# Some New Inclusion and Neighborhood Properties  for

Authors: J. K. Prajapat; R. K. Raina
Journal: International Journal of Mathematics and Mathematical Sciences
 (2008)

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## Abstract

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
associated with the(n,δ)-neighborhoods for these subclasses. Some interesting consequences
of these results are also pointed out.


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## Body

## 1. Introduction and Preliminaries

Let𝒜p(n) denote the
class of functions of the form
(1.1)f(z)=zp+∑k=n∞akzk(p<n;n,p∈ℕ={1,2,…}),
which are analytic and p-valent in the open unit disk
(1.2)𝕌={z;z∈ℂ:|z|<1}.
If f∈𝒜p(n) is given by
(1.1) and g∈𝒜p(n) is given by
(1.3)g(z)=zp+∑k=n∞bkzk,
then the Hadamard product (or convolution) f*g of f and g is defined (as
usual) by
(1.4)(f*g)(z):=zp+∑k=n∞akbkzk:=(g*f)(z).We denote by𝒯p(n) the subclass of 𝒜p(n) consisting of
functions of the form
(1.5)f(z)=zp−∑k=n∞akzk(p<n;ak≥0(k≥n);n,p∈ℕ),
which are p-valent in 𝕌.For a fixed functiong(z)∈𝒜p(n)
defined by
(1.6)g(z)=zp+∑k=n∞bkzk(p<n;bk≥0(k≥n);n,p∈ℕ),
we introduce a new class 𝒮pλ(g;n,b,m) of functions
belonging to the subclass of 𝒯p(n), which consists of functions f(z) of the form (1.5), satisfying the following inequality:
(1.7)|1b(z(f*g)(m+1)(z)+λz2(f*g)(m+2)(z)λz(f*g)(m+1)(z)+(1−λ)(f*g)(m)(z)−(p−m))|<1(z∈𝕌;p∈ℕ;m∈ℕ0;p>m;0≤λ≤1;b∈ℂ∖{0}).We note that there exist several interesting new (or known) subclasses of our function class𝒮pλ(g;n,b,m). For example, if λ=0 in (1.7), we obtain the class 𝒮p(g;n,b,m) studied very
recently by Prajapat et al. [1]. On the other hand, if the coefficients bk in (1.6) are chosen as follows:
(1.8)bk=(k+μp+μ)r(μ≥0;k≥n;r,p,n∈ℕ),
and n is replaced by n+p in (1.4) and
(1.5), then we obtain the class 𝒮n,mp(μ,r,λ,b)  of p-valently
analytic functions (involving the multiplier transformation operator Ip(r,μ) defined in [2])
which was studied recently by Srivastava et al. [3]. Also, if we set λ=0 in (1.7) and if
the arbitrary sequence bk in (1.6) is
selected as follows:
(1.9)bk=(μ+k−1k−p)(μ>−p;k≥n;p,n∈ℕ),
also if n is replaced by n+p in (1.4) and
(1.5), then we obtain the class ℋn,mp(μ,b) of p-valently
analytic functions (involving the familiar Ruscheweyh derivative operator)
investigated by Raina and Srivastava [4]. Further, when
(1.10)λ=0,m=0,b=p(1−α)(p∈ℕ;0≤α<1),
in (1.7), then 𝒮pλ(g;n,b,m) reduces to the
class studied recently by Ali et al. [5]. Moreover, when
(1.11)g(z)=zp+∑k=n∞(α1)k−p⋯(αq)k−p(β1)k−p⋯(βs)k−p(k−p)!zk,(αj∈ℂ(j=1,2,…,q),βj∈ℂ∖{0,−1,−2,…}(j=1,2,…,s))
with the parameters
(1.12)α1,…,αq,β1,…,βs
being so chosen that the coefficients bk in (1.6) satisfy the following condition:
(1.13)bk=(α1)k−p⋯(αq)k−p(β1)k−p⋯(βs)k−p(k−p)!≥0,
then the class 𝒮pλ(g;n,b,m) transforms into
a (presumably) new class 𝒮pλ(n,b,m) defined by
(1.14)𝒮pλ(n,b,m)={f∈𝒯p(n):|1b(z(Hsq[α1]f)(m+1)(z)+λz2(Hsq[α1]f)(m+2)λz(Hsq[α1]f)(m+1)(z)+(1−λ)(Hsq[α1]f)(m)−(p−m))|<1}(z∈𝕌;q≤s+1;m,q,s∈ℕ0;0≤λ≤1,p∈ℕ;b∈ℂ∖{0}).The operator(1.15)(Hsq[α1]f)(z):=Hsq(α1,…,αq;β1,…,βs)f(z),
involved in (1.14), is the Dziok-Srivastava linear
operator (see for details [6]; see also [7, 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 fractional derivative operator. One may refer to [7] or [6] for
further details and references for these operators. The Dziok-Srivastava linear
