# Inclusion Properties for Certain Subclasses of Analytic Functions Defined by a Linear Operator

Authors: Nak Eun Cho
Journal: Abstract and Applied Analysis
 (2008)

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## Abstract

The purpose of the present paper is to investigate some inclusion properties of certain subclasses of analytic functions associated with a family of linear operators, which are defined by means of the Hadamard product (or convolution). Some integral preserving properties are also considered.


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## Body

## 1. Introduction

Let𝒜 denote the
class of functions of the form (1.1)f(z)=z+∑k=2∞akzk which are
analytic in the open unit disk 𝕌={z∈ℂ:|z|<1}. If f and g are analytic in 𝕌, we say that f is subordinate
to g, written f≺g or f(z)≺g(z) if there exists
an analytic function w in 𝕌 with w(0)=0 and |w(z)|<1 for z∈𝕌 such that f(z)=g(w(z)). We denote by 𝒮*, 𝒦, and 𝒞 the subclasses
of 𝒜 consisting of
all analytic functions which are, respectively, starlike, convex, and
close-to-convex in 𝕌.Let𝒩 be the class of
all functions ϕ which are
analytic and univalent in 𝕌 and for which ϕ(𝕌) is convex with ϕ(0)=1 and Re{ϕ(z)}>0 for z∈𝕌.Making use of the principle of subordination between
analytic functions, many authors investigated the subclasses𝒮*(ϕ), 𝒦(ϕ), and 𝒞(ϕ,ψ) of the class 𝒜 for ϕ,ψ∈𝒩 (cf. [1, 2]),
which are defined by (1.2)𝒮*(ϕ):={f∈𝒜:zf′(z)f(z)≺ϕ(z)in𝕌},𝒦(ϕ):={f∈𝒜:1+zf′′(z)f′(z)≺ϕ(z)in𝕌},𝒞(ϕ,ψ):={f∈𝒜:∃g∈𝒮*(ϕ)s.t.zf′(z)g(z)≺ψ(z)in𝕌}. For ϕ(z)=ψ(z)=(1+z)/(1−z) in the
definitions defined above, we have the well-known classes 𝒮*, 𝒦, and 𝒞, respectively. Furthermore, for the function classes 𝒮*[A,B] and 𝒦[A,B] investigated by
Janowski [3] (also see [4]), it is easily seen that (1.3)𝒮*(1+Az1+Bz)=𝒮*[A,B](−1≤B<A≤1),𝒦(1+Az1+Bz)=𝒦[A,B](−1≤B<A≤1).We now define the functionh(a,c)(z) by (1.4)h(a,c)(z):=∑k=0∞(a)k(c)kzk+1,(z∈𝕌;a∈ℝ;c∈ℝ∖ℤ0−;ℤ0−:={0,−1,−2,…}), where (ν)k is the
Pochhammer symbol (or the shifted factorial) defined (in terms of the Gamma
function) by (1.5)(ν)k:=Γ(ν+k)Γ(ν)={1ifk=0,ν∈ℂ∖{0},ν(ν+1)⋯(ν+k−1)ifk∈ℕ:={1,2,…},ν∈ℂ.We also denote byL(a,c): 𝒜→𝒜 the operator
defined by (1.6)L(a,c)f(z)=h(a,c)(z)*f(z)(z∈𝕌;f∈𝒜), where the
symbol (∗) stands for
the Hadamard product (or convolution). Then it is easily observed from
definitions (1.4) and (1.6) that L(2,1)f(z)=zf′(z) and (1.7)z(L(a,c)f(z))′=aL(a+1,c)f(z)−(a−1)L(a,c)f(z). Furthermore, we
note that L(n+1,1)f(z)=Dnf(z)(n>−1), where the
symbol Dn denotes the
familiar Ruscheweyh derivative [5] (also, see [6]) for n∈ℕ0:=ℕ∪{0}. The operator L(a,c) was introduced
and studied by Carlson and Shaffer [7] which has been used widely on the space of
analytic and univalent functions in 𝕌 (see also
[8]).By using the operatorL(a,c), we introduce the following classes of analytic
functions for ϕ,ψ∈𝒩, a∈ℝ and c∈ℝ∖ℤ0−: (1.8)𝒮a,c(ϕ):={f∈𝒜:L(a,c)f(z)∈𝒮*(ϕ)},𝒦a,c(ϕ):={f∈𝒜:L(a,c)f(z)∈𝒦(ϕ)},𝒞a,c(ϕ,ψ):={f∈𝒜:L(a,c)f(z)∈𝒞(ϕ,ψ)}. We also note
that (1.9)f(z)∈𝒦a,c(ϕ)⟺zf′(z)∈𝒮a,c(ϕ). In particular,
we set (1.10)𝒮a,c(1+Az1+Bz)=𝒮a,c[A,B](−1≤B<A≤1),𝒦a,c(1+Az1+Bz)=𝒦a,c[A,B](−1≤B<A≤1).In this paper, we investigate several inclusion
properties of the classes𝒮a,c(ϕ), 𝒦a,c(ϕ), and 𝒞a,c(ϕ,ψ). The integral preserving properties in connection
with the operator L(a,c) are also
considered. Furthermore, relevant connections of the results presented here
with those obtained in earlier works are pointed out.


