POWER SETS OF POLYNOMIALS IN FABER REGIONS

K. A. M. Sayyed, M. S. Metwally, and R. S. Batahan

Abstract. The representation of the regular functions by power sets of polynomials of the single complex variable z in Faber regions is the subject of the present work. Different kinds of order and type of these sets are investigated.

1. Preliminaries. The transformation

z = ϕ(t) = t + ∑_{n=0}^{∞} α_{n}t^{−n} = t + M(\frac{1}{t}),  (1.1)

was introduced and studied by Faber [3], and is called the Faber transformation. Such a transformation is conformal for T_{0} < |t| < ∞. If γ is a fixed number greater than T_{0}, then the circle |t| = γ is mapped onto the simple regular closed curve C in the z–plane. When T_{0} < r < ∞, the circles |t| = r are mapped onto the simple closed regular curves C_{r} such that when r_{1} < r_{2} the curve C_{r_{1}} lies totally inside the curve C_{r_{2}}. The set of curves (C_{r}) are called Faber curves and so the typical representative curve C is the Faber curve C_{γ}. The regions D(C) and D̄(C) which denote the set of points in the interior of C and its closure are called the Faber regions. Let us investigate the expression \frac{dζ}{ζ−z} under the integral sign in Cauchy’s formula where ζ is the integration variable on the z–plane. Substituting ζ = ϕ(t), we get

\frac{dζ}{ζ − z} = \frac{ϕ′(t)dt}{ϕ(t) − z} .  (1.2)

Consider the following function of t:

G(z, t) = \frac{tϕ′(t)dt}{ϕ(t) − z} = ∑_{n=0}^{∞} P_{n}(z)t^{−n}  (1.3)

where P_{n}(z) is a polynomial of degree n. The simple monic set {P_{n}(z)} of polynomials is called the set of Faber polynomials associated with the transformation ϕ(t). The function G(z, t) is called the generating function of Faber polynomials which is regular at the point t = ∞ for any z, G(z, ∞) and the function G(z, t) can have a singularity only at points t satisfying ϕ(t) = z.

Since the transformation z = ϕ(t) is conformal in |z| > T_{0}, then it has an inverse of the form

t = ψ(z) = z + ∑_{n=0}^{∞} β_{n}z^{−n},  (1.4)

which is still a Faber transformation, being conformal in |t| > T̄.

It has been proved by Newns [6], that

P_{n}(z) = \{ψ(z)\}^{n} + \frac{1}{2πi} ∫_{C_{T′}} \frac{\{ψ(ζ)\}^{n}dζ}{ζ − z}; n ≥ 0,  (1.5)

where T′ is any number greater than T_{0} and z ∈ C_{r}, where r > T′ .

The following statement is a consequence of relation (1.5) and has been stated by Ullman [9] that

P_{n}(z) is the polynomial part of {ψ(z)}^{n}.  (1.6)

The inverse transformation ψ(t) has the following relations

\frac{tψ′(t)}{ψ(t) − z} = ∑_{n=0}^{∞} P̄_{n}(z)t^{−n},  (1.7)

P̄_{n}(z) = \{ϕ(z)\}^{n} + \frac{1}{2πi} ∫_{|t|=T′} \frac{\{ϕ(t)\}^{n}dt}{t − z}  (1.8)

and

P̄_{n}(z) is the polynomial part of {ϕ(z)}^{n}  (1.9)

where {P̄_{n}(z)} is the inverse of the set {P_{n}(z)}.

Now, differentiating (1.3) n times with respect to z and then putting z = 0, we obtain

\frac{tϕ′(t)}{\{ϕ(t)\}^{n+1}} = ∑_{k=n}^{∞} \frac{P_{k}^{(n)}(0)}{n!}t^{−k} = ∑_{k=n}^{∞} P_{k,n}t^{−k}.  (1.10)

A sequence p_{0}(z), p_{1}(z), p_{2}(z), …, p_{n}(z), … of polynomials where

p_{n}(z) = ∑_{k} p_{n,k}z^{k}  (1.11)

is said to form a basic set {p_{n}(z)}, (c.f. [10]), if any polynomial, and in particular, the monomial z^{n}, admits a unique finite representation of the form

z^{n} = ∑_{k} π_{n,k}p_{k}(z).  (1.12)

Let z = ϕ(t) be a Faber transformation which is conformal in T_{0} < |t| < ∞, where the circles |t| = r, T_{0} < r < ∞, are mapped in the z–plane on the Faber curves C_{r}, and suppose that {Π_{n}(z)} is the set of Faber polynomials associated with the transformation ϕ(t).