operator defined in [6] has further been generalized by Dziok and Raina [7]
(see also [8, 9]).Following a recent investigation by Frasin and Darus [10], let f(z)∈𝒯p(n),δ≥0, then a (q,δ)-neighborhood
of the function f(z) is defined by
(1.16)𝒩n,δq(f)={h:h∈𝒯p(n):h(z)=zp−∑k=n∞ckzk,∑k=n∞kq+1|ak−ck|≤δ}.
It follows from the definition (1.16) that if
(1.17)e(z)=zp(p∈ℕ),
then
(1.18)𝒩n,δq(e)={h:h∈𝒯p(n):h(z)=zp−∑k=n∞ckzk,∑k=n∞kq+1|ck|≤δ}. We observe that
(1.19)𝒩2,δ0(f)=𝒩δ(f),𝒩2,δ1(f)=ℳδ(f),
where 𝒩δ(f) and ℳδ(f) denote,
respectively, the δ-neighborhoods of the function
(1.20)g(z)=z+∑k=2∞akzk(ak≥0,z∈𝕌),
defined by Ruscheweyh [11] and Silverman [12].Finally, for a fixed function(1.21)g(z)=zp+∑k=n∞bkzk∈𝒜p(n)(p<n;bk>0(k≥n);n,p∈ℕ),
let 𝒫pλ(g;n,b,m) denote the
subclass of 𝒯p(n) consisting of
functions f(z) of the form (1.5) which satisfy the following inequality:
(1.22)|1b{[1−λ(p−m−1)](f*g)(m+1)(z)+λz(f*g)(m+2)(z)−(p−m)}|<p−m(z∈𝕌,m∈ℕ0;p∈ℕ;p>m;0≤λ≤1,b∈ℂ∖{0}).The object of the present paper is to investigate the various properties and characteristics of functions belonging to the above-defined subclasses(1.23)𝒮pλ(g;n,b,m),𝒫pλ(g;n,b,m)
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 functionf(z)∈𝒯p(n) to be in each
of the classes
(2.1)𝒮pλ(g;n,b,m),𝒫pλ(g;n,b,m).Theorem 2.1.
Letf(z)∈𝒯p(n) be given by
(1.5). Then f(z) is in the class
𝒮pλ(g;n,b,m)
if and only if
(2.2)∑k=n∞akbk[λ(k−m−1)+1](k−p+|b|)(km)≤|b|[λ(p−m−1)+1](pm).Proof.
Assume thatf(z)∈Spλ(g;n,b,m). Then, in view
of (1.5)–(1.7), we get
(2.3)ℜ(λz2(f*g)(m+2)(z)+z[1−λ(p−m)](f*g)(m+1)(z)−(1−λ)(p−m)(f*g)(m)(z)λz(f*g)(m+1)(z)+(1−λ)(f*g)(m)(z))>−|b|
which yields
(2.4)ℜ(∑k=n∞akbk(km)(k−p)[λ(k−m−1)+1]zk−p(pm)[λ(p−m−1)+1]zp−m−∑k=n∞akbk(km)[λ(k−m−1)+1]zk−p)<|b|(z∈𝕌).
Putting z=r(0≤r<1) in (2.4), the
denominator expression on the left-hand side of (2.4) remains positive for r=0, and also for
all r∈(0,1). Hence, by letting r→1−, through real values, inequality (2.4) leads to the
desired assertion (2.2) of Theorem 2.1.
Conversely, by applying the hypothesis (2.2) of Theorem 2.1, and letting |z|=1, we find that
(2.5)|z(f*g)(m+1)(z)+λz2(f*g)(m+2)(z)λz(f*g)(m+1)(z)+(1−λ)(f*g)(m)(z)−(p−m)|≤|b|{(pm)[λ(p−m−1)+1]−∑k=n∞akbk[λ(k−m−1)+1](km)}(pm)[λ(p−m−1)+1]−∑k=n∞akbk[λ(k−m−1)+1](km)=|b|.
Hence, by the maximum
modulus principle, we infer that f(z)∈𝒮pλ(g;n,b,m), which completes
the proof of Theorem 2.1.Remark 2.2.
In the special case when(2.6)(i)bk=(k+μp+μ)r(μ≥0;k≥n;r,p,n∈ℕ;n↦n+p).
Theorem 2.1
corresponds to a result given recently by Srivastava et al. [3, Theorem 1,  page 3];
(2.7)(ii)λ=0;bk=(μ+k−1k−p)(μ>−p;k≥n;n,p∈ℕ;n↦n+p).
Theorem 2.1 yields the result given recently by Raina and Srivastava [4, Theorem 1, page 3];
(2.8)(iii)m=0;p=1,bk=kΩ(Ω∈ℕ0;k≥n;n∈ℕ;n↦n+p).
Theorem 2.1 reduces to the result of Orhan and Kamali [13, Lemma 1, page 57];
(2.9)(iv)λ=0,m=0,b=p(1−α)(p∈N;0≤α<1).
Theorem 2.1 gives a recently established result due to Ali et al. [5, Theorem 1, page 181].The following results concerning the class of functions𝒫pλ(g;n,b,m) can be proved
on similar lines as given above for Theorem 2.1.Theorem 2.3.
Letf(z)∈𝒯p(n)
be given by (1.5). Then
f(z) is in the class
𝒫pλ(g;n,b,m)
if and only if
(2.10)∑k=n∞[λ(k−p)+1](k−m)(km)akbk≤(p−m)[|b|−1m!+(pm)].Remark 2.4.
Making use of the same substitutions as mentioned above in (2.6), 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 in (2.8), we get the
result of Orhan and Kamali [13, Lemma 2, page 58].


