R E S E A R C H
Open Access
Harmonic functions with varying
coefficients
Jacek Dziok
1, Jay Jahangiri
2*and Herb Silverman
3*Correspondence:
2Department of Mathematical
Sciences, Kent State University, Burton, OH 44021-9500, USA Full list of author information is available at the end of the article
Abstract
Complex-valued harmonic functions that are univalent and sense preserving in the open unit disk can be written in the formf=h+g, wherehandgare analytic. In this paper we investigate some classes of univalent harmonic functions with varying coefficients related to Janowski functions. By using the extreme points theory we obtain necessary and sufficient convolution conditions, coefficients estimates, distortion theorems, and integral mean inequalities for these classes of functions. The radii of starlikeness and convexity for these classes are also determined.
MSC: Primary 30C45; secondary 30C80
Keywords: S˘al˘agean operator; subordination; extreme points; harmonic functions
1 Introduction
Harmonic functions are famous for their use in the study of minimal surfaces and also play important roles in a variety of problems in applied mathematics (e.g.see Choquet [], Dorff [], Duren [] or Lewy []). A continuous functionf =u+ivis said to be complex-valued harmonic in a domainD⊂Cif bothuandvare real harmonic inD. In any simply connected domain, we can writef =h+g, wherehandgare analytic inD. We shall call
hthe analytic andgthe co-analytic part off. Clunie and Sheil-Small [] pointed out that a necessary and sufficient condition forf to be locally univalent and sense preserving in
Dis that|h(z)|>|g(z)|inD. Note that forf =h+g, harmonic and sense preserving in the open unit diskD={z∈C:|z|< }, the conditionh() = >|g()|implies that the function (f –g()f)/( –|g()|) is also harmonic and sense preserving inD. We letH be the class of functionsf =h+g, harmonic, sense preserving, and univalent in the open unit diskD, for whichfz() =g() = . Such harmonic and sense-preserving functions f=h+g∈Hmay be represented by the power series
h(z) =z+
∞
k=
akzk, g(z) =
∞
k=
bkzk, z∈D. ()
Clunie and Sheil-Small [] proved that the classHis a compact family (with respect to the topology of locally uniform convergence). Note that forg(z)≡, the classHreduces to the classSof normalized analytic functions univalent inD.
For ≤α< we letSH∗(α) andSc
H(α), respectively, denote the subclasses ofSH
con-sisting of harmonic starlike and harmonic convex functions of orderα.
A functionsf of the form () is inSH∗(α) if and only if (e.g.see Clunie and Sheil-Small [] or Duren [])
∂ ∂θ
argfreiθ>α, |z|=r< .
Similarly, a functionf of the form () is inSHc (α) if and only if
∂ ∂θ
arg ∂
∂θf
ρeiθ>α, |z|=r< .
We note that a harmonic functionf ∈SH∗(α) if and only if
ReJHf(z)
f(z) >α, |z|=r< , or
JHf(z) – ( +α)f(z)
JHf(z) + ( –α)f(z)
< , |z|=r< ,
where
JHf(z) :=zh(z) –zg(z).
For ≤α< , it is easy to verify that
f ∈SHc (α) ⇔ JHf ∈SH∗(α).
Forλ∈ {, , , . . .}andf =h+g∈Hof the form (), we consider the linear operatorJλ
H:
H→Hdefined byJ
Hf :=f =h+gand
JHλf(z) :=JHJHλ–f(z)
:=z+
∞
n=
nλa
nzn+ (–)λ ∞
n=
nλb
nzn (z∈D).
For the analytic definition of the above case, see the Sălăgean operator [].
We say that a functionf :D→Cissubordinateto a functiong:D→C, and writef(z)≺
g(z) (or simplyf ≺g), if there exists a complex-valued functionwwhich mapsDinto itself with w() = , such thatf(z) =g(w(z));z∈D. In particular, ifg is univalent inD, then
f() =g() andf(D)⊂g(D).
