Abstract
The paper concerns investigations of holomorphic functions of several complex variables with a factorization of their Temljakov transform. Firstly, there were considered some inclusions between the families \(\mathcal {C}_{\mathcal {G}},\mathcal {M}_{\mathcal {G}},\mathcal {N}_{\mathcal {G}},\mathcal {R}_{\mathcal {G}},\mathcal {V}_{\mathcal {G}}\) of such holomorphic functions on complete n-circular domain \(\mathcal {G}\) of \(\mathbb {C}^{n}\) in some papers of Bavrin, Fukui, Higuchi, Michiwaki. A motivation of our investigations is a condensation of the mentioned inclusions by some new families of Bavrin’s type. Hence we consider some families \(\mathcal {K}_{ \mathcal {G}}^{k},k\ge 2,\) of holomorphic functions f : \(\mathcal {G}\rightarrow \mathbb {C},f(0)=1,\) defined also by a factorization of \( \mathcal {L}f\) onto factors from \(\mathcal {C}_{\mathcal {G}}\) and \(\mathcal {M} _{\mathcal {G}}.\) We present some interesting properties and extremal problems on \(\mathcal {K}_{\mathcal {G}}^{k}\).
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1 Introduction
We say that a domain \(\mathcal {G}\subset \mathbb {C}^{n}\), is complete n-circular if \(z\lambda =(z_{1}\lambda _{1},\ldots ,z_{n}\lambda _{n})\in \mathcal {G}\) for each \(z=(z_{1},\ldots ,z_{n})\in \mathcal {G}\) and every \( \lambda =(\lambda _{1},\ldots ,\lambda _{n})\in \overline{\mathcal {U}^{n}}\), where \(\mathcal {U}\) is the unit disc \(\{\zeta \in \mathbb {C}:|\zeta |<1\}\). From now by \(\mathcal {G}\) will be denoted a bounded complete n-circular domain in \(\mathbb {C}^{n},n\ge 2.\) By \(\mathcal {H}_{\mathcal {G}}\) let us denote the space of all holomorphic functions \(f:\mathcal {G} \longrightarrow \mathbb {C}\) and by \(\mathcal {H}_{\mathcal {G}}(1)\) the collection of all \(f\in \mathcal {H_{G}}\), normalized by \(f(0)=1.\)
Many authors (cf., eg., [1, 2, 5,6,7, 11, 18, 19, 23]) considered some Bavrin’s subfamilies \(\mathcal {X_{G}}\) of the family \(\mathcal {H_{G}} (1).\) In the definitions of these families \(\mathcal {X_{G}}\) the main role play the families \(\mathcal {C_{G}}(\alpha ),\alpha \in [0,1),\)
and the following invertible Temljakov [24] linear operator \(\mathcal {L}:\mathcal {H}_{\mathcal {G}}\longrightarrow \mathcal {H}_{\mathcal {G}}\)
where Df(z) is the Fréchet derivative of f at the point z. By a Bavrin’s family \(\mathcal {X_{G}}\) we mean a collection of functions \(f\in \mathcal {H_{G}}(1)\) whose the Temljakov transform \(\mathcal {L}f\) has a functional factorization \(\mathcal {L}f=p\cdot g\), where \(p\in \mathcal {C_{G}} \equiv \mathcal {C_{G}}(0)\) and g is from a fixed subfamily of \(\mathcal { H_{G}}(1).\) Below, we recall the factorizations which define a few well known Bavrin’s families \(\mathcal {X_{G}},\) like as
It is known that functions of these families were used to construct biholomorphic mappings in \(\mathbb {C}^{n}\) (cf., eg., [10, 13, 20]). Let us note that the above families have geometric interpretation, in particular the functions \(f\in \mathcal {M}_{\mathcal {G}}\) map biholomorphically some planar intersections \(\mathcal {S}\) of \(\mathcal {G}\) onto starlike domains in \(\mathbb {C}\),(see [1]). It is very important, because the starlikeness plays a central role in many different subjects of geometry and topology and in particular, in geometric function theory.
