BOHR RADII OF ELLIPTIC REGIONS

H. Turgay Kaptanoğlu & Nazım Sadık

Abstract. We use Faber series to define a Bohr radius for a simply connected planar domain bounded by an analytic Jordan curve. We estimate the value of the Bohr radius for elliptic domains of small eccentricity and show that these domains do not exhibit Bohr phenomenon when the eccentricity is large. We obtain the classical Bohr radius when the eccentricity tends to 0.

1. Introduction

A classical theorem of Bohr states that if a holomorphic function f with power series f(z) = ∑ a_{n}z^{n} maps the open unit disc 𝔻 to itself, then ∑ |a_{n}z^{n}| < 1 for all |z| < 1/3. The value 1/3 is the best possible and called the Bohr radius of 𝔻. The interest in Bohr phenomena revived in the 1990’s due to the discovery of its generalizations to domains in ℂ^{n} and to more abstract settings; see [BK], [A], [AAD1], [AAD2], [PPS], and [DGM]. But the only domain on which the exact value of the Bohr radius is known is 𝔻. Exceptions are [AT] and [K] where exact values are computed, but they are for harmonic functions.

In this work we return to ℂ and study Bohr phenomena on a planar domain E. There are two problems that need to be resolved. Power series must be replaced by another series that converge globally on E. And certain subsets of E must be associated to discs so that the concept of radius makes sense. Under certain conditions on E, the space of holomorphic functions H(E) on E can be given a basis {F_{n}} made up of polynomials, called the Faber polynomials of E, in the sense that if f ∈ H(E), then f = ∑ a_{n}F_{n} on all of E. Faber polynomials are defined via a mapping of the complement of E onto the complement of a disc. These properties of Faber polynomials allow us to define a Bohr radius for E in a natural way in Section 2.

In Section 3, we concentrate on elliptic regions E. The reasons are that a computation of Bohr radius must be done with a concrete basis, and an ellipse is one of the few regions whose Faber polynomials are known and sufficiently simple; they are essentially the Chebyshev polynomials. We find lower and upper bounds for the Bohr radius of E when the eccentricity of the ellipse is less than about 0.37 and 0.41, respectively. These bounds are the roots of certain quadratic polynomials. Moreover, we show that an ellipse does not have the Bohr property considered

Typeset by AMS-TEX

2000 Mathematics Subject Classification. Primary 30B99, 33C45; Secondary 30C20.

Key words and phrases. Bohr radius, ellipse, Faber polynomial, Faber series, Chebyshev polynomial, capacity, Carathéodory inequality.

The work of the second author is supported by the Research Fund of İstanbul University, project number UDP-259/29032004.

when the eccentricity is more than about 0.41. The precise statements are in Theorems 7 and 8 below, which are our main results. As the eccentricity decreases to 0, however, E becomes 𝔻, and both bounds for the Bohr radius of E increase and converge to the classical Bohr radius of 1/3.

2. Faber Polynomials and Faber Series

We first review the basics of Faber theory enough to let us define a Bohr radius for a sufficiently general domain. Much of this section can be found in standard references mentioned as required.

Let E be a nonempty bounded simply connected domain in ℂ. The its closure \overline{E} is compact, and the complement \overline{E}^{c} of this closure in the extended plane ℂ ∪ \{∞\} is a simply connected set containing ∞. By the Riemann mapping theorem, there is a unique conformal (univalent) map Φ of \overline{E}^{c} onto \overline{𝔻}^{c} satisfying

Φ(∞) = ∞  and  Φ′(∞) = lim_{z→∞} \frac{Φ(z)}{z} = \frac{1}{d} > 0.

Here d is the logarithmic capacity (or transfinite diameter) of \overline{E} and can be defined by d = exp(lim_{z→∞}(log |z| − G(z, ∞))), where G is the Green function for \overline{E}^{c} with singularity at ∞; see [F, pp. 114–119].

