On a certain new subclass of meromorphic close to convex functions

R E S E A R C H

Open Access

On a certain new subclass of meromorphic

close-to-convex functions

Huo Tang

_{, Guan-Tie Deng}

_{and Shu-Hai Li}

*_{Correspondence:}

[email protected] 1_{School of Mathematics and}

Statistics, Chifeng University, Chifeng, Inner Mongolia 024000, China

2_{School of Mathematical Sciences,}

Beijing Normal University, Beijing, 100875, China

Abstract

In this paper, we introduce and investigate a new subclassMK(k)₍

_β

_γ

meromorphic close-to-convex functions. For functions belonging to the class

MK(k)₍

_β

_γ

results presented here would unify and extend some recent work of Wanget al.(Appl. Math. Lett. 25:454-460, 2012).

MSC: 30C45

Keywords: analytic functions; meromorphic close-to-convex functions; coefficient inequality; distortion theorem; subordination

1 Introduction

Letbe the class of functionsf of the form:

f(z) = 

z+ ∞

n=

anzn, (.)

which are analytic in the punctured open unit diskU∗={z∈C:  <|z|< }=U\ {}. LetPdenote the class of functionspgiven by

p(z) =  +

∞

n=

pnzn (z∈U), (.)

which are analytic and convex inUand satisfy the condition(p(z)) >  (z∈U).

A functionf ∈is said to be in the classMS∗(α) of meromorphic starlike functions of orderαif it satisfies the inequality

zf(z)

f(z)

< –α z∈U∗; ≤α< .

In addition, a functionf∈is said to be in the classMCof meromorphic close-to-convex functions if it satisfies the inequality

zf(z)

g(z)

<  z∈U∗;g∈MS∗() =MS∗.

Recently, Srivastavaet al.[] (see also [, ]) introduced and studied the classMS∗_s of meromorphic starlike functions with respect to symmetric points, which satisfies the

dition

zf(z)

f(z) –f(–z)

<  z∈U∗.

More recently, Wanget al.[] discussed a classMK of meromorphic close-to-convex functions, that is, a functionf ∈is said to be in the classMKif it satisfies the inequality

f(z)

g(z)g(–z)

>  z∈U∗,

whereg∈MS∗(_).

Letf(z) =z+az+· · · be analytic inU. If there exists a functiong∈S∗(_), such that

_g(z_z₎f_g(_(–z)_z)+ < z _f(z)

g(z)g(–z)–  + γ

(z∈U),

then we say thatf ∈Ks(γ), ≤γ < , whereS∗(_) denotes the usual class of starlike

func-tions of order /. The function classKs(γ) was introduced and studied recently by

Kowal-czyk and Les-Bomba [] (see also [–]).

For two functionsf andganalytic inU, we say that the functionf(z) is subordinate to

g(z) inU, and we writef(z)≺g(z) (z∈U) if there exists a Schwarz functionw(z), analytic inU with w() =  and|w(z)| ≤, such thatf(z) =g(w(z)) (z∈U). In particular, if the functiongis univalent inU, then we havef() =g() andf(U)⊂g(U) (see, for example, []).

Motivated essentially by the above mentioned function classesMK andKs(γ), we now

introduce a new classMK(k)_{(β,γ) of meromorphic functions.}

Definition  LetMK(k)(β,γ) denote the class of functions insatisfying the inequality

z–_gkk₍f_z(₎z)+ <βz –k_f(z)

gk(z)

+ γ –  z∈U∗;  <β≤; ≤γ< , (.)

whereg∈MS∗(k–_k ),k≥, is a fixed positive integer andgk(z) is defined by the following

equality:

gk(z) = k–

v=

ε–vgεvz εk= . (.)

We note thatMK()_{(, ) =MK (see []), so the classMK}(k)_{(β,γ) is a generation of the} classMK.

2 Main results

First of all, we give two meaningful conclusions about the classMK(k)(β,γ). The proof of Theorem  below is much akin to that of Theorem  in [], so we choose to omit the details involved.

Theorem  A function f∈MK(k)_{(β,γ)if and only if there exists g∈MS}∗₍k–

k )such that

–z –k_f(z)

gk(z) ≺

 + ( – γ)βz

 –βz

z∈U∗. (.)

Remark  From Theorem , we know that

z–k_f(z) gk(z)

<  z∈U∗, (.)

because of(+(–_–βγ_z)βz) >  (z∈U∗).

Lemma  Letϕi∈MS∗(αi),where≤αi<  (i= , , . . . ,k–).Then for k–≤ ki=–αi<k, we have

zk– k–

i=

ϕi(z)∈MS∗

_k–

i=

αi– (k– )

.

Proof Sinceϕi∈MS∗(αi) (i= , , . . . ,k– ), by the definition of meromorphic starlike

functions, we have

zϕ_(z) ϕ(z)

< –α,

zϕ_(z) ϕ(z)

< –α, . . . ,

zϕ_k–(z) ϕk–(z)

< –αk–. (.)

We now let

F(z) =zk–ϕ(z)ϕ(z)· · ·ϕk–(z). (.)

Differentiating (.) with respect tozlogarithmically, we easily get

zF(z)

F(z) =

zϕ_(z) ϕ(z) +

zϕ_(z) ϕ(z) +· · ·+

zϕ_k–(z) ϕk–(z)

+ (k– ). (.)

