R E S E A R C H
Open Access
Notes on analytic functions with a bounded
positive real part
Young Jae Sim and Oh Sang Kwon
*Dedicated to Professor Hari M Srivastava
*Correspondence: [email protected]
Department of Mathematics, Kyungsung University, Busan, 608-736, Republic of Korea
Abstract
For real
α
andβ
such that 0≤α
< 1 <β
, we denote byS(α
,β
) the class of normalized analytic functionsfsuch thatα
< Re{zf(z)/f(z)}<β
inU. We find some properties, including inclusion properties, Fekete-Szegö problem and coefficient problems of inverse functions.MSC: 30C45; 30C55
Keywords: starlike functions; bounded positive real part; Fekete-Szegö problem; bi-univalent functions
1 Introduction
LetHdenote the class of analytic functions in the unit discU={z:|z|< }on the complex planeC. LetAdenote the subclass ofHconsisting of functions normalized byf() = and
f() = . LetSdenote the subclass ofAconsisting of univalent functions. Denote byS∗ andK, the class of starlike functions and convex functions, respectively. It is well-known thatK⊂S∗⊂S.
We say thatf is subordinate toFinU, written asf ≺Fif and only iff(z) =F(w(z)) for some Schwarz function withw() = and|w(z)|< ,z∈U. IfF(z) is univalent inU, then the subordinationf ≺Fis equivalent tof() =F() andf(U)⊂F(U).
We denote byS∗(A,B), the class of Janowski starlike functions, namely, the functions sat-isfying the subordination equation:zf(z)/f(z)≺( +Az)/( +Bz). Note thatS∗(, –) =S∗.
Now, we shall introduce the class of analytic functions used in the sequel.
Definition Letα andβ be real numbers such that ≤α< <β. The functionf ∈A
belongs to the classS(α,β) iff satisfies the following inequality:
α<Re
zf(z)
f(z)
<β (z∈U).
We remark that for givenα,β (≤α< <β),f ∈S(α,β) if and only iff satisfies the following two subordination equations:
zf(z)
f(z) ≺
+ ( – α)z
–z and
zf(z)
f(z) ≺
+ ( – β)z
–z , ()
since the functions ( + ( – α)z)/( –z) and ( + ( – β)z)/( –z) mapUonto the right half plane, having real part greater thanα, and the left half plane, having real part smaller thanβ, respectively. The above classS(α,β) is introduced by Kuroki and Owa []. They investigated coefficient estimates forf ∈S(α,β) and found the necessary and sufficient condition forf ∈S(α,β) using the following subordination.
Lemma (Kuroki and Owa []) Let f ∈Aand≤α< <β.Then f∈S(α,β)if and only if
zf(z)
f(z) ≺ + β–α
π ilog
–eπiβ––ααz
–z
(z∈U).
Lemma means that the functionpdefined by
p(z) = +β–α π ilog
–eπiβ––ααz
–z
()
maps the unit diskUonto the strip domainwwithα<Re(w) <β. We note that the function
f ∈A, given by
f(z) =zexp
β–α
π i
z
tlog
–eπiβ––ααt
–t
dt
,
is in the classS(α,β).
2 Inclusion properties
Theorem For given≤α< <β,let A and B be real numbers such that
–α–β
β–α ≤B<A≤
β– αβ+α
β–α . ()
ThenS∗(A,B)⊂S(α,β).
Proof At first, we note that
– < –α–β
β–α and
β– αβ+α
β–α < .
Forf∈S∗(A,B), we know that the following inequality holds:
–A
–B <Re
zf(z)
f(z)
< +A
+B (z∈U).
Therefore, it suffices to show thatαandβsatisfy the following inequalities:
α≤ –A
–B and
+A
+B ≤β. ()
Using inequality (), we can derive that
+B≥( –α)
β–α and +A≤
β(β–α)
Also,
–B≤(β– )
β–α and –A≥
α(β–a)
β–α . ()
By the above inequalities () and (), we can easily obtain the inequalities (), so the proof
of Theorem is completed.
Lemma (Miller and Mocanu []) LetΞ be a set in the complex planeCand let b be a complex number such thatRe(b) > .Suppose that a functionψ:C×U→Csatisfies the condition
ψ(iρ,σ;z) /∈Ξ
for all realρ,σ≤–|b–iρ|/(Re(b))and all z∈U.If the function p(z)defined by p(z) =
b+bz+bz+· · · is analytic inUand if
ψp(z),zp(z);z∈Ξ,
thenRe(p(z)) > inU.