The function Φ and its nonnegative integer power Φ^{n} have Laurent series at ∞ of the form

Φ(z) = \frac{1}{d}z + γ_{0} + \frac{γ_{−1}}{z} + \frac{γ_{−2}}{z^{2}} + ⋯

and

Φ^{n}(z) = \frac{1}{d^{n}} z^{n} + γ_{n,n−1}z^{n−1} + ⋯ + γ_{n,0} + \frac{γ_{n,−1}}{z} + \frac{γ_{n,−2}}{z^{2}} + · · · .

The principal part at ∞ of the Laurent expansion of Φ^{n}(z) is called the nth Faber polynomial P_{n}(z) of \overline{E} (or for Φ). Hence

P_{0}(z) = 1,  P_{1}(z) = \frac{z}{d} + γ_{0},
P_{n}(z) = \frac{z^{n}}{d^{n}} + γ_{n,n−1}z^{n−1} + ⋯ + γ_{n,1}z + γ_{n,0},

and P_{n} is a polynomial of degree n,

Suppose now the boundary ∂E of E is an analytic Jordan curve, which is the situation originally considered by Faber. This occurs, for example, when E is bounded by an ellipse which is our main case of interest. Let Ψ = Φ^{−1} : \overline{𝔻}^{c} → \overline{E}^{c}. Then Ψ continues holomorphically across ∂𝔻, and there is a least radius R < 1 such that Ψ is univalent for |w| > R; see [C]. For r > R, let C_{r} = { z = Ψ(w) : |w| = r } be a level set of Φ. It is a Jordan curve, and we let E_{r} be the inside of C_{r}. Specifically, C_{1} = ∂E and E_{1} = E. Then the above extension result about Ψ can be restated as that Φ is holomorphic and univalent on \overline{E_{R}}^{c}.

The following analogs of Abel theorems and Cauchy estimates are due to Faber and can be found in [S, Chapter III] and [M, Section 3.14]. If ∂E is not analytic, then R is taken as 1.

Theorem 1. Suppose the series ∑ a_{n}P_{n} satisfies lim sup \sqrt[n]{|a_{n}|} = 1/r < 1/R. Then the series converges absolutely in E_{r} and uniformly on its compact subsets, and thus the sum is in H(E_{r}).

Theorem 2. If r > R and f ∈ H(E_{r}), then f can be represented on E_{r} uniquely as the sum of a series of the form f(z) = ∑ a_{n}P_{n}(z), which converges absolutely on E_{r} and uniformly on its compact subsets.

Thus the Faber polynomials {P_{n}} form a natural basis for the space H(E) when ∂E is analytic. This basis satisfies the necessary condition P_{0} ≡ 1 in Proposition 3.1 of [AAD2] for having a Bohr property.

The series in Theorems 1 and 2 are called Faber series. Let F_{n}(w) = P_{n}(Ψ(w)) for |w| > R. These functions are often easier to work with than P_{n}(z). For f ∈ H(E_{r}), setting F (w) = f(Ψ(w)), we also have F (w) = ∑ a_{n}F_{n}(w) at least for R < |w| < r.

Corollary 3. Under the same conditions as in Theorem 2,

|a_{n}| ≤ \frac{max\{ |f(z)| : z ∈ C_{r} \}}{r^{n}}.

So the curves C_{r} play the same role for Faber series as circles do for power series.

Definition 4. Let E be a bounded simply connected domain in ℂ whose boundary is an analytic Jordan curve. We say E has the Bohr property if there exists an r with R < r < 1 such that if f : E → 𝔻 is holomorphic with Faber expansion f(z) = ∑ a_{n}P_{n}(z), then ∑ |a_{n}| sup_{z∈E_{r}} |P_{n}| < 1. We define the Bohr radius B of E

as the largest of the numbers r.

Equivalently, B is the largest number r satisfying ∑ |a_{n}| sup_{R<|w|<r} |F_{n}(w)| < 1.

We can also say that the basis {P_{n}} has the Bohr property, and B is the Bohr radius of {P_{n}}.

