New Subclasses of Multivalent Analytic Functions Associated with a Linear Operator

Abstract

Making use of a linear operator, which is defined here by means of the Hadamard product (or convolution), we consider two subclasses [subscript]Fp,n[/subscript] (a,c,λ,A,B) and [subscript]Gp,n[/subscript] (a,c,λ,A,B) of multivalent analytic functions with negative coefficients in the open unit disk. Some modified Hadamard products, integral transforms, and the partial sums of functions belonging to these classes are studied.

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Ding-Gong Yang 1 and Jin-Lin Liu 2

Academic Editor:Ngai-Ching Wong

1, School of Mathematical Sciences, Soochow University, Suzhou, Jiangsu 215006, China
2, Department of Mathematics, Yangzhou University, Yangzhou, Jiangsu 225002, China

Received 14 August 2013; Accepted 29 October 2013

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Let _Ap,n denote the class of functions of the form [figure omitted; refer to PDF] which are analytic in the open unit disk U={z:|z|<1} .

For functions f(z) and g(z) analytic in U , we say that f(z) is subordinate to g(z) in U , written f(z)[precedes]g(z) (z∈U) , if there exists an analytic function w(z) in U such that [figure omitted; refer to PDF] Furthermore, if the function g(z) is univalent in U , then [figure omitted; refer to PDF]

In terms of the Pochhammer symbol (b_)n given by (b_)n =b(b+1)...(b+n-1) (n∈N) , we define the function _{[straight phi]p,n} (a,c;z) by [figure omitted; refer to PDF] Corresponding to _{[straight phi]p,n} (a,c;z) , we consider here a linear operator _Lp,n (a,c) on _Ap,n by the following usual Hadamard product (or convolution): [figure omitted; refer to PDF] for f(z) given by (1). For p=n=1 , _L1,1 (a,c) on _A1,1 was first defined by Carlson and Shaffer [1]. Its differential-integral representation can be found in [2]. We remark in passing that a much more general convolution operator than the operator _Lp,1 (a,c) was introduced by Dziok and Srivastava [3].

Let _Tp,n denote the subclass of _Ap,n consisting of functions of the form [figure omitted; refer to PDF] We now consider the following two subclasses of the class _Tp,n .

Definition 1.

A function f(z)∈_Tp,n is said to be in the class _Fp,n (a,c,λ,A,B) if and only if [figure omitted; refer to PDF] where [figure omitted; refer to PDF]

Definition 2.

A function f(z)∈_Tp,n is said to be in the class _Gp,n (a,c,λ,A,B) if and only if [figure omitted; refer to PDF]

For functions _fj (z)∈_Tp,n given by [figure omitted; refer to PDF] we denote by (_f1 ·_f2 )(z) the modified Hadamard (or quasi-Hadamard) product of _f1 (z) and _f2 (z) ; that is, [figure omitted; refer to PDF]

The class [figure omitted; refer to PDF] with n=1 was introduced and studied earlier by Lee et al. [4] (and was further investigated by Aouf and Darwish [5], Aouf et al. [6], and Yaguchi et al. [7]). The class [figure omitted; refer to PDF] with n=1 was studied by Aouf [8] and Aouf et al. [6]. Recently, Aouf [9] investigated the modified Hadamard products of several functions in the classes _Fp (n,λ,α) and _Gp (n,λ,α) for n∈N .

In the present paper, we prove a number of theorems involving the modified Hadamard products, integral transforms, and the partial sums of functions in the classes _Fp,n (a,c,λ,A,B) and _Gp,n (a,c,λ,A,B) . Some of our results are generalizations of the corresponding results in [4-9].

In proving our main results, we need the following lemmas.

Lemma 3 (see [10, 11]).

A function f(z)∈_Tp,n defined by (6) is in the class _Fp,n (a,c,λ,A,B) if and only if [figure omitted; refer to PDF]

Lemma 4 (see [10, 11]).

A function f(z)∈_Tp,n defined by (6) is in the class _Gp,n (a,c,λ,A,B) if and only if [figure omitted; refer to PDF]

Making use of Lemmas 3 and 4, we can show the following two results.

Corollary 5.

Let [figure omitted; refer to PDF] Then [figure omitted; refer to PDF]

Corollary 6.

