Some results of Fekete–Szegö type for Bavrin’s families of holomorphic functions in {\mathbb {C}}^{n}

Abstract

In the paper there is considered a generalization of the well-known Fekete–Szegö type problem onto some Bavrin’s families of complex valued holomorphic functions of several variables. The definitions of Bavrin’s families correspond to geometric properties of univalent functions of a complex variable, like as starlikeness and convexity. First of all, there are investigated such Bavrin’s families which elements satisfy also a ( j , k )-symmetry condition. As application of these results there is given the solution of a Fekete–Szegö type problem for a family of normalized biholomorphic starlike mappings in Cn.{\mathbb {C}}^{n}. C n .

Full text

Vol.:(0123456789)
Annali di Matematica Pura ed Applicata (1923 -)
https://doi.org/10.1007/s10231-021-01094-6
1 3
Some results ofFekete–Szegö type forBavrin’s families
ofholomorphic functions in
ℂn
RenataDługosz1 · PiotrLiczberski2
Received: 2 November 2020 / Accepted: 20 February 2021
© The Author(s) 2021
Abstract
In the paper there is considered a generalization of the well-known Fekete–Szegö type
problem onto some Bavrin’s families of complex valued holomorphic functions of several
variables. The definitions of Bavrin’s families correspond to geometric properties of uni-
valent functions of a complex variable, like as starlikeness and convexity. First of all, there
are investigated such Bavrin’s families which elements satisfy also a (j,k)-symmetry con-
dition. As application of these results there is given the solution of a Fekete–Szegö type
problem for a family of normalized biholomorphic starlike mappings in
ℂn
.
Keywords Holomorphic functions of scv· n-circular domains in
ℂn
· Minkowski function·
(j,k)-symmetry· Fekete–Szegö type estimations
Mathematics Subject Classification 32A30· 30C45
1 Introduction
By
ℂ,ℝ,ℤ,ℕ0,ℕ1,ℕ2
let us denote the sets of complex numbers, real numbers, all integers,
nonnegative integers, positive integers and the integers not smaller than 2, respectively.
We say that a domain
G⊂ℂn,n∈ℕ1,
is complete n-circular if
z𝜆=(z1𝜆1, ..., zn𝜆n)∈G
for each
z=(z1, ..., zn)∈G
and every
𝜆=(𝜆1, ..., 𝜆n)∈U
n , where U is the unit disc
{𝜁∈ℂ∶|𝜁|<1}
. From now on by
G
will be denoted a bounded complete n-circular
domain in
ℂn,n∈
ℕ
1.
Of course, only the open discs with the centre
𝜁=0
and the radius
r>0,
are the bounded complete 1-circular domains
G⊂
ℂ.
In our considerations the Minkowski function
𝜇G∶
ℂ
n
→
[0, ∞)
* Renata Długosz
renata.dlugosz@p.lodz.pl
Piotr Liczberski
piotr.liczberski@p.lodz.pl
1 Centre ofMathematics andPhysics, Lodz University ofTechnology, Ul. Żwirki 36, 90-924Lodz,
Poland
2 Institute ofMathematics, Lodz University ofTechnology, Ul. Żwirki 36, 90-924Lodz, Poland

R.Długosz, P.Liczberski
1 3
will be very useful. It is known (see e.g, [26]) that
𝜇G
is a norm in
ℂn
if
G
is a convex
bounded complete n-circular domain. This function gives the possibility to redefine the
domain
G
and its boundary
𝜕G
as follows:
Now, we recall some information about m-homogeneous polynomials. We say that a func-
tion
Qm∶ℂn
⟶
ℂ,m∈ℕ1,
is an m-homogeneous polynomial if
where
Lm∶(ℂn)m
⟶
ℂ
is a bounded m-linear function (by
Q0
we note a complex con-
stant). For this reason it is very natural to define (see [4]) the following generalization
of the norm of m-homogeneous polynomials
Qm∶ℂn
→
ℂ,
i.e., the
𝜇G
-balance of such
m-homogeneous polynomials
A simple kind of 1-homogeneous polynomials are the linear functionals
J,I∈(
ℂ
n)∗
of the
form
Note that for
m∈ℕ1,
the mapping
is an m-homogeneous polynomial and
𝜇G(Im)=1.
By
F(G,ℂm),m∈ℕ1,
let us denote the space of all functions
f∶G
⟶
ℂm
,
by
HG
the
space of all holomorphic functions
f∈F(G,ℂ)
and by
HG(0),HG(1)
the collection of all
f∈HG
, normalized by
f(0)=0, f(0)=1,
respectively. Let us recall that every function
f∈HG
has a unique power series expansion
where
Qf,m∶ℂn
→
ℂ,m∈ℕ0,
are m-homogeneous polynomials determined uniquelly by
the mth Frechèt differential
Dmf(0)
of f at zero via the formula
Many authors considered some problems connected with m-homogeneous polynomials in
the power series expansion (1.1) of functions from different subfamilies of
HG
(see for
instance [2, 5, 9, 13, 24]). In particular, in the case
G⊂
ℂ
2
Bavrin [2] gives the following
sharp estimates
𝜇
G(z)=inf
{
t>0∶
1
t
z∈G
}
,z∈ℂn
,
G={z∈ℂn∶𝜇G(z)<1},𝜕G={z∈ℂn∶𝜇G(z)=1}.
Qm(z)=Lm(zm)=Lm(z, ..., z),z∈ℂn
,
𝜇
G(Qm)= sup
w∈
ℂn
⧵{0}
|
|
Qm(w)
|
|
(𝜇
G
(w))m=sup
v∈𝜕G
|
|
Qm(v)
|
|
=sup
u∈G
|
|
Qm(u)
|
|.
J
(z)=
n
∑
j=1
zj,z=(z1, ..., zn)∈ℂn,I(z)=
(
𝜇G(J)
)
−1J(z)
.
Im∶
ℂ
n
→ℂ
,Im(z)=(I(z))m
,z∈
ℂ
n
,
(1.1)
f(z)=
∞
∑
m=0
Qf,m(z),z∈G
,
Q
f,m(z)=
1
m!
Dmf(0)(zm)
.