If the number N_{n} of non–zero coefficients in (1.12) is such that lim_{n→∞} Nn^{\frac{1}{n}} = 1, the set {p_{n}(z)} is called a Cannon set (c.f. [10]) of polynomials which admits the unique representation

Π_{n}(z) = ∑_{k=0}^{∞} π_{n,k}p_{k}(z), (n ≥ 0).  (1.13)

If f(z) is an analytic function, which is regular in D̄(C_{r}) ; r > T_{0} , then , according to Faber [3], f(z) will admit the Faber series

f(z) = ∑_{n=0}^{∞} c_{n}P_{n}(z)  (1.14)

in D̄(C_{r}). Introducing (1.13) in (1.14) we get the basic series associated with f(z) in the form

f(z) ∼ ∑_{n=0}^{∞} A_{n}p_{n}(z)  (1.15)

where

A_{n} = ∑_{k=0}^{∞} c_{k}π_{k,n}.

The basic series (1.15) associated with f(z) is said to represent f(z) in D̄(C_{r}) if it converges uniformly to f(z). In this case we say that the set {p_{n}(z)} represents f(z) in D̄(C_{r}).

The basic set {p_{n}(z)} is said to be effective in D̄(C_{r}) for the class H̄(C_{ρ}) of functions regular in D̄(C_{ρ}), where ρ ≥ r, if the set {p_{n}(z)} represents in D̄(C_{r}) every function of the class H̄(C_{ρ}), this is justified by the Cannon function λ{p_{n}; C_{r}} for the Faber regions D̄(C_{r}) in the form

λ\{p_{n}; C_{r}\} = lim sup_{n→∞}\{ω_{n}(C_{r})\}^{\frac{1}{n}}  (1.16)

where

ω_{n}(C_{r}) = ∑_{k=0}^{∞} |π_{n,k}|M(p_{k}; C_{r})  (1.17)

and

M(p_{k}; C_{r}) = max_{z∈C_{r}} |p_{k}(z)|.

According to Newns [5] the necessary and sufficient condition for the Cannon set {p_{n}(z)} to be effective in D̄(C_{r}) for the class H̄(C_{ρ}) ; ρ ≥ r is that

λ{p_{n}; C_{r}} = ρ.  (1.18)

Let {p_{n}(z)} be a given basic set of polynomials, for which the representation (1.13) holds, and suppose further that

p_{n}(z) = ∑_{k} q_{n,k}P_{k}(z); (n ≥ 0).  (1.19)

If we write

q_{n}(z) = ∑_{k} q_{n,k}z^{k},  (1.20)

then {q_{n}(z)} is a basic set and we can write the set {p_{n}(z)} as the product set of two sets {q_{n}(z)} and {P_{n}(z)} as follows:

{p_{n}(z)} = {q_{n}(z)}{P_{n}(z)}.  (1.21)

The set {q_{n}(z)} is called the set associated with {p_{n}(z)} with respect to the transformation ϕ(t). Nassif [4] proved that

λ{p_{n}; C_{r}} = λ{q_{n}; r}  (1.22)

where λ{q_{n}; r} is the Cannon function of {q_{n}(z)} for the region |z| ≤ r.

Thus, to investigate the effectiveness of a given set {p_{n}(z)} in the Faber region D̄(C_{r}) is equivalent to the study of the effectiveness of the associated set {q_{n}(z)} in |z| ≤ r.

The set {p_{n}(z)} of polynomials in which the polynomial p_{n}(z) is of degree n is called a simple set. Such a set for which p_{n,n} = 1 for all n ≥ 0, is called a simple monic set. The simple set for which |p_{n,n}| = 1, for all n ≥ 0, is called a simple absolutely monic set.

2. Power Sets of Faber Polynomials. Let {p_{n}(z)} be a basic set of polynomials of the single complex variable z, where

p_{n}(z) = ∑_{j} p_{n,j}z^{j}; n > 0.

The power set of polynomials {p_{n}^{(μ)}(z)} was given in the form

p_{n}^{(μ)}(z) = ∑_{k} p_{n,k}^{(μ)}z^{k} = ∑_{h_{1},h_{2},...,h_{μ−1},k} p_{n,h_{1}}p_{h_{1},h_{2}} … p_{h_{μ−1},k}z^{k},  (2.1)

where μ is a finite positive integer.

Let ϕ(t) = t + ∑_{n=0}^{∞} α_{n}t^{−n} be a Faber transformation with inverse

ψ(t) = t + ∑_{n=0}^{∞} β_{n}t^{−n}.

