R E S E A R C H
Open Access
Bounds for the second Hankel determinant of
certain univalent functions
See Keong Lee
1*, V Ravichandran
2and Shamani Supramaniam
1Dedicated to Professor Hari M Srivastava
*Correspondence: sklee@cs.usm.my 1School of Mathematical Sciences,
Universiti Sains Malaysia, 11800 USM, Penang, Malaysia Full list of author information is available at the end of the article
Abstract
The estimates for the second Hankel determinanta2a4–a23of the analytic function
f(z) =z+a2z2+a3z3+· · ·, for which eitherzf(z)/f(z) or 1 +zf(z)/f(z) is subordinate
to a certain analytic function, are investigated. The estimates for the Hankel
determinant for two other classes are also obtained. In particular, the estimates for the Hankel determinant of strongly starlike, parabolic starlike and lemniscate starlike functions are obtained.
MSC: 30C45; 30C80
1 Introduction
LetAdenote the class of all analytic functions
f(z) =z+az+az+· · · ()
defined on the open unit diskD:={z∈C:|z|< }. TheHankel determinants Hq(n) (n= , , . . . ,q= , , . . .) of the functionf are defined by
Hq(n) := ⎡ ⎢ ⎢ ⎢ ⎢ ⎣
an an+ · · · an+q– an+ an+ · · · an+q
..
. ... ...
an+q– an+q · · · an+q–
⎤ ⎥ ⎥ ⎥ ⎥
⎦ (a= ).
Hankel determinants are useful, for example, in showing that a function of bounded char-acteristic inD,i.e., a function which is a ratio of two bounded analytic functions with its Laurent series around the origin having integral coefficients, is rational []. For the use of Hankel determinants in the study of meromorphic functions, see [], and various prop-erties of these determinants can be found in [, Chapter ]. In , Pommerenke [] investigated the Hankel determinant of areally meanp-valent functions, univalent func-tions as well as of starlike funcfunc-tions. In [], he proved that the Hankel determinants of univalent functions satisfy
Hq(n)<Kn–(+β)q+ (n= , , . . . ,q= , , . . .),
whereβ> / andKdepends only onq. Later, Hayman [] proved that|H(n)|<An/
(n= , , . . . ;Aan absolute constant) for areally mean univalent functions. In [–], the es-timates for the Hankel determinant of areally meanp-valent functions were investigated. ElHosh obtained bounds for Hankel determinants of univalent functions with a positive Hayman index α [] and of k-fold symmetric and close-to-convex functions []. For bounds on the Hankel determinants of close-to-convex functions, see [–]. Noor stud-ied the Hankel determinant of Bazilevic functions in [] and of functions with bounded boundary rotation in [–]. In the recent years, several authors have investigated bounds for the Hankel determinant of functions belonging to various subclasses of univalent and multivalent functions [–]. The Hankel determinantH() =a–ais the well-known
Fekete-Szegö functional. For results related to this functional, see [, ]. The second Hankel determinantH() is given byH() =aa–a.
An analytic functionfissubordinateto an analytic functiong, writtenf(z)≺g(z), if there is an analytic functionw:D→Dwithw() = satisfyingf(z) =g(w(z)). Ma and Minda [] unified various subclasses of starlike (S∗) and convex functions (C) by requiring that either of the quantityzf(z)/f(z) or +zf(z)/f(z) is subordinate to a functionϕwith a positive real part in the unit diskD,ϕ() = ,ϕ() > ,ϕ mapsDonto a region starlike with respect to and symmetric with respect to the real axis. He obtained distortion, growth and covering estimates as well as bounds for the initial coefficients of the unified classes.
The bounds for the second Hankel determinantH() =aa–aare obtained for
func-tions belonging to these subclasses of Ma-Minda starlike and convex funcfunc-tions in Sec-tion . In SecSec-tion , the problem is investigated for two other related classes defined by subordination. In proving our results, we do not assume the univalence or starlikeness of
ϕas they were required only in obtaining the distortion, growth estimates and the convo-lution theorems. The classes introduced by subordination naturally include several well-known classes of univalent functions and the results for some of these special classes are indicated as corollaries.
LetPbe the class offunctions with positive real partconsisting of all analytic functions
p:D→Csatisfyingp() = andRep(z) > . We need the following results about the functions belonging to the classP.
