Subclass of univalent harmonic functions defined by dual convolution

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R E S E A R C H

Open Access

Subclass of univalent harmonic functions

deﬁned by dual convolution

Rabha M El-Ashwah

*

*_{Correspondence:}

r_elashwah@yahoo.com Department of Mathematics, Faculty of Science, Damietta University, New Damietta, 34517, Egypt

Abstract

In the present paper, we study a subclass of univalent harmonic functions deﬁned by convolution and integral convolution. We obtain the basic properties such as coeﬃcient characterization and distortion theorem, extreme points and convolution condition.

MSC: 30C45; 30C50

Keywords: harmonic function; univalent; sense-preserving; integral convolution

1 Introduction

A continuous functionf =u+ivis a complex-valued harmonic function in a simply con-nected complex domainD⊂Cif bothuandvare real harmonic inD. It was shown by Clunie and Sheil-Small [] that such a harmonic function can be represented byf =h+g, wherehandgare analytic inD. Also, a necessary and suﬃcient condition forfto be locally univalent and sense-preserving inDis that|h(z)|>|g(z)|(see also [–] and []).

Denote bySHthe class of functionsf that are harmonic univalent and sense-preserving in the open unit discU={z∈C:|z|< }, for whichf() =h() =f_z() –  = . Then for f=h+g∈SHwe may express the analytic functionshandgas

h(z) =z+ ∞

n=

anzn, g(z) = ∞

n=

bnzn, |b|< . (.)

Clunie and Sheil-Small [] investigated the classSHas well as its geometric subclasses and obtained some coeﬃcient bounds.

Also, let SH denote the subclass ofSH consisting of functionsf =h+g such that the functionshandgare of the form

h(z) =z– ∞

n=

|an|zn, g(z) = ∞

n=

|bn|zn, |b|< . (.)

Recently Kanas and Wisniowska [] (see also Kanas and Srivastava []) studied the class of k-uniformly convex analytic functions, denoted byk–UCV,k≥, so thatφ∈k–UCV if and only if

Re

 +(z–ζ)φ ₍_z₎

φ(z)

≥ |ζ| ≤k;z∈U. (.)

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Forθ∈R, if we letζ = –kzeiθ_{, then condition (.) can be written as}

Re

 + +keiθzφ ₍_z₎

φ(z)

≥. (.)

Kimet al.[] introduced and studied the classHCV(k,α) consisting of functionsf =h+g, such thathandgare given by (.), and satisfying the condition

Re

 + +keiθz

_h₍_z_{) + }_zg₍_z_{) +}_z_g₍_z₎

zh(z) –zg(z)

≥α (≤α< ;θ∈R;k≥). (.)

Also, the class ofk–USTuniformly starlike functions is deﬁned by using (.) as the class of all functionsψ(z) =zφ(z) such thatφ∈k–UCV, thenψ(z)∈k–USTif and only if

Re +keiθzψ(z)

ψ(z) –ke iθ

≥. (.)

Generalizing the classk–UST to include harmonic functions, we letHST(k,α) denote the class of functionsf =h+g, such thathandgare given by (.), which satisﬁes the condition

Re +keiθzf ₍_z₎

zf(z)–ke iθ

≥α (≤α< ;θ∈R;k≥). (.)

Replacingh+gforf in (.), we have

Re +keiθzh

₍_z_{) –}_zg₍_z₎

h(z) +g(z) –ke iθ

≥α (≤α< ;θ∈R;k≥). (.)

The convolution of two functions of the form

f(z) =z+ ∞

n=

anzn and F(z) =z+ ∞

n= Anzn

is deﬁned as

(f ∗F)(z) =z+ ∞

n=

anAnzn, (.)

while the integral convolution is deﬁned by

(f♦F)(z) =z+ ∞

n= anAn

n z

n_. _(.)

From (.) and (.), we have

(f♦F)(z) =

z 

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Now we consider the subclassHST(φ,ψ,k,α) consisting of functionsf =h+g, such that handgare given by (.), and satisfying the condition

Re +keiθh(z)∗ϕ(z) –g(z)∗χ(z) h(z)♦ϕ(z) +g(z)♦χ(z)–ke

iθ

≥α (≤α< ;k≥;θreal), (.)

where

ϕ(z) =z+ ∞

n=

λnzn (λn≥) and χ(z) =z+ ∞

n=

μnzn (μn≥). (.)

