On a Problem of A. Eremenko

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On a Problem of A. Eremenko

Iossif V. Ostrovskii

(Communicated by Stephan Ruscheweyh)

Abstract. Let Pmn, 0 < m < n − 1, be a polynomial formed by the ﬁrst m terms of the expansion of (1 + z)n _{according to the binomial formula. We}

show that, ifm, n → ∞ in such a way that lim_m,n→∞m/n = α ∈ (0, 1), then the zeros ofP_mntend to a curve which can be explicitly described.

Keywords. Asymptotic formula, conformal mapping, polynomial,

subhar-monic function, Szeg˝o’s method, univalent function, zeros.

2000 MSC. 26C10, 30C15.

1. Introduction and statement of the problem

Consider the polynomials

Pmn(z) := m k=0 n k zk, 0_{< m < n − 1.}

The following problem was posed by A. Eremenko: describe the asymptotic zero

distribution of Pmn as m, n → ∞ in such a way that

(1) lim

m,n→∞

m

n =α ∈ (0, 1).

We give below a solution to this problem. Our method is based on some ideas of the famous paper [4] by G. Szeg˝o (extensive bibliography of works connected with this paper can be found in [2], [3]).

P. B. Borwein and W. Chen [1] have studied Pad´e approximations to powers of 1 +_{z, and especially sequences where the numerator degree divided by the} denominator degree tends to a ﬁnite positive constant. But the case where the denominator is of degree 0 is not treated there; this is precisely the case considered in our paper.

Received December 24, 2003, in revised form March 30, 2004.

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2. Basic formula and restatement of the problem

Lemma 1. _{For 0 < m < n − 1,} (2) Pmn(z) (1 +_z)n_mn(_{n − m)} = ₁ z/(z+1)u m₍₁_{− u)}n−m−1_du.

Proof. It is well known that for a suﬃciently good function _{f one has}

f (z) − p k=0 f(k)(0) k! z k ₌ 1 p! _z 0 (_{z − t)}p_f(p+1)(_{t) dt,} _{p ∈ N.} Applying this to _{f (z) = (1 + z)}n, _{p = n − m − 1, we obtain}

n k=n−m n k zk= n(n − 1) · · · (m + 1) (_{n − m − 1)!} _z 0 (_{z − t)}n−m−1(1 +_t)m_dt.

Since the left hand side of this formula coincides with_zn_P_mn(1_{/z), we derive that}

Pmn(z) = n(n − 1) · · · (m + 1)₍ n − m − 1)! z n 1 z 0 1 z − t _n−m−1 (1 +_t)m_dt. Substituting_{t = −1 + u(1 + z)/z, we obtain (2).}

Let

(3) _Q_mn(_{z) :=}

₁

z u

m₍₁_{− u)}n−m−1_du.

Evidently, _Q_mn is a polynomial in _{z of degree n having a zero of order n − m at}

z = 1. By (2), it is connected with Pmn by the formula

Pmn(z) = n m (n − m)(1 + z)nQmn z z + 1 .

Hence we see that all the zeros of _Q_mn, except the one at _{z = 1, are images of} zeros of _P_mn under bilinear transformation _{z → z/(z + 1). We will call these} zeros non-trivial. Evidently, the problem under consideration is equivalent to the description of the asymptotic behavior of non-trivial zeros of _Q_mn as _{m, n → ∞} under condition (1). The rest of the paper is devoted to the investigation of this problem.

3. A bound for non-trivial zeros

Lemma 2. _{All non-trivial zeros of the function Q}_mn are located in the disc {z ∈ C : |z| < m/(n − 1)}.

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Proof. The change of variable _{u = v/(1 + v) in the right hand side of (3) gives} Qmn z 1 +_z = _∞ z vm (1 +_v)n+1 dv. Integrating by parts _{m + 1 times, we obtain}

Qmn z 1 +_z = 1 n zm (1 +_z)n + m n(n − 1) zm−1 (1 +_z)n−1 +· · · + m! n(n − 1) · · · (n − m) 1 (1 +_z)n−m =: 1 n(1 + z)n−mSmn z 1 +_z ,

where _S_mn(_{ζ) is a polynomial in ζ. Denoting}

Smn(ζ) = n (1− ζ)n−mQmn(ζ) =: m k=0 ckζk, we see that ck = k + 1 n − m + kck+1≤ m n − 1ck+1, k = 0, 1, . . . , m − 1.