The Hadamard product (or convolution) of functionsfandfof the form
fk(z) =z+
∞
n=
ak,nzn+
∞
n=
bk,nzn
z∈D,k∈ {, } ()
is defined by
(f∗f)(z) =z+
∞
n=
a,na,nzn+ ∞
n=
For nonnegative integerλ∈ {, , , . . .}and for –B≤A<B≤ we defineHλ(A,B) to be the class of functionsf ∈Hso that (also see Dziok [, ])
Jλ+
H f(z)
Jλ
Hf(z) ≺
+Az
+Bz ()
andGλ(A,B) to consist of functionsf ∈Hso that
Jλ
Hf(z)
z ≺
+Az
+Bz.
We remark that the classesH(A,B) andG(A,B)for the analytic case,i.e. g≡, were introduced by Janowski [] and the classesSH∗(α) =H(α– , ) andSHc (α) =H(α– , ) for the harmonic case were investigated by Jahangiri [, ] and Silverman []. It is the aim of this paper to obtain necessary and sufficient convolution conditions, coefficient bounds, distortion theorems, radii of starlikeness and convexity, compactness, and ex-treme points for the above defined classesHλ(A,B) andGλ(A,B).
2 Analytic criteria
Our first theorem provides a necessary and sufficient convolution condition for the har-monic functions inHλ(A,B).
Theorem A function f belongs to the classHλ(A,B)if and only if f ∈Hand
Jλ
Hf(z)∗φ(z;ζ)=
ζ∈C,|ζ|= ,
where
φ(z;ζ) =(B–A)ζz+ ( +Aζ)z
( –z) – (–)
λz+ (A+B)ζz– ( +Aζ)z
( –z) (z∈D).
Proof Letf ∈H. Thenf ∈Hλ(A,B) if and only if the condition () holds or equivalently
Jλ+
H f(z)
Jλ
Hf(z)
= +Aζ +Bζ
ζ ∈C,|ζ|= . ()
Now forJλ+
H h(z) =JHλh(z)∗z/( –z)andJHλh(z) =JHλh(z)∗z/( –z), the inequality () yields
( +Bζ)Jλ+
H f(z) – ( +Aζ)JHλf(z)
= ( +Bζ)JHλ+h(z) – ( +Aζ)JHλh(z) – (–)λ( +Bζ)JHλ+g(z) + ( +Aζ)JHλg(z) =JHλh(z)∗
( +Bζ)z
( –z) –
( +Aζ)z
–z
– (–)λ+JHλg(z)∗
( +Bζ)z
( –z) +
( +Aζ)z
–z
=JHλf(z)∗φ(z;ζ)= .
Theorem For z ∈D, the harmonic function f(z) =z+∞n=anzn+∞
n=bnzn is in
Hλ(A,B)if
∞
n=
γn|an|+δn|bn|
≤B–A, ()
where
γn=nλ
n( +B) – ( +A) and δn=nλ
n( +B) + ( +A). ()
Proof Clearly the theorem is true forf(z)≡z. So, we assume that an= orbn= for n≥. Sinceγn≥n(B–A) andδn≥n(B–A) by () we have
h(z)–g(z)≥ –
∞
n=
n|an||z|n– ∞
n=
n|bn||z|n≥ –|z| ∞
n=
n|an|+n|bn|
≥ – |z|
B–A ∞
n=
γn|an|+δn|bn|
≥ –|z|> (z∈D).
Thereforef is sense preserving and locally univalent inD. Further, ifz,z∈Dand we assume thatz=z, then
zn–zn
z–z
= n m=
zm–zn–m ≤ n m=
|z|m–|z|n–m<n (n= , , . . .).
Hence
f(z) –f(z)≥h(z) –h(z)–g(z) –g(z)
≥z–z–
∞
n=
anzn–zn – ∞ n=
bnzn
–zn
≥ |z–z|–
∞
n=
|an|zn–zn–
∞
n=
|bn|zn–zn
=|z–z|
–
∞
n= |an|
zn–zn
z–z
–
∞
n= |bn|
zn –zn
z–z
>|z–z|
–
∞
n=
n|an|–
∞
n=
n|bn|
≥.
This proves thatf is univalent inDi.e. f ∈H.
