# Neighborhoods of Certain Multivalently Analytic Functions

Authors: Serap Bulut
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

We introduce and investigate two new general subclasses of multivalently analytic functions of complex order by making use of the familiar convolution structure of analytic functions. Among the various results obtained here for each of these function classes, we derive the coefficient bounds, distortion inequalities, and other interesting properties and characteristics for functions belonging to the classes introduced here.


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## Body

## 1. Introduction and Definitions

Letℝ=(-∞,∞) be the set of real numbers, and let ℂ be the set of complex numbers,
(1.1)N:={1,2,3,…}=N0∖{0},N*:=N∖{1}={2,3,4,…}.Let𝒯p denote the class of functions of the form
(1.2)f(z)=zp-∑j=k∞ajzj(p<k;aj≥0(j≥k);k,p∈N),
which are analytic and p-valent in the open unit disk
(1.3)U={z:z∈C,|z|<1}.Denote byf*g the Hadamard product (or convolution) of the functions f and g, that is, if f is given by (1.2) and g is given by
(1.4)g(z)=zp+∑j=k∞bjzj(p<k;bj≥0(j≥k);k,p∈N),
then
(1.5)(f*g)(z)∶=zp-∑j=k∞ajbjzj=:(g*f)(z).In [1], the author defined the following general class.Definition 1.1.
Let the functionf∈𝒯p. Then we say that f is in the class 𝒮g(p,k,λ,μ,b,β,m,n) if it satisfies the condition
(1.6)|1b(zn(Fλ,μ*g)(m+n)(z)(Fλ,μ*g)(m)(z)-(p-m)n)|<β,(m+n<p<k;p,n∈N;m∈N0;b∈C∖{0};0≤μ≤λ≤1;0<β≤1;z∈U)
where
(1.7)Fλ,μ(z)=λμz2f′′(z)+(λ-μ)zf′(z)+(1-λ+μ)f(z),g is given by (1.4), and (ν)n denotes the falling factorial defined as follows:
(1.8)(ν)0=1=:(ν0),(ν)n=ν(ν-1)⋯(ν-n+1)=:n!(νn)(n∈N).Various special cases of the class𝒮g(p,k,λ,μ,b,β,m,n) were considered by many earlier researchers on this topic of geometric function theory. For example, 𝒮g(p,k,λ,μ,b,β,m,n) reduces to the function class(i)
𝒮gn(p,λ,b,β) for m=0, n=1, and μ=0, studied by Mostafa and Aouf [2],(ii)
𝒮g(p,k,b,m,n) for λ=μ=0, and β=1, studied by Srivastava et al. [3],(iii)
𝒮g(p,n,b,m) for n=1, λ=μ=0, and β=1, studied by Prajapat et al. [4],(iv)
𝒮n,p(g;λ,μ,α) for m=0, n=1, β=1, and b=p(1-α)(0≤α<1), studied by Srivastava and Bulut [5],(v)
𝒯𝒮g*(p,m,α) for m=0, n=1, λ=μ=0, β=1, and b=p(1-α)(0≤α<1), studied by Ali et al. [6].Definition 1.2.
Let the functionf∈𝒯p. Then we say that f is in the class 𝒦g(p,k,λ,μ,b,β,m,n;q,u) if it satisfies the following nonhomogenous Cauchy-Euler differential equation (see, e.g., [7, page 1360, Equation (9)] and [5, page 6512, Equation (1.9)]):
(1.9)zqdqwdzq+(q1)(u+q-1)zq-1dq-1wdzq-1+⋯+(qq)w∏ɛ=0q-1(u+ɛ)=h(z)∏ɛ=0q-1(u+ɛ+p),
where
(1.10)w=f(z)∈Tp;h∈Sg(p,k,λ,μ,b,β,m,n);q∈N*,u∈(-p,∞).Settingm=0, n=1, μ=0, and q=2 in Definition 1.2, we have the special class introduced by Mostafa and Aouf [2].Following the works of Goodman [8] and Ruscheweyh [9] (see also [10, 11]), Altıntaş [12] defined the δ-neighborhood of a function f∈𝒯(p) by
(1.11)Nkδ(f)={h∈Tp:h(z)=zp-∑j=k∞cjzj,∑j=k∞j|aj-cj|≤δ}.It follows from the definition (1.11) that if
(1.12)e(z)=zp(p∈N),
then
(1.13)Nkδ(e)={h∈Tp:h(z)=zp-∑j=k∞cjzj,∑j=k∞j|cj|≤δ}.The main object of this paper is to investigate the various properties and characteristics of functions belonging to the above-defined classes(1.14)Sg(p,k,λ,μ,b,β,m,n),Kg(p,k,λ,μ,b,β,m,n;q,u).
Apart from deriving coefficient bounds and distortion inequalities for each of these classes, we establish several inclusion relationships involving the δ-neighborhoods of functions belonging to the general classes which are introduced above.