## 2. Inclusion Properties Involving the OperatorL(a,c)

The following
lemmas will be required in our investigation.Lemma 2.1 (See [9, Pages 60-61]).
Leta2≥a1>0. If a2≥2  or a1+a2≥3, then the function h(a1,a2)(z)  defined by (1.4) belongs to the class 𝒦.Lemma 2.2 (See [10]).
Letf∈𝒦  and g∈𝒮*. Then for every analytic function Q  in  𝕌,  (2.1)(f∗Qg)(f∗g)(𝕌)⊂co¯Q(𝕌), where co¯Q(𝕌) denote the
closed convex hull of Q(𝕌).Theorem 2.3.
Leta2≥a1>0, c∈ℝ∖ℤ0−, and ϕ∈𝒩. If  a2≥2  or  a1+a2≥3, then  (2.2)𝒮a2,c(ϕ)⊂𝒮a1,c(ϕ).Proof.
Letf∈𝒮a2,c(ϕ). Then there exists an analytic function w in 𝕌 with |w(z)|<1(z∈𝕌) and w(0)=0 such that (2.3)z(L(a2,c)f(z))′L(a2,c)f(z)=ϕ(w(z))(z∈𝕌). By using (1.6)
and (2.3), we have (2.4)z(L(a1,c)f(z))′L(a1,c)f(z)=z(h(a1,c)(z)∗f(z))′h(a1,c)(z)∗f(z)=z(h(a2,c)(z)∗h(a1,a2)(z)∗f(z))′h(a2,c)(z)∗h(a1,a2)(z)∗f(z)=h(a1,a2)(z)∗z(L(a2,c)f(z))′h(a1,a2)(z)∗L(a2,c)f(z)=h(a1,a2)(z)∗ϕ(w(z))L(a2,c)f(z)h(a1,a2)(z)∗L(a2,c)f(z).  It follows from
(2.3) and Lemma 2.1 that L(a2,c)f(z)∈𝒮* and h(a1,a2)(z)∈𝒦, respectively. Then by applying Lemma 2.2 to (2.4),
we obtain (2.5){h(a1,a2)(z)∗ϕ(w)L(a2,c)f}{h(a1,a2)(z)∗L(a2,c)f}(𝕌)⊂co¯(ϕ(𝕌))⊂ϕ(𝕌), since ϕ is convex
univalent. Therefore, from the definition of subordination and (2.5), we have (2.6)z(L(a1,c)f(z))′L(a1,c)f(z)≺ϕ(z)(z∈𝕌), or,
equivalently, f∈𝒮a1,c(ϕ), which completes the proof of Theorem 2.3.Theorem 2.4.
Leta∈ℝ, c2≥c1>0  and  ϕ∈𝒩. If  c2≥2  or  c1+c2≥3, then (2.7)𝒮a,c1(ϕ)⊂𝒮a,c2(ϕ).Proof.
(f∈𝒮a,c1(ϕ)).  Using a similar argument as in the proof of Theorem
2.3, we obtain (2.8)z(L(a,c2)f(z))′L(a,c2)f(z)=h(a1,a2)(z)∗ϕ(w(z))L(a,c1)f(z)h(a1,a2)(z)∗L(a,c1)f(z), where w is an analytic
function in 𝕌 with |w(z)|<1(z∈𝕌) and w(0)=0. Applying Lemma 2.1 and the fact that L(a,c1)f(z)∈𝒮*, we see that (2.9){h(a1,a2)∗h(w)L(a2,c)f}{h(a1,a2)∗L(a,c1)f}(𝕌)⊂co¯(ϕ(𝕌))⊂ϕ(𝕌), since ϕ is convex
univalent. Thus the proof of Theorem 2.3 is completed.Corollary 2.5.
Leta2≥a1>0, c2≥c1>0, and  ϕ∈𝒩. If  a2≥min{2,3−a1} and c2≥min{2,3−c1}, then  (2.10)𝒮a2,c1(ϕ)⊂𝒮a2,c2(ϕ)⊂𝒮a1,c2(ϕ).Theorem 2.6.