Let us recall also that Bavrin showed the inclusions \(\mathcal {N}_{\mathcal {G}}\varsubsetneq \mathcal {R}_{\mathcal {G}},\mathcal {V}_{\mathcal {G}}\varsubsetneq \mathcal {R}_{\mathcal {G}}\) and pointed that the first of them can be complete to the following double inclusion \(\mathcal {N}_{\mathcal {G} }\varsubsetneq \mathcal {M}_{\mathcal {G}}\varsubsetneq \mathcal {R}_{\mathcal {G}}\). Thus, it is natural to ask whether is possible to do the same in the case of the second above inclusion. In the paper [12] the authors defined a family \(\mathcal {K}_{\mathcal {G}}^{-},\) which satisfies the inclusion \(\mathcal {V}_{\mathcal {G}}\varsubsetneq \mathcal {K}_{\mathcal {G} }^{-}\varsubsetneq \mathcal {R}_{\mathcal {G}}\). An adequate definition of \(\mathcal {K}_{\mathcal {G}}^{-}\) has the form: A function \(f\in \mathcal {H_{G}}(1)\) belongs to \(\mathcal {K}_{\mathcal {G}}^{-}\) if its Temljakov transform \( \mathcal {L}f\) has the factorization
where the family \(\mathcal {M}_{\mathcal {G}}(\alpha ),\alpha \in [0,1), \) is defined similarly as \(\mathcal {M}_{\mathcal {G}},\) but in this case \(p\in \mathcal {C_{G}}(\alpha )\).
In the present paper we consider Bavrin’s type families \(\mathcal {K}_{ \mathcal {G}}^{k},k\ge 2(\mathcal {K}_{\mathcal {G}}^{2}=\mathcal {K}_{\mathcal { G}}^{-})\) separating also the families \(\mathcal {V}_{\mathcal {G}},\mathcal {R }_{\mathcal {G}},\) i.e., satisfying the inclusions \(\mathcal {V}_{\mathcal {G}}\varsubsetneq \mathcal {K}_{\mathcal {G}}^{k}\varsubsetneq \mathcal {R}_{ \mathcal {G}},k\ge 2.\)
The formal definition of such family has the following form.
A function \(f\in \mathcal {H_{G}}(1)\) belongs to \(\mathcal {K}_{\mathcal {G}}^{k}\) if there exist a function \(p\in \mathcal {C_{G}}\) and a function \(h\in \mathcal {M}_{\mathcal {G}}(\frac{k-1}{k})\) such that the Temljakov transform \(\mathcal {L}f\) of f has the factorization
where \(\varepsilon =\varepsilon _{k}=\exp \frac{2\pi i}{k}\) is a generator of the cyclic group of kth roots of unity.
Let us observe that \(\mathcal {K}_{\mathcal {G}}^{k},k\ge 2\) are nonempty families. Indeed, the function \(f=1\) belongs to \(\mathcal {K}_{\mathcal {G}}^{k},\) because it satisfies the factorization (1.1) with \(p=1\in \mathcal {C_{G}}\) and \(h=1\in \mathcal {M}_{\mathcal {G}}(\frac{k-1}{k)}).\)
In the future, we will use a characterization of the family \(\mathcal {K}_{ \mathcal {G}}^{k}\) by a notion of \(\left( j,k\right) \)-symmetry, which is connected with a functional decomposition with respect to the above group.
Let us observe that bounded complete n-circular domains \(\mathcal {G}\) are k-symmetric sets for \(k=2,3,\ldots ,\) that is \(\varepsilon \mathcal {G=G}\). For \( j=0,1,\ldots ,k-1\) we define the collections \(\mathcal {F}_{j,k}(\mathcal {G})\) of functions \(\left( j,k\right) \)-symmetrical, i.e., all functions \(f:\mathcal {G}\rightarrow \mathbb {C}\) such that
If \(n=1\) and \(\mathcal {G}=\mathcal {U},\) then we write \(\mathcal {F}_{j,k}( \mathcal {U})\).
The mentioned functional decomposition appears in the following result from [14].
Theorem A
For every function \(f:\mathcal {G}\rightarrow \mathbb {C}\) there exists exactly one sequence of functions \(f_{j,k}\in \mathcal {F}_{j,k}(\mathcal {G} ), \) \(j=0,1,\ldots ,k-1,\) such that
Moreover,
The functions \(f_{j,k},\) which are uniquely determined by the above decomposition, will be called \(\left( j,k\right) \)-symmetrical components of the function f. Some interesting applications of the above partition may also be found in [15, 16] and [17].
