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Embedding theorems and extremalproblems for holomorphic functions oncircular domains of ℂn

Renata Długosza & Edyta Leśba Institute of Mathematics, ᴌódź University of Technology, Ul.Wólczańska 215, 90-924 ᴌódź, Poland.b Institute of Mathematics, University of Rzeszów, Al. Rejtana 16A2,, 35-310 Rzeszów, Poland.Published online: 26 Jun 2013.

To cite this article: Renata Długosz & Edyta Leś (2014) Embedding theorems and extremal

problems for holomorphic functions on circular domains of ℂn, Complex Variables and EllipticEquations: An International Journal, 59:6, 883-899, DOI: 10.1080/17476933.2013.794139

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Complex Variables and Elliptic Equations, 2014Vol. 59, No. 6, 883–899, http://dx.doi.org/10.1080/17476933.2013.794139

Embedding theorems and extremal problems for holomorphicfunctions on circular domains of C

n

Renata Długosza∗ and Edyta Lesb

aInstitute of Mathematics, Łódz University of Technology, Ul. Wólczanska 215, 90-924Łódz, Poland; bInstitute of Mathematics, University of Rzeszów, Al. Rejtana 16 A2, 35-310

Rzeszów, Poland

Communicated by J. Leiterer

(Received 1 December 2012; final version received 4 April 2013)

The paper concerns complex-valued functions which are holomorphic in boundedcomplete n-circular domains G ⊂ Cn and fulfil some geometric conditions. Thefamilies XG of such kind of functions were considered for instance by Bavrin[1,2], Dobrowolska and Liczberski [4], Dziubinski and Sitarski [5], Fukui [6],Higuchi [8], Jakubowski and Kaminski [9], Liczberski and Wrzesien [14], March-lewska [15,16], Michiwaki [17], and Stankiewicz [22]. The above functions wereapplied later to research some families of locally biholomorphic mappings in Cn

(see for instance Pfaltzgraff and Suffridge [19], Liczberski [12], Hamada, Hondaand Kohr [7]). In this paper, we consider an interesting family MS

G of the typeXG which separates two Bavrin’s families NG , RG . These families correspondto the well-known families of convex univalent and close-to-convex univalentfunctions of one variable, respectively. We define MS

G using the property of

evenness of functions. We obtain for MSG some embedding theorems relevant to

the mentioned separation question. Applying the Minkowski function of G, wesolve also some extremal problems for functions from MS

G . As an application,

we give a topologic property of the family MSG .

Keywords: holomorphic functions of several complex variables; n-circulardomains in Cn ; Minkowski function; Bavrins families; evenness property offunctions; embedding theorems; growth theorems; Taylor series developmentof functionsAMS Subject Classifications: MSC2010: 32A30; 30C45

1. Introduction

Poincare [20] pointed that the Riemann mapping theorem is false in Cn, n > 1. For thisreason it is very natural to consider the holomorphicity inCn on domains from a sufficientlywide class. In the paper, we work with bounded complete circular domains, because suchdomains play the same role for Taylor series in Cn as open discs in one dimensional case.

A domain G ⊂ Cn , n ≥ 2, containing the origin is called complete n-circular, if z� =(z1λ1, . . . , znλn) ∈ G for each z = (z1, . . . , zn) ∈ G and every � = (λ1, . . . , λn) ∈ U n ,where U is the unit disc {ζ ∈ C : |ζ | < 1}.∗Corresponding author. Email: renata.dlugosz@p.lodz.pl

© 2013 Taylor & Francis

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We use the following numerical characterization of bounded complete n- circulardomains

� = �(G) = supz=(z1,...,zn)∈G

∣∣∣∣∣∣n∑

j=1

z j

∣∣∣∣∣∣ .By HG(1) and HG, let us denote the families of all holomorphic functions f : G −→ C,

normalized by f (0) = 1 and unnormalized, respectively. Then, the Taylor series develop-ment of every function f ∈ HG(1) around zero is given by

f (z) = 1 +∞∑

k=1

Q f, k(z), z ∈ G, (1)

where Q f, k, k ∈ N, are the k−homogeneous polynomials, i.e.

Q f, k(z) =∑

α1+...+αn=k

cα1...αn zα11 . . . zαn

n , z = (z1, . . . , zn) ∈ Cn,

and the coefficients cα1...αn , α j ∈ N ∪ {0} for j = 1, . . . , n, are defined by the partialderivatives

cα1...αn = 1

α1! · . . . · αn !∂α1+...+αn f

∂zα11 . . . ∂zαn

n(0) .

Using the Minkowski function μG : Cn → [0,∞)

μG(z) = in f

{t > 0 : 1

tz ∈ G

}, z ∈ Cn,

we obtain for every bounded complete n- circular domain G and its boundary ∂G theformulas:

G = {z ∈ Cn : μG(z) < 1}, ∂G = {z ∈ Cn : μG(z) = 1}.Let us remind that μG is a seminorm in Cn for such domains and μG is a norm in Cn in thecase if G is also convex.