On the other hand,f ∈Hλ(A,B) if and only if there exists a complex-valued functionw,
w() = ,|w(z)|< (z∈D) such that
Jλ+
H f(z)
Jλ
Hf(z)
= +Aw(z)
or equivalently
JHλ+f(z) –JHλf(z)
BDλ+
H f(z) –ADλHf(z)
< (z∈D). ()
The above inequality () holds since for|z|=r( <r< ) we obtain
JHλ+f(z) –JHλf(z)–BDλH+f(z) –ADλHf(z)
=
∞
n=
nλ(n– )a
nzn– (–)λ ∞
n=
nλ(n+ )b
nzn
–
(B–A)z+ ∞
n=
nλ(Bn–A)a
nzn+ (–)λ ∞
n=
nλ(Bn+A)b
nzn
≤
∞
n=
nλ(n– )|an|rn+
∞
n=
nλ(n+ )|bn|rn– (B–A)r
+
∞
n=
nλ(Bn–A)|an|rn+
∞
n=
nλ(Bn+A)|bn|rn
≤r ∞
n=
γn|an|+δn|bn|
rn–– (B–A)
< ,
and thereforef ∈Hλ(A,B).
Next we show that the condition () is also necessary for the functionsf ∈Hto be in the classHλ
T(A,B) :=Tλ∩Hλ(A,B) whereTλis the class of functionsf =h+g∈Hwith
varying coefficients (see [, ] or []) for which there exists a real numberφso that
f =h+g=z–
∞
n=
ei(–n)φ|a
n|zn+ (–)λ ∞
n=
ei(n–)φ|b
n|zn (z∈D). ()
Theorem Let f=h+g be defined by().Then f ∈Hλ
T(A,B)if and only if the condition
()holds.
Proof The ‘if ’ part follows from Theorem . For the ‘only-if ’ part, assume that f ∈ Hλ
T(A,B), then by () we have
∞n=nλ[(n– )anzn–+ (–)λ(n+ )bnzn–]
B–A–∞n=nλ[(Bn–A)a
nzn–+ (–)λ(Bn+A)bnzn–]
< (z∈D).
For|z|=r< we obtain
∞
n=nλ[(n– )|an|+ (n+ )|bn|]rn– (B–A) –∞n=nλ[(Bn–A)|a
n|+ (Bn+A)|bn|]rn–
< .
Thus, forγnandδnas defined by (), we have ∞
n=
γn|an|+δn|bn|
Let{σn}be the sequence of partial sums of the series ∞
n=(γn|an|+δn|bn|). Then{σn}is
a nondecreasing sequence and by () it is bounded above byB–A. Thus, it is convergent and
∞
n=
γn|an|+δn|bn|
= lim
n→∞σn≤B–A.
This gives the condition ().
A similar argument can be used to prove the following.
Theorem Let f =h+g∈Hbe a function of the form().Then f ∈Gλ
T(A,B) :=Tλ∩
Gλ(A,B)if and only if
∞
n=
nλ|an|+|bn|≤B–A
+B.
For special cases, Theorems , , and yield the following corollaries.
Corollary Let f =h+g∈H.Then f ∈Hλ:=Hλ(–, )if and only if
JHλf(z)∗φ(z;ζ)= |ζ|= ,
where
φ(z;ζ) =ζz+ ( –ζ)z
( –z) – (–)
λz+ ( –ζ)z
( –z) (z∈D).
Corollary Let f =h+g∈Hbe a function of the form().Then f ∈Hλ
T(α) :=HTλ(α–, )
if and only if
∞
n=
nλ(n–α)|an|+ (n+α)|bn|≤.
Corollary Let f =h+g∈Hbe a function of the form().Then f ∈Hλ
T :=HλT()if and
only if
∞
n=
nλ+|an|+|bn|≤,
i.e.
Hλ
T ≡GλT+:=GTλ+(–, ).
3 Extreme points
We say thatFislocally uniformly boundedif for eachr, <r< , there is a real constant
M=M(r) so that|f(z)| ≤Mwheref ∈Fand|z| ≤r.