2 Results
Now we can present a characterization of \(f\in \mathcal {K}_{\mathcal {G}}^{k}, \) simpler than (1.1).
Theorem 1
A function \(f\in \mathcal {H_{G}}(1)\) belongs to the family \( \mathcal {K}_{\mathcal {G}}^{k},k\ge 2\) if and only if there exists a function \(g\in \mathcal {M}_{\mathcal {G}}\cap \mathcal {F}_{0,k}(\mathcal {G})\) and a function \(p\in \mathcal {C_{G}}\) such that
Proof
Let \(f\in \) \(\mathcal {K}_{\mathcal {G}}^{k}.\) Then there exists \(p\in \mathcal {C_{G}}\) and \(h\in \mathcal {M}_{\mathcal {G}}(\frac{k-1}{k})\) such that
where
It is obvious that \(g\in \mathcal {F}_{0,k}(\mathcal {G})\). We show that \(g\in \mathcal {M}_{\mathcal {G}}.\) To do it, using the differentiation product rule and the form of the operator \(\mathcal {L},\) we have at \(z\in \mathcal {G}\)
Hence and by the fact that \(h\in \mathcal {M}_{\mathcal {G}}(\frac{k-1}{k}),\) we obtain that \({\text {Re}}\frac{\mathcal {L}g(z)}{g(z)}>1-k+k\frac{k-1}{k}=0.\) Thus \(g\in \mathcal {M}_{\mathcal {G}}.\)
Now, let us suppose that f satisfies the equality (2.1), with a \( p\in \mathcal {C_{G}}\) and a \(g\in \mathcal {M}_{\mathcal {G}}\cap \mathcal {F} _{0,k}(\mathcal {G}).\) Let us put \(h(z)=\left( g(z)\right) ^{\frac{1}{k}},z\in \mathcal {G},\) with the power function taking the value 1 at the point 1. Since \(g(z)\ne 0\) (see [1]), the function h is holomorphic. It remains to show that \(h\in \mathcal {M}_{\mathcal {G}}(\frac{ k-1}{k})\) and the equality (1.1) is fulfilled. To this end we compute step by step
The formula (1.1) follows from the definition of the function h. Indeed,
because \(h\in \mathcal {F}_{0,k}(\mathcal {G}).\)
The proof is complete. \(\square \)
Now we consider an extremal problem for \(f\in \mathcal {K}_{\mathcal {G}}^{k}.\) More precisely, we look for some estimates for \(\mathcal {G}\)-balances of m -homogeneous polynomias \(Q_{f,m}\) of its unique power series expansion
In our considerations the Minkowski function
will be very useful. This function gives a possibility to redefine the domain \(\mathcal {G}\) and its boundary \(\partial \mathcal {G}\) as follows:
The notion of \(\mathcal {G}\)-balance of m-homogeneous polynomial \(Q_{m}: \mathbb {C}^{n}\rightarrow \mathbb {C},\) \(m\in \mathbb {N\cup }\{0\},\) was defined in [3] as the quantity
The \(\mathcal {G}\)-balance \(\mu _{\mathcal {G}}(Q_{m})\) generalizes the norm \(\left\| Q_{m}\right\| \) of the polynomial \(Q_{m}\) and if \(\mathcal {G}\) is convex , then \(\mu _{\mathcal {G}}(Q_{m})\) reduces to \(\left\| Q_{m}\right\| ,\) because
and for bounded convex complete n-circular domains \(\mathcal {G}\) also \(\mu _{\mathcal {G}}(w)=||w||\) (see, e.g., [21]).
We present the announced estimates of \(\mathcal {G}\)-balances \(\mu _{\mathcal { G}}(Q_{f,m})\) of m-homogeneous polynomials \(Q_{f,m}\) from the Taylor series of \(f\in \mathcal {M}_{\mathcal {G}}^{k}\) in the following theorem.
Theorem 2
If the expansion of the function \(f\in \mathcal {K}_{\mathcal {G} }^{k}\), \(k\ge 2,\) into a series of m-homogenous polynomials \( Q_{f,m}\) has the form (2.2), then for the \(\mathcal {G}\)-balances \(\mu _\mathcal {G}(Q_{f,m})\) of polynomials \(Q_{f,m}\) the following sharp estimate hold:
where \(\left\lfloor q\right\rfloor \) means the integral part of the number q. We use a standard convention that the product \(\prod \limits _{l=l_{1}}^{l_{2}}a_{l}\) is equal to 1 for \(l_{2}<l_{1}\).