From this reason, we will use a generalization μG(Q f, k) of the norm of homogeneouspolynomials Q f, k . In view of the k-homogeneity of Q f, k , the formula for ∂G and themaximum principle for modulus of holomorphic functions of several variables, we can putfor k ∈ N

μG(Q f, k) = supw∈Cn\{0}

∣∣Q f, k(w)∣∣

(μG(w))k= sup

v∈∂G

∣∣Q f, k(v)∣∣ = sup

u∈G

∣∣Q f, k(u)∣∣ .

It is easy to see that for every k ∈ N, the quantity μG(Q f, k) has the following basic property∣∣Q f, k(w)∣∣ ≤ μG(Q f, k)(μG(w))k, w ∈ Cn,

which generalizes the well-known inequality

|Q f,k(w)| ≤ ||Q f,k || · ||w||k, w ∈ Cn .

By the above and in view of the fact that every complete n-circular domain is balanced, thequantities μG(z) and μG(Q f, k) will be called G-balance of the point z and G-balance ofk−homogeneous polynomials Q f, k , respectively.

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Complex Variables and Elliptic Equations 885

Analytic definitions of Bavrin’s families XG are based on a Temljakov linear operator[24] L : HG −→ HG defined by

L f (z) = f (z) + D f (z)(z), z ∈ G,

where D f (z) is the Frechet derivative of f at the point z, i.e. the row vector[

∂ f (z)∂z1

,. . .,∂ f (z)∂zn

].

It is known (see e.g. [1]) that the inverse operator to L have the form

L−1 f (z) =1∫0

f (t z)dt, z ∈ G.

Bavrin [1] considered the class NG consisting of all functions f ∈ HG(1) fulfilling thecondition

ReLL f (z)

L f (z)> 0, z ∈ G. (2)

Note that the above family has an interesting geometric characterization. For n = 2 afunction f ∈ HG(1) belongs to NG if there hold the following both conditions:

(i) the function z1 f (z1, αz1) of one variable z1 is convex univalent in the disc which isthe projection onto the plane z2 = 0 of the intersection of the domain G and everyanalytic plane

{(z1, z2) ∈ C2 : z2 = αz1

}, α ∈ C;

(ii) the function z2 f (0, z2) is convex univalent on the intersection G and the plane z1 = 0.Therefore, we say that the class NG corresponds to the well-known class Sc of holo-

morphic convex univalent functions. In his papers, I. Bavrin examined also a family RG offunctions in a way “close” to functions of the family NG . We say that f ∈ HG(1) belongsto RG if there exists ϕ ∈ NG such that

ReL f (z)

Lϕ(z)> 0, z ∈ G. (3)

The family RG corresponds (similarly as in the above) to the well-known class Scc ofclose-to-convex functions in the unit disc U (considered by Kaplan [10] and Lewandowski[11]). Bavrin showed that NG � RG and looked for a subclass MG of the class HG(1)

such that NG �MG � RG . He defined the class MG as the class containing all functionsf ∈ HG(1) which fulfil the condition

ReL f (z)

f (z)> 0, z ∈ G. (4)

From (4) it follows that the class MG corresponds (similarly as in the above) to the well-known class S∗ of holomorphic starlike univalent functions. Bavrin showed (see e.g. [1])that each of the conditions (2),(4) implies the relation f (z)L f (z) = 0 for z ∈ G. He provedalso the following higher dimensional version of the well-known Alexander’s theorem: iff ∈ NG , then L f ∈ MG and conversely, if f ∈ MG , then L−1 f ∈ NG .

Functions of the families NG , MG , RG and of other similar families were consideredalso in the papers [1,2,4–6,8,9,14–17,22]. The functions of the class MG were appliedlater to research some families of locally biholomorphic mappings in Cn (see for instance)[7,12,19].

It is very natural to ask whether there exists another interesting class MSG which

separates the classes NG,RG . In the next part of the paper, we will show that the answer ispositive.

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Our definition of MSG is based on the condition (4) and uses an even function from

HG(1). The definition of the family MSG will be preceded by an introduction.

It is easy to verify that for every complex-valued function f on a symmetric set G ⊂Cn(−1G = G), there exists exactly one function fE from the class FE (G) of even functionsand exactly one function fO from the class FO(G) of odd functions on G, such that

f = fE + fO . (5)

Moreover,

fE (z) = 1

2[ f (z) + f (−z)], fO(z) = 1

2[ f (z) − f (−z)], z ∈ G.

In view of the uniqueness of this partition, the functions fE and fO will be called the evenand odd part of f, respectively.

Note that this unique decomposition of every real-valued function f, on a symmetricsubset of Rn , onto its even and odd part was used for instance in a proof of the BorsukTheorem.[3] Many different applications of such decomposition for complex-valued func-tions can be find for instance in ref.[13].