Lemma [] If the function p∈Pis given by the series
p(z) = +cz+cz+cz+· · ·, ()
then the following sharp estimate holds:
|cn| ≤ (n= , , . . .). ()
Lemma [] If the function p∈Pis given by the series(),then
c=c+x –c
, ()
c=c+ –c
cx–c –c
x+ –c –|x|z, ()
2 Second Hankel determinant of Ma-Minda starlike/convex functions
Subclasses of starlike functions are characterized by the quantityzf(z)/f(z) lying in some domain in the right half-plane. For example,f is strongly starlike of orderβifzf(z)/f(z) lies in a sector|argw|<βπ/, while it is starlike of orderα ifzf(z)/f(z) lies in the half-planeRew>α. The various subclasses of starlike functions were unified by subordination in []. The following definition of the class of Ma-Minda starlike functions is the same as the one in [] except for the omission of starlikeness assumption ofϕ.
Definition Letϕ:D→Cbe analytic, and let the Maclaurin series ofϕbe given by
ϕ(z) = +Bz+Bz+Bz+· · · (B,B∈R,B> ). ()
The classS∗(ϕ) ofMa-Minda starlike functions with respect toϕ consists of functions
f ∈Asatisfying the subordination
zf(z)
f(z) ≺ϕ(z).
For the functionϕgiven byϕα(z) := ( + ( – α)z)/( –z) , <α≤, the classS∗(α) :=
S∗(ϕα) is the well-known class of starlike functions of orderα. Let
ϕPAR(z) := +
π
log +
√ z
–√z
.
Then the class
S∗
P:=S∗(ϕPAR) =
f ∈A:Re
zf(z)
f(z)
>zf (z)
f(z) –
is theparabolic starlikefunctions introduced by Rønning []. For a survey of parabolic starlike functions and the related class of uniformly convex functions, see []. For <β≤, the class
S∗ β:=S∗
+z
–z
β
=
f ∈A:arg
zf(z)
f(z)
<βπ
is the familiar class ofstrongly starlike functions of orderβ. The class
S∗
L:=S∗( √
+z) =
f∈A:
zf(z)
f(z)
– <
is the class oflemniscate starlikefunctions studied in [].
Theorem Let the function f ∈S∗(ϕ)be given by(). . IfB,BandBsatisfy the conditions
then the second Hankel determinant satisfies
aa–a≤ B
.
. IfB,BandBsatisfy the conditions
|B| ≥B, BB–B – B–B|B|– B≥,
or the conditions
|B| ≤B, BB–B – B– B≥,
then the second Hankel determinant satisfies
aa–a≤
BB–B
– B.
. IfB,BandBsatisfy the conditions
|B|>B, BB–B – B–B|B|– B≤,
then the second Hankel determinant satisfies
aa–a≤ B
|BB–B – B|– B|B|+ B–B |BB–B – B|– B|B|–B
.
Proof Sincef∈S∗(ϕ), there exists an analytic functionwwithw() = and|w(z)|< inD such that
zf(z)
f(z) =ϕ w(z)
. ()
Define the functionspby
p(z) :=
+w(z)
–w(z)= +cz+cz
+· · ·,
or, equivalently,
w(z) = p(z) –
p(z) +
=
cz+
c– c
z+· · ·
. ()
Thenpis analytic inDwithp() = and has a positive real part inD. By using () together
with (), it is evident that
ϕ
p(z) – p(z) +
= +
Bcz+
B
c– c
+ Bc
z+· · ·. ()
Since
zf(z)
f(z) = +az+ –a
+ a
z+ a– aa+a
it follows by (), () and () that
a= Bc
,
a=
B
–B+B
c+ Bc
,
a=
–B+ B+B
– B+ BB+ B
c
+ B– B+ B
cc+ Bc
.
Therefore
aa–a= B
c
–B
+
B
–B+ B– B
B
+ cc(B–B) + Bcc– Bc
.
Let
d= B, d= (B–B),
d= –B, d= – B
+
B
–B+ B– B
B
, ()
T=B .
Then
aa–a=Tdcc+dcc+dc+dc. ()
Since the functionp(eiθz) (θ∈R) is in the classPfor anyp∈P, there is no loss of gener-ality in assumingc> . Writec=c,c∈[, ]. Substituting the values ofcandc
respec-tively from () and () in (), we obtain
aa–a= T
c
(d
+ d+d+ d) + xc –c
(d+d+d)
+ –cx –dc+d –c
+ dc –c –|x|
z.