The composite transformation

ϕ_{μ}(t) = ϕ(ϕ(… (ϕ(t)) …)),  (2.2)

is still a Faber transformation of the form

ϕ_{μ}(t) = t + ∑_{n=0}^{∞} α_{n}^{∗}t^{−n}.  (2.3)

Suppose that {Π_{n}(z)} is the set of Faber polynomials associated with the transformation ϕ(t).

Let {P_{n}^{(μ)}(z)} be the power set of Faber polynomials given in the form

{P_{n}^{(μ)}(z)} = {P_{n}(z)}{P_{n}^{(μ−1)}(z)}  (2.4)

where

P_{n}^{(μ)}(z) = ∑_{h_{1}=0}^{n} ∑_{h_{2}=0}^{h_{1}} ⋯ ∑_{h_{μ−1}=0}^{h_{μ−2}} ∑_{k=0}^{h_{μ−1}} P_{n,h_{1}}P_{h_{1},h_{2}} … P_{h_{μ−1},k}z^{k},

P_{n}(z) = z^{n} + ∑_{k=0}^{n−1} P_{n,k}z^{k},  (2.5)

P̄_{n}(z) = z^{n} + ∑_{k=0}^{n−1} P̄_{n,k}z^{k},  (2.6)

P_{n}^{(μ)}(z) = z^{n} + ∑_{k=0}^{n−1} P_{n,k}^{(μ)}z^{k}  (2.7)

and

P_{n,k}^{(μ)} = ∑_{i} P_{n,i}P_{i,k}^{(μ−1)}; (n ≥ 0).  (2.8)

The following result concerning Faber power sets of polynomials is proved by using mathematical induction.

\underline{Theorem 2.1}. The power set as given in (2.4) is the set of Faber polynomials associated with the composite transformation (2.2).

\underline{Proof}. From (2.4) and (2.9) we have

∑_{n=0}^{∞} P_{n}^{(μ)}t^{−n} = ∑_{n=0}^{∞} [ ∑_{h_{1}=0}^{n} ∑_{h_{2}=0}^{h_{1}} ⋯ ∑_{h_{μ−1}=0}^{h_{μ−2}} ∑_{k=0}^{h_{μ−1}} P_{n,h_{1}}P_{h_{1},h_{2}} … P_{h_{μ−1},k}z^{k}]t^{−n}

= ∑_{n=0}^{∞} [ ∑_{h_{1}=0}^{n} P_{n,h_{1}}t^{−n}] ∑_{h_{2}=0}^{h_{1}} ⋯ ∑_{h_{μ−1}=0}^{h_{μ−2}} ∑_{k=0}^{h_{μ−1}} P_{h_{1},h_{2}} … P_{h_{μ−1},k}z^{k}

= ∑_{h_{1}=0}^{∞} [ ∑_{h_{1}=n}^{∞} P_{n,h_{1}}t^{−n}] ∑_{h_{2}=0}^{h_{1}} ⋯ ∑_{h_{μ−1}=0}^{h_{μ−2}} ∑_{k=0}^{h_{μ−1}} P_{h_{1},h_{2}} … P_{h_{μ−1},k}z^{k}

= \frac{tϕ′(t)}{ϕ(t)} ∑_{h_{1}=0}^{∞} ∑_{h_{2}=0}^{h_{1}} ⋯ ∑_{h_{μ−1}=0}^{h_{μ−2}} ∑_{k=0}^{h_{μ−1}} P_{h_{1},h_{2}} … P_{h_{μ−1},k}z^{k}\{ϕ(t)\}^{−h_{1}}

= \frac{tϕ′(t)ϕ′(ϕ(t))}{ϕ(ϕ(t))} ∑_{h_{2}=0}^{∞} ∑_{h_{3}=0}^{h_{2}} ⋯ ∑_{h_{μ−1}=0}^{h_{μ−2}} ∑_{k=0}^{h_{μ−1}} P_{h_{2},h_{3}} … P_{h_{μ−1},k}z^{k}\{ϕ(ϕ(t))\}^{−h_{2}}

. . .

\begin{matrix}= \frac{tϕμ^{′}(t)}{ϕ_{μ}(t)} ∑_{k=0}^{∞} \{\frac{z}{ϕ_{μ}(t)}\}^{k} \\ = \frac{tϕμ^{′}(t)}{ϕ_{μ}(t) − z} . \end{matrix}

That is,

∑_{n=0}^{∞} P_{n}^{(μ)}(z)t^{−n} = \frac{tϕμ^{′}(t)}{ϕ_{μ}(t) − z} .  (2.9)

Therefore the set {P_{n}^{(μ)}(z)} is a set of Faber polynomials associated with the composite transformation (2.2), and this completes the proof of Theorem 2.1.