Let us observe that every bounded complete n- circular domain G ⊂ Cn is symmetricand for every function f ∈ HG(1)

(L f )E = L( fE ). (6)

2. Relations between a family MSG and families MG, NG, RG

Before we give the definition MSG , we will prove the following theorem:

Theorem 1 Let G be a bounded complete n-circular domain in Cn. A function f ∈HG(1) belongs to MG ∩ FE (G) if

L f (z) = h(z) f (z), z ∈ G, (7)

where h ∈ CG ∩ FE (G) and

CG = { f ∈ HG(1) : Re f (z) > 0, z ∈ G}.Proof From the definition of the family MG it follows that a function f ∈ HG(1) belongsto MG if there exists a function h ∈ CG such that the relation (7) holds. Let us supposethat f ∈ MG ∩ FE (G).1 Then, in view of the uniqueness of the decomposition (5), wehave f = fE and by (6) also L f = (L f )E . Hence, the function h from (7) belongs toCG ∩ FE (G).

Supposing now that h from (7) belongs to CG ∩FE (G) and using to the functions f andL f the partition (5), we have

(L f )E + (L f )O = h fE + h fO .

Since h fE ∈ FE (G), h fO ∈ FO(G) and the decomposition (5) is unique, we have (L f )E =h fE and by (6) also

L( fE )(z) = h(z) fE (z), z ∈ G. (8)

This and the relation that fE ∈ HG(1) imply the relation fE ∈ MG .

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Complex Variables and Elliptic Equations 887

Now we will show that f = fE . The equality f (0) = fE (0) is obvious. Let usz ∈ G \ {0} be an arbitrarily fixed point. Then, its G-balance μG(z) is different from zeroand ζ z

μG(z) ∈ G for every ζ ∈ U . Therefore, and by (7), we obtain

D f

(ζ

z

μG(z)

)(ζ

z

μG(z)

)= f

(ζ

z

μG(z)

)(h

(ζ

z

μG(z)

)− 1

), ζ ∈ U.

Since for ζ ∈ U

1

ζ

(h

(ζ

z

μG(z)

)− 1

)= 1

ζ

∞∑k=1

ζ k

(μG(z))kQh,k(z)−−→

ζ→0

Qh,1(z)

μG(z),

the function U � ζ −→ 1ζ

(h(ζ z

μG(z)

)− 1)

is holomorphic. Therefore, and by the factthat f (z) = 0 for z ∈ G (see Section 1), the formula for the differential D f can be rewrittenin the form

d

dζlog f

(ζ

z

μG(z)

)= 1

ζ

(h

(ζ

z

μG(z)

)− 1

),

where the branch of the logarithm of the function of the variable ζ ∈ U is such that log 1 = 0.Integrating this equality on the segment line from ζ = 0 to ζ = μG(z), we obtain

log f (z) =∫ μG(z)

0

1

ζ

(h

(ζ

z

μG(z)

)− 1

)dζ, z ∈ G \ {0},

hence

f (z) = exp

{∫ μG(z)

0

1

ζ

(h

(ζ

z

μG(z)

)− 1

)dζ

}, z ∈ G \ {0}.

Similarly, using the fact that fE (z) = 0 for z ∈ G ( fE ∈ MG), we obtain from (8) that

fE (z) = exp

{∫ μG(z)

0

1

ζ

(h

(ζ

z

μG(z)

)− 1

)dζ

}, z ∈ G \ {0}.

The above integral formulas for f and fE imply the equality f = fE , i.e. the relationf ∈ FE (G). Thus, f ∈ MG ∩ FE (G). �

Let us remind that in the proof of the necessary condition of Theorem 1 we showedthat the assumption f ∈ MG ∩ FE (G) implies the relations f = fE ,L f = (L f )E andh ∈ CG ∩ FE (G). Therefore, the Equation (7) can be rewritten in the form (8) and in thefollowing forms

L( fE )(z) = h(z) f (z), z ∈ G, (9)

L( f )(z) = h(z) fE (z), z ∈ G (10)

with h ∈ CG ∩ FE (G) in all Equations (7)–(10).Now, we will solve a converse problem which is more general than the sufficient

condition of Theorem 1. We consider the question if the function f ∈ HG(1) fulfillingwhatever equation of (7)–(10), with h ∈ CG ∩ FE (G), belongs to MG ∩ FE (G).

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A positive answer gives the sufficient condition of Theorem 1 in the case ofEquation (7). Unfortunately, the answer is negative in the case of Equation (8). Indeed,putting

f (z) = 1 − 1

�

n∑j=1

z j , z ∈ G,

we have that f ∈ HG(1), fE = 1, L( fE ) = 1 and the function fE fulfils Equation (8) withthe function h = 1, which belongs to CG ∩ FE (G). On the other hand f /∈ MG , because

L f (◦z) = 1 − 2

�

n∑j=1

◦z j = 0,

in every point◦z ∈ G fulfilling the condition

∑nj=1

◦z j = 1

2�. This contradicts the fact thatf (z)L f (z) = 0 for every f ∈ MG and every z ∈ G.