Replacing|x|byμand substituting the values ofd,d,danddfrom () yield
aa–a≤ T
c–B+ B– B
B
+ |B|μc –c
+μ –c Bc+ B
+ Bc –c –μ
=T
c
–B+ B– B
B
+ Bc –c
+|B| –c
μc
+B μ
–c(c– )(c– )
Note that for (c,μ)∈[, ]×[, ], differentiatingF(c,μ) in () partially with respect to
μyields
∂F ∂μ=T
|B| –c
+Bμ –c
(c– )(c– ). ()
Then, for <μ< and for any fixedcwith <c< , it is clear from () that∂∂μF > , that is,
F(c,μ) is an increasing function ofμ. Hence, for fixedc∈[, ], the maximum ofF(c,μ) occurs atμ= , and
maxF(c,μ) =F(c, )≡G(c).
Also note that
G(c) = B
c
–B+ B– B B
– |B|– B
+ c |B|–B
+ B
.
Let
P=
–B+ B– B B
– |B|– B
,
Q= |B|–B
, ()
R= B.
Since
max
≤t≤ Pt
+Qt+R=
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
R, Q≤,P≤–Q;
P+ Q+R, Q≥,P≥–Q orQ≤,P≥–Q;
PR–Q
P , Q> ,P≤– Q ,
()
we have
aa–a≤ B
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
R, Q≤,P≤–Q;
P+ Q+R, Q≥,P≥–Q orQ≤,P≥–Q;
PR–Q
P , Q> ,P≤– Q ,
whereP,Q,Rare given by ().
Remark WhenB=B=B= , Theorem reduces to [, Theorem .].
Corollary
. Iff ∈S∗(α),then|aa–a| ≤( –α).
. Iff ∈SL∗,then|aa–a| ≤/ = ..
. Iff ∈SP∗,then|aa–a| ≤/π≈..
Definition Letϕ:D→Cbe analytic, and letϕ(z) be given as in (). The classC(ϕ) of
Ma-Minda convex functions with respect toϕconsists of functionsf satisfying the subor-dination
+zf (z)
f(z) ≺ϕ(z).
Theorem Let the function f ∈C(ϕ)be given by(). . IfB,BandBsatisfy the conditions
B+ |B|– B≤, BB+BB–B– B– B≤,
then the second Hankel determinant satisfies
aa–a≤ B
.
. IfB,BandBsatisfy the conditions
B+ |B|– B≥, BB+BB–B– B–B– B|B|– B≥,
or the conditions
B+ |B|– B≤, BB+BB–B– B– B≥,
then the second Hankel determinant satisfies
aa–a≤
BB+B
B–B– B.
. IfB,BandBsatisfy the conditions
B+ |B|– B> , BB+BB–B– B–B– B|B|– B≤,
then the second Hankel determinant satisfies
aa–a≤ B
⎛ ⎜ ⎜ ⎜ ⎜ ⎝
|BB+BB–B– B|– B– B|B|
– B
–B– B|B|– B
|BB+BB–B – B|–B– B|B|– B
⎞ ⎟ ⎟ ⎟ ⎟ ⎠.
Proof Sincef ∈C(ϕ), there exists an analytic functionwwithw() = and|w(z)|< inD such that
+zf (z)
f(z) =ϕ w(z)
. ()
Since
+zf (z)
f(z) = + az+ –a
+ a
z+ a– aa+ a
equations (), () and () yield
a= Bc
,
a=
B
–B+B
c+ Bc
,
a=
–B+ B+B
– B+ BB+ B
c
+ B– B+ B
cc+ Bc
.
Therefore
aa–a= B
c
–
B+ B–
B – B +
BB+ B– B B + cc
B– B+ B
+ Bcc–
Bc
.
By writing
d= B, d=
B
– B+ B
,
d= –
B, d= – B+
B–
B – B +
BB+ B– B B , ()
T= B ,
we have
aa–a=Tdcc+dcc+dc+dc. ()
Similar as in Theorems , it follows from () and () that
aa–a= T
c
(d
+ d+d+ d) + xc –c
(d+d+d)
+ –cx –dc+d –c
+ dc –c –|x|
z.