We further consider the subclassHST(φ,χ,k,α) ofHST(φ,χ,k,α) forhandggiven by (.).

We note that

(i) HST(φ,χ, ,α) =HS(φ,χ,α)(see Dixitet al.[]); (ii) HST(_(–z_z₎,_(–z_z₎, ,α) =GH(α)(see Rosyet al.[]);

(iii) HST(_(–z+z_z₎, z+z 

(–z),k,α) =HCV(k,α)(see Kimet al.[]);

(iv) HST( z

(–z),_(–z_z₎, ,α) =TH∗(α)(see Jahangiri [], see also Joshi and Darus []);

(v) HST(_(–z+z_z₎, z+z 

(–z), ,α) =CH(α)(see Jahangiri [], see also Joshi and Darus []).

In this paper, we extend the results of the above classes to the classesHST(φ,χ,k,α) and HST(φ,χ,k,α), we also obtain some basic properties for the classHST(φ,χ,k,α).

2 Coefﬁcient characterization and distortion theorem

Unless otherwise mentioned, we assume throughout this paper thatϕ(z) and χ(z) are given by (.), ≤α< ,k≥ andθ is real. We begin with a suﬃcient condition for functions in the classHST(φ,χ,k,α).

Theorem  Let f =h+g be such that h and g are given by(.).Furthermore,let

∞

n= λn

n

( +k)n– (k+α) ( –α) |an|+

∞

n= μn

n

( +k)n+ (k+α)

( –α) |bn| ≤, (.)

where

n( –α)≤λn

( +k)n– (k+α) and n( –α)≤μn

( +k)n+ (k+α) for n≥.

Then f is sense-preserving,harmonic univalent in U and f ∈HST(φ,χ,k,α).

Proof First we note thatf is locally univalent and sense-preserving inU. This is because

h(z)≥ – ∞

n=

n|an|rn–>  – ∞

n=

n|an| ≥ – ∞

n= λn

n

( +k)n– (k+α) ( –α) |an|

≥

∞

n= μn

n

( +k)n+ (k+α) ( –α) |bn| ≥

∞

n= n|bn| ≥

∞

n=

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To show thatf is univalent inU, supposez,z∈Uso thatz=z, then

_hf(₍z_z) –__{) –}f_h(₍z_z)_₎≥ –g(z) –g(z) h(z) –h(z)

=  –

∞

n=bn(zn–zn) (z–z) +

∞

n=an(zn –zn)

≥ –

∞ n=n|bn|  –∞_n_=n|an|>  –

_∞

n=

μn

n(

(+k)n+(k+α) (–α) )|bn|  –∞_n_=λn

n(

(+k)n–(k+α) (–α) )|an|

≥.

Now, we prove thatf ∈HST(φ,ψ,k,α), by definition, we only need to show that if (.) holds, then condition (.) is satisfied. From (.), it suffices to show that

Re

( +keiθ₎₍_h₍_z₎_∗_ϕ₍_z_{) –}_g₍_z₎_∗_χ₍_z_{)) – (}_keiθ₊_α₎₍_h₍_z₎_♦_ϕ₍_z_{) +}_g₍_z₎_♦_χ₍_z₎₎

h(z)♦ϕ(z) +g(z)♦χ(z)

≥. (.)

Substituting forh,g,ϕandχin (.) and dividing by ( –α)z, we obtainReA(z)

B(z) ≥, where

A(z) =  + ∞

n= λn

n

( +keiθ₎_n_{– (}_keiθ₊_α₎

( –α) anz

n–

–

z z

∞

n= μn

n

( +keiθ₎_n_{+ (}_keiθ₊_α₎

( –α) bnz

n–

and

B(z) =  + ∞

n= λn

nanz n–₊

z z

∞

n= μn

n bnz n–

.