We say that a classFisconvexifμf + ( –μ)g∈Ffor allf andginFand ≤μ≤. Moreover, we definethe closed convexhull ofF, denoted bycoF, as the intersection of all closed convex subsets ofH(with respect to the topology of locally uniform convergence) that containF.
A real-valued functionalJ :H→Ris calledconvexon a convex classF⊂HifJ(μf+ ( –μ)g)≤μJ(f) + ( –μ)J(g) for allf andginFand ≤μ≤.
The Krein-Milman theorem (see []) is fundamental in the theory of extreme points. In particular, it implies the following.
Lemma IfFis a non-empty compact subclass of the classH,then EFis non-empty and
coEF=coF.
Lemma [] LetFbe a non-empty compact convex subclass of the classHandJ :H→R
be a real-valued,continuous,and convex functional onF.Then
maxJ(f) :f ∈F=maxJ(f) :f∈EF.
SinceHis a complete metric space, Montel’s theorem [] implies the following.
Lemma A classF ⊂H is compact if and only if F is closed and locally uniformly
bounded.
Now, we are ready to state and prove our next theorem.
Theorem The classHλ
T(A,B)is a convex and compact subset ofH.
Proof For ≤μ≤, letf,f∈HλT(A,B) be defined by (). Then
μf(z) + ( –μ)f(z) =z+
∞
n=
μa,n+ ( –μ)a,n
zn+μb,n+ ( –μ)b,n
zn
and
∞
n=
γnμa,n+ ( –μ)a,n+δnμb,n+ ( –μ)b,nzn
≤μ
∞
n=
γn|a,n|+δn|b,n|
+ ( –μ)
∞
n=
γn|a,n|+δn|b,n|
≤μ(B–A) + ( –μ)(B–A) =B–A.
Thus, the functionφ=μf+ ( –μ)fbelongs to the classHλT(A,B). This means that the classHλ
T(A,B) is convex.
On the other hand, forf ∈Hλ
T(A,B),|z| ≤rand <r< , we have
f(z)≤r+
∞
n=
|an|+|bn|
rn≤r+
∞
n=
γn|an|+δn|bn|
Therefore,Hλ
T(A,B) is locally uniformly bounded. Let
fk(z) =z+ ∞
n=
ak,nzn+ ∞
n=
bk,nzn (z∈D,k∈N)
and letf=h+gbe given by (). Using Theorem we have
∞
n=
γn|ak,n|+δn|bk,n|
≤B–A (k∈N). ()
If we assume thatfk→f, then we conclude that|ak,n| → |an|and|bk,n| → |bn|ask→ ∞
(n∈N). Let{σn}be the sequence of partial sums of the series ∞
n=(γn|an|+δn|bn|). Then
{σn}is a nondecreasing sequence and by () it is bounded above byB–A. Thus, it is
convergent and
∞
n=
γn|an|+δn|bn|
= lim
n→∞σn≤B–A.
Therefore,f ∈Hλ
T(A,B), and therefore the classHλT(A,B) is closed. In consequence, by
Lemma , the classHλ
T(A,B) is compact subset ofH, which completes the proof.
Our next theorem is on the extreme points ofHλ
T(A,B).
Theorem Extreme points of the classHλ
T(A,B)are the functions f of the form()where
h=hnand g=gnare of the form
h(z) =z, hn(z) =z– B–A
γn
ei(–n)φzn, ()
gn(z) = (–)λB–A
δn
ei(n–)φzn z∈D,n∈ {, , . . .}.
Proof Letgn=μf+ ( –μ)f where <μ< andf,f∈STλ(A,B) are functions of the form (). Then, by (), we have |b,n|=|b,n|= Bδ–nA, and therefore a,k =a,k = for k∈ {, , . . .}andb,k=b,k= fork∈ {, , . . .}{n}. It follows thatgn=f=fand conse-quentlygn∈EST∗(A,B). Similarly, we can verify that the functionshnof the form () are the extreme points of the classSλ
T(A,B).
Now, suppose that a functionf of the form () belongs to the setEHλ
T(A,B) andf is not
of the form (). Then there existsm∈ {, , . . .}such that
<|am|<B–A
αm
or <|bm|<B–A
βm
.
