R E S E A R C H
Open Access
An investigation on a new class of harmonic
mappings
Emel Yavuz Duman
1*, Ya¸sar Polato ˘glu
1and Yasemin Kahramaner
2*Correspondence:
1Department of Mathematics and
Computer Science, ˙Istanbul Kültür University, Ataköy Campus, Bakırköy, ˙Istanbul, 34156, Turkey
Full list of author information is available at the end of the article
Abstract
In the present paper, we give an extension of the idea which was introduced by Sakaguchi (J. Math. Soc. Jpn. 11:72-75, 1959), and we give some applications of this extended idea for the investigation of the class of harmonic mappings.
MSC: Primary 30C45; 30C55
Keywords: harmonic mappings; distortion theorem; grow theorem
1 Introduction
LetD={z| |z|< }be the open unit disc in the complex planeC. A complex valued har-monic functionf :D→Chas the representation
f =h(z) +g(z), (.)
whereh(z) andg(z) are analytic inDand have the following power series expansions:
h(z) =
∞
n=
anzn, g(z) = ∞
n=
bnzn, z∈D,
wherean,bn∈C,n= , , , . . . . Chooseg() = (i.e.,b= ), so the representation (.) is unique inDand is called the canonical representation off inD. It is convenient to make further normalization (without loss of generality)h() = (i.e.,a= ) andh() = (i.e.,
a= ). The family of such functionsf is denoted bySH[]. It is known that iff is a sense-preserving harmonic mapping ofDonto some other region, then by Lewy’s theorem its Jacobian is strictly positive,i.e.,
Jf(z)=h(z)–g(z)> .
Equivalently, the inequality|g(z)|<|h(z)|holds for allz∈D. The family of all functionsf ∈ SHwith the additional property thatg() = (i.e.,b= ) is denoted bySH []. Observe that the classical family of univalent functionsSconsists of all functionsf ∈S
Hsuch that g(z) = for allz∈D. Thus, it is clear thatS⊂SH ⊂SH.
Let be the family of functionsφ(z) regular in the open unit discDand satisfy the conditionsφ() = ,|φ(z)|< for allz∈D.
Denote byPthe family of functionsp(z) = +pz+pz+· · ·regular inDsuch thatp(z) inPif and only if
p(z) = +φ(z)
–φ(z) (.)
for someφ(z)∈and everyz∈D.
Ifs(z) =z+cz+· · ·is regular in the open unit discDand satisfies the condition
Re
eiαzs (z)
s(z)
> , z∈D (.)
for some realα,|α|<π/, thens(z) is said to be anα-spirallike function inD[, ]. Such functions are known to be univalent []. The class of such functions is denoted bySα∗.
LetF(z) =z+αz+αz+· · ·andF(z) =z+βz+βz+· · ·be analytic functions inD. If there exists a functionφ(z)∈such thatF(z) =F(φ(z)) for everyz∈D, then we say thatF(z) is subordinate toF(z) and we writeF≺F. We also note that ifF≺F, then
F(D)⊂F(D) [, ]. We also note thatw(z) =g(z)/h(z) is the analytic second dilatation off and|w(z)|< for everyz∈D.
In this paper we investigate the class of harmonic mappings defined by
SH(α)=
f =h(z) +g(z)w(z)≺ +z
–z,h(z)∈S ∗ α
.
For this aim we need the following theorems and lemmas.
Theorem .[] Let s(z)∈Sα∗,then
rF(cosα, –r)≤s(z)≤rF(cosα,r) (.)
and
( –r)cosα– ( +r)sinα F(cosα, –r)
≤s(z)≤( +r)cosα+ ( –r)sinα F(cosα,r), (.)
where
F(cosα,r) =
( +r)cosα(cosα–)( –r)cosα(cosα+) (.)
for all|z|=r< and|α|<π/.
Lemma .[] Letφ(z)be regular in the unit discDwithφ() = ,then if|φ(z)|attains its maximum value on the circle|z|=r at the point z,one has zφ(z) =kφ(z)for some
k≥.
Lemma .[] Let f =h(z) +g(z)be an element ofSHPST(α)∗,then
|b|–r –|b|r
≤g(z)
h(z)
≤ |b|+r +|b|r
(.)
for all|z|=r< .This inequality is sharp because the extremal function is
eiαg (z)
h(z)=
z+b
+bz¯ ,
where b=eiαb
.
2 Main results
Theorem . s(z)∈Sα∗if and only if
zs (z)
s(z) – ≺
cosαe–iαz
–z (.)
for all z∈D.
Proof Lets(z) inSα∗, then we have
eiαzs (z)
s(z) =cosα +φ(z) –φ(z)+isinα
or
eiαzs (z)
s(z) =e
iα +φ(z)
–φ(z)
for someφ(z)∈and allz∈D. Thus
zs (z)
s(z) – =
+e–iαφ(z)
–φ(z) –
= + (cosα–isinα)φ(z) – +φ(z) –φ(z)
=cosαe –iαφ(z)
–φ(z)
for someφ(z)∈and allz∈D. Sinceφ(z)∈, we have that (.) is true. The sufficient part of the proof can be seen by following the above steps in the opposite direction by considering the subordination principle.