However, we have the following result.

Theorem 2 Let G ⊂ Cn be a bounded complete n-circular domain and let f ∈ HG(1).If there exists a function h ∈ CG ∩ FE (G) such that the equality (9) or (10) holds, thenf ∈ MG ∩ FE (G).

Proof Let us consider firstly the Equation (9). Replacing here z by −z and using theevenness of both functions h,L( fE ) we obtain

L( fE )(z) = h(z) f (−z), z ∈ G.

This and (9) implies the equality

0 = h(z)( f (−z) − f (z)), z ∈ G.

Thus, f (−z) = f (z) for z ∈ G, because h(z) = 0 for z ∈ G. Hence, fO = 0 and bythe uniqueness of the partition (5) f = fE . Using this equality in (9) we obtain (8) andconsequently fE ∈ MG . This fact and (9) imply the relation f ∈ MG ∩ FE (G).

Now, let us consider the Equation (10). Replacing here z by −z and using the evennessof both functions h, fE we obtain

L( f )(−z) = h(z) fE (z), z ∈ Gand hence by (10) also

L( f )(−z) − L( f )(z) = 0, z ∈ G,

i.e. L f ∈ FE (G). Thus, applying the uniqueness of the partition (5) to the function L f ,we have that L f = (L f )E . Using this equality in (10), we obtain (9) which implies therelation fE ∈ MG . Now, let us observe that

f =(LL−1

)f = L−1(L f ) = L−1(L( fE )) =

(L−1L

)( fE ) = fE ,

because L−1L is the identity operator. Hence, f = fE and by the fact that fE ∈ MG, wehave finally the thesis f ∈ MG ∩ FE (G). �

In view of Theorem 2, a natural problem is to characterize the family MSG of all functions

f ∈ HG(1) which fulfil the Equation (10) with h ∈ CG without the assumption thath ∈ FE (G), that is all functions f ∈ HG(1), which fulfil the condition

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Complex Variables and Elliptic Equations 889

ReL f (z)

fE (z)> 0, z ∈ G, (11)

where fE is the even part of the function f.The family MS

G corresponds (in a similar sense as in the above) to the well-known classS∗∗ (see [21]) of normalized univalent functions, starlike with respect to two symmetricpoints, that is the class of all holomorphic functions F : U −→ C, F(0) = 0, F ′(0) = 1,which fulfil the condition

Reζ F ′(ζ )

FO(ζ )> 0, ζ ∈ U, (12)

where FO is the odd part of function F, i.e. FO(ζ ) = 12 (F(ζ ) − F(−ζ )), ζ ∈ U (see the

decomposition (5) for n = 1).Now, we give two embedding theorems for the family MS

G .

Theorem 3 Let G ⊂ Cn be a bounded complete n-circular domain. There holds theinclusions

(MG ∩ FE (G)) �MSG ,(MS

G ∩ FE (G))�MG . (13)

Proof The inclusions (MG ∩ FE (G)) ⊂ MSG ,(MS

G ∩ FE (G)) ⊂ MG follow from the

definition of the family MSG , from Theorem 2 and the fact that if f belongs to the set of the

left-hand side of whichever inclusion of (13), then f = fE .To prove that the equalities in (13) does not hold, we consider the function

f (z) = �

� −∑nj=1 z j

, z ∈ G. (14)

Since f ∈ HG(1) and for z ∈ G

L f (z) = �2(� −∑n

j=1 z j

)2, fE (z) = �2

�2 −(∑n

j=1 z j

)2,

we have that

ReL f (z)

fE (z)= Re

� +∑nj=1 z j

� −∑nj=1 z j

> 0, z ∈ G

and consequently, f ∈ MSG . On the other hand, f ∈ MG , because f ∈ NG (see e.g.

[1]). Hence, f belongs to every of the sets on the right-hand side of both inclusions (13).However, f /∈ MG ∩ FE (G) and f /∈ MS

G ∩ FE (G), because f /∈ FE (G)( f = fE ). Theproof is complete. �

From Theorem 3 it follows.

Corollary There holds the following equality

MSG ∩ FE (G)= MG ∩ FE (G).

The Corollary, Theorems 1 and 2 imply the following observation.

Observation 1 A function f ∈ HG(1) belongs to MSG ∩FE (G) if there exists a function

h ∈ CG ∩ FE (G) such that whichever of the equalities (7),(9) and (10) holds.

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890 R. Długosz and E. Les

Theorem 4 Let G ⊂ Cn be a bounded complete n-circular domain. There hold theinclusions

NG �MSG � RG . (15)

Proof We start with the proof of the inclusion MSG ⊂ RG . First, let us observe that if

f ∈ MSG , then fE ∈ MG . Indeed, we have that fE ∈ HG(1) and, by (6) and (11), we

obtain for z ∈ G

ReL( fE )(z)

fE (z)= 1

2

(Re

L f (z)

fE (z)+ Re

L f (−z)

fE (−z)

)> 0.

