# Quasi-Hadamard Product of Certainω -Starlike and  ω-Convex Functions with respect to Symmetric Points

Authors: Huo Tang; Guan-Tie Deng
Journal: Journal of Mathematics
 (2013)

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## Abstract

The main purpose of this paper is to derive some results associated with the quasi-Hadamard  product of certainω-starlike and ω-convex univalent analytic functions with respect to symmetric points.


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## Body

## 1. Introduction and Definitions

LetA(ω) denote the class of functions of the form(1)F(z)=(z-ω)+∑n=2∞‍an(z-ω)n,
which are analytic and univalent in the open unit disk U={z:|z|<1} and normalized with F(ω)=F′(ω)-1=0, where ω is a fixed point in U.Throughout this paper, let the functions of the forms(2)f(z)=a1(z-ω)+∑n=2∞‍an(z-ω)n(a1>0,an≥0),(3)fi(z)=a1,i(z-ω)+∑n=2∞‍an,i(z-ω)n(a1,i>0,an,i≥0),(4)g(z)=b1(z-ω)+∑n=2∞‍bn(z-ω)n(b1>0,bn≥0),(5)gj(z)=b1,j(z-ω)+∑n=2∞‍bn,j(z-ω)n(b1,j>0,bn,j≥0)
be regular and univalent in the unit disk U.Letω be a fixed point in U, and S(ω)={F∈A(ω):FisunivalentinU}.In [1], Kanas and Ronning defined the following classes of functions of ω-starlike and ω-convex, respectively,
(6)STω=Sω*={F∈S(ω):Re((z-ω)F′(z)F(z))>0,z∈U},CVω=Sωc={F∈S(ω):1+Re((z-ω)F′′(z)F′(z))>0,z∈U}.
Recently, Acu and Owa [2], Oladipo [3], and Oladipo and Breaz [4] have studied the previous classes extensively.LetSs* be the subclass of S(ω) consisting of functions given by (1) with ω=0 and satisfying the condition
(7)Re(zF′(z)F(z)-F(-z))>0(z∈U).
These functions are called starlike with respect to symmetric points and were introduced by Sakaguchi [5].Motivated by the previous classesSs*, Oladipo [6] (see also [7]) defined the following classes of functions with respect to symmetric points.Definition 1.
(i) LetSs*(ω) be the subclass of S(ω) consisting of functions given by (1) satisfying the condition
(8)Re((z-ω)F′(z)F(z)-F(-z))>0(z∈U),
and ω is a fixed point in U. These functions are called ω-starlike with respect to symmetric points.
(ii) LetSsc(ω) be the subclass of S(ω) consisting of functions given by (1) satisfying the condition
(9)Re(((z-ω)F′(z))′(F(z)-F(-z))′)>0(z∈U),
and ω is a fixed point in U. These functions are called ω-convex with respect to symmetric points.
Suppose thatf and g are two analytic functions in U. Then, we say that the function g is subordinate to the function f, and we write
(10)g(z)≺f(z)(z∈U),
if there exists a Schwarz function ϖ(z) with ϖ(0)=0 and |ϖ(z)|<1 such that