Proof
Let \(f\in \) \(\mathcal {K}_{\mathcal {G}}^{k}\) be arbitrarily fixed. Then, by Theorem 1, the factorization (2.1) holds with a function \(p\in \mathcal {C_{G}}\) of the form
and a function \(g\in \mathcal {M}_{\mathcal {G}}\cap \mathcal {F}_{0,k}( \mathcal {G})\) of the form
From the above, by the series expansion of \(\mathcal {L}f\)
and by the equalities \(Q_{f,0}=Q_{p,0}=Q_{g,0}=1,\) we obtain the recursive formula for \(m\in \mathbb {N}\)
Hence
Since
(see [1]) we need some bounds for \(\left| Q_{g,k\mu }(z)\right| \). We show that for \(g\in \mathcal {M}_{\mathcal {G}}\cap \mathcal {F}_{0,k}(\mathcal {G})\) and \(\mu \in \mathbb {N}\) there hold the inequalities
For this purpose let us observe that for each \(z\in \mathcal {G},\) the function
belongs to the family \(\mathcal {S}^{*}\cap \mathcal {F}_{1,k}(\mathcal {U} ) \) of (1, k)-symmetric univalent starlike mappings (in the unit disc \( \mathcal {U)}\) and its Taylor series has the form
Thus, in view of the estimates [25] of the coefficients of functions from \(\mathcal {S}^{*}\cap \mathcal {F}_{1,k}(\mathcal {U})\) we get the announced bounds (2.6).
In two next parts of the proof we use also the fact [4] that for every k, \(s\in \mathbb {N{\setminus } }\left\{ 1\right\} \) there holds the identity:
Now, we will estimate the quantities \(\left| Q_{f,m}(z)\right| ,z\in \mathcal {G},\) using all the conditions (2.4)-(2.7).
First let us assume that \(m=ks,\) where \(s\in \mathbb {N}.\) Since \( Q_{p,m-kl}(z)=1\) for \(l=s,\) we get from (2.4) that
Thus for \(z\in \mathcal {G}\), in view of (2.6) and (2.7),
Hence, for \(m=k,2k,3k,\ldots \)
Now let us consider the case \(m=ks+r,\) where \(s\in \mathbb {N}\cup \{0\}\) and \(r\in \{1,2,\ldots ,k-1\}.\) In this case we apply in (2.4) the inequality \( \left| Q_{p,m-kl}(z)\right| \le 2,l=0,\ldots ,s=\left\lfloor \frac{m}{k} \right\rfloor ,\) which follows from estimates (2.5), because \(m-kl>0\). Thus, in view of (2.6) and (2.7) we get step by step
Summing up the results of both cases we get
and consequently
These inequalities and the definition of \(\mathcal {G}\)-balances \(\mu _{ \mathcal {G}}(Q_{f,m})\) of m-homogeneous polynomials imply the estimates from the statement of the theorem.
Now, we will show the sharpness of the above estimates.
For the linear functional \(I=\left( \mu _{\mathcal {G}}(J)\right) ^{-1}J,\) with
let us denote by \(\mathcal {Z}\) an analytic set \(\mathcal {\ G}\cap \, I^{-1}\{0\}\) and let \(I^{m}(z)=\left( Iz\right) ^{m},z\in \mathcal {G},m\in \mathbb {N}\cup \{0\}.\) The equalities in our estimates are achieved for the following function \(f\in \mathcal {K}_{\mathcal {G}}^{k},\,k\ge 2,\)
where \(H(a,b,c,\zeta ):\mathcal {U}\rightarrow \mathbb {C}\) is a hypergeometric function
defined by Pochhamer symbols \(\ \left( a\right) _{\nu },\left( b\right) _{\nu },\left( c\right) _{\nu }:\)
and the branch of the power function \(x^{\frac{2}{k}}\) takes the value 1 at the point \(x=1.\) In the case \(k=2,3\) we use a standard convention that the sum
is equal to zero, if the superscript of the sum is smaller than the subscript.