Some results ofFekete–Szegö type forBavrin’s families of…
1 3
for the homogeneous polynomials
Qf,m
of functions belonging to the family
Note, that the sharpness in the Bavrin’s result was understood as follows: There exists
a bounded complete 2-circular domain
G⊂
ℂ2
and a function
f∈BG
which realizes the
equality in the above inequality. Let us observe that in general case
G⊂ℂn,n∈ℕ1,
the
above inequality takes currently the following form
by the definition of the
𝜇G
-balance
𝜇G(Qm).
In the paper, we solve for
𝜆∈ℂ,k∈ℕ2,
the problem of the sharp upper estimate
for the pairs
Qf,k,Qf,2k
homogeneous polynomials of functions belonging to some Bavrin’s
subfamilies of the families
HG(0),HG(1)
(the case
k=1
see the section Final Remarks).
Moreover, here the sharpness is understand more generally. It means that for every bounded
complete n-circular domain
G⊂
ℂn
there exists a function f, which belongs to a mentioned
Bavrin’s subfamily and realizes the equality in the above inequality.
Note that the afore-mentioned estimate is a generealization of the well known planar
Fekete–Szegö [8] result onto the s.c.v. case.
In the sequel we use a special kind of functions symmetry. Let us observe that bounded
complete n-circular domains
G⊂ℂn
are k-symmetric sets,
k∈ℕ2,
that is
𝜀G
=
G,
where
𝜀
=𝜀k=exp
2𝜋i
k
is a generator of the cyclic group of kth roots of unity. For
k∈ℕ2,j∈ℤ
we define the collections
Fj,k(G,ℂm)
of functions
f∈F(G,ℂm),(j,k)
-symmetrical, i.e.,
Let us observe that
Fj,k(G,ℂm)≠Fl,k(G,ℂm)
for different
j,l∈{0, 1, ..., k−1}.
Moreover,
the intersections
Fj,k(G,
ℂ
m)∩Fl,k(G,
ℂ
m)
are the singleton
{0}
for such j,l.
Now we present a functions decomposition theorem [19].
Theorem A For every function
f∈F(G,ℂm)
and every
k∈ℕ2
there exists exactly one
sequence of functions
fj,k∈Fj,k(G,
ℂ
m),
j=0, 1, ..., k−1,
such that
Moreover,
By the uniqueness of this decomposition, the functions
fj,k,j=0, 1, ..., k−1,
will be
called
(j,k)−
symmetrical components of the function f.
sup
z∈G
|
|
|
Qf,m(z)
|
|
|≤
1−
|
|
|
Qf,0
|
|
|
2
,m∈ℕ1
,
BG
=
{
f∈H
G
∶
|
f(z)
|
<1, z∈G
}.
𝜇
G
(
Qf,m
)≤
1−
|
|
|
Qf,0
|
|
|
2
,m∈ℕ1
,
𝜇
G

Qf,2k−𝜆

Qf,k

2
≤
M(𝜆,k
)
f(𝜀z)=𝜀jf(z),z∈
G
.
(1.2)
f=
k−1
∑
j=0
fj,k
.
fj,k(z)=1
k
k−1
∑
l=0
𝜀−jlf
(
𝜀lz
)
,z∈G
.

R.Długosz, P.Liczberski
1 3
In the next sections of the paper will be very useful the fact that
f0,k∈HG(1)
for
f∈HG(1).
Note that
Note that the above unique decomposition (1.2) of functions was used in [20] to solve some
functional equations, in [21] to construction a semi power series and in [22] to obtain a
uniqueness theorem of Cartan type for holomorphic mappings in
ℂn
.
We close this section with the following Golusin’s [10] result, very useful in the proof
of the first result of Fekete–Szegö type for holomorphic functions of several complex
variables.
Lemma 1.1 Let
Φ∶U→U
be a holomorphic function of the form
Then
for every
m∈ℕ1,p∈ℕ0,
satisfying the condition
0≤2p<m.
The estimates are optimal.
Proof Let us recall that the simplest case
follows from the well-known inequality
For another m,p we use a Krzyż’s idea [17, Chapt. 6.2] and some properties of (j,k)-sym-
metrical functions.
Let us take
Then
Ψ∶U→U
and is holomorphic. Thus the
(0, m−p)
-part
Ψ0,m−p
of
Ψ
transforms U
into itself, is holomorphic and
Hence, by the Schwarz Lemma (in version with the zero
𝜁=0
of multiplicity
m−p>0
) it
fulfils the inequality
f0,k(z)=1+
∞
∑
m=1
Q(f0,k),m(z)=1+
∞
∑
s=1
Qf,sk(z),z∈G
.
Φ
(𝜁)=
∞
∑
𝜈=0
a𝜈𝜁𝜈,𝜁∈U
.
|
|
am
|
|≤
1−
|
|
|
ap
|
|
|
2
,
|
|
a
1|
|≤
1−
|
|
a
0|
|
2
|
Φ�(𝜁)
|≤
1−
|
Φ(𝜁)
|2
1−|𝜁|
2,𝜁∈U
.
Ψ
(𝜁)=𝜁m−2pΦ(𝜁)=
∞
∑
s=0
as𝜁s+m−2p,𝜁∈U
.
Ψ
0,m−p(𝜁)=
∞
∑
s=0
ap+s(m−p)𝜁(s+1)(m−p),𝜁∈U
.