## 3. Inclusion Properties

We now obtain some inclusion relationships for the function classes(3.1)𝒮pλ(g;n,b,m),𝒫pλ(g;n,b,m),
involving the (n,δ)-neighborhood defined by (1.18).Theorem 3.1.
Ifbk≥bn(k≥n)
and
(3.2)δ:=n[λ(p−m−1)+1]|b|(pm)(n−p+|b|)[λ(n−m−1)+1](nm)bn(p>|b|),
then
(3.3)𝒮pλ(g;n,b,m)⊂𝒩n,δ0(e).Proof.
Letf(z)∈𝒮pλ(g;n,b,m). Then, in view of assertion (2.2) of Theorem 2.1, and
the given condition bk≥bn(k≥n),  we get
(3.4)[λ(n−m−1)+1](n−p+|b|)(nm)bn∑k=n∞ak≤∑k=n∞akbk[λ(k−m−1)+1](k−p+|b|)(km)≤|b|[λ(p−m−1)+1](pm),
which implies that
(3.5)∑k=n∞ak≤|b|[λ(p−m−1)+1](pm)(n−p+|b|)[λ(n−m−1)+1](nm)bn.
Applying the assertion (2.2) of Theorem 2.1 again (in conjunction with (3.5)), we obtain
(3.6)(nm)[λ(n−m−1)+1]bn∑k=n∞kak≤|b|[λ(p−m−1)+1](pm)+(p−|b|)[λ(n−m−1)+1](nm)bn∑k=n∞ak≤|b|[λ(p−m−1)+1](pm)+(p−|b|)[λ(n−m−1)+1](nm)bn⋅|b|[λ(p−m−1)+1](pm)(n−p+|b|)[λ(n−m−1)+1](nm)bn=n|b|[λ(p−m−1)+1](pm)(n−p+|b|).
Hence,
(3.7)∑k=n∞kak≤n[λ(p−m−1)+1]|b|(pm)(n−p+|b|)[λ(n−m−1)+1](nm)bn:=δ(p>|b|),
which by virtue of (1.18) establishes the inclusion relation (3.3) of Theorem 3.1.In the analogous manner, by applying the assertion (2.10) of Theorem 2.3 instead of the
assertion (2.2) of Theorem 2.1 to the functions in the class 𝒫pλ(g;n,b,m), we can prove the following inclusion relationship.Theorem 3.2.
Ifbk≥bn(k≥n)
and
(3.8)δ:=(p−m)[(|b|−1)/m!+(pm)][λ(n−p)+1](n−1m)bn,
then
(3.9)𝒫pλ(g;n,b,m)⊂Nn,δ0(e).Remark 3.3.
Applying the parametric substitutions listed in (2.6), Theorems 3.1 and 3.2 would
yield the known results due to Srivastava et al. [3, Theorem 3, page 4; Theorem 4, page 5].  Also,  using substitutions (as mentioned above in (2.8)) in
Theorems 3.1 and 3.2, we get the results due to Orhan and Kamali [13, Theorem 1, page 58; Theorem 2, page 59].