Let f(z)∈_Tp,n . Then f(z)∈_Fp,n (_a0 ,_c0 ,_λ0 ,A,B) (_Gp,n (_a0 ,_c0 ,_λ0 ,A,B)) if and only if (f*g)(z)∈_Fp,n (_a1 ,_c1 ,_λ1 ,A,B) (_Gp,n (_a1 ,_c1 ,_λ1 ,A,B)) , where _λ0 ...5;0 , _λ1 ...5;0, _aj >0 and _cj >0 (j=0,1) , and [figure omitted; refer to PDF]

If we let [figure omitted; refer to PDF] then Lemmas 3 and 4 reduce to the following result.

Corollary 7.

Let f(z)∈_Tp,n be defined by (6). Then

(i) f(z) is in the class _Fp (n,λ,α) if and only if [figure omitted; refer to PDF]

(ii) f(z) is in the class _Gp (n,λ,α) if and only if [figure omitted; refer to PDF]

2. Modified Hadamard Products

Hereafter in this paper we assume that (8) is satisfied: [figure omitted; refer to PDF]

Theorem 8.

Let _fj (z)∈_Fp,n (a,c,λ,_Aj ,_Bj ) (j=1,2,...,m) and a...5;c>0 . Then [figure omitted; refer to PDF] where m...5;2 and [figure omitted; refer to PDF] The result is sharp; that is, A(B) cannot be decreased for each B∈[-1,0] .

Proof.

By (24) we have B<A(B)...4;1 for a...5;c>0 . Let [figure omitted; refer to PDF] Then Lemma 3 gives [figure omitted; refer to PDF] and hence [figure omitted; refer to PDF] Also, using Lemma 3, [figure omitted; refer to PDF] if and only if [figure omitted; refer to PDF]

To prove the result of Theorem 8, it follows from (27) and (29) that we need to find the smallest A such that [figure omitted; refer to PDF] that is, that [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] we see that the right-hand side of (31) is a decreasing function of k . Consequently, taking k=p+n in (31), we have h(z)∈_Fp,n (a,c,λ,A(B),B) , where A(B) is given by (24).

Furthermore, by considering the functions _fj (z) defined by: [figure omitted; refer to PDF] we have _fj (z)∈_Fp,n (a,c,λ,_Aj ,_Bj ) and [figure omitted; refer to PDF] Noting that [figure omitted; refer to PDF] we conclude that A(B) cannot be decreased for each B .

By using Lemma 4 instead of Lemma 3, the following theorem can be proved on the lines of the proof of Theorem 8. We omit the details involved.

Theorem 9.

Let _fj (z)∈_Gp,n (a,c,λ,_Aj ,_Bj ) (j=1,2,...,m) and a...5;c>0 . Then [figure omitted; refer to PDF] where m...5;2 and [figure omitted; refer to PDF] The result is sharp for the functions _fj (z) defined by [figure omitted; refer to PDF]

Theorem 10.

Let _fj (z)∈_Fp,n (a,c,λ,_Aj ,_Bj ) (j=1,2,...,m) with a...5;c>0 and λ...5;1 . Then [figure omitted; refer to PDF] where m...5;2 and [figure omitted; refer to PDF] The result is sharp for the functions _fj (z) (j=1,2,...,m) defined by (33).

Proof.

Obviously, B<A(B)...4;1 for m...5;2 , a...5;c>0 , and λ...5;1 . By applying Lemma 4, we know that [figure omitted; refer to PDF] if and only if [figure omitted; refer to PDF] Proceeding as in the proof of Theorem 8, we need to find the smallest A such that [figure omitted; refer to PDF] Defining the function [straight phi](x) by [figure omitted; refer to PDF] we see that [figure omitted; refer to PDF] for m...5;2 and λ...5;1 . Hence, the right-hand side of (43) is a decreasing function of k . Thus, we arrive at (_f1 ·_f2 ·...·_fm )(z)∈_Gp,n (a,c,λ,A(B),B) , where A(B) is given by (40).

Sharpness can be verified easily.

By putting [figure omitted; refer to PDF] Theorem 10 reduces to the following.

Corollary 11.