If <|am|<Bα–mA, then putting
μ=|am|αm
B–A , φ=
–μ(f –μhm),
Thus,f ∈/EHλ
T(A,B). Similarly, if <|bm|<Bδ–nA, then putting
μ=|bm|βm
B–A , φ=
–μ(f–μgm),
we have <μ< ,gm=φ, and
f =μgm+ ( –μ)φ.
It follows thatf ∈/EHλ
T(A,B), and so the proof is completed.
It is clear that if the classF={fn∈H:n∈N}is locally uniformly bounded, then
coF=
∞
n=
μnfn: ∞
n=
μn= ,μn≥ (n∈N)
.
Thus, by Theorem , we have the following.
Corollary Let hn,gnbe defined by().Then
Hλ
T(A,B) =
∞
n=
(μnhn+δngn) : ∞
n=
(μn+δn) = ,δ= ,μn,δn≥ (n∈N)
.
For all fixed values ofm,n,λ∈N,z∈D, the following real-valued functionals are con-tinuous and convex onH:
J(f) =|an|, J(f) =|bn|, J(f) =f(z), J(f) =JHλf(z) (f∈H).
Moreover, forμ> , <r< , the real-valued functional
J(f) =
π
π
freiθμdθ
/μ
(f ∈H)
is continuous onH. Forμ≥, by Minkowski’s inequality it is also convex onH. Therefore, by Lemma and Theorem , we have the following corollaries.
Corollary Let f ∈Hλ
T(A,B)be a function of the form().Then
|an| ≤B–A
γn
, |bn| ≤B–A
δn
(n= , , . . .),
whereγn,δnare defined by().The result is sharp and the functions hn,gnof the form() are the extremal functions.
Corollary Let f ∈Hλ
T(A,B)and|z|=r< .Then
r– B–A λ( + B–A)r
≤f(z)≤r+ B–A λ( + B–A)r
r– B–A + B–Ar
≤Jλ
Hf(z)≤r+ + B–BA–Ar (λ= , , , . . .).
The result is sharp and the function hof the form()is the extremal function.
The following covering result follows from Corollary.
Corollary If f ∈Hλ
T(A,B)thenD(r)⊂f(D)where
r= – B–A λ( + B–A).
We also conclude to the following.
Corollary Let <r< andμ≥.If f ∈Hλ
T(A,B)then
π
π
freiθμdθ≤
π
π
h
reiθμdθ,
π
π
Jλ
Hf(z)
μ
dθ≤
π
π
Jλ
Hh
reiθμdθ (μ= , , . . .).
4 Radii of starlikeness and convexity
LetB⊆Hand letD(r) :={z∈C:|z|<r≤}. We define the radius of starlikeness and the radius of convexity of the classB, respectively, by
R∗α(B) :=inf
f∈B
supr∈(, ] :f is starlike of orderαinD(r),
Rcα(B) :=inf
f∈B
supr∈(, ] :f is convex of orderαinD(r).
At this point, for the caseα= , it is easy to verify that
Hλ
(A,B)⊂Hλ(–, )⊂Hλ–(–, )⊂SH∗()⊂H,
and consequently
R∗Hλ(A,B)=R∗HλT(A,B)=RcHλ(A,B)=RcHλT(A,B)= (λ= , , . . .). In the following theorem we determineR∗α(HλT(A,B)) for ≤α< .
Theorem Let≤α< andγnandδnbe defined by().Then
R∗α
Hλ
T(A,B)
=inf
n≥
–α
B–Amin
γn n–α,
δn n+α
n–
. ()
Proof Letf ∈Hλ
T(A,B) be of the form (). Then, for|z|=r< , we have
JHf(z) – ( +α)f(z)
JHf(z) + ( –α)f(z)
= –αz+
∞
n=((n– –α)anzn– (n+ +α)bnz
n
) ( –α)z+∞n=((n+ –α)anzn– (n– +α)bnzn)
≤ α+
∞
Note (see Jahangiri [], Theorem ) thatf is starlike of orderαinD(r) if and only if
JHf(z) – ( +α)f(z)
JHf(z) + ( –α)f(z)
< z∈D(r)
or
∞
n=
n–α
–α|an|+
n+α
–α|bn|
rn–≤. ()
Also, by Theorem , we have
∞
n=
γn B–A|an|+
δn B–A|bn|
≤,
whereγnandδnare defined by ().