Some results ofFekete–Szegö type forBavrin’s families of…
1 3
Therefore, the function
maps U into itself, is holomorphic and has the expansion
Consequently, replacing
𝜁m−p∈U
by
𝜉∈
U, we get that the function
transforms holomorphically U into itself. Hence, by the first part of the proof, we get the
thesis.
Note that the equality in the inequality is attained by the functions
F(𝜁)=
𝜁m
,𝜁∈U
and
F(𝜁)=
𝜁p
,𝜁∈U.
This completes the proof
◻
2 Main results
We start this section with an n-dimensional Fekete–Szegö type theorem for bounded holo-
morphic functions on bounded complete n-circular domains in
ℂn
.
Note that it is a gener-
alization of a 1-dimensional result given by Keogh and Merkes [14].
Theorem2.1 Let
𝜑
be a function from the family
and has the form
Then, for every
k∈ℕ1
and every
𝛾∈ℂ,
there holds the inequality
The estimate is sharp.
Proof For every arbitrarily fixed
z∈G
we define the function



Ψ0,m−p(𝜁)


≤
𝜁

m−p,𝜁∈U
.
Θ
(𝜁)=
1
𝜁
m−pΨ0,m−p(𝜁),𝜁∈U
,
Θ
(𝜁)=
∞
∑
s=0
ap+s(m−p)𝜁s(m−p),𝜁∈U
.
∞
∑
s=0
bs𝜉s=
∞
∑
s=0
ap+s(m−p)𝜉s,𝜉∈
U
BG
(0)=
{
𝜑∈H
G
(0)∶
|
𝜑(z)
|
<1, z∈G
}
𝜑
(z)=
∞
∑
l=1
Q𝜑,l(z),z∈G
.
(2.1)
𝜇
G

Q𝜑,2k−𝛾

Q𝜑,k

2
≤
max{1,

𝛾

}
.
Φ
(𝜁)=
{𝜑(𝜁z)
𝜁,𝜁∈U�{0}
Q𝜑,1(z)=lim𝜁→0
𝜑(𝜁z)
𝜁
,𝜁=0
.

R.Długosz, P.Liczberski
1 3
Then
Φ
is holomorphic,
al−1=Q𝜑,l(z),l∈ℕ1,
and
|Φ(0)|<1.
Moreover, applying a
ℂn
-version of Schwarz Lemma,
i.e.,
|𝜑(
z
)|≤𝜇G(
z
)
,z
∈G,
for
𝜑∈BG(
0
),
we conclude also that for
𝜁∈U�{0}
hence
Φ∶U⟶U.
Now let us observe that from the Lemma 1.1 we get
for arbitrarily fixed
m,p∈ℕ1,
satisfying the condition
0≤2(p−1)<m−1.
Thus
Therefore, for
z∈G
and every
𝛾∈ℂ
because
Consequently, by the arbitrarinnes of
z
∈
G,
also
Putting in the above
p=k
and
m=2k,k∈ℕ1,
we obtain
m−1>2(p−1)≥1≥0
and
Finally, by the fact that
Q𝜑,2k
−𝛾
(
Q
𝜑,k)2
is a 2k-homogeneous polynomial for every
𝛾∈ℂ
and by the definition of its
𝜇G
-balance, we get the statements of the Theorem2.1.
Now, we will analyse the sharpness of the above estimate.
First, we prove that in the case
|𝛾|≥1,
the equality in (2.1) is attained by the func-
tion
𝜑
=̃𝜑 ∈B
G
(0),̃𝜑 =I
k,
more precisely
̃𝜑
=I
k
|G
, i.e.,
̃𝜑 (z)=Ik(z),z∈
G
.
Indeed, since
Q̃𝜑 ,k
=I
k
,Q
̃𝜑 ,2k
=
0
and
𝜇G
(I
2k
)=
1,
we have
Φ
(𝜁)=
∞
∑
𝜈=0
a𝜈𝜁𝜈,𝜁∈U
,
|
Φ(𝜁)
|
=
|𝜑
(
𝜁
z)
|
|𝜁|≤
𝜇
G
(𝜁z)
|𝜁|
=
|𝜁|𝜇
G
(z)
|𝜁|
<
1,
|
|
am−1
|
|≤
1−
|
|
|
ap−1
|
|
|
2
,
|
|
|
Q𝜑,m(z)
|
|
|≤
1−
|
|
|
Q𝜑,p(z)
|
|
|
2
,z∈G
.