(11)g(z)=f(ϖ(z))(z∈U).
By applying the previous subordination definition, we define the following subclasses ofSs*(ω) and Ssc(ω).Definition 2.
(i) LetSs*(ω,A,B) be the subclass of S(ω) consisting of functions given by (1) satisfying the condition
(12)2(z-ω)F′(z)F(z)-F(-z)≺1+A(z-ω)1+B(z-ω),
where -1≤B<A≤1, z∈U, and ω is a fixed point in U.
(ii) LetSsc(ω,A,B) be the subclass of S(ω) consisting of functions given by (1) satisfying the condition
(13)2((z-ω)F′(z))′(F(z)-F(-z))′≺1+A(z-ω)1+B(z-ω),
where -1≤B<A≤1, z∈U, and ω is a fixed point in U.
Forω=0, we have
(14)Ss*(0,A,B)=Ss*(A,B),Ssc(0,A,B)=Ssc(A,B)
which were introduced by Goel and Mehrok [8] and Selvaraj and Vasanthi [9], respectively.
Using (12) and (13), we can easily obtain the characterization properties for the classes Ss*(ω,A,B) and Ssc(ω,A,B) as follows.Lemma 3.
A functionf defined by (2) belongs to the class Ss*(ω,A,B), if it satisfies the condition
(15)∑n=2∞‍([2n-(1-(-1)n)]+|2Bn+A(1-(-1)n)|)an≤2(A-B)a1.Lemma 4.
A functionf defined by (2) belongs to the class Ssc(ω,A,B), if it satisfies the condition
(16)∑n=2∞‍n([2n-(1-(-1)n)]+|2Bn+A(1-(-1)n)|)an≤2(A-B)a1.Now, we introduce the following class of analytic functions inU.Definition 5.
A functionf of the form (2), which is analytic in U, belongs to the class Ss,k*(ω,A,B), if it satisfies the condition
(17)∑n=2∞‍nk([2n-(1-(-1)n)]+|2Bn+A(1-(-1)n)|)an≤2(A-B)a1,
where -1≤B<A≤1 and k is any fixed nonnegative real number.
We note that, for any nonnegative real numberk, the class Ss,k*(ω,A,B) is nonempty as the functions of the form
(18)f(z)=a1(z-ω)+∑n=2∞‍2(A-B)a1nk([2n-(1-(-1)n)]+|2Bn+A(1-(-1)n)|)×λn(z-ω)n,
where a1>0, λn≥0, and ∑n=2∞λn≤1, satisfy inequality (17).
Clearly, we have the following relationships:(i)
Ss,0*(ω,A,B)≡Ss*(ω,A,B) and Ss,1*(ω,A,B)≡Ssc(ω,A,B);(ii)
Ss,k1*(ω,A,B)⊂Ss,k2*(ω,A,B)(k1>k2≥0);(iii)
Ss,k*(ω,A,B)⊂Ss,k-1*(ω,A,B)⊂⋯⊂Ssc(ω,A,B)⊂Ss*(ω,A,B)(k∈N).
Let us define the quasi- Hadamard  product of the functionsf(z) and g(z) by
(19)f*g(z)=a1b1(z-ω)+∑n=2∞‍anbn(z-ω)n.Similarly, we can define the quasi-Hadamard product of more than two functions; for example,(20)f1*f2*⋯*fm(z)=(∏i=1m‍a1,i)(z-ω)+∑n=2∞‍(∏i=1m‍an,i)(z-ω)n,
where the functions fi(i=1,2,…,m) are given by (3).In this paper, we derive certain results associated with the quasi-Hadamard  product of functions in the classesSs,k*(ω,A,B), Ss*(ω,A,B), and Ssc(ω,A,B), which extend the results obtained by Kumar [10, 11], Darwish [12], and Aouf [13].