Sinceγn<δn(n= , , . . .), the condition () is true if n–α
–αr
n–≤ γn B–A and
n+α
–αr
n–≤ δn
B–A (n= , , . . .),
or if
r≤
–α
B–Amin
γn n–α,
δn n+α
n–
(n= , , . . .).
It follows that the functionf is starlike of orderαin the diskU(r∗) where
r∗:=inf
n≥
–α
B–Amin
γn n–α,
δn n+α
n– .
The function
fn(z) =hn(z) +gn(z) =z–B–A
γn
ei(–n)φzn+ (–)λB–A
δn
ei(n–)φzn
proves that the radiusr∗cannot be any larger. Thus we have ().
Using a similar argument as above we obtain the following.
Theorem Let≤α< andγnandδnbe defined by().Then
RcαHλ
T(A,B)
=inf
n≥
–α
B–Amin
γn n(n–α),
δn n(n+α)
n– .
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Author details
1Faculty of Mathematics and Natural Sciences, University of Rzeszów, ul. Prof. Pigonia 1, Rzeszów, 35-310, Poland. 2Department of Mathematical Sciences, Kent State University, Burton, OH 44021-9500, USA.3Department of
Mathematics, College of Charleston, Charleston, SC 29424, USA.
Received: 17 March 2016 Accepted: 29 April 2016
References
1. Choquet, G: Sur un type de transformation analytique généralisant la représentation conforme et définie au moyen de fonctions harmoniques. Bull. Sci. Math.89, 156-165 (1945)
2. Dorff, M: Minimal graphs inR3over convex domains. Proc. Am. Math. Soc.132(2), 491-498 (2003)
3. Duren, P: Harmonic Mappings in the Plane. Cambridge Tracts in Mathematics, vol. 156. Cambridge University Press, Cambridge (2004)
4. Lewy, H: On the non-vanishing of the Jacobian in certain one-to-one mappings. Bull. Am. Math. Soc.42, 689-692 (1936)
5. Clunie, J, Sheil Small, TB: Harmonic univalent functions. Ann. Acad. Sci. Fenn., Ser. A 1 Math.9, 3-25 (1948) 6. S˘al˘agean, GS: Subclasses of univalent functions. In: Complex Analysis - Fifth Romanian-Finnish Seminar. Lecture
Notes in Math., vol. 1013, pp. 362-372. Springer, Berlin (1983)
7. Dziok, J: On Janowski harmonic functions. J. Appl. Anal.21(2), 99-107 (2015)
8. Dziok, J: Classes of harmonic functions defined by subordination. Abstr. Appl. Anal.2015, Article ID 756928 (2015) 9. Janowski, W: Some extremal problems for certain families of analytic functions. Ann. Pol. Math.28, 297-326 (1973) 10. Jahangiri, JM: Coefficient bounds and univalence criteria for harmonic functions with negative coefficients. Ann.
Univ. Mariae Curie-Skłodowska, Sect. A52(2), 57-66 (1998)
11. Jahangiri, JM: Harmonic functions starlike in the unit disk. J. Math. Anal. Appl.235, 470-477 (1999)
12. Silverman, H: Harmonic univalent functions with negative coefficients. J. Math. Anal. Appl.220, 283-289 (1998) 13. Jahangiri, JM, Silverman, H: Harmonic univalent functions with varying arguments. Int. J. Appl. Math.8, 267-275
(2002)
14. Ahuja, OP, Jahangiri, JM: Certain harmonic univalent functions with varying arguments. Int. J. Math. Sci.2, 9-16 (2003) 15. Murugusundaramoorthy, G, Vijaya, K, Raina, RK: A subclass of harmonic functions with varying arguments defined by
Dziok-Srivastava operator. Arch. Math.45(1), 37-46 (2009)
16. Krein, M, Milman, D: On the extreme points of regularly convex sets. Stud. Math.9, 133-138 (1940)