Let _fj (z)∈_Fp (n,λ,_αj ) (j=1,2,...,m) . Then (_f1 ·_f2 ·...·_fm )(z)∈_Gp (n,λ,α) , where m...5;2 , λ...5;1 , and [figure omitted; refer to PDF] The result is sharp for the functions [figure omitted; refer to PDF]

Theorem 12.

Let _fj (z)∈_Fp,n (a,c,λ,_Aj ,_Bj ) (j=1,2,...,m) , [figure omitted; refer to PDF] and a...5;c>0 . Then one has

(i) ^∑j=1m (_fj ·_gj )(z) -(m-1)^zp ∈_Fp,n (a,c,λ,A(B),B) , where [figure omitted; refer to PDF]

: provided that A(B)...4;1 .

(ii) (1/m)^∑j=1m (_fj ·_gj )(z)∈_Fp,n (a,c,λ,A~(B),B) , where [figure omitted; refer to PDF]

Proof.

It is clear that -1...4;_B0 ...4;_Bj <_Aj ...4;_A0 ...4;1 , _B0 ...4;0 , -1...4;_D0 ...4;_Dj <_Cj ...4;_C0 ...4;1 , _D0 ...4;0 , [figure omitted; refer to PDF] for j=1,2,...,m . Let [figure omitted; refer to PDF] Then Lemma 3 gives [figure omitted; refer to PDF] if and only if [figure omitted; refer to PDF] Also, by Lemma 3 and (52), we deduce that [figure omitted; refer to PDF] and hence [figure omitted; refer to PDF]

To prove Theorem 12(i), it follows from (55) and (57) that we need to find the smallest A such that [figure omitted; refer to PDF] for a...5;c>0 . This leads to the assertion of Theorem 12(i).

Analogously, we can prove Theorem 12(ii).

In the special case when [figure omitted; refer to PDF] Theorem 12 reduces to the following.

Corollary 13.

Let _fj (z)∈_Fp (n,λ,_αj ) (j=1,2,...,m) . Then

(i) ^∑j=1m (_fj ·_fj )(z) -(m-1)^zp ∈_Fp (n,λ,α) , where [figure omitted; refer to PDF]

: provided that α...5;0 .

(ii) (1/m)^∑j=1m (_fj ·_fj )(z)∈_Fp (n,λ,α~) , where [figure omitted; refer to PDF]

Replacing Lemma 3 by Lemma 4 in the proof of Theorem 12, one can prove the following.

Theorem 14.

Let _fj (z)∈_Gp,n (a,c,λ,_Aj ,_Bj ) (j=1,2,...,m) , [figure omitted; refer to PDF] and a...5;c>0 . Then

(i) ^∑j=1m ...(_fj ·_gj )(z)-(m-1)^zp ∈_Gp,n (a,c,λ,A(B),B) , where [figure omitted; refer to PDF]

: and _A0 ,_B0 ,_C0 , and _D0 are given as in Theorem 12, provided that A(B)...4;1 .

(ii) (1/m)^∑j=1m (_fj ·_gj )(z)∈_Gp,n (a,c,λ,A~(B),B) , where [figure omitted; refer to PDF]

As a special case of Theorem 14, one has the following.

Corollary 15.

Let _fj (z)∈_Gp (n,λ,_αj ) (j=1,2,...,m) . Then

(i) ^∑j=1m ...(_fj ·_fj )(z)-(m-1)^zp ∈_Gp (n,λ,α) , where [figure omitted; refer to PDF]

: provided that α...5;0 .

(ii) (1/m)^∑j=1m (_fj ·_fj )(z)∈_Gp (n,λ,α~) , where [figure omitted; refer to PDF]

3. Integral Operator

Theorem 16.

Let [figure omitted; refer to PDF] If f(z)∈_Fp,n (_a0 ,_c0 ,_λ0 ,A,B) , then the function I(z) defined by [figure omitted; refer to PDF] belongs to _Fp,n (_a1 ,_c1 ,_λ1 ,C(D),D) , where -1...4;D...4;0 and [figure omitted; refer to PDF] The result is sharp; that is, the number C(D) cannot be decreased for each D .

Proof.

Note that D<C(D)<1 . For [figure omitted; refer to PDF] it follows from (68) that [figure omitted; refer to PDF]

To prove the result of Theorem 16, we need to find the smallest C such that [figure omitted; refer to PDF] where we have used Lemma 3. In view of (67), it is easy to know that the right-hand side of (72) is a decreasing function of k . Therefore, we conclude that [figure omitted; refer to PDF] where C(D) is given by (69).