3. Effectiveness of Power Sets of Polynomials in Faber Regions. Let {p_{n}(z)} be a simple absolutely monic set of polynomials, and {p_{n}^{(μ)}(z)} be the power set of polynomials of the form

{p_{n}^{(μ)}(z)} = {p_{n}(z)}{p_{n}^{(μ−1)}(z)}.  (3.1)

Hence, we have

p_{n}^{(μ)}(z) = ∑_{i} p_{n,i}^{(μ)}z^{i} = ∑_{i} ∑_{k} p_{n,i}p_{i,k}^{(μ−1)}z^{k}; n ≥ 0.

Suppose that the Faber transformation

z = ϕ(t) = t + ∑_{n=0}^{∞} α_{n}t^{−n}  (3.2)

is conformal in |t| ≥ T, then the inverse transformation is such that

z = ψ(t) = t + ∑_{n=0}^{∞} β_{n}t^{−n}  (3.3)

will also be conformal in |t| ≥ T̄.

The composite transformation (2.2) is conformal in |t| ≥ T_{0}. Take the number γ such that

γ > (T, T̄, T_{0})  (3.4)

and suppose that the Faber curves C, C̄ and C^{μ} are the respective images, in the z plane, of the circle |t| = γ by the transformations (2.2), (3.2) and (3.3). The curve C^{μ} is said to be the power curve of C.

Now, we study effectiveness of the power set {P_{n}^{(μ)}(z)} in the Faber region D̄(C). For this purpose consider the following

G(r) = sup_{|t|=r} |ϕ(t)|, r > T  (3.5)

and

H(r) = sup_{|t|=r} |ψ(t)|, r > T.  (3.6)

Let {q_{n}(z)}, {h_{n}(z)} and {g_{n}(z)} be the sets of polynomials associated respectively with sets {p_{n}(z)}, {p_{n}^{(μ−1)}(z)} and {p_{n}^{(μ)}(z)} with the respective transformations ϕ(t), ϕ_{μ−1}(t), and ϕ_{μ}(t).

From (1.21) one gets

{p_{n}(z)} = {q_{n}(z)}{P_{n}(z)},  (3.7)
{p_{n}^{(μ−1)}(z)} = {q_{n}(z)}{P_{n}^{(μ−1)}(z)},  (3.8)

and

{p_{n}^{(μ)}(z)} = {g_{n}(z)}{P_{n}^{(μ)}(z)}.  (3.9)

For μ = 2, from (3.7) and (3.9) we get

{g_{n}(z)} = {p_{n}^{(2)}(z)}{P̄_{n}^{(2)}(z)}
= {p_{n}(z)}{p_{n}(z)}{P̄_{n}(z)}{P̄_{n}(z)}
= {q_{n}(z)}{P_{n}(z)}{q_{n}(z)}{P̄_{n}(z)}  (3.10)

and

{ḡ_{n}(z)} = {P_{n}(z)}{q̄_{n}(z)}{P̄_{n}(z)}{q̄_{n}(z)}.  (3.11)

Suppose that the set {p_{n}(z)} is effective in the Faber region D̄(C), then the set {p̄_{n}(z)} is effective in the same region (c.f. [1]), and its associated sets {q_{n}(z)} and {q̄_{n}(z)} are effective in |z| ≤ γ. Since effectiveness of simple sets in |z| ≤ γ implies effectiveness in |z| ≤ r for all r ≥ γ; this happens for absolutely simple monic sets, so the sets {q_{n}(z)} and {q̄_{n}(z)} are effective in |z| ≤ r, for all r ≥ γ.

We shall use an increasing sequence (r_{j}); j = 1, 2, … of positive numbers greater than γ, i.e., r_{j+1} > r_{j}, for all j, r_{1} > γ.