Hence, fE ∈ MG . From this, in view of Bavrin’s version of Alexander’s theorem (see theintroduction), the function ϕ = L−1( fE ) belongs to NG . Since simultaneously Lϕ = fE ,we can rewrite the inequality (11) in the form (3) with ϕ ∈ NG . Thus, f ∈ RG andMS

G ⊂ RG .Now, we will show that MS

G = RG . To achieve this, let us consider the function

f (z) = �2(� −∑n

j=1 z j

)2, z ∈ G.

This function belongs to MG (see [1]), hence it belongs also to RG , because MG ⊂ RG .We will prove that f /∈ MS

G . To do it, let us observe that for z ∈ G

L f (z) =�2(� +∑n

j=1 z j

)(� −∑n

j=1 z j

)3, fE (z) =

�2(

�2 +(∑n

j=1 z j

)2)

(�2 −

(∑nj=1 z j

)2)2

and consequently

L f (z)

fE (z)=

(� +∑n

j=1 z j

)3

(� −∑n

j=1 z j

)(�2 +

(∑nj=1 z j

)2) , z ∈ G.

Hence,

ReL f ( z )

fE ( z )= r4 − 6r2 + 1

1 − r4< 0

at every point z ∈ G such that∑n

j=1 z j = ri� and r ∈(√

3 − 2√

2, 1)

. Thus by (11),

f /∈ MSG .

Now, we will prove that NG ⊂ MSG . To do it, let f ∈ NG and z ∈ G�{0} be arbitrarily

fixed. It is obvious that the function F : U −→ C

F(ζ ) = ζ f

(ζ

z

μG(z)

), ζ ∈ U (16)

is holomorphic and normalized by the conditions F(0) = 0, F ′(0) = 1. Moreover forζ ∈ U

F ′(ζ ) = L f

(ζ

z

μG(z)

), F ′(ζ ) + ζ F ′′(ζ ) = LL f

(ζ

z

μG(z)

).

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Complex Variables and Elliptic Equations 891

The above equalities, the assumption that f ∈ NG and (2) give the inequality

Re

(1 + ζ F ′′(ζ )

F ′(ζ )

)> 0, ζ ∈ U,

which is a sufficient condition for the relation F ∈ Sc. Consequently, by a result ofSuffridge,[23] we obtain the relation

Re

{2ζ F ′(ζ )

F(ζ ) − F(ξ)− ζ + ξ

ζ − ξ

}≥ 0, ζ, ξ ∈ U,

which implies the inequality

Re2ζ F ′(ζ )

F(ζ ) − F(−ζ )≥ 0, ζ ∈ U.

This and the extremum principle for harmonic functions give inequality (12). Hence andby the formula (16), we have

ReζL f

(ζ z

μG(z)

)ζ fE

(ζ z

μG(z)

) > 0, ζ ∈ U,

because the odd part FO of the function F has the form FO(ζ ) = ζ fE

(ζ z

μG(z)

). Putting

in the above inequality ζ = μG(z) ∈ (0, 1), we obtain the inequality (11) at all pointsz ∈ G�{0}. This and the obvious inequality ReL f (0)

fE (0)> 0 give the relation f ∈ MS

G .

It remains to prove that NG = MSG . To do it let us consider the function

f (z) = �2

�2 −(∑n

j=1 z j

)2, z ∈ G.

For this function, fE = f , because f is even. Since simultaneously

L f (z) =�2(

�2 +(∑n

j=1 z j

)2)

(�2 −

(∑nj=1 z j

)2)2

, z ∈ G,

we have that

ReL f (z)

fE (z)= Re

�2 +(∑n

j=1 z j

)2

�2 −(∑n

j=1 z j

)2> 0, z ∈ G,

which implies the relation f ∈ MSG . However f /∈ NG . Indeed Since

LL f (z) =�2(

�4 +(∑n

j=1 z j

)4 + 6�2(∑n

j=1 z j

)2)

(�2 −

(∑nj=1 z j

)2)3

, z ∈ G,

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we have

ReLL f (z)

L f (z))= Re

⎧⎪⎨⎪⎩�4 +

(∑nj=1 z j

)4 + 6�2(∑n

j=1 z j

)2

�4 −(∑n

j=1 z j

)4

⎫⎪⎬⎪⎭ , z ∈ G.

Thus, ReLL f (z)L f (z)) = −1 < 0 at all points z ∈ G such that

∑nj=1 z j = �

√3

3 i . Therefore, thecondition (2) does not hold and consequently f /∈ NG . The proof is complete. �

3. Extremal problems in MSG

We start with estimates of G-balances of k−homogeneous polynomials Q f,k in MSG .