## 2. Main Results

Unless otherwise mentioned, we will assume throughout the following results that-1≤B<A≤1, z∈U, k is any fixed nonnegative real number, and ω is a fixed point in U.Theorem 6.
Let the functionsfi(z) defined by (3) be in the class Ssc(ω,A,B) for every i=1,2,…,m, and let the functions gj(z) defined by (5) be in the class Ss,k*(ω,A,B) for every j=1,2,…,q. Then, the quasi-Hadamard product f1*f2*⋯*fm*g1*g2*⋯*gq(z) belongs to the class Ss,2m+(k+1)q-1*(ω,A,B).Proof.
LetG(z)=f1*f2*⋯*fm*g1*g2*⋯*gq(z); then,
(21)G(z)=(∏i=1m‍a1,i∏j=1q‍b1,j)(z-ω)+∑n=2∞‍(∏i=1m‍an,i∏j=1q‍bn,j)(z-ω)n.
It is sufficient to show that
(22)∑n=2∞‍[(∏i=1m‍an,i∏j=1q‍bn,j)n2m+(k+1)q-1([2n-(1-(-1)n)]+|2Bn+A(1-(-1)n)|)×(∏i=1m‍an,i∏j=1q‍bn,j)]≤2(A-B)(∏i=1m‍a1,i∏j=1q‍b1,j).
Since fi∈Ssc(ω,A,B), by Lemma 4, we have
(23)∑n=2∞‍n([2n-(1-(-1)n)]+|2Bn+A(1-(-1)n)|)an,i≤2(A-B)a1,i,
for every i=1,2,…,m. Thus,
(24)n([2n-(1-(-1)n)]+|2Bn+A(1-(-1)n)|)an,i≤2(A-B)a1,i,
or
(25)an,i≤2(A-B)n([2n-(1-(-1)n)]+|2Bn+A(1-(-1)n)|)a1,i,
for every i=1,2,…,m. The right-hand expression of the last inequality is not greater than n-2a1,i. Therefore,
(26)an,i≤n-2a1,i,
for every i=1,2,…,m. Also, since gj∈Ss,k*(ω,A,B), we find from (17) that
(27)∑n=2∞‍nk([2n-(1-(-1)n)]+|2Bn+A(1-(-1)n)|)bn,j≤2(A-B)b1,j,
for every j=1,2,…,q. Hence, we obtain
(28)bn,j≤n-(k+1)b1,j,
for every j=1,2,…,q.
Using (26)–(28) for i=1,2,…,m, j=q, and j=1,2,…,q-1, respectively, we have
(29)∑n=2∞[(∏i=1m‍an,i∏j=1q‍bn,j)n2m+(k+1)q-1([2n-(1-(-1)n)]+|2Bn+A(1-(-1)n)|)‍×(∏i=1m‍an,i∏j=1q‍bn,j)]≤∑n=2∞‍[n2m+(k+1)q-1·n-2m·n-(k+1)(q-1)(∏i=1m‍a1,i∏j=1q-1‍b1,j)×([2n-(1-(-1)n)]+|2Bn+A(1-(-1)n)|)bn,q(∏i=1m‍a1,i∏j=1q-1‍b1,j)]≤(∏i=1m‍a1,i∏j=1q-1‍b1,j)×[∑n=2∞‍nk([2n-(1-(-1)n)]+|2Bn+A(1-(-1)n)|)bn,q]≤2(A-B)(∏i=1m‍a1,i∏j=1q‍b1,j).
Thus, we have G(z)∈Ss,2m+(k+1)q-1*(ω,A,B). This completes the proof of Theorem 6.Upon settingk=1 in Theorem 6, we obtain the following result.Corollary 7.
Let the functionsfi(z) defined by (3) and the functions gj(z) defined by (5) belong to the class Ssc(ω,A,B) for every i=1,2,…,m and j=1,2,…,q. Then, the quasi-Hadamard product f1*f2*⋯*fm*g1*g2*⋯*gq(z) belongs to the class Ss,2m+2q-1*(ω,A,B).Theorem 8.
Let the functionsfi(z) defined by (3) be in the class Ss,k*(ω,A,B) for every i=1,2,…,m, and let the functions gj(z) defined by (5) be in the class Ss*(ω,A,B) for every j=1,2,…,q. Then, the quasi-Hadamard product f1*f2*⋯*fm*g1*g2*⋯*gq(z) belongs to the class Ss,(k+1)m+q-1*(ω,A,B).Proof.
Suppose thatG(z) is defined as (21). To prove the theorem, we need to show that
(30)∑n=2∞[(∏i=1m‍an,i∏j=1q‍bn,j)n(k+1)m+q-1([2n-(1-(-1)n)]+|2Bn+A(1-(-1)n)|)‍×(∏i=1m‍an,i∏j=1q‍bn,j)]≤2(A-B)(∏i=1m‍a1,i∏j=1q‍b1,j).