Q𝜑,m(z)−𝛾

Q𝜑,p(z)
2


≤


Q𝜑,m(z)


+

𝛾




Q𝜑,p(z)2


≤1−


Q𝜑,p(z)


2
+

𝛾




Q𝜑,p(z)


2
=1+(

𝛾

−1)



Q𝜑,p(z)



2
≤max{1,

𝛾

},





(

𝛾

−1)


Q𝜑,p(z)


2
≤
0, if

𝛾

<1,
0≤(

𝛾

−1)



Q𝜑,p(z)



2
≤

𝛾

−1, if

𝛾

≥1
.
sup
z∈G


Q𝜑,m(z)−𝛾

Q𝜑,p(z)
2

≤
max{1,

𝛾

}
.
sup
z∈G


Q𝜑,2k(z)−𝛾

Q𝜑,k(z)
2

≤
max{1,

𝛾

}
.

Some results ofFekete–Szegö type forBavrin’s families of…
1 3
Now, we show that in the case
|𝛾|<1,
the equality in (2.1) realizes the function
𝜑
=̂𝜑 ∈B
G
(0),̂𝜑 =I
2k,
more precisely
̂𝜑
=I
2k
|
G , i.e.,
̂𝜑 (z)=I2k(z),z∈
G
.
Indeed, since
Q̂𝜑 ,k
=0, Q
̂𝜑 ,2k
=I
2k,
we get
This completes the proof.
◻
Note that Theorem2.1 generalizes an earlier result given by authors in [7].
In the sequel we apply Theorem2.1 to study two Bavrin’s families
MG
,
NG
of functions
f∈HG(1)
. These families are defined by the following family
CG,
and by the following Temljakov [29] linear operator
L∶HG
⟶
HG
where Df(z) means the Fréchet derivative of f at the point z. Note that the operator
L
is
invertible and
It is obvious also that for the transforms
Lf,LLf
of the functions
f∈HG(1)
we have
Moreover,
We say that a function
f∈HG(1)
belongs to the Bavrin’s family
MG
(NG)
if it satisfies the
factorization
together with a function
h∈CG
and the transform
Lf,
(LLf),
respectively. Note that the
families
MG,
NG
correspond with the well-known families of normalized univalent starlike
(convex) functions in the disc U [2] and the family
MG
can be used to construction biho-
lomorphic starlike mappings in
ℂn
(see [6, 18], compare also [12, 25]). Between functions
𝜇
G

Q𝜑,2k−𝛾

Q𝜑,k

2

=𝜇G

−𝛾

Q̃𝜑 ,k

2

=

𝛾

𝜇G

Q̃𝜑 ,k

2

=

𝛾

=max{1,

𝛾

}
.
𝜇
G

Q𝜑,2k−𝛾

Q𝜑,k

2

=𝜇G

Q̂𝜑 ,2k

=1=max{1,

𝛾

}
.
CG={
f
∈HG(
1
)∶
Ref
(
z
)>
0, z
∈G}
Lf(z)=f(z)+Df (z)(z),z∈G,
L
−1f(z)=
1
∫
0
f(zt)dt,z∈G
.
(2.2)
L
f(z)=1+
∞
∑
m=1
QLf,m(z)=1+
∞
∑
m=1
(m+1)Qf,m(z),z∈G
,
(2.3)
LL
f(z)=1+
∞
∑
m=1
QLLf,m(z)=1+
∞
∑
m=1
(m+1)2Qf,m(z),z∈G
.
(2.4)
L
−1f(z)=1+
∞
∑
m=1
QL−1f,m(z)=1+
∞
∑
m=1
1
m+1Qf,m(z),z∈G
.
(2.5)
L
f
(
z
)=
f
(
z
)
h
(
z
)
,z
∈G
,
LLf(z)=Lf(z)h(z),z∈G,

R.Długosz, P.Liczberski
1 3
from
MG
,
NG
there holds a relationship, corresponding to the well-known Alexander
type connexion [1], for univalent starlike and convex mappings in the unit disc. Here, this
relationship is the following: if f∈
NG,
then
Lf∈MG
and conversely, if
f∈MG,
then
L−1
f∈M
G
[2].
We begin the presentation of some Fekete–Szegö type results in Bavrin’s families with
the following theorem.
Theorem 2.2 Let
G⊂
ℂ
n
be a bounded complete n-circular domain and let
f∈
NG
∩
F0,k
(
G
,ℂ)
,
k∈ℕ2.
If the expansion of the function f into a series of m-homoge-
nous polynomials
Qf,m
has the form (1.1), with
Qf,0 =1,
then for the homogeneous polyno-
mials
Qf,2k,Qf,k
and every
𝜆∈ℂ
there holds the following sharp estimate:
Proof Let us recall that between the functions
p∈
CG
and
𝜑∈BG(0),
there holds the fol-
lowing relation [2]:
Let
k∈ℕ2
be arbitrarily fixed and let the function f belongs to
N
G
∩F0,k(G
,ℂ
).
Then, by
the definition of the family
NG
and by the above relation between the families
CG,BG(0),
we
get
where
𝜑∈BG(0)∩F0,k(G,ℂ).
On the other hand, from (1.1) we have for
z∈G
:
and from (2.2) , (2.3) also
Inserting the above expansion of functions into (2.7) , we receive after computations
(2.6)
𝜇
G