Using the fact thatRe(w)≥ if and only if| +w| ≥ | –w|inU, it suﬃces to show that

|A(z) +B(z)|–|A(z) –B(z)| ≥. Substituting forA(z) andB(z) gives

A(z) +B(z)–A(z) –B(z)

=

 +

∞

n= λn

n

( +keiθ₎_n_{– (}_keiθ_{+ }_α_{– )}

( –α) anz

n–

–

z z

∞

n= μn

n

( +keiθ₎_n_{+ (}_keiθ_{+ }_α_{– )}

( –α) bnz

n–

–

∞

n= λn

n

( +keiθ₎_n_{– ( +}_keiθ₎

( –α) anz n–

–

z z

∞

n= μn

n

( +keiθ₎_n_{+ ( +}_keiθ₎

( –α) bnz

n–

≥ – ∞

n= λn

n

( +k)n– (k+ α– ) ( –α) |an||z|

n–

– ∞

n= μn

n

( +k)n+ (k+ α– ) ( –α) |bn||z|

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– ∞

n= λn

n

( +k)n– ( +k) ( –α) |an+||z|

n–

– ∞

n= μn

n

( +k)n+ ( +k) ( –α) |bn||z|

n–

≥

 – ∞

n= λn

n

( +k)n– (k+α) ( –α) |an|–

∞

n= μn

n

( +k)n+ (k+α) ( –α) |bn|

≥ by (.).

The harmonic functions

f(z) =z+ ∞

n= n λn

( –α)

( +k)n– (k+α)xnz n

+ ∞

n= n μn

( –α)

( +k)n+ (k+α)ynz

n_, _(.)

where∞_n_=|xn|+∞_n_=|yn|= , show that the coeﬃcient bound given by (.) is sharp. The functions of the form (.) are in the classHST(φ,χ,k,α) because

∞

n=

λn n

( +k)n– (k+α) ( –α) |an|+

∞

n= μn

n

( +k)n+ (k+α) ( –α) |bn|

= ∞

n=

|xn|+ ∞

n=

|yn|= .

This completes the proof of Theorem .

In the following theorem, it is shown that condition (.) is also necessary for functions f =h+g, wherehandgare given by (.).

Theorem  Let f =h+g be such that h and g are given by(.).Then f ∈HST(φ,χ,k,α) if and only if

∞

n= λn

n

( +k)n– (k+α) ( –α) |an|+

∞

n= μn

n

( +k)n+ (k+α)

( –α) |bn| ≤. (.)

Proof SinceHST(φ,χ,k,α)⊂HST(φ,χ,k,α), we only need to prove the ‘only if ’ part of the theorem. To this end, we notice that the necessary and suﬃcient condition forf ∈ HST(φ,χ,k,α) is that

Re +keiθh(z)∗ϕ(z) –g(z)∗χ(z) h(z)♦ϕ(z) +g(z)♦χ(z)–ke

iθ

≥α.

This is equivalent to

Re

( +keiθ₎₍_h₍_z₎_∗_ϕ₍_z_{) –}_g₍_z₎_∗_χ₍_z_{)) – (}_keiθ₊_α₎₍_h₍_z₎_♦_ϕ₍_z_{) +}_g₍_z₎_♦_χ₍_z₎₎

h(z)♦ϕ(z) +g(z)♦χ(z)

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which implies that

Re

( –α)z–∞_n_=λn

n[( +kei

θ₎_n_{– (}_keiθ₊_α_)]_|_an_|_zn

z–∞_n_=λn

n|an|zn+

∞ n=

μn

n|bn|z n

–

∞ n=

μn

n[( +kei

θ₎_n_{+ (}_keiθ₊_α_)]_|_bn_|_zn

z–∞_n_=λn

n|an|zn+

∞ n=

μn

n|bn|z n

=Re

( –α) –∞_n_=λn

n[( +kei

θ₎_n_{– (}_keiθ₊_α_)]_|_an_|_zn–

 –∞_n_=λn

n|an|zn–+ ( z z)

∞ n=

μn

n|bn|z n–

–( z z)

∞ n=

μn

n[( +ke

iθ₎_n_{+ (}_keiθ₊_α_)]_|_b

n|zn–  –∞_n_=λn

n|an|zn+ ( z z)

∞ n=

μn

n|bn|z n–

> , (.)

sinceRe(eiθ₎_{≤ |}_eiθ_|_{= , the required condition (.) is equivalent to}

_{ –}∞ n=

λn

n

(+k)n–(k+α) (–α) |an|rn–  –∞_n_=λn

n|an|rn–+

_∞

n=

μn

n|bn|rn– –

_∞

n=

μn

n

(+k)n+(k+α) (–α) |bn|rn–  –∞_n_=λn

n|an|rn–+

_∞

n=

μn

n|bn|rn–

≥. (.)