Theorem . Let f =h(z) +g(z)be an element ofSH(α).If w(z) =g (z)
h(z) ∈P,then
g(z)
h(z)∈P
for all z∈D.
Proof A version of this theorem was proved by Sakaguchi for a univalent starlike func-tion [, ]. Sincef =h(z) +g(z)∈SH(α), thenh(z) andg(z) are regular inDandh() = g() = . On the other hand, we have
w(z) = g
(z)
h(z)∈P if and only if
g(z)
h(z)≺ +z
for allz∈D. Geometrically, this means that gh((zz))mapsDinside the open disc centered on
the real axis with diameter end points–+rr and+–rr. Now we define a functionφ(z) by
g(z)
h(z)= +φ(z)
–φ(z) (z∈D). (.)
Thenφ(z) is analytic inD, andφ() = . On the other hand,
w(z) = g
(z)
h(z)= zφ(z) –φ(z)
eiα( +e–iα)φ(z)+
+φ(z)
–φ(z) (z∈D). (.)
Now, it is easy to realize that the subordination (.) is equivalent to|φ(z)|< in (.) for allz∈D. Indeed, assume to the contrary that there existsz∈Dsuch that|φ(z)|= . Then by Jack’s lemma (Lemma .),zφ(z) =kφ(z),k≥, for suchzwe have
w(z) = g
(z
)
h(z)=
kφ(z) –φ(z)
eiα( +e–iα)φ(z)+
+φ(z) –φ(z)=w
φ(z)∈/w(D),
since|φ(z)|= andk≥. But this is a contradiction to the conditionw(z) =g
(z) h(z) ≺
+z
–z,
and so the assumption is wrong,i.e.,|φ(z)|< for allz∈D.
Remark . Theorem . is an extension of Lemma . to the harmonic mappings.
Corollary . Let f =h(z) +g(z)be an element ofSH(α),then
r( –r)
+r F(cosα, –r)≤g(z)≤ r( +r)
–r F(cosα,r) (.)
and
( –r)cosα– ( +r)sinα –r
+rF(cosα, –r)
≤g(z)≤( +r)cosα+ ( –r)sinα +r
–rF(cosα,r), (.)
where F(cosα,r)is given by(.)for all|z|=r< and|α|<π/.
Proof The proof of this theorem is a simple consequence of Theorem . and Theorem . since
Rew(z) =Reg
(z)
h(z) ⇒
g(z)
h(z)≺ +z
–z ⇒
–r
+r≤
hg((zz))
≤ +r
–r,
then
h(z) –r +r≤g
(z)≤h(z) +r
–r
and
Reg(z)
h(z)> ⇒
g(z)
h(z)≺ +z
–z ⇒
–r
+r≤
then
h(z) –r
+r≤g(z)≤h(z)
+r
–r.
Corollary . Let f =h(z) +g(z)be an element ofSH(α),then
[( –r)cosα– ( +r)sinα](F(cosα, –r))( –r)( –|b |) ( +|b|r)
≤Jf(z)≤
[( +r)cosα+ ( –r)sinα](F(cosα,r))( –r)( –|b |) ( –|b|r)
,
where F(cosα,r)is given by(.)for all|z|=r< ,|α|<π/and Jf(z)is the Jacobian of f
defined by Jf(z)=|h(z)|–|g(z)|for all z∈D.
Proof The proof of this corollary is a simple consequence of Lemma . and Lemma ..
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.
Author details
1Department of Mathematics and Computer Science, ˙Istanbul Kültür University, Ataköy Campus, Bakırköy, ˙Istanbul, 34156,
Turkey.2Department of Mathematics, ˙Istanbul Commerce University, Üsküdar Campus, Üsküdar, ˙Istanbul, 34672, Turkey.
Received: 28 December 2012 Accepted: 28 August 2013 Published:07 Nov 2013 References
1. Duren, P: Harmonic Mappings in the Plane. Cambridge University Press, Cambridge (2004) 2. Goodman, AW: Univalent Functions, vol. I. Mariner Pub. Comp. Inc., Tampa (1983) 3. Goodman, AW: Univalent Functions, vol. II. Mariner Pub. Comp. Inc., Tampa (1983)
4. ˘Spa˘cek, L: Contribution à la théorie des fonctions univalentes. ˇCas. Pˇest. Mat.62, 12-19 (1932) 5. Polato ˘glu, Y: Grow and distortion theorem for Janowskiα-spirallike functions in the unit disc. Stud. Univ.
Babe¸s-Bolyai, Math.57(2), 255-259 (2012)
6. Jack, IS: Functions starlike and convex of orderα. J. Lond. Math. Soc.3, 469-474 (1971) 7. Libera, RJ: Some classes of regular univalent functions. Proc. Am. Math. Soc.16, 755-758 (1965)
8. Aydo ˘gan, M, Yavuz Duman, E, Polato ˘glu, Y, Kahramaner, Y: Harmonic function for which the second dilatation is α-spiral. J. Inequal. Appl.2012, 262 (2012)
9. Sakaguchi, K: On a certain univalent mappings. J. Math. Soc. Jpn.11, 72-75 (1959)
10.1186/1029-242X-2013-478