Furthermore, it can easily be verified that the result is sharp, with the extremal function [figure omitted; refer to PDF]

With the aid of Lemma 4 (instead of Lemma 3) and using the same steps as in the proof of Theorem 16, we can prove the following.

Theorem 17.

Let (67) in Theorem 16 be satisfied. If f(z)∈_Gp,n (_a0 ,_c0 ,_λ0 ,A,B) , then the function I(z) defined by (68) belongs to _Gp,n (_a1 ,_c1 ,_λ1 ,C(D),D) , where C(D) (-1...4;D...4;0) is the same as in Theorem 16. The result is sharp for the function [figure omitted; refer to PDF]

If we let [figure omitted; refer to PDF] then Theorem 17 yields the following.

Corollary 18.

Let f(z)∈_Gp (n,λ,α) . Then the function I(z) defined by (68) belongs to _Gp (n,λ,β(α)) , where [figure omitted; refer to PDF] The number β(α) cannot be increased for each α∈[0,p) .

4. Partial Sums

In this section, we let f(z)∈_Tp,n be given by (6) and define the partial sums _s1 (z) and _sm (z) by [figure omitted; refer to PDF] Also we make the notation simple by writing [figure omitted; refer to PDF]

Theorem 19.

Let f(z)∈_Fp,n (a,c,λ,A,B) and a...5;c>0 . Then for z∈U , one has the following. [figure omitted; refer to PDF] The results are sharp for each m∈N .

Proof.

Let a...5;c>0 and _βk be given by (79). Then _βk+1 ...5;_βk ...5;1 for k...5;p+n , and so it follows from Lemma 3 that [figure omitted; refer to PDF] for f(z)∈_Fp,n (a,c,λ,A,B) .

If we put [figure omitted; refer to PDF] then p(0)=1 and [figure omitted; refer to PDF] because of (82). Hence, we have Re_p1 (z)>0 for z∈U , which implies that (80) holds true for m...5;2 .

Similarly, by setting [figure omitted; refer to PDF] it follows from (82) that [figure omitted; refer to PDF] Therefore, we see that Re_p2 (z)>0 for z∈U , that is, that (81) holds for m...5;2 .

For m=1 , replacing (82) by [figure omitted; refer to PDF] and proceeding as the above, we know that (80) and (81) are also true.

Furthermore, by taking the function [figure omitted; refer to PDF] we find that _sm (z)=^zp , [figure omitted; refer to PDF] The proof of Theorem 19 is thus completed.

By virtue of Theorem 19 and Definition 2, we easily have the following.

Corollary 20.

Let f(z)∈_Gp,n (a,c,λ,A,B) and a...5;c>0 . Then we have [figure omitted; refer to PDF] The results are sharp for each m∈N .

Theorem 21.

Let f(z)∈_Fp,n (a,c,λ,A,B) with a...5;c>0 and λ...5;1 . Then one has [figure omitted; refer to PDF] The results are sharp for each m∈N .

Proof.

Let a...5;c>0 , λ...5;1 and _βk be given by (79). Then it is easy to verify that [figure omitted; refer to PDF] and hence we deduce from Lemma 3 that [figure omitted; refer to PDF] for f(z)∈_Fp,n (a,c,λ,A,B) .

Defining the function _q1 (z) by [figure omitted; refer to PDF] it follows from (94) that [figure omitted; refer to PDF] This leads to the inequality (91) for m...5;2 .

Similarly, for the function _q2 (z) defined by [figure omitted; refer to PDF] we deduce from (94) that [figure omitted; refer to PDF] This yields the inequality (92) for m...5;2 .

For m=1 , replacing (94) by [figure omitted; refer to PDF] we know that (91) and (92) are also true.

Furthermore, the bounds in (91) and (92) are the best possible for the function f(z) defined by (88).

Finally, Theorem 21 yields the following.

Corollary 22.

Let f(z)∈_Gp,n (a,c,λ,A,B) with a...5;c>0 and λ...5;1 . Then [figure omitted; refer to PDF] The results are sharp for each m∈N .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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