Replacing|x|byμand then substituting the values ofd,d,danddfrom () yield
aa–a≤ T
c– B
+
BB+ B–
B B
+ μc –c
B
+
|B|
+μ –c
Bc
+
B
+ Bc –c –μ
=T c
–B+BB+ B– B
B
+ Bc –c
+ μc
–c B + |B|
+B μ
–c(c– )(c– )
≡F(c,μ). ()
Again, differentiatingF(c,μ) in () partially with respect toμyields
∂F ∂μ=T
c
–c
B + |B|
+B
μ –c
(c– )(c– )
. ()
It is clear from () that ∂∂μF > . ThusF(c,μ) is an increasing function ofμfor <μ< and for any fixedcwith <c< . So, the maximum ofF(c,μ) occurs atμ= and
maxF(c,μ) =F(c, )≡G(c).
Note that
G(c) =T
c
–B+BB+ B– B B
–B– |B|– B
+ c
B
+ |B|– B
+
B
.
Let
P=
–B+BB+ B– B
B
–B– |B|– B
,
Q=
B
+ |B|– B
, ()
R= B.
By using (), we have
aa–a≤ B
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
R, Q≤,P≤–Q;
P+ Q+R, Q≥,P≥–Q orQ≤,P≥–Q;
PR–Q
P , Q> ,P≤– Q
,
whereP,Q,Rare given in ().
Remark For the choice ofϕ(z) = ( +z)/( –z), Theorem reduces to [, Theorem .].
3 Further results on the second Hankel determinant
Definition Letϕ:D→Cbe analytic, and letϕ(z) be as given in (). Let ≤γ ≤ and
τ ∈C\{}. A functionf ∈Ais in the classRτ
γ(ϕ) if it satisfies the following subordination:
+
τ f
Theorem Let≤γ ≤,τ∈C\ {},and let the function f as in()be in the classRτ γ(ϕ).
Also,let
p=
( +γ)( + γ) ( + γ) .
. IfB,BandBsatisfy the conditions
|B|( –p) +B( – p)≤, BB–pB–pB ≤,
then the second Hankel determinant satisfies
aa–a≤
|τ|B
( + γ).
. IfB,BandBsatisfy the conditions
|B|( –p) +B( – p)≥, BB–pB– ( –p)B|B|–B≥,
or the conditions
|B|( –p) +B( – p)≤, BB–pB–B≥,
then the second Hankel determinant satisfies
aa–a≤
|τ|
( +γ)( + γ)BB–pB
.
. IfB,BandBsatisfy the conditions
|B|( –p) +B( – p) > , BB–pB– ( –p)B|B|–B≤,
then the second Hankel determinant satisfies
aa–a≤
|τ|B
( +γ)( + γ)
×
⎛ ⎜ ⎜ ⎜ ⎜ ⎝
p|BB–pB|– ( –p)B[|B|( – p) +B]
– B( –p)–B( – p)
|BB–pB|– ( –p)B(|B|+B)
⎞ ⎟ ⎟ ⎟ ⎟ ⎠.
Proof Forf ∈Rτ
γ(ϕ), there exists an analytic functionwwithw() = and|w(z)|< inD such that
+
τ f
Sincef has the Maclaurin series given by (), a computation shows that
+
τ f
(z) +γzf(z) –
= +a( +γ)
τ z+
a( + γ) τ z
+a( + γ) τ z
+· · ·. ()
It follows from (), () and () that
a= τBc
( +γ),
a= τB
( + γ)
c+c
B B – ,
a= τ
( + γ)
B c– cc+c
+ Bc c–c
+Bc
.
Therefore
aa–a
= τ
B c
( +γ)( + γ)
B c– cc+c
+ Bc c–c
+Bc
– τ
B
( + γ)
c+c
B B
–
+ cc
B B – = τ
B
( +γ)( + γ) cc– c
c+c
+Bc
B
c–c
+B
B c
–
( +γ)( + γ) ( + γ)
c+c
B B
–
+ cc
B B – , which yields
aa–a=T
cc+c
– B
B –p B B –
+B
B
– pc
– cc
–B
B +p B B – , () where
T= |τ|
B
( +γ)( + γ) and p=
( +γ)( + γ) ( + γ) .
It can be easily verified thatp∈[,] for ≤γ≤. Let
d= , d= –
–B
B +p B B – ,
d= –p, d= – B B –p B B –
+B
B
.