From (1.5), (1.8), (3.5), and (3.6) we have

M(P_{n}; r_{3}) < k{H(r_{3})}^{n}  (3.12)

and

M(P̄_{n}; γ) < k{G(r)}^{n}; n ≥ 0.  (3.13)

Write

G(γ) < r_{5}, H(r_{3}) < r_{4}.  (3.14)

Since the sets {q_{n}(z)} and {q̄_{n}(z)} are simple absolutely monic, then we have

M(q_{n}; r_{4}) < kr_{7}^{n}  (3.15)

and

M(q̄_{n}; r_{2}) < kr_{5}^{n}.  (3.16)

By using Cauchy’s inequality and the relations (3.12), (3.13), (3.14), and (3.15) we get

M(g_{n}; γ) = max_{|z|=γ} |g_{n}(z)|

≤ ∑_{i_{1},i_{2},i_{3},i_{4}} |q_{n,i_{1}}||P_{i_{1},i_{2}}||q_{i_{2},i_{3}}||P̄_{i_{3},i_{4}}|γ^{i_{4}}

< K(n + 1)^{4}r_{7}^{n}(\frac{r_{7}}{r_{3}} )^{n}(\frac{r_{5}}{r_{4}} )^{n}.  (3.17)

The inverse set {ḡ_{n}(z)}, also has the following relations

M(P_{n}; r_{10}) < k{H(r_{10})}^{n},  (3.18)
M(P̄_{n}; r_{5}) < k{G(r_{5})}^{n},  (3.19)
G(r_{5}) < r_{9}  (3.20)

and

M(q̄_{n}; r_{4}) < kr_{11}; n ≥ 0.  (3.21)

It follows from (3.11), (3.13), (3.20), (3.21), and by using Cauchy’s inequality that

M(ḡ_{n}; r_{4}) = max_{|z|=r_{4}} |ḡ_{n}(z)|

≤ ∑_{i_{1},i_{2},i_{3},i_{4}} |P_{n,i_{1}}||q̄_{i_{1},i_{2}}||P̄_{i_{2},i_{3}}||q̄_{i_{3},i_{4}}|r_{4}^{i_{4}}

< K(n + 1)^{4}(\frac{r_{11}}{r_{10}} )^{n}(\frac{r_{9}}{r_{4}} )^{n}(\frac{r_{11}}{r_{5}} )^{n}\{H(r_{10})\}^{n}.  (3.22)

Inserting (3.17) and (3.22) in the Cannon sum ω_{n}(γ) of the set {g_{n}(z)} we obtain

ω_{n}(γ) = ∑_{k} |ḡ_{n,k}|M(g_{k}; γ)  (3.23)

≤ ∑_{k} \frac{M(ḡ_{n}; r_{4})}{r_{4}^{k}}M(g_{k}; γ)

< K(n + 1)^{8}(\frac{r_{11}}{r_{10}} )^{n}(\frac{r_{9}}{r_{4}} )^{n}(\frac{r_{11}}{r_{5}} )^{n}(\frac{r_{7}}{r_{4}} )^{n}(\frac{r_{7}}{r_{3}} )^{n}(\frac{r_{5}}{r_{4}} )^{n}\{H(r_{10})\}^{n}.

The Cannon function λ(g_{n}; γ) of the set {g_{n}(z)} is

λ(g_{n}; γ) = lim sup_{n→∞}\{ω_{n}(γ)\}^{\frac{1}{n}}  (3.24)

≤ K(\frac{r_{11}}{r_{10}} )(\frac{r_{9}}{r_{4}} )(\frac{r_{11}}{r_{5}} )(\frac{r_{7}}{r_{4}} )(\frac{r_{7}}{r_{3}} )(\frac{r_{5}}{r_{4}} )\{H(r_{10})\}.

Since r_{11} can be taken arbitrarily close to γ, we get

λ(g_{n}; γ) ≤ H(γ) = A.  (3.25)

Applying (1.22) we get

λ(p_{n}; C^{2}) = λ(g_{n}; γ) ≤ A

where λ(p_{n}; C^{2}) is the Cannon function of set {p_{n}^{(2)}(z)} for the region D̄(C^{2}).

Therefore the set {p_{n}^{(2)}(z)} is effective in the region D̄(C^{2}) for the class of functions H̄(C_{A}^{2}).

Now, suppose that the set {p_{n}^{(μ−1)}(z)} is effective in the region D̄(C^{(μ−1)}) for the class of functions H̄(C_{A}^{μ−1}).

We shall prove that {p_{n}^{(μ)}(z)} is effective in the same region for the same class, when the set {p_{n}(z)} is a simple absolutely monic set which is effective in the region D̄(C^{μ}).

