhoods are studied.

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

2000 Mathematics Subject Classification. 30C45.

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

155-09

is defined as

(f ∗g)(z) = z^p+

∞

X

k=n+p

a_kb_kz^k.

Definition 1.1. Fora ∈R, c ∈ R\Z⁻0,where Z⁻0 ={...,−2,−1,0}andη ∈ R(η ≥ 0),the operatorLp(a, c, η) :Ap(n)→ Ap(n), is defined as

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

Dηf(z) = (1−η)f(z) + η

pzf⁰(z), (η≥0, z∈ U) 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+pakz^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) Nn,δ(f) =

(

g ∈ Tp(n) :g(z) = z^p−

∞

X

k=n+p

bkz^k and

∞

X

k=n+p

k|ak−bk| ≤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] .

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)

(Lp(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(Lp(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.

2. COEFFICIENTBOUNDS

In this section, we determine the coefficient inequalities for functions to be in the subclasses H^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^p,b_n,m(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)ak ≤ |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 = 0 and for allr, 0 ≤ r < 1.Hence, by letting r 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

≤

|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

η

i_(a)

k−p

(c)k−p

k m

ak

=|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

.

3. INCLUSIONRELATIONSHIPS 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_n,m^p,b (a, c, η)⊂ N_n,δ(h).

Proof. Let f ∈ H^p,b_n,m(a, c, η). By Theorem 2.1, we have, (n+|b|)

1 + n

pη (a)_n

(c)_n

n+p m

^∞ X

k=n+p

ak ≤ |b|

p m

,

which implies (3.2)

∞

X

k=n+p

a_k≤ |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

≤ |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).

4. FURTHER NEIGHBORHOODPROPERTIES

In this section, we determine the neighborhood properties of functions belonging to the sub- classesH_n,m^p,b,α(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 class H^p,b,α_n,m (a, c, η) if there exists a function g ∈ H^p,b_n,m(a, c, η) such that

(4.1)

f(z) g(z) −1

< p−α.

For 0 ≤ α < p and z ∈ U, a function f is 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|a_k−b_k| ≤δ,

which yields the coefficient inequality, (4.4)

∞

X

k=n+p

|a_k−b_k| ≤ δ

n+p, (n∈N).

Since g ∈ H_n,m^p,b (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−

δ[µ(n+p−1) + 1]

1 + ⁿ_pη

(a)n

(c)n

n+p−1 m

(n+p)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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