Theorem 5 Let G ⊂ Cn be a bounded complete n-circular domain and let f ∈ MSG . If

the expansion of the function f into a series of homogenous polynomials Q f,k has the form(1), then for the G−balances μG(Q f,k) of polynomials Q f,k there hold the following sharpbounds

μG(Q f,k) ≤ 1, k ∈ N. (17)

Proof By the definition of the family MSG , there exists a function h ∈ CG , such that f

fulfils the Equation (10). Putting into (10) the development

h(z) =∞∑

k=0

Qh, k(z), Qh, 0 = 1, z ∈ G

of the above function h ∈ CG and the developments

L f (z) =∞∑

k=0

(k + 1)Q f, k(z), fE (z) =∞∑

k=0

Q f, 2k(z), Q f, 0 = 1, z ∈ G

of L f and of the even part fE of the above f ∈ MSG , we obtain for k ∈ N the recursive

formula

(k + 1)Q f, k(z) =

⌊12 k⌋∑

l=0

Q f, 2l(z)Qh, k−2l(z), z ∈ G,

where �x� means the integral part of x.Now for every k ∈ N, by mathematical induction, we show the relation

∣∣Q f, k(z)∣∣ ≤

1, z ∈ G.Step 1 Using the fact that

∣∣Qh, m(z)∣∣ ≤ 2 for h ∈ CG and m ∈ N (see e.g. [1]) and

choosing k = 1 in the above recursive formula, we obtain that

2∣∣Q f, 1(z)

∣∣ = ∣∣Q f, 0(z)∣∣ ∣∣Qh, 1(z)

∣∣ ≤ 2, z ∈ G.

Hence,∣∣Q f, 1(z)

∣∣ ≤ 1 for z ∈ G.

Step 2 We will show that for every k ∈ N there holds the following implication∣∣Q f, 1(z)∣∣ ≤ 1, . . . ,

∣∣Q f, k(z)∣∣ ≤ 1, z ∈ G =⇒ ∣∣Q f, k+1(z)

∣∣ ≤ 1, z ∈ G.

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Complex Variables and Elliptic Equations 893

To do it, let us fix arbitrarily k ∈ N and observe that from the above recursive formula bythe induction supposition and the estimates

∣∣Qh, m(z)∣∣ ≤ 2 for h ∈ CG and m ∈ N, we

obtain

∣∣Q f, k+1(z)∣∣ ≤ 1

k + 2

⌊12 (k+1)

⌋∑l=0

∣∣Q f, 2l(z)∣∣ ∣∣Qh, k−2l+1(z)

∣∣ ≤ 2

k + 2

(1

2k + 1

)≤ 1, z ∈ G.

The principle of the mathematical induction completes this part of the proof, by 1 and 2.To obtain the inequality (17), it suffices to use the definition of the G-balance of Q f,k .It remains to show the sharpness of estimation (17). Let us observe that the function f

given by (14) is an extremal function. Indeed, it belongs to MSG (see the proof of Theorem

3) and the homogeneous polynomials Q f,k in its development (1), have the form

Q f,k(z) =⎛⎝ 1

�

n∑j=1

z j

⎞⎠k

, z ∈ G,

hence

μG(Q f, k) = supu∈G

∣∣∣∣∣∣ 1

�

n∑j=1

z j

∣∣∣∣∣∣k

= 1

�k�k = 1.

�Next, we give two results of the type “growth theorems” in MS

G .

Theorem 6 Let G ⊂ Cn be a bounded complete n-circular domain and μG its Minkowskifunction. If f ∈ MS

G , then there hold the following sharp estimates

1

(1 + r)2≤ |L f (z)| ≤ 1

(1 − r)2, μG(z) ≤ r ∈ [0, 2 − √

3],

4(1 − r2)

3√

3(1 + r2)2≤ |L f (z)| ≤ 1

(1 − r)2, μG(z) ≤ r ∈ [2 − √

3, 1),

Proof The first bound is obvious for r = 0. Let us fix arbitrarily r ∈ (0, 1) and z ∈ Gwith μG(z) = r . Then, the function F : U −→ C defined in (16) belongs to S∗∗ andF ′(ζ ) = L f

(ζ z

μG(z)

)for ζ ∈ U (see the proof of Theorem 3). Therefore, using the

bounds of |F ′(ζ )| in the class S∗∗ (see [18]), we obtain

1

(1 + |ζ |)2≤∣∣∣∣L f

(ζ

z

μG(z)

)∣∣∣∣ ≤ 1

(1 − |ζ |)2, |ζ | ∈ [0, 2 − √

3],

4(1 − |ζ |2)3√

3(1 + |ζ |2)2≤∣∣∣∣L f

(ζ

z

μG(z)

)∣∣∣∣ ≤ 1

(1 − |ζ |)2, |ζ | ∈ [2 − √

3, 1).

Putting in the above ζ = μG(z) = r and applying the maximum principle for modulus ofholomorphic functions of several variables, we obtain the required bounds.

In order to prove the sharpness of the upper estimations, let us consider the function

f (z) = �

� − eiα∑n

j=1 z j, z ∈ G,

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with arbitrarily fixed α ∈ R. Then

L f (z) = �2(� − eiα

∑nj=1 z j

)2, z ∈ G.