In the paper [4], it was proven that the above function gives the equalities in the bounds from the statement of the theorem. It remains to show that \(f\in \mathcal {K}_{\mathcal {G}}^{k}\) for \(k\ge 2.\) To do it, let us observe that as shown in [4]
This implies, in view of Theorem 1, the relation \(f\in \mathcal {K}_{\mathcal { G}}^{k},\) because the functions
belong to \(\mathcal {C}_{\mathcal {G}}\) and to \(\mathcal {M}_{\mathcal {G}}\cap \mathcal {F}_{0,k}(\mathcal {G}),\) respectively. \(\square \)
We use the estimates of \(\mathcal {G}\)-balances \(\mu _{\mathcal {G}}(Q_{f,m})\) of polynomials \(Q_{f,m}\) to solve the mentioned separation problem for the families \(\mathcal {V}_{\mathcal {G}},\mathcal {K}_{\mathcal {G}}^{k},\mathcal {R} _{\mathcal {G}}\). We prove the following theorem:
Theorem 3
For every \(k\ge 2\) there holds the double inclusion
Proof
We start with the inclusion \(\mathcal {V}_{\mathcal {G}}\subset \mathcal {K}_{ \mathcal {G}}^{k}.\) To do it, let us assume that \(f\in \mathcal {V}_{\mathcal {G }},\) then \(\mathcal {L}f\) \(\in \mathcal {C_{G}}.\) Putting \(p=\mathcal {L}f\) and \(h=1,\) we obtain the factorization (1.1) with \(p\in \mathcal {C_{G}}\) and \(g = 1 \in \mathcal {M}_{\mathcal {G}}\cap \mathcal {F}_{0,k}( \mathcal {G}).\) Hence \(f\in \mathcal {K}_{\mathcal {G}}^{k}.\) It remains to show the relation \(\mathcal {V}_{\mathcal {G}}\ne \mathcal {K}_{\mathcal {G} }^{k}.\) To do it, let us observe that for \(f\in \mathcal {V}_{\mathcal {G}}\) there hold the sharp estimates \(\mu _{\mathcal {G}}(Q_{f,m})\le \) \(\frac{2}{ m+1},m\in \mathbb {N}\) (cf., eg., [1]), while for \(f\in \,\mathcal {K}_{\mathcal {G}}^{k}\) the sharp estimates \(\mu _{\mathcal {G} }(Q_{f,m})\le \) B(m) (Theorem 2.), with the obvious bound \(B(m)>\frac{ 2}{m+1},m\in \mathbb {N}\diagdown \{1\}.\) Hence, the extremal function \(f\in \mathcal {K}_{\mathcal {G}}^{k}\) does not belong to \(\mathcal {V}_{\mathcal {G} }. \)
Now we prove that \(\mathcal {K}_{\mathcal {G}}^{k}\subset \mathcal {R}_{ \mathcal {G}}.\)To this end, let us suppose that \(f\in \mathcal {K}_{\mathcal {G} }^{k}.\) Then there exist functions \(p\in \mathcal {C_{G}},\) \(g\in \mathcal {M} _{\mathcal {G}}\cap \mathcal {F}_{0,k}(\mathcal {G})\) such that \(\mathcal {L} f=p\cdot g.\) Denoting \(\varphi =\mathcal {L}^{-1}g,\) we have that \(\varphi \in \mathcal {N}_{\mathcal {G}}\) (by the Aleksander type theorem [1]) and \(\mathcal {L}f=p\mathcal {L}\varphi .\) Thus \(f\in \mathcal {R}_{ \mathcal {G}}\). It remains to show the relation \(\mathcal {K}_{\mathcal {G} }^{k}\ne \mathcal {R}_{\mathcal {G}}.\) For this purpose, let us observe that in the above estimates \(\mu _{\mathcal {G}}(Q_{f,m})\le \) \(B(m),m\in \mathbb { N},\) we have \(B(m)\le 1,m\in \mathbb {N}\) (see below), while for \(f\in \mathcal {R}_{\mathcal {G}}\) there hold the sharp estimates \(\mu _{\mathcal {G} }(Q_{f,m})\le m+1(\)see for instance [1]). Therefore, the extremal function \(f\in \mathcal {R}_{\mathcal {G}}\) does not belong to \(\mathcal {K}_{ \mathcal {G}}^{k}.\)
To complete the proof, we show that \(B(m)\le 1,\) \(m\in \mathbb {N}.\) To do it, we consider two cases, according to the partition \(m=ks+r,\) \(r\in \{0,1,\ldots ,k-1\},\) from the proof of Theorem 2.