Since fi∈Ss,k*(ω,A,B), from (17), we have
(31)∑n=2∞‍nk([2n-(1-(-1)n)]+|2Bn+A(1-(-1)n)|)an,i≤2(A-B)a1,i,
for every i=1,2,…,m. Hence, we get
(32)an,i≤n-(k+1)a1,i,
for every i=1,2,…,m. Further, since gj∈Ss*(ω,A,B), by Lemma 3, we have
(33)∑n=2∞‍([2n-(1-(-1)n)]+|2Bn+A(1-(-1)n)|)bn,j≤2(A-B)b1,j,
for every j=1,2,…,q. Whence, we obtain
(34)bn,j≤n-1b1,j,
for every j=1,2,…,q.
Using (32)–(34) for i=m, i=1,2,…,m-1, and j=1,2,…,q, respectively, we get
(35)∑n=2∞[(∏i=1m‍an,i∏j=1q‍bn,j)n(k+1)m+q-1([2n-(1-(-1)n)]+|2Bn+A(1-(-1)n)|)‍×(∏i=1m‍an,i∏j=1q‍bn,j)]≤∑n=2∞‍[n(k+1)m+q-1·n-(k+1)(m-1)·n-q(∏i=1m-1‍a1,i∏j=1q‍b1,j)×([2n-(1-(-1)n)]+|2Bn+A(1-(-1)n)|)an,m(∏i=1m-1‍a1,i∏j=1q‍b1,j)]≤(∏i=1m-1‍a1,i∏j=1q‍b1,j)×[∑n=2∞‍nk([2n-(1-(-1)n)]+|2Bn+A(1-(-1)n)|)an,m]≤2(A-B)(∏i=1m‍a1,i∏j=1q‍b1,j).
Therefore, we have G(z)∈Ss,(k+1)m+q-1*(ω,A,B). We complete the proof.By takingk=0 in Theorem 8, we get the following result.Corollary 9.
Let the functionsfi(z) defined by (3) and the functions gj(z) defined by (5) belong to the class Ss*(ω,A,B) for every i=1,2,…,m and j=1,2,…,q. Then, the quasi-Hadamard  product f1*f2*⋯*fm*g1*g2*⋯*gq(z) belongs to the class Ss,m+q-1*(ω,A,B).By puttingk=0 in Theorem 6, or k=1 in Theorem 8, we obtain the following result.Corollary 10.
Let the functionsfi(z) defined by (3) be in the class Ssc(ω,A,B) for every i=1,2,…,m, and let the functions gj(z) defined by (5) be in the class Ss*(ω,A,B) for every j=1,2,…,q. Then, the quasi-Hadamard product f1*f2*⋯*fm*g1*g2*⋯*gq(z) belongs to the class Ss,2m+q-1*(ω,A,B).Next, we discuss some applications of Theorems6 and 8.Taking into account the quasi-Hadamard product of functionsf1(z),f2(z),…,fm(z) only, in the proof of Theorem 6, and using (23) and (26) for i=m and i=1,2,…,m-1, respectively, we are led to the following.Corollary 11.
Let the functionsfi(z) defined by (3) belong to the class Ssc(ω,A,B) for every i=1,2,…,m. Then, the quasi- Hadamard product f1*f2*⋯*fm(z) belongs to the class Ss,2m-1*(ω,A,B).Also, taking into account the quasi-Hadamard product of functionsg1(z),g2(z),…,gq(z) only, in the proof of Theorem 8, and using (33) and (34) for j=q and j=1,2,…,q-1, respectively, we are led to the following.Corollary 12.
Let the functionsgj(z) defined by (5) belong to the class Ss*(ω,A,B) for every j=1,2,…,q. Then, the quasi-Hadamard product g1*g2*⋯*gq(z) belongs to the class Ss,q-1*(ω,A,B).Remark 13.
By takingω=0 in the previous results and making use of relationship (14), we obtain the corresponding results.



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