Example 5. We work out a test case. Let E = D(c, d) be the disc of radius d centered at c. Then Φ : \overline{D(c, d)}^{c} → \overline{𝔻}^{c} is given by Φ(z) = (z − c)/d and shows that the capacity of a disc is its radius. The inverse of Φ is Ψ(w) = dw + c. Both Φ and Ψ are holomorphic and univalent on ℂ. Then P_{n}(z) = Φ^{n}(z) = (z − c)^{n}/d^{n} and F_{n}(w) = w^{n}. For any r > 0, we have E_{r} = { z ∈ E : |z − c| < rd } and sup{ |P_{n}(z)| : z ∈ E_{r} } = r^{n}. Now if f(z) = ∑ a_{n}P_{n}(z) maps E to 𝔻, then ∑ |a_{n}| sup_{E_{r}} |P_{n}| = ∑ |a_{n}|r^{n}, and this is the series used in the classical Bohr theorem. Therefore E_{1/3} = D(c, d/3) is the largest set that is the inverse image under Φ of a disc on which ∑ |a_{n}| sup_{E_{r}} |P_{n}| < 1.

We need the following lemma which is a Carathéodory inequality for a function with a singularity at 0.

Lemma 6. If

G(w) = b_{0} + ∑_{n=1}^{∞}(b_{n}w^{n} + \frac{c_{n}}{w^{n}} )

is holomorphic on R < |w| < 1 for some R ≥ 0 and has positive real part, then ||b_{n}| − |c_{n}|| ≤ 2 Re b_{0} for n = 1, 2, . . ..

Proof. By considering radii increasing to 1, we may assume that G is holomorphic in a neighborhood of ∂𝔻. We take n = 1, 2, … and compute

\frac{1}{2π} ∫_{0}^{2π} e^{−inθ} G(e^{iθ}) dθ = b_{n}  and  \frac{1}{2π} ∫_{0}^{2π} e^{−inθ} \overline{G(e^{iθ})} dθ = \overline{c_{n}}.

Adding yields

||b_{n}| − |c_{n}|| ≤ |b_{n} + \overline{c_{n}}| ≤ \frac{1}{2π} ∫_{0}^{2π} |e^{−inθ}| |G(e^{iθ}) + \overline{G(e^{iθ})}| dθ

= \frac{1}{2π} ∫_{0}^{2π} 2 Re G(e^{iθ}) dθ = 2 Re b_{0}.

When G is holomorphic in 𝔻, all c_{n} = 0, and the result reduces to the Carathéodory inequality. □

3. Elliptic Regions

In the rest of the paper, E is the domain bounded by the ellipse x^{2}/a^{2}+y^{2}/b^{2} = 1, where a > b > 0, and c = \sqrt{a^{2} − b^{2}} is the focal distance. The Zhukovski map

Ψ(w) = \frac{1}{2}((a + b)w + \frac{a − b}{w})

takes the unit circle onto ∂E and hence \overline{𝔻}^{c} onto \overline{E}^{c}. We obtain the well-known fact that

d = lim_{w→∞} \frac{Ψ(w)}{w} = \frac{a + b}{2}.

The derivative Ψ′(w) is equal to 0 when w = ±R, where

(1)  R = \sqrt{\frac{a − b}{a + b}} = \frac{c}{2d} < 1.

Thus although Ψ is holomorphic for all |w| > 0, it is univalent only for |w| > R. The circle |w| = R is mapped onto the line segment [−c, c] ⊂ E in a two-to-one manner. The inverse of Ψ is

Φ(z) = \frac{z + \sqrt{z^{2} − c^{2}}}{a + b} ,

and is holomorphic and univalent on E \ [−c, c].

The eccentricity ε < 1 of an ellipse is the ratio c/a = \sqrt{1 − (b/a)^{2}}. Solving for the ratio b/a in (1), we obtain ε in terms of R; and then inverting it, we obtain R in terms of ε; so

(2)  ε = \frac{2R}{1 + R^{2}}  and  R = \frac{1 − \sqrt{1 − ε^{2}}}{ε} .