Leta2≥a1>0, c2≥c1>0  and ϕ∈𝒩. If  a2≥min{2,3−a1} and c2≥min{2,3−c1}, then  (2.11)𝒦a2,c1(ϕ)⊂𝒦a2,c2(ϕ)⊂𝒦a1,c2(ϕ).Proof.
Applying (1.9) and Corollary 2.5,
we observe that (2.12)f(z)∈𝒦a2,c1(ϕ)⟺L(a2,c1)f(z)∈𝒦(ϕ)⟺z(L(a2,c1)f(z))′∈𝒮*(ϕ)⟺L(a2,c1)(zf′(z))∈𝒮*(ϕ)⟺zf′(z)∈𝒮a2,c1(ϕ)⇒zf′(z)∈𝒮a2,c2(ϕ)⟺L(a2,c2)(zf′(z))∈𝒮*(ϕ)⟺z(L(a2,c2)f(z))′∈𝒮*(ϕ)⟺L(a2,c2)f(z)∈𝒦(ϕ)⟺f(z)∈𝒦a2,c2(ϕ),f(z)∈𝒦a2,c2(ϕ)⟺L(a2,c2)f(z)∈𝒦(ϕ)⟺L(a2,c2)(zf′(z))∈𝒮*(ϕ)⇒zf′(z)∈𝒮a1,c2(ϕ)⟺z(L(a1,c2)f(z))′∈𝒮*(ϕ)⟺f(z)∈𝒦a1,c2(ϕ), which evidently
proves Theorem 2.6.Takingϕ(z)=(1+Az)/(1+Bz)(−1≤B<A≤1;z∈𝕌) in Corollary
2.5 and Theorem  2.6, we have the following corollary.Corollary 2.7.
Leta2≥a1>0  and  c2≥c1>0. If  a2≥min{2,3−a1}  and  c2≥min{2,3−c1}, then  (2.13)𝒮a2,c1[A,B]⊂𝒮a2,c2[A,B]⊂𝒮a1,c2[A,B](−1≤B<A≤1),𝒦a2,c1[A,B]⊂𝒦a2,c2[A,B]⊂𝒦a1,c2[A,B](−1≤B<A≤1).To prove the
theorems below, we need the following lemma.Lemma 2.8.
Letϕ∈𝒩. If f∈𝒦  and  q∈𝒮*(ϕ), then  f∗q∈𝒮*(ϕ).Proof.
Letq∈𝒮*(ϕ). Then (2.14)zq′(z)=q(z)ϕ(ω(z))(z∈𝕌), where ω is an analytic
function in 𝕌 with |w(z)|<1(z∈𝕌) and w(0)=0. Thus we have (2.15)z(f(z)∗q(z))′f(z)∗q(z)=f(z)∗zq′(z)f(z)∗q(z)=f(z)∗ϕ(ω(z))q(z)f(z)∗q(z)(z∈𝕌). By using
similar arguments to those used in the proof of Theorem 2.3, we conclude that
(2.15) is subordinated to ϕ in 𝕌 and so f∗q∈𝒮*(ϕ).Theorem 2.9.
Leta2≥a1>0,  c2≥c1>0  and  ϕ,ψ∈𝒩. If  a2≥min{2,3−a1} and c2≥min{2,3−c1}, then  (2.16)𝒞a2,c1(ϕ,ψ)⊂𝒞a2,c2(ϕ,ψ)⊂𝒞a1,c2(ϕ,ψ).Proof.
First of all, we show that(2.17)𝒞a2,c1(ϕ,ψ)⊂𝒞a2,c2(ϕ,ψ). Let f∈𝒞a2,c1(ϕ,ψ). Then there exists a function q2∈𝒮*(ϕ) such that (2.18)z(L(a2,c1)f(z))′q2(z)≺ψ(z)(z∈𝕌). From (2.18), we
obtain (2.19)z(L(a2,c1)f(z))′=ψ(w(z))(z∈𝕌), where w is an analytic
function in 𝕌 with |w(z)|<1(z∈𝕌) and w(0)=0. By virtue of Lemmas 2.1 and 2.8, we see that h(a1,a2)(z)∗q2(z)≡q1(z) belongs to 𝒮*(ϕ). Then we have (2.20)z(L(a2,c2)f(z))′q1(z)=h(c1,c2)(z)∗z(L(a2,c1)f(z))′h(c1,c2)(z)∗q2(z)=h(c1,c2)(z)∗ψ(w(z))q2(z)h(c1,c2)(z)∗q2(z)≺ψ(z)(z∈𝕌), which implies
that f∈𝒞a1,c(ϕ,ψ).
Moreover, the proof of the second part is similar to that
of the first part and so we omit the details involved.