Similarly, as in the case of the function defined by the formula (14), we have that f ∈ MSG .

If we choose a point z ∈ G�{0}, μG( z ) = r ∈ (0, 1) and α = α ∈ R fulfilling theequalities

�μG( z ) =∣∣∣∣∣∣

n∑j=1

z j

∣∣∣∣∣∣ = ei αn∑

j=1

z j ,

then

|L f ( z )| =∣∣∣∣ �2

(� − �μG( z ))2

∣∣∣∣ = 1

(1 − r)2.

This proves the sharpness of both upper bounds.In order to show the sharpness of the lower bound for r ∈ [0, 2 − √

3] it suffices, toconsider the same function f and choose a point z ∈ G�{0}, μG( z ) = r ∈ (0, 2 − √

3]and α = α ∈ R such that there hold the equalities

�μG( z ) =∣∣∣∣∣∣

n∑j=1

z j

∣∣∣∣∣∣ = −ei αn∑

j=1

z j .

Now, we will show the sharpness of the lower bound for r ∈ [2 − √3, 1). To obtain this,

for α ∈ R, b ∈ [−1, 1] and A ={

z ∈ G :∑nj=1 z j = 0

}, let us consider the following

function

f (z) =

⎧⎪⎪⎨⎪⎪⎩�

2beiα∑n

j=1 z j

⎛⎝√√√√�2+2�beiα∑n

j=1 z j +(

eiα∑n

j=1 z j

)2

�2−2�beiα∑n

j=1 z j +(

eiα∑n

j=1 z j

)2 − 1

⎞⎠1

,z ∈ G�Az ∈ A

.

First, let us observe that f ∈ HG(1), because it can be extended holomorphically fromG�A onto G (the set A is closed and nowhere dense in G). Now we check that

L f (z)=�2(

�2 −(

eiα∑nj=1 z j

)2)

(�2 − 2�beiα∑n

j=1 z j

)√(�2 +

(eiα∑n

j=1 z j

)2)2

− 4�2b2(

eiα∑nj=1 z j

)2, z ∈G,

fE (z) = �2√(�2 +

(eiα∑n

j=1 z j

)2)2

− 4�2b2(

eiα∑nj=1 z j

)2, z ∈ G.

Therefore,

ReL f (z)

fE (z)= Re

�2 −(

eiα∑nj=1 z j

)2

�2 − 2�beiα∑n

j=1 z j +(

eiα∑n

j=1 z j

)2> 0, z ∈ G

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Complex Variables and Elliptic Equations 895

(we obtain the above inequality by replacing ζ by eiα

�

∑nj=1 z j in the well-known relation

Re 1−ζ 2

1−2bζ+ζ 2 > 0, ζ ∈ U ). Consequently, f ∈ MSG .

Now, choosing a point z ∈ G�{0}, μG( z ) = r ∈ [2 − √3, 1) and α = α ∈ R such

that

�μG( z ) =∣∣∣∣∣∣

n∑j=1

z j

∣∣∣∣∣∣ = −ei αn∑

j=1

z j

and putting b = 1+r2

4r , we have

|L f ( z )| = 4(1 − r2)

3√

3(1 + r2)2,

that is the equality in the lower bound for r ∈ [2 − √3, 1). �

Theorem 7 Let G be a bounded complete n-circular domain inCn. If f ∈ MSG , then the

following estimates hold

1

1 + r≤ | f (z)| ≤ 1

1 − r, μG(z) ≤ r ∈ [0, 2 − √

3],1

2r− 5

6√

3r+ 4

3√

3(1 + r2)< | f (z)| ≤ 1

1 − r, μG(z) ≤ r ∈ (2 − √

3, 1),

where μG is the Minkowski function of domain G. The both upper estimates and the lowerestimation in the first case are sharp.

Proof The proof of the above estimates runs similarly as the proof of the estimates in theprevious theorem.

Now, we prove the sharpness of the both upper estimates and the first lower estimation.To do it, let us consider the function

f (z) = �

� − eiα∑n

j=1 z j, z ∈ G, α ∈ R

belonging to the family MSG (see the first part of the proof of the sharpness in the previous

theorem). If we choose a point z ∈ G�{0}, μG( z ) = r ∈ (0, 1) and α = α ∈ R such that

�μG( z ) =∣∣∣∣∣∣

n∑j=1

z j

∣∣∣∣∣∣ = ei αn∑

j=1

z j ,

then

| f ( z )| =∣∣∣∣ �

(� − �μG( z ))2

∣∣∣∣ = 1

1 − r.

Hence, we obtain the equalities in the upper bounds.On the other hand, choosing a point z ∈ G, μG( z ) = r ∈ (0, 2 − √

3] and α = α suchthat

�μG( z ) =∣∣∣∣∣∣

n∑j=1

z j

∣∣∣∣∣∣ = −ei αn∑

j=1

z j ,

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we obtain

| f ( z )| =∣∣∣∣ �

(� + �μG( z ))2

∣∣∣∣ = 1

1 + r.

