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R E S E A R C H

Open Access

Some properties of meromorphically multivalent

functions

Yi-Hui Xu

1

, Qing Yang

2

and Jin-Lin Liu

2*

* Correspondence: jlliu@yzu.edu.cn

2

Department of Mathematics, Yangzhou University, Yangzhou, 225002, PR China

Full list of author information is available at the end of the article

Abstract

By using the method of differential subordinations, we derive certain properties of meromorphically multivalent functions.

2010 Mathematics Subject Classification:30C45; 30C55.

Keywords:analytic function, meromorphically multivalent function, subordination

1 Introduction

LetΣ(p) denotes the class of meromorphically multivalent functionsf(z) of the form

f(z)=zp+

k=1

akpzkp

pN={1, 2, 3,. . .}, (1:1)

which are analytic in the punctured unit disk

U∗=z:zCand 0<|z|<1=U\ {0}.

Letf(z) andg(z) be analytic inU. Then, we say thatf(z) is subordinate tog(z) in U, writtenf(z)≺g(z), if there exists an analytic functionw(z) inU, such that |w(z)|≤ |z| andf(z) =g(w(z)) (zÎ U). Ifg(z) is univalent inU, then the subordinationf(z)≺g(z) is equivalent tof(0) =g(0) andf(U)⊂g(U).

Letp(z) = 1 +p1z+ ... be analytic in U. Then for -1≤B<A≤1, it is clear that

p(z)≺ 1 +Az

1 +Bz (zU) (1:2) if and only if

p(z)−1−AB

1−B2

< AB

1−B2 (−1<B<A≤1;zU) (1:3) and

Rep(z) > 1−A

2 (B=−1;zU). (1:4)

Recently, several authors (see, e.g., [1-7]) considered some interesting properties of meromorphically multivalent functions. In the present article, we aim at proving some subordination properties for the classΣ(p).

To derive our results, we need the following lemmas.

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Lemma 1 (see [8]. Leth(z) be analytic and starlike univalent inUwithh(0) = 0. Ifg (z) is analytic in Uand zg’(z)≺h(z), then

g(z)g(0)+

z

0 h(t)

t dt.

Lemma 2(see [9]. Let p(z) be analytic and nonconstant inU withp(0) = 1. If 0 < |

z0| < 1 andRe p(z0)= min|z|≤|z0|Rep(z), then

z0p(z0)≤ −

1−p(z0)2

21−Rep(z0) .

2 Main results

Our first result is contained in the following.

Theorem1. Letα∈(0,12]andbÎ(0,1). Iff(z)ÎΣ(p) satisfiesf(z)≠0 (zÎU*) and

fz(zp) zff(z(z)) +p< δ (zU), (2:1)

whereδis the minimum positive root of the equation

αsin πβ 2

x2−x+(1−α)sin πβ 2

= 0, (2:2)

then

arg f(z)

zpα

< π

2β (zU). (2:3)

The bound bis the best possible for eachα∈(0,12]. Proof. Let

g(x)=αsin πβ 2

x2−x+(1−α)sin πβ 2

. (2:4)

We can see that the Equation (2.2) has two positive roots. Sinceg(0) > 0 and g(1) < 0, we have

0< α

1−αδδ <1. (2:5)

Put

f(z)

zp =α+(1−α)p(z). (2:6)

Then from the assumption of the theorem, we see that p(z) is analytic inUwithp(0) = 1 and a+ (1 - a)p(z)≠0 for allz Î U. Taking the logarithmic differentiations in both sides of (2.6), we get

zf(z) f(z) +p=

(1−α)zp(z)

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and

zp f(z)

zf(z) f(z) +p

= (1−α)zp

(z)

α+(1−α)p(z)2 (2:8)

for all zÎU. Thus the inequality (2.1) is equivalent to

(1−α)zp(z)

α+(1−α)p(z)2 ≺δz. (2:9)

By using Lemma 1, (2.9) leads to

z

0

(1−α)p(t)

α+(1−α)p(t)2dtδz or to

1− 1

α+(1−a)p(z)δz. (2:10)

In view of (2.5), (2.10) can be written as

p(z)≺ 1 + α 1−αδz

1−δz . (2:11)

Now by taking A= 1ααδand B= -δin (1.2) and (1.3), we have

arg f(z)

zpα

=argp(z) <arcsin δ

1−α+αδ2

= π 2β

for all zÎUbecause ofg(δ) = 0. This proves (2.3). Next, we consider the functionf(z) defined by

f(z)= z

p

1−δz

zU∗.

It is easy to see that

fz(zp) zff((zz)) +p=|δz|< δ (zU).

Since

f(z)

zpα=(1−α)

1 + 1ααδz

1−δz , it follows from (1.3) that

sup

zU

arg f(z)

zpα

= arcsin δ 1−α+αδ2

= π 2β.