## 2. Coefficient Bounds and Distortion Theorems

Lemma 2.1 (see [1]).
Let the functionf∈𝒯p be given by (1.2). Then f is in the class 𝒮g(p,k,λ,μ,b,β,m,n) if and only if
(2.1)∑j=k∞(j)m[(j-m)n-(p-m)n+β|b|]ψ(j)ajbj≤β|b|(p)mψ(p),(m+n<p<k;p,n∈N;m∈N0;b∈C∖{0};0<β≤1;z∈U),
where
(2.2)ψ(s)=(s-1)(λμs+λ-μ)+1(0≤μ≤λ≤1).Remark 2.2.
If we setm=0, n=1, and μ=0 in Lemma 2.1, then we have [2, Lemma 1].Lemma 2.3 (See[1]).
Let the functionf∈𝒯p given by (1.2) be in the class 𝒮g(p,k,λ,μ,b,β,m,n). Then, for bj≥bk(j≥k), one has
(2.3)∑j=k∞aj≤β|b|(p)mψ(p)(k)m[(k-m)n-(p-m)n+β|b|]ψ(k)bk,(2.4)∑j=k∞jaj≤(k-m)!β|b|(p)mψ(p)(k-1)![(k-m)n-(p-m)n+β|b|]ψ(k)bk(p>|b|),
where ψ is defined by (2.2).Remark 2.4.
If we setm=0, n=1, and μ=0 in Lemma 2.3, then we have [2, Lemma  2].The distortion inequalities for functions in the class𝒮g(p,k,λ,μ,b,β,m,n) are given by the following Theorem 2.5.Theorem 2.5.
Let a functionf∈𝒯p be in the class 𝒮g(p,k,λ,μ,b,β,m,n). Then
(2.5)|f(z)|≤|z|p+β|b|(p)mψ(p)(k)m[(k-m)n-(p-m)n+β|b|]ψ(k)bk|z|k,(2.6)|f(z)|≥|z|p-β|b|(p)mψ(p)(k)m[(k-m)n-(p-m)n+β|b|]ψ(k)bk|z|k,
and in general
(2.7)|f(r)(z)|≤(p)r|z|p-r+β|b|(p)m(k)rψ(p)(k)m[(k-m)n-(p-m)n+β|b|]ψ(k)bk|z|k-r,|f(r)(z)|≥(p)r|z|p-r-β|b|(p)m(k)rψ(p)(k)m[(k-m)n-(p-m)n+β|b|]ψ(k)bk|z|k-r,(p>r;r∈N0;z∈U),
where ψ is defined by (2.2).Proof.
Suppose thatf∈𝒮g(p,k,λ,μ,b,β,m,n). We find from the inequality (2.3) that
(2.8)|f(z)|≤|z|p+|z|k∑j=k∞‍aj≤|z|p+β|b|(p)mψ(p)(k)m[(k-m)n-(p-m)n+β|b|]ψ(k)bk|z|k,
which is equivalent to (2.5) and
(2.9)|f(z)|≥|z|p-|z|k∑j=k∞‍aj≥|z|p-β|b|(p)mψ(p)(k)m[(k-m)n-(p-m)n+β|b|]ψ(k)bk|z|k,
which is precisely the assertion (2.6).If we setm=0, n=1, and μ=0 in Theorem 2.5, then we get the following.Corollary 2.6.
Let a functionf∈𝒯p be in the class 𝒮gn(p,λ,b,β). Then
(2.10)|f(z)|≤|z|p+β|b|[1+λ(p-1)][k-p+β|b|][1+λ(k-1)]bk|z|k,|f(z)|≥|z|p-β|b|[1+λ(p-1)][k-p+β|b|][1+λ(k-1)]bk|z|k,
and in general