Let n = 1, 2, . . .. A computation shows that R^{2n}/Φ^{n}(z) has no principal part at ∞. Then Φ^{n}(z) and Φ^{n}(z) + R^{2n}/Φ^{n}(z) have the same principal part at ∞. But

(3)  Φ^{n}(z) + \frac{R^{2n}}{Φ^{n}(z)} = \frac{1}{(a + b)^{n}} [(z + \sqrt{z^{2} − c^{2}})^{n} + (z − \sqrt{z^{2} − c^{2}})^{n}]

is a polynomial of degree n since the odd powers of the radical cancel out and its even powers are polynomials. Thus this sum is the nth Faber polynomial P_{n}(z) of \overline{E}. An explicit expression for P_{n}(z) is given in [C] as

P_{n}(z) = \frac{2}{(a + b)^{n}} ∑_{k=0}^{⌊n/2⌋} \binom{n}{2k} z^{n−2k} (z^{2} − c^{2})^{k}.

On the other hand,

F_{n}(w) = P_{n}(Ψ(w)) = Φ^{n}(Ψ(w)) + \frac{R^{2n}}{Φ^{n}(Ψ(w))} = w^{n} + \frac{R^{2n}}{w^{n}} .

Of course, P_{0}(z) = F_{0}(w) = 1. Substituting z = c cos t in (3), we also obtain that P_{n}(z) = 2R^{n} cos(n cos^{−1}(z/c)). Thus the Faber polynomials of E are the suitably normalized Chebyshev polynomials.

Let R < r < 1. We have

max_{z∈C_{r}} |P_{n}(z)| = max_{|w|=r} |F_{n}(w)| = max_{θ} |r^{n}e^{inθ} + \frac{R^{2n}}{r^{n}e^{inθ}} | = r^{n} + \frac{R^{2n}}{r^{n}}

> 2R^{n} = max_{|w|=R} |F_{n}(w)|.

Thus

(4)  sup_{z∈E_{r}} |P_{n}(z)| = sup_{R<|w|<r} |F_{n}(w)| = max_{|w|=r} |F_{n}(w)| = r^{n} + \frac{R^{2n}}{r^{n}} .

The polynomials P_{n} have the property that P_{n}(0) = 0 when n is odd, and

P_{n}(0) = 2(−1)^{n/2}R^{n} = 2(−1)^{n/2} (\frac{a − b}{a + b})^{n/2} ≠ 0 (2 ≤ n ∈ 2ℤ).

This property of Chebyshev polynomials persists also at other points the interval [−c, c] ⊂ E_{r}. Thus the natural basis {P_{n}} for H(E) does not satisfy the sufficient condition of vanishing at some fixed point given in Theorem 3.3 of [AAD2] for having a Bohr property. Yet it has the Bohr property defined above, at least for certain values of R, as the following theorem shows.

Theorem 7. Let E be a domain bounded by an ellipse of eccentricity ε. There is a constant ε_{1} ≈ 0.373814 such that if ε < ε_{1}, then a lower bound for the Bohr radius of E is the larger root of the polynomial q(r) = (3−R^{2})r^{2}−(1+4R^{2}−R^{4})r+R^{2}(3−R^{2}), where R and ε are related by (2).

Proof. Let f(z) = ∑ a_{n}P_{n}(z) be the Faber expansion of a nonconstant holomorphic f : E → 𝔻. Substituting z = Ψ(w), we obtain the nonconstant holomorphic function F : 𝔻 \ \overline{D(0, R)} → 𝔻 with expansion

F (w) = a_{0} + ∑_{n=1}^{∞} a_{n}(w^{n} + \frac{R^{2n}}{w^{n}} ) (|w| = r > R).

By Corollary 3, |a_{0}| < 1 and |a_{n}| ≤ 1 for n = 1, 2, . . ..