Qf,2k−𝜆

Qf,k

2
≤
1
k(2k+1)max

1,

4(2k+1)𝜆−(k+1)2(k+2)
k(k+1)2
.
p−1
p+1
=𝜑∈BG(0)⟺p∈CG
.
(2.7)
LLf(z)−Lf(z)
LLf(z)+
L
f(z)
=𝜑(z),z∈G
.
𝜑
(z)=
∞
∑
s=1
Q𝜑,sk(z),
f(z)=1+
∞
∑
s=1
Qf,sk(z
)
Lf(z)=1+
∞
∑
s=1
(sk +1)Qf,sk(z),
LL
f(z)=1+
∞
∑
s=1
(sk +1)2Qf,sk(z)
.
∞
∑
s=1
sk(1+sk)Qf,sk (z)=
(∞
∑
s=1
Q𝜑,sk(z)
)(
2+
∞
∑
s=1
(sk +1)(sk +2)Qf,sk(z)
).

Some results ofFekete–Szegö type forBavrin’s families of…
1 3
Then, comparing the m-homogeneous polynomials of the same degree on both sides of the
above equality, we can determine homogeneous polynomials
Q𝜑,k,Q𝜑,2k,
as follows
Putting the above equalities into Theorem2.1 and using the fact that the mapping
(
Q
f,k)2,
is a 2k-homogenous polynomial, we obtain
Denoting
we obtain
Now, we show the sharpness of our estimate. To do it, let us consider two cases.
At the beginning, we prove that, in the case
the equality in (2.6) is attained by the function
f=L
−1
̃
f,
with
where the branch of the function
(1−𝜉)
−2
k takes value 1 at the point
𝜉=0.
Indeed. Function
f belongs to
F0,k(G,ℂ),
because
̃
f∈F
0,k
(G,ℂ)
.
On the other hand, the function
f=
L−1
̃
f
belongs to
NG,
because the function
𝜑
from (2.7) has the form
𝜑=̃𝜑 =Ik
and belongs to
BG(0).
Therefore, we can write equalities (2.8 ) in the following form
From this, by the case condition for
𝜆,
we have step by step:
(2.8)
Q𝜑,k=
1
2k(k+1)Qf,k,
Q
𝜑,2k=k(2k+1)Qf,2k−1
4
k(k+1)2(k+2)
(
Qf,k
)
2
.
𝜇
G

k(2k+1)Qf,2k−
1
4
k(k+1)2(k+2)+𝛾
1
4
k2(k+1)2

Qf,k

2
≤
max{1,

𝛾

}
.
𝜆
=(k+1)
2
4(2k+1)
(k+2+𝛾k)
,
𝜇
G

Qf,2k−𝜆

Qf,k

2
≤
1
k(2k+1)max

1,

4(2k+1)𝜆−(k+1)2(k+2)
k(k+1)2
.
|
|
|
4(2k+1)𝜆−(k+1)2(k+2)
|
|
|
k(k+1)
2≥
1
(2.9)
̃
f(z)=
(
1−Ik(z)
)−2
k,z∈G
,
(2.10)
Q
f,k=
2
k(k+1)
Ik,Qf,2k=
k+2
k
2
(2k+1)(
Ik
)
2
.

R.Długosz, P.Liczberski
1 3
Now, we show that, in the case
the equality in (2.6) realizes the function
f=
L−1
̂
f,
with
where the branch of the function
(1−𝜉)
−1
k takes value 1 at the point
𝜉=0.
The function
f=
L−1
̂
f
belongs to
F0,k(G,ℂ),
because
̂
f∈F
0,k
(G,ℂ)
.
Also
f∈
NG,
because the func-
tion
𝜑
from (2.7) has the form
𝜑=̂𝜑 =I2k
and belongs to
BG(0).
Therefore, we can write
equalities (2.8) in the following form
From this, by the case condition for
𝜆,
we have:
This completes the proof.
◻
We continue the presentation of some Fekete–Szegö type results in Bavrin’s families
with the following theorem:
Theorem2.3 Let
G⊂
ℂn
be a bounded complete n-circular domain and let
k∈ℕ2.
If the
expansion of the function
f∈MG∩F0,k(G,ℂ)
, into a series of m-homogenous polynomi-
als
Qf,m
has the form (1.1), with
Qf,0 =1,
then for the homogeneous polynomials
Qf,2k,Qf,k
and
𝜆∈ℂ
there holds the following sharp estimate:
Proof Let
k∈ℕ2
be arbitrarily fixed. Then, it is obvious that
L−1f
∈F0,k(G,ℂ),
because
f∈F0,k(G,ℂ).
Also the assumption that f
∈MG,
by the relationship of the Alexander type,
gives that
L−1f
∈NG.
Hence, we have that
L−1f
∈NG∩F0,k(G,ℂ).
On the other hand, its
expansion into a series of m-homogenous polynomials
Qf,m,
by ( 2.4), has the form
𝜇
G(Qf,2k−𝜆
(
Qf,k
)
2)=𝜇G
[
(k+1)
2
(k+2)−4(2k+1)𝜆
k2(k+1)2(2k+1)
(
Ik
)
2
]
=|||||
(k+1)2(k+2)−4(2k+1)𝜆
k2(k+1)2(2k+1)|||||
𝜇G(I2k)
=1
k(2k+1)max
{
1,
|||||
4(2k+1)𝜆−(k+1)2(k+2)
k(k+1)2
|||||}.
|
|
|
4(2k+1)𝜆−(k+1)2(k+2)
|
|
|
k(k+1)
2<
1
(2.11)
̂
f(z)=
(
1−I2k(z)
)−1
k,z∈G
,
(2.12)
Q
f,k=0, Qf,2k=
1
k(2k+1)
I2k
.
𝜇
G(Qf,2k−𝜆