Then () becomes
aa–a=Tdcc+dcc+dc+dc. ()
It follows that
aa–a= T
c
(d
+ d+d+ d) + xc –c
(d+d+d)
+ –cx –dc+d –c
+ dc –c –|x|
z.
Application of the triangle inequality, replacement of|x|byμand substituting the values ofd,d,danddfrom () yield
aa–a≤ T
cB B
–pB B
+ B
B
μc –c( –p)
+ –cμ c+ p –c+ c –c –μ
=T
cB B
–pB B
+ c –c+ μB B
c –c( –p)
+μ –c( –p)(c–α)(c–β)
≡F(c,μ), ()
whereα= ,β= p/( –p) > .
Similarly as in the previous proofs, it can be shown thatF(c,μ) is an increasing function ofμfor <μ< . So, for fixedc∈[, ], let
maxF(c,μ) =F(c, )≡G(c),
which is
G(c) =T
cB B
–pB B
– ( –p)
B
B
+
+ c
B
B
( –p) + – p
+ p
.
Let
P=B
B
–pB B
– ( –p)
B
B
+
,
Q=
B
B
( –p) + – p
, ()
Using (), we have
aa–a≤T
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
R, Q≤,P≤–Q;
P+ Q+R, Q≥,P≥–Q orQ≤,P≥–Q;
PR–Q
P , Q> ,P≤– Q ,
whereP,Q,Rare given in ().
Remark For the choiceϕ(z) := ( +Az)/( +Bz) with –≤B<A≤, Theorem reduces to [, Theorem .].
Definition Letϕ:D→Cbe analytic, and letϕ(z) be as given in (). For a fixed real numberα, the functionf ∈Ais in the classGα(ϕ) if it satisfies the following subordination:
( –α)f(z) +α
+zf
(z)
f(z)
≺ϕ(z).
Al-Amiri and Reade [] introduced the classGα:=Gα(( +z)/( –z)) and they showed thatGα⊂Sforα< . Univalence of the functions in the classGαwas also investigated in [, ]. Singhet al.also obtained the bound for the second Hankel determinant of func-tions inGα. The following theorem provides a bound for the second Hankel determinant of the functions in the classGα(ϕ).
Theorem Let the function f given by()be in the classGα(ϕ), ≤α≤.Also,let
p=
( + α) ( +α).
. IfB,BandBsatisfy the conditions
Bα( – p) + |B|( +α–p) +B( +α– p)≤,
Bα(α– –pα) +αBB( – p) + (α+ )BB–pB–pB≤,
then the second Hankel determinant satisfies
aa–a≤ B
( +α).
. IfB,BandBsatisfy the conditions
Bα( – p) + |B|( +α–p) +B( +α– p)≥,
Bα(α– –pα) +αBB( – p) + (α+ )BB–pB–Bα( – p)
– ( +α–p)B|B|– (α+ )B≥,
or
Bα( – p) + |B|( +α–p) +B( +α– p)≤,
then the second Hankel determinant satisfies
aa–a≤ |B
α(α– –pα) +αBB( – p) + (α+ )BB–pB|
( +α)( + α) .
. IfB,BandBsatisfy the conditions
Bα( – p) + |B|( +α–p) +B( +α– p) > ,
Bα(α– –pα) +αBB( – p) + (α+ )BB–pB–Bα( – p)
– ( +α–p)B|B|– (α+ )B≤,
then the second Hankel determinant satisfies
aa–a
≤ B
( +α)( + α)
×
⎡ ⎢ ⎢ ⎢ ⎢ ⎣p–
[Bα( – p) + |B|( +α–p) +B( +α– p)] |B
α(α– –pα) +αBB( – p) + (α+ )BB–pB|
–Bα( – p) – ( +α–p)B(|B|+B)
⎤ ⎥ ⎥ ⎥ ⎥ ⎦.
Proof Forf ∈Gα(ϕ), a calculation shows that aa–a
=T( +α)Bcc+c
–αB+α(α– )B+B( +α) + αBB
+ ( +α)(B– B) –p
(αB
–B+B) B
– pBc
+ cc
–( +α)B+ αB+ ( +α)B– p αB–B+B, ()
where
T= B
( +α)( + α) and p=
( + α) ( +α).