Choose ϕ so that e^{iϕ}a_{0} = |a_{0}| and put G(w) = 1 − e^{iϕ}F (w). Then G is holomorphic where F is, and its real part is positive. Applying Lemma 6 to G for which b_{0} = 1 − e^{iϕ}a_{0}, b_{n} = a_{n}, and c_{n} = a_{n}R^{2n}, we obtain

|a_{n}| ≤ \frac{2(1 − |a_{0}|)}{1 − R^{2n}} ≤ \frac{2(1 − |a_{0}|)}{1 − R^{2}}.

Then using (4) gives

∑_{n=0}^{∞} |a_{n}| sup_{z∈E_{r}} |P_{n}(z)| ≤ |a_{0}| + \frac{2(1 − |a_{0}|)}{1 − R^{2}} ∑_{n=1}^{∞}(r^{n} + \frac{R^{2n}}{r^{n}} )

= |a_{0}| + \frac{2(1 − |a_{0}|)}{1 − R^{2}}(\frac{r}{1 − r} + \frac{R^{2}/r}{1 − R^{2}/r} )

= |a_{0}| + (1 − |a_{0}|)[\frac{2}{1 − R^{2}} \frac{r^{2} − 2R^{2}r + R^{2}}{−r^{2} + (1 + R^{2})r − R^{2}} ].

The last expression is less than 1 if and only if the product of the two fractions in brackets is less than 1 which holds if and only if

\frac{(3 − R^{2})r^{2} − (1 + 4R^{2} − R^{4})r + R^{2}(3 − R^{2})}{2(1 − r)(r − R^{2})} = \frac{q(r)}{2(1 − r)(r − R^{2})} < 0.

The denominator and the coefficient of r^{2} in q(r) are positive, so the fraction takes negative values if and only if q(r) has two real roots which occurs if and only if its discriminant δ(R) = (1 − R^{2})(1 − 27R^{2} + 11R^{4} − R^{6}) is positive. The polynomial 1 − 27x + 11x^{2} − x^{3} has two real roots that are greater than 1 and of no interest to us, and a smallest real root between 0 and 1. Then this polynomial is positive for x between 0 and the smallest root. Let r_{0} be the square root of the smallest root; we compute r_{0} ≈ 0.193937. Then δ(r) is positive and q(r) takes negative values under the condition R < r_{0}. The corresponding condition on eccentricity is ε < ε_{1} ≈ 0.373814. If r_{1} < r_{2} are the roots of q(r), then q(r) < 0 for r_{1} < r < r_{2}. After some tedious work simplifying polynomial expression, we show that 0 < r_{1} < R < r_{2} < 1. Since r > R, we have q(r) < 0, which implies ∑ |a_{n}| sup_{E_{r}} |P_{n}| < 1, if and only if R < r < r_{2}. Thus the Bohr radius of E satisfies B ≥ r_{2}. □

Theorem 8. Let E be a domain bounded by an ellipse of eccentricity ε. There exists a constant ε_{2} ≈ 0.408804 with the following property. If ε < ε_{2}, then an upper bound for the Bohr radius of E is the larger root of the quadratic polynomial Q(r) = (3 + 5R^{2})r^{2} − (1 + 8R^{2} + 7R^{4})r + R^{2}(3 + 5R^{2}), where R and ε are related by (2). If ε ≥ ε_{2}, then E does not have the Bohr property.

Proof. To obtain the upper bound in the classical Bohr theorem, one function that is used is the disc automorphism

h_{t}(z) = \frac{t − z}{1 − tz} = t − \frac{1 − t^{2}}{t} ∑_{n=1}^{∞} t^{n}z^{n},

where 0 < t < 1 (see [B]). In analogy with this, we let

G_{t}(w) = t − \frac{1 − t^{2}}{t} ∑_{n=1}^{∞} t^{n}(w^{n} + \frac{R^{2n}}{w^{n}} ) (R < |w| < 1)

with the same values of t. The corresponding function on E is

g_{t}(z) = G_{t}(Φ(z)) = t − \frac{1 − t^{2}}{t} ∑_{n=1}^{∞} t^{n}(Φ^{n}(z) + \frac{R^{2n}}{Φ^{n}(z)}) = t − \frac{1 − t^{2}}{t} ∑_{n=1}^{∞} t^{n}P_{n}(z).