Qf,k

2)=𝜇G

1
k(2k+1)I2k

=1
k(2k+1)𝜇G

I2k

=1
k(2k+1)max of

1,

4(2k+1)𝜆−(k+1)2(k+2)
k(k+1)2
.
(2.13)
𝜇
G

Qf,2k−𝜆

Qf,k

2
≤
1
kmax

1,

2+k−4𝜆

k
.

Some results ofFekete–Szegö type forBavrin’s families of…
1 3
Thus, in view of (2.6) from Theorem2.2, we get that for every
𝛿∈ℂ
Hence,
and denoting
𝛿2k+1
(k+1)
2
=
𝜆,
finally
which is the same as (2.13).
It remains to show the sharpness of the estimate (2.13).
First, we prove that, in the case
the equality in (2.13) is attained by the function
f=
̃
f
defined in (2.9). Of course
̃
f∈F
0,k
(G,ℂ
)
and, by the Alexander type relationship and the fact that
L
−1
̃
f∈
NG
(see the
proof of the sharpness in Theorem2.2), the function
̃
f
belongs to
MG.
Therefore, in view
of (2.2) and (2.10), we achieve
From this (see the proof of the sharpness in Theorem2.2) and by the case condition for
𝜆,
we have step by step:
Now, we show that, in the case
the equality in (2.13) realizes the function
f=
̂
f
defined in (2.11). Of course
̂
f∈F
0,k
(G,ℂ
)
and, by the Alexander type relationship and the fact that
L
−1
̂
f∈
NG
(see the proof of the
L
−1f(z)=1+
∞
∑
s=1
1
sk +1Qf,sk(z),z∈G
.
𝜇
G

1
2k+1Qf,2k−𝛿

1
k+1Qf,k
2
≤1
k(2k+1)max

1,

4(2k+1)𝛿−(k+1)2(k+2)
k(k+1)2
.
𝜇
G

Qf,2k−𝛿2k+1
(k+1)2

Qf,k

2
≤
1
kmax

1,

4(2k+1)𝛿−(k+1)2(k+2)
k(k+1)2

𝜇
G

Qf,2k−𝜆

Qf,k

2
≤
1
kmax

1,

4
𝜆
−(k+2)
k
,
|2+k−4𝜆|
k≥1
Q
f,k=Q̃
f,k=
2
k
Ik,Qf,2k=Q̃
f,2k=
k+2
k
2
(
Ik
)
2
.
𝜇
G(Qf,2k−𝜆

Qf,k

2)=𝜇G

(k+2)−4
𝜆
k2

Ik

2

=

(k+2)−4𝜆
k2

𝜇G

I2k

=1
kmax

1,

2+k−4𝜆

k
.
|2+k−4𝜆|
k
<
1

R.Długosz, P.Liczberski
1 3
sharpness in Theorem2.2), the function
̂
f
belongs to
MG.
Therefore, in view of ( 2.2) and
(2.12) we achieve
From this (see the proof of the sharpness in Theorem2.2) and by the case condition for
𝜆,
we conclude that:
This completes the proof.
◻
Now, we transfer the statement of Theorem2.3, onto a family
Mk
G
∩F
0,k
(G,ℂ),k∈ℕ
2.
Here,
Mk
G
is defined by the factorization similar as for the elements from
MG.
More pre-
cisely, the function f of the right hand side in (2.5) is replaced by a function from
F0,k(G,ℂ),
generated by f. Formally, we say that a function
f∈HG(1)
belongs to
Mk
G
,k∈ℕ2
,
(see [3,
5]) if there exists a function
h∈CG
such that
where
f0,k∈F0,k(G,
ℂ
)
is the
(0, k)
- symmetrical component of the function f in the decom-
position (1.2) from Theorem A. This family for
k=2
corresponds to a well-known Saka-
guchi family [27] of a complex variable functions; strictly speaking of functions univalent
starlike with respect to two symmetric points. In the paper [5] it was shown that for
k∈ℕ2
the inclusions
M
G⊂M
k
G,
Mk
G
⊂M
G
do not hold, but
This identity and Theorem 2.3 implies directly the next result of Fekete–Szegö type in
Bavrin’s families:
Theorem 2.4 Let
G⊂
ℂn
be a bounded complete n-circular domain and let
f∈M
k
G∩F0,k(G,ℂ),k∈ℕ2
.
If the expansion of the function f into a series of m-homoge-
nous polynomials
Qf,m
has the form (1.1), with
Qf,0 =1,
then for the homogeneous polyno-
mials
Qf,2k,Qf,k
and
𝜆∈ℂ
there holds the following sharp estimate:
The equality in the above inequality realize the same functions
f=
̃
f,f=
̂
f
as in the previ-
ous Theorem 2.2.
3 Applications
In this section we apply Theorems 2.3 and 2.4, to obtain a Fekete–Szegö type results
for two families of biholomorphic mappings in
ℂn
.
By
S∗(𝔹n)
let us denote the family
of biholomorphic mappings
F∈F(𝔹n
,ℂn),
F(0)=0,
DF(0)=I
onto starlike domains
Q
f,k=Q̂
f,k=0, Qf,2k=Q̂
f,2k=
1
k
I2k
.
𝜇
G(Qf,2k−𝜆