From (.) together with (.), we obtain

zF(z)

F(z)

=

zϕ_(z) ϕ(z)

+

zϕ_(z) ϕ(z)

+· · ·+

zϕ_k–(z) ϕk–(z)

+ (k– )

< –

k–

i=

αi+ (k– ) = –

_k–

i=

αi– (k– )

by noting that ≤ k_i=–αi– (k– ) < , which implies that

F(z) =zk–

k–

i=

ϕi(z)∈MS∗

_k–

i=

αi– (k– )

.

Theorem  Let g(z) =_z+ ∞_n=bnzn∈MS∗(k–_k ),then zk–gk(z)∈MS∗.

Proof From (.), we know

zk–gk(z) =zk– k–

v=

ε–vgεvz=zk– k–

v= ε–v

 εv_z+

∞

n=

bn

εvzn

=zk– k–

v=

 εv_z+

∞

n=

bnε(n–)vzn

. (.)

Sinceg(z) =_z+ ∞_n=bnzn∈MS∗(k–_k ), by Lemma  and (.), we can easily get the

asser-tion of Theorem .

Remark  From Theorem  and the inequality (.), we see that iff ∈MK(k)_{(β,γ), then}

f(z) is a meromorphic close-to-convex function. So,MK(k)(β,γ) is a subclass of the class

MCof meromorphic close-to-convex functions.

Next, we give some coefficient inequalities for functions belonging to the class

MK(k)_(β,γ).

Theorem  Let f(z) = _z + ∞_n=anznand g(z) = _z + ∞n=bnznbe analytic in U∗. If for

 <β≤and≤γ < ,we have

∞

n=

n( +β)|an|+

∞

n=

β| – γ|+ |Bn| ≤β( –γ), (.)

where the coefficients Bn(n= , , . . .)are given by(.),then f ∈MK(k)(β,γ).

Proof Suppose that

Gk(z) =zk–gk(z). (.)

By Theorem , we know thatGk∈MS∗. Hence, equality (.) can be written as

Gk(z) =zk–gk(z) =



z+ ∞

n=

Bnzn∈MS∗. (.)

To provef ∈MK(k)(β,γ), it suffices to show that

zf

_(z)

Gk(z)+ 

zf(z)

Gk(z)+ γ – 

<β,

whereGkis given by (.). From (.), we know that

β( – γ) –

∞

n=

nβ|an|–

∞

n=

β| – γ||Bn| ≥ ∞

n=

n|an|+

∞

n=

Now, by the maximum modulus principle, we deduce from (.), (.) and (.) that

zf(z)

Gk(z)+ 

zf(z)

Gk(z)+ γ – 

=

∞

n=nanzn++ ∞n=Bnzn+ ∞

n=nanzn+– ∞n=( – γ)Bnzn+– ( – γ)

<

∞

n=n|an|+

∞ n=|Bn|

( – γ) – ∞_n=n|an|– ∞_n=| – γ||Bn|≤β.

This evidently completes the proof of Theorem .

Theorem  Let f(z) = _z+ ∞_n=anzn∈MK(k)(β,γ).Then

 +

∞

n=

n( –βeiθ₎ ( –γ)βeiθanz

n+₊

∞

n=

 + ( – γ)βeiθ ( –γ)βeiθ Bnz

n+_{= }

z∈U∗;  <θ< π, (.)

where the coefficients Bn(n= , , . . .)are given by(.).

Proof Suppose thatf∈MK(k)(β,γ). Then we know that

–zf

_(z)

Gk(z)

= + ( – γ)βe

iθ

 –βeiθ

z∈U∗;  <θ< π, (.)

whereGk is given by (.). After a simple computation, the inequality (.) is equivalent

to

zf(z) –βeiθ+Gk(z)

 + ( – γ)βeiθ=  z∈U∗;  <θ< π. (.)

By substituting (.) and (.) into (.), we obtain the desired assertion (.) of Theo-rem .

Finally, we provide the following distortion theorem for the considered class of functions

MK(k)_(β,γ).

Theorem  If f ∈MK(k)(β,γ),then

( –r)_{( – ( – γ)βr)}

r_{( +βr)} ≤f

_(z)

≤( +r)( + ( – γ)βr) r_{( –βr)}

|z|=r;  <r< . (.)

Proof Iff ∈MK(k)_{(β,γ), then there exists a functiong∈MS}∗₍k–

k ) such that (.) holds

true. It follows from Theorem  that the functionGk given by (.) is a meromorphic

starlike function. Hence, we have (see [])

( –r)

r ≤Gk(z)≤

( +r)

r

Let us definep(z) by

–zf

_(z)

Gk(z)

=p(z) z∈U∗, (.)

where

p(z)≺ + ( – γ)βz  –βz .

Then, by using a similar method as in [, p.], we have

 – ( – γ)βr

 +βr ≤p(z)≤

 + ( – γ)βr

 –βr

|z|=r;  <r< . (.)

Thus, from (.), (.) and (.), we readily get the inequality (.). The proof of

The-orem  is thus completed.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors jointly worked on the results and they read and approved the final manuscript.

Acknowledgements

The present investigation was partly supported by the Natural Science Foundation of China under Grant 11271045, the Higher School Doctoral Foundation of China under Grant 2010000311000 4 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.

Received: 23 September 2012 Accepted: 14 March 2013 Published: 10 April 2013 References

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