## 4. Neighborhood Properties

This concluding section determines the neighborhood properties for each of the
classes(4.1)𝒮p(λ,α)(g;n,b,m),𝒫p(λ,α)(g;n,b,m)
which are defined as follows.A functionf(z)∈𝒯p(n) is said to be
in the class 𝒮p(λ,α)(g;n,b,m) if there exists
a function h(z)∈𝒮pλ(g;n,b,m) such that
(4.2)|f(z)h(z)−1|<p−α(z∈𝕌;0≤α<p).
Analogously, a function f(z)∈𝒯p(n) is said to be
in the class 𝒫p(λ,α)(g;n,b,m) if there exists
a function h(z)∈𝒫pλ(g;n,b,m) such that
inequality (4.2) holds true.Theorem 4.1.
Ifh(z)∈𝒮pλ(g;n,b,m)
and
(4.3)α=p−δnq+1.(n−p+|b|)[λ(n−m−1)+1](nm)bn[(n−p+|b|)[λ(n−m−1)+1](nm)bn−|b|[λ(p−m−1)+1](pm)],
then
(4.4)𝒩n,δq(h)⊂𝒮p(λ,α)(g;n,b,m).Proof.
Suppose thatf(z)∈𝒩n,δq(h). We then find
from (1.16) that
(4.5)∑k=n∞kq+1|ak−ck|≤δ,
which readily implies that
(4.6)∑k=n∞|ak−ck|≤δnq+1(n∈ℕ).
Next, sinceh(z)∈𝒮pλ(g;n,b,m), we have in view of (3.5) that
(4.7)∑k=n∞ck≤|b|[λ(p−m−1)+1](pm)(n−p+|b|)[λ(n−m−1)+1](nm)bn,
so that
(4.8)|f(z)h(z)−1|≤∑k=n∞|ak−ck|1−∑k=n∞ck≤δnq+111−|b|[λ(p−m−1)+1](pm)/(n−p+|b|)[λ(n−m−1)+1](nm)bn≤δnq+1(n−p+|b|)[λ(n−m−1)+1](nm)bn[(n−p+|b|)[λ(n−m−1)+1](nm)bn−|b|[λ(p−m−1)+1](pm)]=p−α,
provided that α is given by
(4.3). Thus, by the above definition, f∈𝒮p(λ,α)(g;n,b,m) where α is given by
(4.3), which proves Theorem 4.1.The proof of Theorem4.2 below is similar to that of Theorem 4.1 above, and its proof details
are, therefore, omitted here.Theorem 4.2.
Ifh(z)∈𝒫pλ(g;n,b,m)
and
(4.9)α=p−δnq+1[λ(n−p)+1)](n−m)(nm)bn[[λ(n−p)+1](n−m)(nm)bn−(p−m){(|b|−1)/m!+(pm)}],
then
(4.10)𝒩n,δq(h)⊂𝒫p(λ,α)(g;n,b,m).Remark 4.3.
Applying the parametric substitutions listed in (2.6), Theorems 4.1 and 4.2 would yield the corresponding results of Srivastava et al. [3, Theorems 5 and 6, page 6]. Also using substitutions as mentioned above in (2.8), we get the
results due to Orhan and Kamali [13, Theorem 3, page 60; Theorem 4,  page 61].



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