Qf,k

2)=𝜇G

1
kI2k

=1
k=1
kmax

1,

2+k−4
𝜆
k
.
Lf(z)=f0,k(z)h(z),z∈G,
M
G∩F
0,k
(G,ℂ)=M
k
G
∩F
0,k
(G,ℂ)
.
𝜇
G

Qf,2k−𝜆

Qf,k

2
≤
1
k
max

1,

2+k−4
𝜆
k.

Some results ofFekete–Szegö type forBavrin’s families of…
1 3
F(𝔹n).
For a wide collection of references in this area see the monographs [11, 16]. In
a Kikuchi–Matsuno–Suffridge characterization [15, 23, 28] of the family
S∗(𝔹n)
, the col-
lection
P(𝔹n)
of all holomorphic mappings
H∈F(𝔹n,ℂn),H(0)=0, DH(0)=I,
such that
Re⟨H(z),z⟩>0, z∈𝔹n�{0},
plays the main role (here
⟨
⋅
,
⋅
⟩
means the Euclidean inner
product). This characterization is included in the following theorem:
Theorem B A locally biholomorphic mapping
F∈F(𝔹n
,ℂn)
normalized by the conditions
F(0)=0, DF(0)=I,
belongs to
S∗(𝔹n),
iff there exist a mapping
H∈P(𝔹n)
such that
Let
̃
S
∗
(
𝔹n
)
be the family of mappings
F∈S∗(𝔹n)
with the factorization
where
f∈H𝔹
n
(1)
.
In the paper [6] the authors considered a family of biholomorphic mappings
Sk(
𝔹
n),
k
∈
ℕ
2,
defined by an equation similar to (3.1). More precisely, the mapping F of
the left hand side in (3.1) is replaced by a function from
F1,k(𝔹n,ℂn),
generated by F.
Formally, we say that a locally biholomorphic mapping
F∈F(𝔹n
,ℂn),
normalized by
F(0)=0, DF(0)=I,
belongs to the family
Sk(
𝔹
n),k∈
ℕ
2,
if it satisfies the equation
where
H∈P(𝔹n)
and
F1,k∈F1,k(𝔹n,ℂn)
is the
(1, k)
-symmetrical part of F in the decom-
position (1.2). Also in the paper [6] the authors proved, that for every
k∈ℕ2
any inclu-
sions
Sk(
𝔹
n)⊂S∗(
𝔹
n),
S∗(
𝔹
n)⊂Sk(
𝔹
n)
do not holds. However, for the same k there holds
the following identity:
Let
̃
S
k
(
𝔹n
),k∈
ℕ
2,
be the family of mappings
F∈Sk(
𝔹
n)
with the factorization (3.2).
Now, we present the main theorem, in this section of the paper. It is a Fekete–Szegö type
result for locally biholomorphic mappings in
ℂn
,
compare [30].
Theorem3.1 For mappings
F∈
̃
S
k
(
𝔹n
)
∩F1,k(𝔹n,ℂn),k∈ℕ2,
the parameter
𝜆∈ℂ
and
points
z∈
𝔹
n�{0}
there holds the following sharp estimate
where
Tz∈(
ℂ
n)∗,z∈
𝔹
n�{0},
is arbitrary functional satisfying the conditions
‖
‖
T
z‖
‖
=
1,
Tz(
z
)=‖
z
‖.
Proof We start with a few facts, very useful in the proof:
1. A mapping F, satisfying (3.2) , belongs to
Sk(
𝔹
n),
iff f∈M
k
𝔹n
(see [6, 18]).
2. A mapping F of the form (3.2) belongs to
F1,k(𝔹n,ℂn),
iff
f∈
F0,k(𝔹n,ℂ).
3. For
z∈𝔹n
there hold (see for instance [30]) the equalities:
(3.1)
F(z)=DF(z)H(z),z∈
𝔹
n
.
(3.2)
F(z)=zf (z),z∈𝔹n
,
F1,k(z)=DF(z)H(z),z∈𝔹n,
(3.3)
Sk
(𝔹
n
)∩F
1,k
(𝔹
n
,ℂ
n
)=S
∗
(𝔹
n
)∩F
1,k
(𝔹
n
,ℂ
n
)
.
(3.4)

Tz

D2k+1F(0)(z2k+1)

(2k+1)!

z

2k+1−𝜆

Tz

Dk+1F(0)(zk+1)

(k+1)!