1. Let us suppose that \(r=0.\) Then, if \(s=\frac{m}{k}=1,\) we see that the superscript \(s-1\) of the first product in Theorem 2 is smaller than its subscript 1. Hence, we replace the referred product by 1 and consequently, we get \(\mu _{\mathcal {G}}(Q_{f,m})\le \frac{2}{m}\le 1,\) because \( m=k\ge 2\). Next, if \(s\ge 2,\) then from Theorem 2, by the inequality \(1+ \frac{2}{\nu k}\le \frac{\nu +1}{\nu },\nu \in \mathbb {N},k\in \mathbb {N} \diagdown \{1\},\) we obtain
2. Let us suppose that \(r\in \{1,\ldots ,k-1\}.\) Then, if \( s=\left\lfloor \frac{m}{k}\right\rfloor =0,\) we see that the superscript of the second product in Theorem 2 is smaller than its subscript. Hence we replace the referred product by 1 and consequently, we get \(\mu _{G}(Q_{f,m})\le \frac{2}{m+1}\le 1,\) because \(m\le k-1.\) Next, if \( s=\left\lfloor \frac{m}{k}\right\rfloor \ge 1,\) then similarly as in step 1, we obtain
\(\square \)
Now, we give a growth theorem for \(f\in \mathcal {K}_{\mathcal {G}}^{k}\) and its Temljakov transform \(\mathcal {L}f.\)
Theorem 4
For functions \(f\in \mathcal {K}_{\mathcal {G}}^{k}\) there follow the following sharp estimates
Proof
First, let us observe that the above estimates are true for \(z=0\) (in (2.10) the values at \(r=0,\) of the left and right hand sides, mean the limit if \(r\rightarrow 0^{+}).\) Thus, in the sequel we will assume that \( z\in \mathcal {G\diagdown }\{0\}.\) We start with the estimates (2.9). Since \(f\in \mathcal {K}_{\mathcal {G}}^{k},\) there exist a function \(p\in \mathcal {C_{G}}\) and a function \(g\in \mathcal {M}_{\mathcal {G}}\cap \mathcal { F}_{0,k}(\mathcal {G})\) such that the factorization (2.1) holds. Therefore, we show for such functions g the following inequalities
To this aim, let us fix arbitrarily a point \(z\in \mathcal {G}\) such that \( \mu _{\mathcal {G}}(z)=r\in (0,1)\) and let us consider the function
Then \(\ G\) is (1, k)-symmetric, holomorphic, normalized and satisfies the condition
Hence \(G\in \mathcal {S}^{*}\cap \mathcal {F}_{1,k}(\mathcal {U})\) and by [9, Thm. 2.2.13]
Putting \(\zeta =\mu _{\mathcal {G}}(z)\) in the above we obtain, by the definition of the function G, the announced inequality.
On the other hand, there hold for \(p\in \mathcal {C_{G}}\) the following estimates [1]
Using the estimates of \(\ \left| p(z)\right| \) and \(\left| g(z)\right| \) we get the estimates (2.9). The sharpness of the upper bounds (2.9) confirms the function given by (2.8). Indeed, for \(r\in (0,1)\) and function \(f\in \mathcal {K}_{\mathcal {G}}^{k}\) given by (2.8), we get
at points \(z\in \mathcal {G},\) \(\mu _{\mathcal {G}}(z)=r\in (0,1)\) such that \(I(z)=r\) (this condition is fulfilled by the points \(z=rz^{*},\) where \( z^{*}\in \partial \mathcal {G}\) and \(I(z^{*})=1).\)
The sharpness of the lower bounds (2.9) can be proven in a similar way.