Theorem Let f ∈A, /≤α< andRe{zf(z)/f(z)}>αinU.Then
Re
f(z)
z
>γ(α) :=
– α (z∈U). ()
Proof Writeγ(α) :=γ and note that ≤γ < for ≤α< . Letpbe defined by
p(z) = –γ
zf(z)
f(z) –γ
.
Thenpis analytic inU,p() = and
zf(z)
f(z) = +
( –γ)zp(z) ( –γ)p(z) +γ =ψ
p(z),zp(z),
where
ψ(r,s) = + ( –γ)s
( –γ)r+γ. ()
Also,
ψp(z),zp(z):z∈U⊂ w∈C:Re(w) >α:=Ωα.
Now for all realρ,σ≤–( +ρ),
Reψ(iρ,σ)=Re
+ ( –γ)σ ( –γ)iρ+γ
= + γ( –γ)σ γ+ ( –γ)ρ
≤ –
γ( –γ)
Now, we let
g(ρ) = +ρ
γ+ ( –γ)ρ. ()
Then
g(ρ) = (γ– )ρ
(γ+ ( –γ)ρ),
henceg(ρ) = occurs at onlyρ= andgsatisfies
g() = γ
and
lim ρ→∞g(ρ) =
( –γ).
Since /≤γ < , we have
γ ≤g(ρ) < ( –γ),
hence we get
Reψ(iρ,σ)≤ –
γ( –γ) γ =
γ–
γ =α.
This shows thatRe{ψ(iρ,σ)}∈/Ωα. By Lemma , we getRe(p(z)) > inU, and this shows
that inequality () holds and the proof of Theorem is completed.
Theorem Let f ∈A, <β< /andRe{zf(z)/f(z)}<βinU.Then
Re
f(z)
z
<δ(β) :=
– β (z∈U). ()
Proof Note thatδ:=δ(β) = –β > forβ> . Letpbe defined by
p(z) = –δ
zf(z)
f(z) –δ
.
Thenpis analytic inU,p() = and
zf(z)
f(z) =ψ
p(z),zp(z),
whereψis given in (). Also
Now, for all realρ,σ≤–( +ρ),
Reψ(iρ,σ)≥ –
δ( –δ)g(ρ),
whereg(ρ) is given (). Since
( –δ) <g(ρ)≤ δ
for allδ> , we have
Reψ(iρ,σ)≥δ–
δ =β.
This shows that Re{ψ(iρ,σ)}∈/ Ωβ. By Lemma , we getRe(p(z)) > inU, and this is equivalent to
Re
f(z)
z
<δ (z∈U),
and the proof of Theorem is completed.
Combining the above Theorems and , we can obtain the following result:
Theorem Let f ∈A, /≤α< <β< /andα<Re{zf(z)/f(z)}<βinU.Then
γ(α) <Re
f(z)
z
<δ(β) (z∈U),
whereγ(α)andδ(β)is given()and().
3 Some coefficient problems
In this section, we investigate coefficient problems for functions in the classS(α,β). In [], Kuroki and Owa investigated the coefficient of the functionpgiven by (); the functionp
can be written as
p(z) = +
∞
n=
Bnzn,
where
Bn=
(β–α)
nπ sin
–α β–απ
.
We denote bySσ(α,β) the class of bi-univalent functionsf such thatf ∈S(α,β) and the inverse function f–∈S(α,β). Srivastavaet al.investigated the estimates on the initial coefficient for certain subclasses of analytic and bi-univalent functions in [, ]. Aliet al.
have studied similar problems in [].
Lemma (Keogh and Merkers []) Let p(z) = +cz+cz+· · ·be a function with positive
real part inU.Then,for any complex numberν,
c–νc≤max ,| – ν|
.
The following result holds for the coefficient off∈S(α,β).
Theorem Let≤α< <βand let the function f given by f(z) =z+∞n=anznbe in the
classS(α,β).Then,for a complex numberμ,
a–μa≤ β–α
π
– cos
–α β–α ·π
·max
;
+ ( – μ) β–α
π i+
– ( – μ) β–α
π i
eπiβ––αα
. ()
Proof Let us consider a functionqgiven byq(z) =zf(z)/f(z). Then, sincef ∈S(α,β), we haveq(z)≺p(z), where
p(z) = +β–α π ilog
–eπiβ––ααz
–z
.