If condition (.) does not hold, then the numerator in (.) is negative forz=rsuﬃciently close to . Hence there existsz=rin (, ) for which the quotient in (.) is negative. This contradicts the required condition forf∈HST(φ,χ,k,α), and so the proof of Theorem 

is completed.

Theorem  Let f ∈HST(φ,χ,k,α).Then,for|z|=r< ,|b|<__k–₊αα+and

Dn≤λn

n, En≤ μn

n for n≥ and C=min{D,E}, (.)

we have

f(z)≤ +|b|

r+

( –α) C( +k–α)–

k+  +α C( +k–α)|b|

r

and

f(z)≥ –|b|

r–

( –α) C( +k–α)–

k+  +α C( +k–α)|b|

r.

The results are sharp.

Proof We prove the left-hand side inequality for|f|. The proof for the right-hand side inequality can be done by using similar arguments.

Letf ∈HST(φ,χ,k,α), then we have

f(z)=

z–

∞

n=

|an|zn+ ∞

n=

|bn|zn

≥r–|b|r– ∞

n=

|an|+|bn|

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≥r–|b|r

– ( –α) C( +k–α)

∞

n=

C(( +k)n– (k+α)) ( –α)

|an|+|bn|r

≥r–|b|r

– ( –α) C( +k–α)

∞

n=

C(( +k)n– (k+α)) ( –α) |an|

+C(( +k)n+ (k+α)) ( –α) |bn|

r

≥ –|b|

r– ( –α) C( +k–α)

 –k+  +α ( –α) |b|

r

≥ –|b|

r–

( –α) C( +k–α)–

k+  +α C( +k–α)|b|

r.

The bounds given in Theorem  are respectively attained for the following functions:

f(z) =z+|b|z+

( –α) C( +k–α)–

k+  +α C( +k–α)|b| z



and

f(z) = –|b|

z–

( –α) C( +k–α)–

k+  +α C( +k–α)|b| z

_.

The following covering result follows from the left side inequality in Theorem .

Corollary  Let f ∈HST(φ,χ,k,α),then for|b|<__k–₊αα+ the set

w:|w|<  – ( –α) C( +k–α)–

 – k+  +α C( +k–α) |b|

is included in f(U),where C is given by(.).

3 Extreme points

Our next theorem is on the extreme points of convex hulls of the classHST(φ,χ,k,α), denoted byclcoHST(φ,χ,k,α).

Theorem  Let f =h+g be such that h and g are given by(.).Then f∈clcoHST(φ,χ,k,α) if and only if f can be expressed as

f(z) = ∞

n=

Xnhn(z) +Yngn(z), (.)

where

h(z) =z,

hn(z) =z– n( –α) λn(( +k)n– (k+α))

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gn(z) =z+ n( –α) μn(( +k)n+ (k+α))

zn (n≥),

Xn≥, Yn≥, ∞

n=

[Xn+Yn] = .

In particular,the extreme points of the class HST(φ,χ,k,α)are{hn}and{gn},respectively.

Proof For functionsf(z) of the form (.), we have

f(z) = ∞

n=

[Xn+Yn]z– ∞

n=

n( –α) λn(( +k)n– (k+α))

Xnzn+ ∞

n=

n( –α) μn(( +k)n+ (k+α))

Ynzn.

Then

∞

n=

λn(( +k)n– (k+α)) n( –α)

n( –α) λn(( +k)n– (k+α))

Xn

+ ∞

n=

μn(( +k)n+ (k+α)) n( –α)

n( –α) μn(( +k)n+ (k+α))

Yn

= ∞

n= Xn+

∞

n=

Yn=  –X≤,

and sof(z)∈clcoHST(φ,χ,k,α). Conversely, suppose thatf(z)∈clcoHST(φ,χ,k,α). Set

Xn=λn(( +k)n– (k+α))

n( –α) |an| (n≥)

and

Yn=

μn(( +k)n+ (k+α))

n( –α) |bn| (n≥),

then note that by Theorem , ≤Xn≤ (n≥) and ≤Yn≤ (n≥). Consequently, we obtain

f(z) = ∞

n=

Xnhn(z) +Yngn(z).