It can be easily verified that for ≤α≤,p∈[,]. Let
d= ( +α)B,
d=
–( +α)B+ αB+ ( +α)B– p αB–B+B
,
d= –pB,
d= –αB+α(α– )B+B( +α) + αBB
+ ( +α)(B– B) –p
(αB
–B+B) B
.
Then
aa–a=Tdcc+dcc+dc+dc. ()
Similarly as in earlier theorems, it follows that
aa–a= T
c
(d
+ d+d+ d) + xc –c
(d+d+d)
+ –cx –dc+d –c
+ dc –c –|x|
z
≤T
cBα(α– –pα) +αBB( – p)
+ (α+ )B–p B
B
+μc –cBα( – p)
+ |B|( +α–p)
+ c –cB( +α)
+μ –cB( +α–p)(c– )
c– p
+α–p
≡F(c,μ), ()
and for fixedc∈[, ],maxF(c,μ) =F(c, )≡G(c) with
G(c) =T
cBα(α– –pα) +αBB( – p) + (α+ )B–p B
B
–Bα( – p) – ( +α–p)(|B|+B)
+ cBα( – p)
+ |B|( +α–p) +B( +α– p)
+ pB
.
Let
P=Bα(α– –pα) +αBB( – p) + (α+ )B–p B
B
–Bα( – p) – ( +α–p) |B|+B
,
Q= Bα( – p) + |B|( +α–p) +B( +α– p)
,
R= pB.
()
By using (), we have
aa–a≤T
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
R, Q≤,P≤–Q;
P+ Q+R, Q≥,P≥–Q orQ≤,P≥–Q;
PR–Q
P , Q> ,P≤– Q ,
Remark Forα= , Theorem reduces to Theorem . For ≤α< , letϕ(z) := ( + ( – α)z)/( –z). For this functionϕ,B=B=B= ( –α). In this case, Theorem reduces
to [, Theorem .].
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
1School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM, Penang, Malaysia.2Department of Mathematics,
University of Delhi, Delhi, 110007, India.
Acknowledgements
The work presented here was supported in part by research grants from Universiti Sains Malaysia (FRGS grants) and University of Delhi as well as MyBrain MyPhD programme of the Ministry of Higher Education, Malaysia.
Received: 11 December 2012 Accepted: 11 March 2013 Published: 5 June 2013 References
1. Cantor, DG: Power series with integral coefficients. Bull. Am. Math. Soc.69, 362-366 (1963) 2. Wilson, R: Determinantal criteria for meromorphic functions. Proc. Lond. Math. Soc.4, 357-374 (1954) 3. Vein, R, Dale, P: Determinants and Their Applications in Mathematical Physics. Applied Mathematical Sciences,
vol. 134. Springer, New York (1999)
4. Pommerenke, C: On the coefficients and Hankel determinants of univalent functions. J. Lond. Math. Soc.41, 111-122 (1966)
5. Pommerenke, C: On the Hankel determinants of univalent functions. Mathematika14, 108-112 (1967) 6. Hayman, WK: On the second Hankel determinant of mean univalent functions. Proc. Lond. Math. Soc.18, 77-94
(1968)
7. Noonan, JW, Thomas, DK: On the Hankel determinants of areally meanp-valent functions. Proc. Lond. Math. Soc.25, 503-524 (1972)
8. Noonan, JW: Coefficient differences and Hankel determinants of areally meanp-valent functions. Proc. Am. Math. Soc.46, 29-37 (1974)
9. Noonan, JW, Thomas, DK: On the second Hankel determinant of areally meanp-valent functions. Trans. Am. Math. Soc.223, 337-346 (1976)
10. Elhosh, MM: On the second Hankel determinant of univalent functions. Bull. Malays. Math. Soc.9(1), 23-25 (1986) 11. Elhosh, MM: On the second Hankel determinant of close-to-convex functions. Bull. Malays. Math. Soc.9(2), 67-68
(1986)
12. Noor, KI: Higher order close-to-convex functions. Math. Jpn.37(1), 1-8 (1992)
13. Noor, KI: On the Hankel determinant problem for strongly close-to-convex functions. J. Nat. Geom.11(1), 29-34 (1997)
14. Noor, KI: On certain analytic functions related with strongly close-to-convex functions. Appl. Math. Comput.197(1), 149-157 (2008)
15. Noor, KI, Al-Bany, SA: On Bazilevic functions. Int. J. Math. Math. Sci.10(1), 79-88 (1987)
16. Noor, KI: On analytic functions related with functions of bounded boundary rotation. Comment. Math. Univ. St. Pauli