From (3.1), (3.8), (3.9), and (3.10) we have

{g_{n}(z)} = {p_{n}^{(μ)}(z)}{P̄_{n}^{(μ)}(z)}
= {p_{n}(z)}{p_{n}^{(μ−1)}(z)}{P̄_{n}^{(μ−1)}(z)}{P̄_{n}(z)}
= {q_{n}(z)}{P_{n}(z)}{h_{n}(z)}{P̄_{n}(z)}  (3.26)

and

{ḡ_{n}(z)} = {P_{n}(z)}{h̄_{n}(z)}{P̄_{n}(z)}{q̄_{n}(z)}.  (3.27)

Proceeding similar to the case when μ = 2 above, we see directly that the set {p_{n}^{(μ)}(z)} is effective in the region D̄(C^{μ}) for the class of functions H̄(C_{A}^{μ}).

This completes the proof of the following main result.

\underline{Theorem 3.1}. Let \{p_{n}(z)\} be a simple absolutely monic set of polynomials, effective in the Faber region D̄(C) where C is the Faber curve. The number γ > (T, T_{0}) can be chosen such that

G(γ) > T and G(γ) > T_{0}.  (3.28)

Then the power set {p_{n}^{μ}(z)} = {p_{n}(z)}{p_{n}^{(μ−1)}(z)} will be effective in the Faber region D̄(C^{μ}) where C^{μ} is the power curve of C, for the class of functions H̄(C_{A}^{μ}), where

A = H(γ).  (3.29)

4. Mode of Increase of Power Sets of Polynomials. We give, for the sake of complement, some results concerning some orders and types of the power sets of polynomials.

Recall the definitions of the order ω and the type τ [4], the logarithmic order ρ^{∗} and logarithmic type Γ^{∗} [8], and the q–order ρ_{(q)} and the q–type Γ_{(q)} [7] of the basic set {p_{n}(z)}, respectively, in the forms

ω = lim_{r→∞} lim sup_{n→∞} \frac{ln ω_{n}(r)}{n ln n},  (4.1)

if 0 < ω < ∞, then

τ = lim_{r→∞} \frac{e}{ω} [ lim sup_{n→∞}\{ω_{n}(r)\}^{\frac{1}{n}} n^{−ω}]^{\frac{1}{ω}} ,  (4.2)

ρ^{∗} = 1 + lim_{r→∞} lim sup_{n→∞} \frac{ln\{(ω_{n}(r))^{\frac{1}{n}} \}}{ln ln n},  (4.3)

if 1 < ρ^{∗} < ∞, then

Γ^{∗} = \frac{1}{ρ^{∗}} (\frac{1}{ρ^{∗} − 1})^{\frac{ρ^{∗}−1}{ρ^{∗}}} lim_{r→∞} lim sup_{n→∞} \frac{\{ω_{n}(r)\}^{\frac{1}{n(ρ^{∗}−1)}}}{ln ln n} ,  (4.4)

ρ_{(q)} = lim_{r→∞} lim sup_{n→∞} \frac{ln ω_{n}(r)}{n ln^{[q−1]} n}  (4.5)

if 0 < ρ_{(q)} < ∞, then

Γ_{(q)} = lim_{r→∞} lim sup_{n→∞} \frac{\{ω_{n}(r)\}^{\frac{1}{nρ_{(q)}}}}{ln^{[q−2]} n} ,  (4.6)

where

ω_{n}(r) = ∑_{k=0}^{∞} |π_{n,k}|M(p_{k}; r),  (4.7)

and M(p_{k}; r) = max_{|z|=r} |p_{k}(z)|.

We have the following results.

\underline{Theorem 4.1}. Let \{p_{n}(z)\} be a simple monic set of polynomials whose coefficients satisfy the condition of the form

|p_{n,k}| ≤ Mn^{λ(n−k)}, 0 ≤ k ≤ n − 1.  (4.8)

Then the power set {p_{n}^{(μ)}(z)} is of order not exceeding λ.

\underline{Theorem 4.2}. Let \{p_{n}(z)\} be a simple monic set of polynomials of order λ and of finite type. Then the power set {p_{n}^{(μ)}(z)} is of order ρ and does not exceed (2μ − 1)λ and is of type zero if the equality sign happens.

\underline{Theorem 4.3}. Let \{p_{n}(z)\} be a simple monic set of polynomials of logarithmic order λ^{∗} and of finite logarthmic type. Then the power set {p_{n}^{(μ)}(z)} is of logarithmic order

ρ^{∗} ≤ (2μ − 1)λ^{∗} − (2μ − 2),  (4.9)

and is of logarithmic type zero whenever ρ^{∗} = (2μ − 1)λ^{∗} − (2μ − 2).

\underline{Theorem 4.4}. Let \{p_{n}(z)\} be a simple monic set of polynomials of q−order ρ_{(q)} and of finite q−type. Then the power set {p_{n}^{(μ)}(z)} is of q−order

ρ_{(q)} ≤ (2μ − 1)λ_{(q)},  (4.10)

and is of q−type zero whenever ρ_{(q)} = (2μ − 1)λ_{(q)}.