Hence, we have the equality in the lower bound for r ∈ (0, 2 − √3].

The above estimation for | f (z)| , r ∈ (2 − √3, 1) is not sharp. �

4. A topologic property of the family MSG

We start with the following observation.

Observation 2 The family MSG is not convex.

Proof Let us consider the function f = 12 (g + h), where

g(z) = �2

�2 −(∑n

j=1 z j

)2, h(z) = �2

�2 +(∑n

j=1 z j

)2, z ∈ G.

Then, g, h ∈ MSG (see the second part of the proof of Theorem 4) and

f (z) = �4

�4 −(∑n

j=1 z j

)4, L f (z) =

�8 + 3�4(∑n

j=1 z j

)4

(�4 −

(∑nj=1 z j

)4)2

, z ∈ G.

Since fE = f , we have from the above

L f (z)

fE (z)=

�4 + 3�4(∑n

j=1 z j

)4

�4 −(∑n

j=1 z j

)4, z ∈ G.

Therefore, choosing a point z ∈ G, μG( z ) = r ∈(

13 , 1)

such that

n∑j=1

z j = �4√−r ,

we have L f ( z )

fE ( z )= 1 − 3r

1 + r< 0.

Thus, the condition (11) does not hold and consequently, f /∈ MSG . Hence, MS

G isnon-convex. �

However MSG has the following property:

Theorem 8 The family MSG is a path connected set, hence a connected set.

Proof In order to prove the path connectivity of MSG , it suffices to show that for every

functions f , f ∈ MSG there exits a one-parameter family of functions f ∈ MS

G , ∈ [a, b],

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Complex Variables and Elliptic Equations 897

such that fa = f , fb = f and for every sequence of numbers υ ∈ [a, b], 1 = a,limν→∞ υ = b an adequate sequence of functions fυ ∈ MS

G converges almost uniformlyto f on the set G. To do it, let us consider firstly the family of functions f, ∈ [0, 1],defined by f ∈ MS

G as follows

f(z) = f (z), z ∈ G.

Then, f0 = 1 and f1 = f.Using for every r ∈ (0, 1) the relation rG = {z ∈ Cn : μG(z) ≤ r} ⊂ G, we will show

that

∀ε > 0∃δ > 0∀1, 2 ∈ [0, 1]∀z ∈ rG {|2 − 1| < δ =⇒ ∣∣ f2(z) − f1(z)∣∣ < ε

}.

(18)To this aim, let us observe that from the definition of f(z) and by the form of the

development of the function f onto a series of k− hogeneous polynomials, we have forz ∈ rG∣∣ f2(z) − f1(z)

∣∣ = ∣∣∣∣∣∞∑

k=1

Q f, k(2z) −∞∑

k=1

Q f, k(1z)

∣∣∣∣∣ ≤∞∑

k=1

∣∣∣k2 − k

1

∣∣∣ ∣∣Q f, k(z)∣∣ .

Therefore, by the basic property of G−balance μG(Q f, k), its estimate (7) for f ∈ MSG

and the inequality∣∣k

2 − k1

∣∣ ≤ k |2 − 1|, we have

∣∣ f2(z) − f1(z)∣∣ ≤ |2 − 1|

∞∑k=1

krk = |2 − 1| r

(1 − r)2.

From this there follows the condition (18) with δ = ε(1−r)2

r .Now, we will show that for every sequence of numbers υ ∈ [0, 1], 1 = 0,

limν→∞ υ = 1, an adequate sequence of functions fυ ∈ MSG converges almost

uniformly to f on the set G. In order to do it, let us consider an arbitrarily fixed compact setG ⊂ G. Then, there exists an r ∈ (0, 1) such that G ⊂ rG ⊂ G. Thus, by (18) there existsthe limit

limν→∞ fν (z) = f1(z) = f (z), z ∈ G

and the convergence is uniform in G. Hence, by the arbitrariness of G, almost uniform inG.

Similarly, for every sequence of numbers υ ∈ [0, 1], 1 = 1, limν→∞ υ = 0, anadequate sequence of functions fυ ∈ MS

G converges almost uniformly on G to the functionf0 = 1.

Finally, for every functions f , f ∈ MSG the announced one-parameter family of

functions f ∈ MSG , ∈ [0, 1] is the union of the following families

f(z) = f ((1 − 2)z), ∈[

0,1

2

], f0 = f , f 1

2= 1

f(z) = f ((1 − 2)z), ∈[

1

2, 1

], f 1

2= f , f1 = f .

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The family constructed above joins (in the sense of almost uniformly convergence) thefunctions f , f trough the function f = 1.

This gives the path connectivity of the family MSG and as result, the connectivity of

MSG . �

Note1. In view of the normalization f (0) = 1 in MG the case if f ∈ MG ∩ FO (G) is impossible.

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