(2.11)|f(r)(z)|≤p!(p-r)!|z|p-r+k!β|b|[1+λ(p-1)](k-r)![k-p+β|b|][1+λ(k-1)]bk|z|k-r,|f(r)(z)|≥p!(p-r)!|z|p-r-k!β|b|[1+λ(p-1)](k-r)![k-p+β|b|][1+λ(k-1)]bk|z|k-r,(p>r;r∈N0;z∈U),
where ψ is defined by (2.2).The distortion inequalities for functions in the class𝒦g(p,k,λ,μ,b,β,m,n;q,u) are given by Theorem 2.7 below.Theorem 2.7.
Let a functionf∈𝒯p be in the class 𝒦g(p,k,λ,μ,b,β,m,n;q,u). Then
(2.12)|f(z)|≤|z|p+β|b|(p)mψ(p)∏ɛ=0q-1(u+ɛ+p)(k)m[(k-m)n-(p-m)n+β|b|]ψ(k)(q-1)∏ɛ=0q-2(u+ɛ+k)bk|z|k,(2.13)|f(z)|≥|z|p-β|b|(p)mψ(p)∏ɛ=0q-1(u+ɛ+p)(k)m[(k-m)n-(p-m)n+β|b|]ψ(k)(q-1)∏ɛ=0q-2(u+ɛ+k)bk|z|k,
and in general
(2.14)|f(r)(z)|≤(p)r|z|p-r+(k)rβ|b|(p)mψ(p)∏ɛ=0q-1(u+ɛ+p)(k)m[(k-m)n-(p-m)n+β|b|]ψ(k)(q-1)∏ɛ=0q-2(u+ɛ+k)bk|z|k-r,|f(r)(z)|≥(p)r|z|p-r-(k)rβ|b|(p)mψ(p)∏ɛ=0q-1(u+ɛ+p)(k)m[(k-m)n-(p-m)n+β|b|]ψ(k)(q-1)∏ɛ=0q-2(u+ɛ+k)bk|z|k-r,(p>r;r∈N0;z∈U),
where ψ is defined by (2.2).Proof.
Suppose that a functionf∈𝒯p is given by (1.2), and also let the function h∈𝒮g(p,k,λ,μ,b,β,m,n) be occurring in the nonhomogenous Cauchy-Euler differential equation (1.9) with of course
(2.15)cj≥0(j≥k).
Then we readily see from (1.9) that
(2.16)aj=∏ɛ=0q-1(u+ɛ+p)∏ɛ=0q-1(u+ɛ+j)cj(j≥k),
so that
(2.17)f(z)=zp-∑j=k∞ajzj=zp-∑j=k∞∏ɛ=0q-1(u+ɛ+p)∏ɛ=0q-1(u+ɛ+j)cjzj,(2.18)|f(z)|≤|z|p+|z|k∑j=k∞∏ɛ=0q-1(u+ɛ+p)∏ɛ=0q-1(u+ɛ+j)cj.
Moreover, since h∈𝒮g(p,k,λ,μ,b,β,m,n), the first assertion (2.3) of Lemma 2.3 yields the following inequality:
(2.19)cj≤β|b|(p)mψ(p)(k)m[(k-m)n-(p-m)n+β|b|]ψ(k)bk,
and together with (2.19) and (2.18) it yields that
(2.20)|f(z)|≤|z|p+β|b|(p)mψ(p)∏ɛ=0q-1(u+ɛ+p)(k)m[(k-m)n-(p-m)n+β|b|]ψ(k)bk|z|k∑j=k∞1∏ɛ=0q-1(u+ɛ+j).
Finally, in view of the following sum:
(2.21)∑j=k∞‍1∏ɛ=0q-1(j+u+ɛ)=∑j=k∞‍(∑ɛ=0q-1‍(-1)ɛ(q-1-ɛ)!ɛ!(j+u+ɛ))=1(q-1)∏ɛ=0q-2(u+ɛ+k),(u∈R-{-k,-k-1,-k-2,…}),
the assertion (2.12) of Theorem 2.7 follows at once from (2.20) together with (2.21). The assertion (2.13) can be proven by similarly applying (2.17), and (2.19)–(2.21).Remark 2.8.
If we setm=0, n=1, μ=0, and q=2 in Theorem 2.7, then we have [2, Theorem 1].