Using Theorem , it is easily seen that the classHST(φ,χ,k,α) is convex and closed and soclcoHST(φ,χ,k,α) =HST(φ,χ,k,α).

4 Convolution result

For harmonic functions of the form

f(z) =z– ∞

n=

|an|zn+ ∞

n=

|bn|zn (.)

and

G(z) =z– ∞

n= Anzn+

∞

n=

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we deﬁne the convolution of two harmonic functionsf andGas

(f ∗G)(z) =f(z)∗G(z) =z– ∞

n=

anAnzn+ ∞

n= bnBnzn.

Using this deﬁnition, we show that the classHST(φ,χ,k,α) is closed under convolution.

Theorem  For≤α< ,let f ∈HST(φ,χ,k,α)and G∈HST(φ,χ,k,α).Then f(z)∗ G(z)∈HST(φ,χ,k,α).

Proof Let the functionsf(z) defined by (.) be in the classHST(φ,χ,k,α), and let the functionsG(z) defined by (.) be in the classHST(φ,χ,k,α). Obviously, the coefficients off andGmust satisfy a condition similar to inequality (.). So, for the coefficients of f(z)∗G(z), we can write

∞

n= λn

n

( +k)n– (k+α)

( –α) |an|An+ ∞

n= μn

n

( +k)n+ (k+α) ( –α) |bn|Bn

≤

∞

n=

λn n

( +k)n– (k+α) ( –α) |an|+

∞

n= μn

n

( +k)n+ (k+α) ( –α) |bn|

,

the right-hand side of this inequality is bounded by  becausef ∈HST(φ,χ,k,α). Then f(z)∗G(z)∈HST(φ,χ,k,α).

Finally, we show thatHST(φ,χ,k,α) is closed under convex combinations of its mem-bers.

Theorem  The class HST(φ,χ,k,α)is closed under convex linear combination.

Proof Fori= , , , . . . , letfi∈HST(φ,χ,k,α), where the functionsfiare given by

fi(z) =z– ∞

n=

|an,i|zn+ ∞

n=

|bn,i|zn.

For∞_i_=ti= ; ≤ti≤, the convex linear combination offimay be written as ∞

i=

tifi(z) =z– ∞

n=

_∞

i= ti|an,i|

zn+ ∞

n=

_∞

i= ti|bn,i|

zn,

then by (.) we have

∞

n= λn

n

( +k)n– (k+α) ( –α)

∞

i=

ti|an,i|+ ∞

n= μn

n

( +k)n+ (k+α) ( –α)

∞

i= ti|bn,i|

= ∞

i= ti

_∞

n=

λn n

( +k)n– (k+α) ( –α) |an,i|+

∞

n= μn

n

( +k)n+ (k+α) ( –α) |bn,i|

≤

∞

(10)

This condition is required by (.) and so∞_i_=tifi(z)∈HST(φ,χ,k,α). This completes the

proof of Theorem .

Remarks

(i) Puttingk= in our results, we obtain the results obtained by Dixitet al.[]; (ii) Puttingϕ(z) =χ(z) = _(–z_z₎ andk= in our results, we obtain the results obtained

by Rosyet al.[];

(iii) Puttingϕ(z) =χ(z) = _(–z+z_z₎ in our results, we obtain the results obtained by Kimet al.[];

(iv) Puttingϕ(z) =χ(z) = z

(–z) andk= in our results, we obtain the results obtained

by Jahangiri [];

(v) Puttingϕ(z) =χ(z) = _(–z+z_z₎ andk= in our results, we obtain the results obtained

by Jahangiri [].

Competing interests

The author declares that they have no competing interests.

Acknowledgements

The author would like to express her sincere gratitude to Springer Open Accounts Team for their kind help.

Received: 6 December 2012 Accepted: 14 October 2013 Published:12 Nov 2013

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