## 3. Inclusion Properties Involving Various Operators

The next
theorem shows that the classes𝒮a,c(ϕ), 𝒦a,c(ϕ), and 𝒞a,c(ϕ,ψ) are invariant
under convolution with convex functions.Theorem 3.1.
Leta>0,  c∈ℝ∖ℤ0−,  ϕ,ψ∈𝒩  and let  g∈𝒦. Then (i)
f∈𝒮a,c(ϕ)⇒g∗f∈𝒮a,c(ϕ),(ii)
f∈𝒦a,c(ϕ)⇒g∗f∈𝒦a,c(ϕ),(iii)
f∈𝒞a,c(ϕ,ψ)⇒g∗f∈𝒞a,c(ϕ,ψ).Proof.
(i) Letf∈𝒮a,c(ϕ). Then we have (3.1)z(L(a,c)(g∗f)(z))′L(a,c)(g∗f)(z)=g(z)∗z(L(a,c)f(z))′g(z)∗L(a,c)f(z). By using the
same techniques as in the proof of Theorem 2.3, we  obtain (i).
(ii) Letf∈𝒦a,c(ϕ). Then, by (1.9), zf′(z)∈𝒮a,c(ϕ) and hence from
(i),  g(z)∗zf′(z)∈𝒮a,c(ϕ). Since (3.2)g(z)∗zf′(z)=z(g∗f)′(z), we have (ii)
applying (1.9) once again.
(iii) Letf∈𝒞a,c(ϕ,ψ). Then there exists a function q∈𝒮*(ϕ) such that (3.3)z(L(a,c)f(z))′=ψ(w(z))q(z)(z∈𝕌), where w is an analytic
function in 𝕌 with |w(z)|<1(z∈𝕌) and w(0)=0. From Lemma  2.8, we have that g∗q∈𝒮*(ϕ). Since (3.4)z(L(a,c)(g∗f)(z))′(g∗q)(z)=g(z)∗z(L(a,c)f(z))′g(z)∗q(z)=g(z)∗ψ(w(z))q(z)g(z)∗q(z)≺ψ(z)(z∈𝕌), we obtain
(iii).Now we
consider the following operators [5, 11] defined by (3.5)Ψ1(z)=∑k=1∞1+ck+czk(Re{c}≥0;z∈𝕌),Ψ2(z)=11−xlog[1−xz1−z](log1=0;|x|≤1,x≠1;z∈𝕌). It is well
known ([12], see also [5]) that the operators Ψ1 and Ψ2 are convex
univalent in 𝕌. Therefore, we have the following result, which can
be obtained from Theorem 3.1 immediately.Corollary 3.2.
Leta>0,  c∈ℝ∖ℤ0−,  ϕ,ψ∈𝒩  and let  Ψi(i=1,2)  be defined by (3.5). Then (i)
f∈𝒮a,c(ϕ)⇒Ψi∗f∈𝒮a,c(ϕ),(ii)
f∈𝒦a,c(ϕ)⇒Ψi∗f∈𝒦a,c(ϕ),(iii)
f∈𝒞a,c(ϕ,ψ)⇒Ψi∗f∈𝒞a,c(ϕ,ψ).



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