## 3. Neighborhoods for the ClassesSg(p,k,λ,μ,b,β,m,n) and 𝒦g(p,k,λ,μ,b,β,m,n;q,u)

In this section, we determine inclusion relations for the classes(3.1)Sg(p,k,λ,μ,b,β,m,n),Kg(p,k,λ,μ,b,β,m,n;q,u),
involving δ-neighborhoods defined by (1.11) and (1.13).Theorem 3.1 (see [1]).
Ifbj≥bk(j≥k) and
(3.2)δ=(k-m)!β|b|(p)mψ(p)(k-1)![(k-m)n-(p-m)n+β|b|]ψ(k)bk(p>|b|),
then
(3.3)Sg(p,k,λ,μ,b,β,m,n)⊂Nkδ(e),
where e and ψ are given by (1.12) and (2.2), respectively.Remark 3.2.
If we setm=0, n=1, and μ=0 in Theorem 3.1, then we have [2, Theorem 2].Theorem 3.3.
Ifbj≥bk(j≥k) and
(3.4)δ=(k-m)!β|b|(p)mψ(p)(k-1)![(k-m)n-(p-m)n+β|b|]ψ(k)bk(1+∏ɛ=0q-1(u+ɛ+p)(q-1)∏ɛ=0q-2(u+ɛ+k))(p>|b|),
then
(3.5)Kg(p,k,λ,μ,b,β,m,n;q,u)⊂Nkδ(h),
where h and ψ are given by (1.11) and (2.2), respectively.Proof.
Suppose thatf∈𝒦g(p,k,λ,μ,b,β,m,n;q,u). Then, upon substituting from (2.16) into the following coefficient inequality:
(3.6)∑j=k∞j|cj-aj|≤∑j=k∞jcj+∑j=k∞jaj(cj≥0;aj≥0),
we obtain
(3.7)∑j=k∞j|cj-aj|≤∑j=k∞jcj+∑j=k∞‍∏ɛ=0q-1(u+ɛ+p)∏ɛ=0q-1(u+ɛ+j)jcj.
Since h∈𝒮g(p,k,λ,b,β,m,n), the assertion (2.4) of Lemma 2.3 yields
(3.8)jcj≤(k-m)!β|b|(p)mψ(p)(k-1)![(k-m)n-(p-m)n+β|b|]ψ(k)bk(p>|b|).
Finally, by making use of (2.4) as well as (3.8) on the right-hand side of (3.7), we find that
(3.9)∑j=k∞j|cj-aj|≤(k-m)!β|b|(p)mψ(p)(k-1)![(k-m)n-(p-m)n+β|b|]ψ(k)bk×(1+∑j=k∞∏ɛ=0q-1(u+ɛ+p)∏ɛ=0q-1(u+ɛ+j)),
which, by virtue of the sum in (2.21), immediately yields
(3.10)∑j=k∞j|cj-aj|≤(k-m)!β|b|(p)mψ(p)(k-1)![(k-m)n-(p-m)n+β|b|]ψ(k)bk×(1+∏ɛ=0q-1(u+ɛ+p)(q-1)∏ɛ=0q-2(u+ɛ+k))=:δ.
Thus, by applying the definition (1.11), we complete the proof of Theorem 3.3.Remark 3.4.
If we setm=0, n=1, μ=0, and q=2 in Theorem 3.3, then we have [2, Theorem  3].



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