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Theorem 2. Iff(z)Î Σ(p) satisfiesf(z)≠0 (zÎU*) and

Re

zp f(z)

zf(z) f(z) +p

< γ (zU), (2:12)

where

0< γ < 1

2 log 2, (2:13)

then

Re z

p

f(z) >1−2γ log 2 (zU). (2:14) The bound in (2.14) is sharp.

Proof. Let

p(z)= f(z)

zp. (2:15)

Then p(z) is analytic inUwithp(0) = 1 andp(z)≠0 forzÎ U. In view of (2.15) and (2.12), we have

1− zp

(z)

γp2(z)

1 +z

1−z, i.e.,

z 1

p(z)

≺ 2γz

1−z.

Now by using Lemma 1, we obtain

1

p(z) ≺1−2γ log(1−z). (2:16)

Since the function 1 - 2glog(1 -z) is convex univalent inUand

Re 1−2γ log(1−z)>1−2γ log 2 (zU),

from (2.16), we get the inequality (2.14).

To show that the bound in (2.14) cannot be increased, we consider

f(z)= z

p

1−2γ log(1−z)

zU∗.

It is easy to verify that the functionf(z) satisfies the inequality (2.12). On the other hand, we have

Re z

p

f(z) →1−2γlog 2

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Theorem 3. Letf(z)ÎΣ(p) withf(z)≠0 (zÎ U*). If

Im

zf(z) f(z)

f(z) zp −λ

<λλ+ 2p (zU) (2:17)

for some l(l> 0), then

Re f(z)

zp >0 (zU). (2:18)

Proof. Let us define the analytic functionp(z) inUby

f(z)

zp =p(z).

Then p(0) = 1,p(z)≠0 (zÎU) and

zf(z) f(z)

f(z) zp −λ

=p(z)−λ zp

(z)

p(z)p

(zU). (2:19)

Suppose that there exists a pointz0(0 < |z0| < 1) such that

Rep(z) >0 (|z|<|z0|) and p(z0)=, (2:20) wherebis real andb≠0. Then, applying Lemma 2, we get

z0p(z0)≤ −

1 +β2

2 . (2:21)

Thus it follows from (2.19), (2.20), and (2.21) that

I0= Im

z0f(z0) f(z0)

f(z0) z0p −λ

=−

βz0p(z0). (2:22)

In view ofl> 0, from (2.21) and (2.22) we obtain

I0≥ −λ

+λ+ 22

2β

λλ+ 2p (β <0) (2:23)

and

I0≤ −

λ+λ+ 22

2β ≤ −

λλ+ 2p (β >0). (2:24)

But both (2.23) and (2.24) contradict the assumption (2.17). Therefore, we have Rep (z) > 0 for allzÎU. This shows that (2.18) holds true.

Author details

1Department of Mathematics, Suqian College, Suqian 223800, PR China2Department of Mathematics, Yangzhou

University, Yangzhou, 225002, PR China

Authorscontributions

All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

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References

1. Ali, RM, Ravichandran, V, Seenivasagan, N: Subordination and superordination of the Liu-Srivastava linear operator on meromorphically functions. Bull Malays Math Sci Soc.31, 193–207 (2008)

2. Aouf, MK: Certain subclasses of meromprphically multivalent functions associated with generalized hypergeometric function. Comput Math Appl.55, 494–509 (2008)

3. Cho, NE, Kwon, OS, Srivastava, HM: A class of integral operators preserving subordination and superordination for meromorphic functions. Appl Math Comput.193, 463–474 (2007)

4. Liu, J-L, Srivastava, HM: A linear operator and associated families of meromorphically multivalent functions J. Math Anal Appl.259, 566–581 (2001)

5. Liu, J-L, Srivastava, HM: Classes of meromorphically multivalent functions associated with the generalized hypergeometric function. Math Comput Model.39, 21–34 (2004)

6. Wang, Z-G, Jiang, Y-P, Srivastava, HM: Some subclasses of meromorphically multivalent functions associated with the generalized hypergeometric function Comput. Math Appl.57, 571586 (2009)

7. Wang, Z-G, Sun, Y, Zhang, Z-H: Certain classes of meromorphically multivalent functions. Comput Math Appl.58, 14081417 (2009)

8. Suffridge, TJ: Some remarks on convex maps of the unit disk. Duke Math J.37, 775–777 (1970)

9. Miller, SS, Mocanu, PT: Second order differential inequalities in the complex plane. J Math Anal Appl.65, 289305 (1978)

doi:10.1186/1029-242X-2012-86

Cite this article as:Xuet al.:Some properties of meromorphically multivalent functions.Journal of Inequalities and

Applications20122012:86.

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