\underline{Theorem 4.5}. Let \{p_{n}(z)\} be a simple set of polynomials of order λ and finite type which satisfies the following condition:

\{ \begin{matrix}lim sup_{n→∞} \frac{ln |p_{n,n}|}{n ln n} = b \\ lim inf_{n→∞} \frac{ln |p_{n,n}|}{n ln n} = a. \end{matrix}  (4.11)

Then the power set {p_{n}^{(μ)}(z)} is of order

ρ ≤ (2μ − 1)λ + (μ − 1)(b − a).  (4.12)

If ρ = (2μ − 1)λ + (μ − 1)(b − a), then the type of the power set is zero.

\underline{Theorem 4.6}. Let \{p_{n}(z)\} be a simple set of polynomials of logarithmic order λ^{∗} and finite logarithmic type which satisfies the following condition:

\{ \begin{matrix}lim sup_{n→∞} ^{\frac{ln |p_{n,n}|^{\frac{1}{n}}}{ln ln n}} = b \\ lim inf_{n→∞} ^{\frac{ln |p_{n,n}|^{\frac{1}{n}}}{ln ln n}} = a. \end{matrix}  (4.13)

Then the power set {p_{n}^{(μ)}(z)} is of logarithmic order

ρ^{∗} ≤ (2μ − 1)λ^{∗} + (μ − 1)(b − a − 2)  (4.14)

and logarithmic type zero whenever ρ^{∗} = (2μ − 1)λ^{∗} + (μ − 1)(b − a − 2).

\underline{Theorem 4.7}. Let \{p_{n}(z)\} be a simple set of polynomials of q–order λ_{(q)} and finite q–type which satisfies the following condition:

\{ \begin{matrix}lim sup_{n→∞} \frac{ln |p_{n,n}|}{n ln^{[q−1]} n} = b \\ lim inf_{n→∞} \frac{ln |p_{n,n}|}{n ln^{[q−1]} n} = a. \end{matrix}  (4.15)

Then the power set {p_{n}^{(μ)}(z)} is of q−order

ρ_{(q)} ≤ (2μ − 1)λ_{(q)} + (μ − 1)(b − a).  (4.16)

These upper bounds in the above theorems are attainable.

\underline{Proof of Theorem 4.6}. We give as a model the proof of Theorem 4.6 as follows. Let {p_{n}(z)} be a simple set of polynomials of logarithmic order λ^{∗} and of finite

logarithmic type which satisfies (4.13). Suppose that σ is a positive finite number such that σ > λ^{∗} − 1. From (1.6), we have

ω_{n}(r) < K(ln n)^{σn}; n ≥ 1.  (4.17)

From (4.13) we get

K(ln n)^{n(a−ϵ)} < |p_{n,n}| < K(ln n)^{n(b+ϵ)}; ϵ > 0.  (4.18)

Since the set {p_{n}(z)} is simple, then p_{n,n}p̄_{n,n} = 1. Using Cauchy’s inequality and from (4.17) and (4.18) we have

|p_{n,k}| < K \frac{(ln n)^{(σ+b+ϵ)n}}{r^{k}}; k ≥ 0,  (4.19)

and

|p̄_{n,k}| < K \frac{(ln n)^{(σ−a+ϵ)n}}{r^{k}}; k ≥ 0.  (4.20)

Hence,

M(p_{n}^{(μ)}; r) = max_{|z|=r} |p_{n}^{(μ)}(z)|  (4.21)

< KM(p_{n}; r)(ln n)^{(μ−1)(σ+b+ϵ)n} \frac{1}{r^{(μ−1)n}} .

If ϕ_{n}(r) is the Cannon sum of the power set {p_{n}^{(μ)}(z)}, then by using (4.20) and (4.21) we find

ϕ_{n}(r) = ∑_{k} |p̄_{n,k}^{(μ)}|M(p_{k}^{(μ)}; r)

< K(ln n)^{(2μ−1)nσ+(μ−1)(b−a)n+2(μ−1)ϵ} \frac{1}{r^{(2μ−3)n}} .  (4.22)

Then the logarithmic order ρ^{∗} of the power set {p_{n}^{(μ)}(z)} is such that

ρ^{∗} = 1 + lim_{r→∞} lim sup_{n→∞} \frac{ln\{(ϕ_{n}(r))^{\frac{1}{n}} \}}{ln ln n}

≤ 1 + (2μ − 1)σ^{∗} + (μ − 1)(b − a).