z

k+1
2
≤
1
kmax

1,

2+k−4𝜆

k
,

R.Długosz, P.Liczberski
1 3
4. For
∈ℂ
the mapping
Q
f,2k−𝜆Q
2
f,k
∈F(𝔹
n
,ℂ)
,
is a 2kth homogeneous polynomial.
5. There hold the identity
𝜇𝔹
n
(
⋅
)=||
⋅
||
in
ℂn
.
Applying the above facts and Theorem2.5, we get step by step
It remains to show the sharpness of the estimate (3.4).
To do it, observe first that there exist points
z0∈
𝔹
n�{0}
such that
�
I(z
0
‖
z0
‖
)
�
=
1.
It fol-
lows from the maximum principle for the modulus of holomorphic functions of several
complex variables, because
‖I‖=𝜇𝔹
n
(I)=1.
We will show that the equality in (3.4) in
such points are attained by the mappings
̃
F,
̂
F
of the form (3.2), where
f=
̃
f,f=
̂
f
are
defined in
G=𝔹n
by ( 2.9), (2.11) in the cases
|2+k−4𝜆|≥k
and
|2+k−4𝜆|<k,
respectively. The mappings
̃
F,
̂
F
belong to
F1,k(𝔹n,ℂn),
by the enumerate above fact 2,
because
̃
f,
̂
f∈F
0,k
(𝔹n,ℂ)
.
Also
̃
F,
̂
F∈S
k
(
𝔹n
),
by the enumerated above fact l and the rela-
tions
̃
f,
̂
f∈M
k
𝔹
n
.
First, we assume that
|2+k−4𝜆|≥k,
i.e.,
f=
̃
f.
It is easy to check that in this case
For z
=z0
and
F=
̃
F,
i.e.,
(f=
̃
f),
we obtain (for the first below equality see the previous
part of the proof)
D2k+1
F(0)(z
2k+1
)
(2k+1)!=zD
2k
f(0)(z
2k
)
(2k)!
,
Dk+1F(0)(zk+1)
(k+1)!
=zDkf(0)(zk)
(k)!
.

Tz

D2k+1F(0)(z2k+1)

(2k+1)!z2k+1−𝜆

Tz

Dk+1F(0)(zk+1)

(k+1)!zk+1
2

=
Tz(z)D2kf(0)(z2k)
(2k)!z2k+1−𝜆Tz(z)Dkf(0)(zk)
(k)!zk+12
=
D2kf(0)(z2k)
(2k)!z2k−𝜆Dkf(0)(zk)
(k)!zk2
=
Qf,2k(z)
z2k−𝜆Qf,k(z)
zk2
≤sup
z∈𝔹nQf,2k(z)−𝜆Qf,k(z)2
z2k=𝜇𝔹nQf,2k−𝜆Qf,k2

≤1
kmax

1,

2+k−4𝜆

k

.
Q
f,k=Q̃
f,k=
2
k
Ik,Qf,2k=Q̃
f,2k=
k+2
k
2
(
Ik
)
2
.

Some results ofFekete–Szegö type forBavrin’s families of…
1 3
Now, we assume that
|2+k−4𝜆|<k,
i.e.,
f=
̂
f.
In this case it is easy to check that
For z
=z0
and
F=
̂
F,
i.e.,
(f=
̂
f),
we obtain similarly
◻
The identity (3.3) and Theorem3.1 implies the following result of Fekete–Szegö type
for starlike biholomorphic mappings in
ℂn
.
Theorem 3.2 For mappings
F∈
̃
S
∗
(
𝔹n
)
∩F1,k(𝔹n,ℂn),k∈ℕ2,
points
z∈𝔹n�{0}
and
parameter
𝜆∈ℂ,
there holds the following sharp estimate
where
Tz∈(ℂn)∗,z∈𝔹n�{0},
is arbitrary functional satisfying the conditions:
‖
‖
T
z‖
‖
=
1,
Tz(z)=‖z‖.
In the other way a similar theorem was proved by Xu [30].
4 Final remarks
It is possible to allow also
k=1
in definition of
(j,k)
-symmetrical,
j∈ℤ
, functions, from
F(G,ℂm).
Then
𝜀=1
and we should take the convention
Fj,1(G,ℂm)=F(G,ℂm)
for
j∈ℤ.
Consequently, in the case
m=1

Tz

D2k+1F(0)(z2k+1)

(2k+1)!z2k+1−𝜆

Tz

Dk+1F(0)(zk+1)

(k+1)!zk+1
2

=1
z2kQf,2k(z)−𝜆Qf,2k(z)2
=1
z2k
k+2
k2Ik(z)2−𝜆4
k2Ik(z)2=

Ik(z
z)2


k+2
k2−𝜆4
k2

=1
kmax

1,

2+k−4𝜆

k

.
Q
f,k=Q̂
f,k=0, Qf,2k=Q̂
f,2k=
1
k
I2k
.

Tz

D2k+1F(0)(z2k+1)

(2k+1)!z2k+1−𝜆

Tz

Dk+1F(0)(zk+1)

(k+1)!zk+1
2


=1
z2k
1
kI2k(z)

=1
k

I2k(z

z

)

=max

1,

2+k−4𝜆

k

.

Tz

D2k+1F(0)(z2k+1)

(2k+1)!

z

2k+1−𝜆

Tz

Dk+1F(0)(zk+1)

(k+1)!

z

k+1
2
≤
1
kmax

1,

2+k−4𝜆

k
,

R.Długosz, P.Liczberski
1 3
while in the case
m=n
Therefore, for
𝜆∈ℂ,
we obtain the following sharp estimates
with
Tz∈(
ℂ
n)∗
,
�
�
T
z�
�=
1, T
z(
z
)=‖
z
‖,
for the families
N
G,MG=M1
G
,
̃
S∗(𝔹n)=
̃
S1(𝔹n)
,
respectively.
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