Let
h(z) = +p –(q(z))
–p–(q(z))= +hz+hz
+· · · (z∈U).
Thenhis analytic and has positive real part in the open unit diskU. We also have
q(z) =p
h(z) –
h(z) +
(z∈U). ()
We find from equation () that
a= Bh
and
a= Bh–
Bh
+
Bh
+
B
h,
which imply that
a–μa= B
h–νh
,
where
ν=
–B
B
–B+ μB
Applying Lemma , we can obtain
a–μa=
|B|h–νh
≤
·max ;| – ν|
. ()
And substituting
B= β–α
π i
–eπiβ––αα ()
and
B= β–α
π i
–eπiβ––αα ()
in (), we can obtain the result as asserted.
Using the above Theorem , we can get the following result.
Corollary Let the function f,given by f(z) =z+n=∞ anzn,be in the classS(α,β).Also
let the function f–,defined by
f–f(z)=z=ff–(z), ()
be the inverse of f.If
f–(w) =w+
∞
n=
bnwn
|w|<r;r>
, ()
then
|b| ≤
(β–α)
π sin
–α β–απ
and
|b| ≤ β–α
π
– cos
–α β–α ·π
·max
; –
β–α π i+
+
β–α
π i
eπiβ––αα
.
Proof Relations () and () give
b= –a and b= a–a.
Thus, we can get the estimate for|b|by
|b|=|a| ≤
(β–α)
π sin
–α β–απ
immediately. An application of Theorem (withμ= ) gives the estimates for|b|, hence
the proof of Corollary is completed.
Next, we shall estimate on some initial coefficient for the bi-univalent functions f ∈ Sσ(α,β).
Theorem Let f be given by f(z) =z+∞n=anznbe in the classSσ(α,β).Then
|a| ≤ |
B| √
|B| |B+B–B|
()
and
|a| ≤ |B|+|B–B|, ()
where Band Bare given by()and().
Proof Iff∈Sσ(α,β), thenf∈S(α,β) andg∈S(α,β), whereg=f–. Hence
Q(z) :=zf
(z)
f(z) ≺p(z) and L(z) :=
zg(z)
g(z) ≺p(z),
wherep(z) is given by (). Let
h(z) = +p –(Q(z))
–p–(Q(z))= +hz+hz +· · ·
and
k(z) = +p –(L(z))
–p–(L(z))= +kz+kz +· · ·.
Thenhandkare analytic and have positive real part inU. Also, we have
Q(z) =p
h(z) –
h(z) +
and L(z) =p
k(z) –
k(z) +
.
By suitably comparing coefficients, we get
a=
Bh, ()
a–a= Bh–
Bh
+
Bh
, ()
–a=
Bk ()
and
a– a= Bk–
Bk
+
Bk
whereB andBare given by () and (), respectively. Now, considering () and (), we get
h= –k. ()
Also, from (), (), () and (), we find that
a= B
(h+k) (B
+B–B)
. ()
Therefore, we have
a≤ |B|
|B
+B–B|
|h|+|k|
≤ |B| |B
+B–B| .
This gives the bound on|a|as asserted in (). Now, further computations from (), (), () and () lead to
a=
B(h+ k) + h(B–B)
.
This equation, together with the well-known estimates:
|h| ≤, |h| ≤ and |k| ≤
lead us to inequality (). Therefore, the proof of Theorem is completed.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The corresponding author, OSK carried out the subclasses of analytic functions studies and conceived of the study. YJS participated in the sequence alignment and drafted the manuscript. All authors read and approved the final manuscript.
Acknowledgements
The research was supported by Kyungsung University Research Grants in 2013.
Received: 13 December 2012 Accepted: 23 July 2013 Published: 7 August 2013 References
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3. Srivastava, HM, Mishra, AK, Gochhayat, P: Certain subclasses of analytic and bi-univalent functions. Appl. Math. Lett. 23, 1188-1192 (2010)
4. Xu, QH, Gui, YC, Srivastava, HM: Coefficient estimates for a certain subclass of analytic and bi-univalent functions. Appl. Math. Lett.25, 990-994 (2012)
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doi:10.1186/1029-242X-2013-370