30(2), 113-118 (1981)
17. Noor, KI: On meromorphic functions of bounded boundary rotation. Caribb. J. Math.1(3), 95-103 (1982)
18. Noor, KI: Hankel determinant problem for the class of functions with bounded boundary rotation. Rev. Roum. Math. Pures Appl.28(8), 731-739 (1983)
19. Noor, KI, Al-Naggar, IMA: On the Hankel determinant problem. J. Nat. Geom.14(2), 133-140 (1998)
20. Arif, M, Noor, KI, Raza, M: Hankel determinant problem of a subclass of analytic functions. J. Inequal. Appl.2012, Article ID 22 (2012)
21. Hayami, T, Owa, S: Generalized Hankel determinant for certain classes. Int. J. Math. Anal.4(49-52), 2573-2585 (2010) 22. Hayami, T, Owa, S: Applications of Hankel determinant forp-valently starlike and convex functions of orderα. Far East
J. Appl. Math.46(1), 1-23 (2010)
23. Hayami, T, Owa, S: Hankel determinant forp-valently starlike and convex functions of orderα. Gen. Math.17(4), 29-44 (2009)
24. Janteng, A, Halim, SA, Darus, M: Hankel determinant for starlike and convex functions. Int. J. Math. Anal.1(13-16), 619-625 (2007)
25. Mishra, AK, Gochhayat, P: Second Hankel determinant for a class of analytic functions defined by fractional derivative. Int. J. Math. Math. Sci.2008, Article ID 153280 (2008)
26. Mohamed, N, Mohamad, D, Cik Soh, S: Second Hankel determinant for certain generalized classes of analytic functions. Int. J. Math. Anal.6(17-20), 807-812 (2012)
27. Murugusundaramoorthy, G, Magesh, N: Coefficient inequalities for certain classes of analytic functions associated with Hankel determinant. Bull. Math. Anal. Appl.1(3), 85-89 (2009)
29. Ali, RM, Ravichandran, V, Seenivasagan, N: Coefficient bounds forp-valent functions. Appl. Math. Comput.187(1), 35-46 (2007)
30. Ma, WC, Minda, D: A unified treatment of some special classes of univalent functions. In: Proceedings of the Conference on Complex Analysis, Tianjin, 1992. Conf. Proc. Lecture Notes Anal., vol. I, pp. 157-169. International Press, Cambridge (1922)
31. Duren, PL: Univalent Functions. Grundlehren der Mathematischen Wissenschaften, vol. 259. Springer, New York (1983)
32. Grenander, U, Szegö, G: Toeplitz Forms and Their Applications. California Monographs in Mathematical Sciences. University of California Press, Berkeley (1958)
33. Rønning, F: Uniformly convex functions and a corresponding class of starlike functions. Proc. Am. Math. Soc.118(1), 189-196 (1993)
34. Ali, RM, Ravichandran, V: Uniformly convex and uniformly starlike functions. Math. News Lett.21(1), 16-30 (2011) 35. Sokół J, Stankiewicz, J: Radius of convexity of some subclasses of strongly starlike functions. Zeszyty Nauk. Politech.
Rzeszowskiej Mat.19, 101-105 (1996)
36. Bansal, D: Upper bound of second Hankel determinant for a new class of analytic functions. Appl. Math. Lett.26(1), 103-107 (2013)
37. Al-Amiri, HS, Reade, MO: On a linear combination of some expressions in the theory of the univalent functions. Monatshefte Math.80(4), 257-264 (1975)
38. Singh, S, Gupta, S, Singh, S: On a problem of univalence of functions satisfying a differential inequality. Math. Inequal. Appl.10(1), 95-98 (2007)
39. Singh, V, Singh, S, Gupta, S: A problem in the theory of univalent functions. Integral Transforms Spec. Funct.16(2), 179-186 (2005)
40. Verma, S, Gupta, S, Singh, S: Bounds of Hankel determinant for a class of univalent functions. Int. J. Math. Math. Sci.
2012, Article ID 147842 (2012) doi:10.1186/1029-242X-2013-281