The main purpose of this paper is to derive some results associated with the quasi-Hadamard product of certain ω-starlike and ω-convex univalent analytic functions with respect to symmetric points.
1. Introduction and Definitions
Let A(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.
Let ω 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.
Let Ss* 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 classes Ss*, Oladipo [6] (see also [7]) defined the following classes of functions with respect to symmetric points.
Definition 1.
(i) Let Ss*(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) Let Ssc(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 that f 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 of Ss*(ω) and Ssc(ω).
Definition 2.
(i) Let Ss*(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) Let Ssc(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.
For ω=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 function f 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 function f 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 in U.
Definition 5.
A function f 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 number k, 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:
Ss,0*(ω,A,B)≡Ss*(ω,A,B) and Ss,1*(ω,A,B)≡Ssc(ω,A,B);
Let us define the quasi- Hadamard product of the functions f(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 classes Ss,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 functions fi(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.
Let G(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 setting k=1 in Theorem 6, we obtain the following result.
Corollary 7.
Let the functions fi(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 functions fi(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 that G(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 taking k=0 in Theorem 8, we get the following result.
Corollary 9.
Let the functions fi(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 putting k=0 in Theorem 6, or k=1 in Theorem 8, we obtain the following result.
Corollary 10.
Let the functions fi(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 Theorems 6 and 8.
Taking into account the quasi-Hadamard product of functions f1(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 functions fi(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 functions g1(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 functions gj(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 taking ω=0 in the previous results and making use of relationship (14), we obtain the corresponding results.
Acknowledgments
The present investigation is partly supported by the Natural Science Foundation of China under Grant 11271045, the Higher School Doctoral Foundation of China under Grant 20100003110004 and the Natural Science Foundation of Inner Mongolia of China under Grant 2010MS0117. The authors are thankful to the referees for their careful reading and making some helpful comments which have essentially improved the presentation of this paper.
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