Theorem 1 shows that g_{t} converges and is holomorphic on E_{1/t} ⊃ E. This can also be seen from the fact that G_{t} converges and is holomorphic for tR^{2} < |w| < 1/t which contains the annulus R ≤ |w| < 1, even when t = 1. Summing the series, we obtain

G_{t}(w) = \frac{w^{2} − t(1 + 2R^{2} − t^{2}R^{2})w + R^{2}}{tw^{2} − (1 + t^{2}R^{2})w + tR^{2}} = \frac{1}{t} − \frac{(1 − t^{2})(1/t − tR^{2})}{t(1/t − w)(1 − tR^{2}/w)}.

Now it is clear that

∥g_{t}∥ = ∥G_{t}∥ = sup_{R<|w|<1} |G_{t}(w)| = G_{t}(−1) = \frac{1 + R^{2} + tR^{2}(1 − t)}{1 + tR^{2}}.

We set F_{t}(w) = G_{t}(w)/∥G_{t}∥ and f_{t}(z) = g_{t}(z)/∥g_{t}∥ = ∑ a_{n}P_{n}(z).

By virtue of (4),

∑_{n=0}^{∞} |a_{n}| sup_{z∈E_{r}} |P_{n}(z)| = \frac{2t − G_{t}(r)}{∥G_{t}∥} = \frac{1}{∥G_{t}∥}[2t − \frac{r^{2} − t(1 + 2R^{2} − t^{2}R^{2})r + R^{2}}{−tr^{2} + (1 + t^{2}R^{2})r − tR^{2}} ]

= \frac{1}{∥G_{t}∥} \frac{(1 − 2t^{2})r^{2} + t(1 − 2R^{2} + 3t^{2}R^{2})r + R^{2}(1 − 2t^{2})}{(1 − tr)(r − tR^{2})}.

The last expression is greater than or equal to 1 if and only if the final fraction is greater than or equal to ∥G_{t}∥ which holds if and only if

\frac{(1 − t)Q_{t}(r)}{(1 − tr)(r − tR^{2})(1 + tR^{2})} ≥ 0,

where Q_{t}(r) is the quadratic polynomial

[(1+2t)+tR^{2}(2+3t)]r^{2}−[1+R^{2}(1+4t+3t^{2})+t^{2}R^{4}(3+4t)]r+R^{2}[(1+2t)+tR^{2}(2+3t)]

obtained after a very long computation. All factors multiplying Q_{t} in the above inequality are positive. We cancel them and then let t → 1^{−}. Consequently ∑ |a_{n}| sup_{E_{r}} |P_{n}| ≥ 1 if and only if

Q(r) = (3 + 5R^{2})r^{2} − (1 + 8R^{2} + 7R^{4})r + R^{2}(3 + 5R^{2}) ≥ 0.

The discriminant of Q is Δ(R) = (1 − R^{2})(1 − 19R^{2} − 61R^{4} − 49R^{6}). The polynomial 1 − 19x − 61x^{2} − 49x^{3} has one positive root between 0 and 1 and two complex roots, and hence is nonnegative for x between 0 and the positive root. Let R_{0} be the square root of the positive root; we compute R_{0} ≈ 0.213740. The corresponding eccentricity is ε_{2} ≈ 0.408804.

First suppose ε < ε_{2} or equivalently R < R_{0}. Then Δ(R) > 0 and Q(r) has two real roots. If R_{1} < R_{2} are these roots, after more tedious work, we show that 0 < R_{1} < R < R_{2} < 1. Then Q(r) ≥ 1, which implies ∑ |a_{n}| sup_{E_{r}} |P_{n}| ≥ 1, if and only if r ≥ R_{2}. Thus the Bohr radius of E satisfies B ≤ R_{2}.