Since σ can be taken very close to λ^{∗} − 1, then

ρ^{∗} ≤ (2μ − 1)λ^{∗} + (μ − 1)(b − a − 2).  (4.23)

Suppose that ρ^{∗} = (2μ − 1)λ^{∗} + (μ − 1)(b − a − 2). Then for (4.22), we get the logarithmic type as given:

Γ^{∗} = \frac{1}{ρ^{∗}} (\frac{1}{ρ^{∗} − 1})^{\frac{ρ^{∗}−1}{ρ^{∗}}} lim_{r→∞} lim sup_{n→∞} \frac{\{ω_{n}(r)\}^{\frac{1}{n(ρ^{∗}−1)}}}{ln ln n} = 0.

The following example shows that the upper bound in (4.23) is attainable.

\underline{Example 4.1}. Let \{p_{n}(z)\} be the simple set given by

p_{n}(z) = \{ \begin{matrix}b_{n}z^{n} + α_{n}(z^{n−1} + z^{n−2}); & n even \\ a_{n}(z^{n} + z^{n−1}); & n odd \end{matrix}

where a_{n} = (ln n)^{an+2}, b_{n} = (ln n)^{bn+3}, and α_{n} = (ln n)^{(α+b−1)n}. Hence, λ^{∗} = α. For μ = 2, we construct the power set {p_{n}^{(2)}(z)} as follows:

\begin{aligned}p_{n}^{(2)}(z) & = b_{n}^{2}z^{n} + α_{n}(b_{n} + a_{n−1})z^{n−1} \\ & + α_{n}(b_{n} + b_{n−2} + a_{n−1})z^{n−2} + α_{n}α_{n−1}(z^{n−3} + z^{n−4}); n even \end{aligned}

and

p_{n}^{2}(z) = a_{n}^{2}z^{n} + a_{n}(a_{n} + b_{n−1})z^{n−1} + a_{n}α_{n−1}(z^{n−2} + z^{n−3}); n odd.

Then the logarithmic order ρ^{∗} of the power set {p_{n}^{(2)}(z)} is such that

ρ^{∗} = 1 + lim_{r→∞} lim sup_{n→∞} \frac{ln\{(ϕ_{n}(r))^{\frac{1}{n}} \}}{ln ln n} = 3α + b − a − 2

where a, b and α are real positive numbers. Hence, the proof of Theorem 4.6 is completed.

\underline{References}

1. F. R. Barsoum and M. Nassif, “Inverse of Simple Sets of Polynomials Effective in Faber Region,” Proceedings of the Royal Netherlands Academy of Sciences; Ser. A, 65 (1962), 38–46.

2. B. Cannon, “On the Convergence of Series of Polynomials,” Proc. London Math. Soc., Ser. 2, 43 (1937), 348–364.

3. G. Faber, “Uber die Polynomische Entwickelungs,” Mathemtische Annalen, 57 (1903), 398–408.

4. M. Nassif, Product and Zeros of Basic Sets, Assiut University, Assiut, Egypt, 1958.

5. W. F. Newns, “On the Representation of Analytic Functions by Infinite Series,” Philosophical Transactions of the Royal Society of London, Ser. A, 245 (1953), 429–468.

6. W. F. Newns, “The Range of Effectiveness of a Basic Set,” Proceedings of the London Mathematical Society, Ser. 3, 18 (1968), 745–767.

7. K. A. Sayyed and M. S. Metwally, “q–order and q–type of Functions and Sets of Polynomials of Two Complex Variables,” Nat. Acad. Sci. Lett., 16 (1993), 247–252.

8. K. A. Sayyed and M. S. Metwally, “Logarithmic Order and Logarithmic Type of Functions and Sets of Polynomials of Two Complex Variables,” Bull. Fac. Sci. Qena (Egypt), 2 (1994), 127–140.

9. J. L. Ullman, “Studies in Faber Polynomials I,” Trans. Amer. Math. Soc., 94 (1960), 515–528.

10. J. M. Whittaker, Sur les Series de Polynmes Quelconques, Gauthier-Villars, Paris, 1949.

K. A. M. Sayyed Department of Mathematics Faculty of Science University of Assiut Assiut 71516, EGYPT

M. S. Metwally Department of Mathematics Faculty of Science University of Assiut Assiut 71516, EGYPT email: met641958@yahoo.com

R. S. Batahan Department of Mathematics Faculty of Science University of Assiut Assiut 71516, EGYPT