Now, we prove the estimates (2.10). To obtain the upper bound (2.10), we use the proved above upper bound (2.9) and the fact that the Temljakov operator \(\mathcal {L}\) is invertible and
Indeed, we have for \(f\in \mathcal {K}_{\mathcal {G}}^{k}\) and \(z\in \mathcal {G},\,\mu _{\mathcal {G}}(z)=r\in (0,1),\)
To prove the lower bound (2.10) let us consider the function
with arbitrarily fixed \(f\in \mathcal {K}_{\mathcal {G}}^{k}\) and \(z\in \mathcal {G},\,\mu _{\mathcal {G}}(z)=r\in (0,1).\) Since
we get, by Theorem 1, that there exist functions \(g\in \mathcal {M}_{ \mathcal {G}}\cap \mathcal {F}_{0,k}(\mathcal {G})\) and \(p\in \mathcal {C_{G}}\) such that the factorization (2.1) is true. Thus
where for \(\zeta \in \mathcal {U}\)
Moreover, \(G\in \mathcal {S}^{*}\cap \mathcal {F}_{1,k}(\mathcal {U})\) ( see the proof of the estimates (2.9)) and \(P:\mathcal {U}\rightarrow \mathbb {C},P(0)=1,\) is a holomorphic function with a positive real part. Therefore, \(\ F\) belongs to a subclass \(\mathcal {K}^{(k)}\) (considered in [22] and for \(k=2\) in [8]) of the class of close-to-convex functions. Hence, F is univalent in the disc \(\mathcal {U}\).
On the other hand, by the lower bound (2.9), we have that
because \(r=\mu _{\mathcal {G}}\left( \zeta \frac{z}{\mu _{\mathcal {G}}(z)} \right) =|\zeta |.\) Now we show that
To this aim, it is sufficient to show that it holds for the nearest point \( F(\zeta _{0})\) from zero \((\left| \zeta _{0}\right| =r\in (0,1)),\) otherwise, we have \(|F(\zeta )|\ge |F(\zeta _{0})|,\left| \zeta \right| =r.\) Since F is univalent in the disc \(\mathcal {U},\) the original image of the line segment \(\overline{0,F(\zeta _{0})}\) is a piece of arc \(F^{-1}\left( \overline{0,F(\zeta _{0})}\right) \) in the disc \(r \overline{\mathcal {U}}.\) Thus
Thus, by the definition of F, we get
Hence, putting \(\zeta =\) \(\mu _{\mathcal {G}}\left( z\right) \) \(=r\in (0,1),\) we have the lower bound (2.10).
Finally, let us note that we obtain the equalities in the inequalities (2.10) for the function (2.8) in adequate points \(z\in \mathcal {G}.\) \(\square \)
We close the paper with a sufficient condition guaranteeing that a function \( f\in \mathcal {H_{G}}(1)\) belongs to \(\mathcal {K}_{\mathcal {G}}^{k}.\) We formulate it in the term of \(\mathcal {G}\)-balances of m-honogeous polynomials in developments of functions from \(\mathcal {H_{G}}(1).\)
Theorem 5
Let \(f\in \mathcal {H_{G}}(1)\) has the form (2.2). If there exists a function \(g\in \mathcal {M}_{\mathcal {G}}\cap {\mathcal {F}}_{0,k}( \mathcal {G})\) of the form (2.3) such that
then \(f\in \mathcal {K}_{\mathcal {G}}^{k}.\)
Proof
Since g, as a function from \(\mathcal {M}_{\mathcal {G}}\) omits zero [1], we will prove that
To do it, we compute step by step
Thus
and hence
This gives the mentioned inequality by a maximum principle for pluriharmonic functions of several complex variables. Putting \(p(z)=\frac{\mathcal {L}f(z)}{ g(z)},\) \(z\in \mathcal {G},\) we obtain that the transform \(\mathcal {L}f\) has the factorization (1) with \(g\in \mathcal {M}_{\mathcal {G}}\cap \mathcal {F} _{0,k}(\mathcal {G})\) and \(p\in \mathcal {C_{G}}.\ \)Consequently, \(f\in \mathcal {K}_{\mathcal {G}}^{k}\). \(\square \)
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Communicated by David Shoikhet.
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Długosz, R., Liczberski, P. & Trybucka, E. Bavrin’s Type Factorization of the Temljakov Operator for Holomorphic Functions in Circular Domains of \(\mathbb {C}^{n}\). Complex Anal. Oper. Theory 12, 1321–1335 (2018). https://doi.org/10.1007/s11785-018-0770-0
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DOI: https://doi.org/10.1007/s11785-018-0770-0
Keywords
- Holomorphic functions on n-circular domains in \(\mathbb {C}^{n}\)
- Minkowski function
- Estimates of homogeneous polynomials of Taylor series
- Temljakov operator
- Bavrin’s families of functions