Next suppose ε ≥ ε_{2} or equivalently R ≥ R_{0}. Then Δ(R) ≤ 0, and Q(r) ≥ 0 for all r. This implies that ∑ |a_{n}| sup_{E_{r}} |P_{n}| ≥ 1 for all r > R. Thus there is no Bohr phenomenon for {P_{n}} in this case. □

Remark 9. By (2), R is an increasing function of ε. Moreover, one can show that both r_{2} and R_{2} are decreasing functions of R whenever they exist. We conclude that the upper and lower bounds for B increase as ε → 0^{+}.

For example, R = 0.19 corresponds to ε ≈ 0.367, where we have r_{2} ≈ 0.225790 and R_{2} ≈ 0.278434. Also R = 0.1 corresponds to ε ≈ 0.198, where we have r_{2} ≈ 0.316163 and R_{2} ≈ 0.323407. In each case, r_{2} ≤ B ≤ R_{2}.

In the case ε = 0, we have b = a, c = 0, R = 0, and E becomes a disc of radius d = a = b. Both q(r) and Q(r) turn into the polynomial 3r^{2} −r whose only relevant root is r = r_{2} = R_{2} = 1/3, which is the Bohr radius. Thus this special case turns out to be what is considered in Example 5.

References

[A] L. Aizenberg, Multidimensional Analogues of Bohr’s Theorem on Power Series, Proc. Amer. Math. Soc. 128 (2000), 1147–1155.

[AAD1] L. Aizenberg, A. Aytuna, & P. Djakov, An Abstract Approach to Bohr’s Phenomenon, Proc. Amer. Math. Soc. 128 (2000), 2611–2619.

[AAD2] L. Aizenberg, A. Aytuna, & P. Djakov, Generalization of a Theorem of Bohr for Bases in Spaces of Holomorphic Functions of Several Complex Variables, J. Math. Anal. Appl. 258 (2001), 429–447.

[AT] L. Aizenberg & N. Tarkhanov, A Bohr Phenomenon for Elliptic Equations, Proc. London Math. Soc. 82 (2001), 385–401.

[B] H. P. Boas, Majorant Series, J. Korean Math. Soc. 37 (2000), 321–337.

[BK] H. P. Boas & D. Khavinson, Bohr’s Power Series Theorem in Several Variables, Proc. Amer. Math. Soc. 125 (1997), 2975–2979.

[C] J. H. Curtiss, Faber Polynomials and the Faber Series, Amer. Math. Monthly 78 (1971), 577–596.

[DGM] A. Defant, D. Garcia, & M. Maestre, Bohr’s Power Series Theorem and Local Banach Space Theory, J. Reine Angew. Math. 557 (2003), 173–197.

[F] W. H. J. Fuchs, Topics in the Theory of Functions of One Complex Variable, Van Nostrand Math. Stud., vol. 12, Van Nostrand, Princeton, 1967.

[K] H. T. Kaptanoğlu, Bohr Phenomena for Laplace-Beltrami Operators, preprint.

[M] A. I. Markushevich, Theory of Functions of a Complex Variable, 2nd ed., vol. III, Chelsea, New York, 1977.

[PPS] V. I. Paulsen, G. Popescu, & D. Singh, On Bohr’s Inequality, Proc. London Math. Soc. 85 (2002), 493–512.

[S] P. K. Suetin, Series of Faber Polynomials, Gordon & Breach, Amsterdam, 1998.

Mathematics Department, Middle East Technical University, Ankara 06531, Turkey

   E-mail address: kaptan@math.metu.edu.tr
   URL: http://www.math.metu.edu.tr/∼kaptan/

Mathematics Department, İstanbul University, Vezneci̇ler, İstanbul 34459, Turkey

   E-mail address: sadnaz@istanbul.edu.tr
   URL: http://www.istanbul.edu.tr/fen/matematik/nazsad.htm