In the last inequality, equality occurs only for _{k = m − 1. Applying the well} known Kakeya Theorem, we obtain the assertion of the lemma.

Further, we ﬁx a sequence of pairs (_{m, n) such that (1) is satisﬁed. Denote by Z} the set of all accumulation points of

mn

{z ∈ C : z = 1, Qmn(z) = 0}.

The following lemma is an immediate corollary of Lemma 2.

Lemma 3. The set Z is a subset of {z ∈ C : |z| ≤ α}.

This lemma shows that we need to study the polynomials _Q_mn only in the disc

{z ∈ C : |z| ≤ α}.

4. The main asymptotic formula

Consider the curve

z ∈ C : |z||1 − z|1/α−1 =_b_α_, where

bα := max

x∈[0,1]x(1 − x)

1/α−1 ₌_{α(1 − α)}1/α−1_.

This curve consists of two loops surrounding 0 and 1 with _{z = α as the only} common point. Only the left loop will be of interest and it will be denoted

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-0.5 0.5 1 1.5 -0.5 0.5 α = 0.3 -0.5 0.5 1 1.5 -0.5 0.5 α = 0.5 -0.5 0.5 1 1.5 -0.5 0.5 α = 0.7

Figure 1. The curve {z ∈ C : |z||1 − z|1/α−1 = b_α} and its relevant part_C_α inside the disk {z ∈ C : |z| ≤ α}.

by _C_α. Note that _C_α ⊆ {z ∈ C : |z| ≤ α} and C_α lies in the interior of _C_β for

α < β.

For{z ∈ C : |z| ≤ α} consider the analytic function

Wα(z) := 1

bαz(1 − z)

1/α−1_,

where the branch assumes positive values on the interval (0_{, 1).}

Lemma 4. _{The function W}_α is univalent in {z ∈ C : |z| ≤ α} and maps the region bounded by Cα conformally onto the unit disc.

Proof. For_{θ ∈ (0, 2π) we have}

∂ ∂θ arg(Wα(αe iθ_{)) = 1 +} 1 α − 1 Im ∂ ∂θ log(1− αe iθ₎ = (1 +α)(1 − cos θ) 1 +_α2− 2α cos θ > 0.

Therefore, arg(_W_α(_{z)) increases when z traverses {z ∈ C : |z| = α} in the} positive direction. Since the origin is the only zero of _W_α, the total increment of the argument is 2_{π by the Argument Principle. So, the ﬁrst assertion of the} lemma is a consequence of a well known fact of conformal mapping theory. The second assertion is its immediate corollary because |W_α(_{z)| = 1 for z ∈ C}_α.

Lemma 5. The asymptotic formula

(4) 1

qmnQmn(z) = 1 − (1 +O(1)) m z

0

[_W_α(_u)]m_du _{as m → ∞,}

holds uniformly in {z ∈ C : |z| ≤ α}, where qmn := n 1

m

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Proof. Using (3), we have Qmn(z) = ₁ 0 u m₍₁_{− u)}n−m−1_{du −} z 0 u m₍₁_{− u)}n−m−1_du = _q_mn− q_mn _z 0 [_W_mn(_u)]m_du, where Wmn(u) = qmn−1/mu(1 − u)(n−1)/m−1. Noting that Wmn(z) = (1 +O(1))Wα(z) asm → ∞, uniformly in{z ∈ C : |z| ≤ α}, we obtain (4).

Now we can prove the main asymptotic formula for _Q_mn.

Lemma 6. _{For any δ > 0, the asymptotic formula}

(5) 1

qmnQmn(z) = 1 −

(1 +O(1))m[_W_α(_z)]m+1

(_{m + 1)W}_α(_z) as m → ∞.

holds uniformly in {z ∈ C : |z| ≤ α} \ {z ∈ C : |z − α| < δ}.

Proof. Formula (4) of Lemma 5 implies that it suﬃces to show that _z 0 [_W_α(_u)]m_{du =} [Wα(z)] m+1 (_{m + 1)W}_α(_z)(1 +O(1)), asm → ∞, holds uniformly in{z ∈ C : |z| ≤ α} \ {z ∈ C : |z − α| < δ}.

Note that, according to Lemma 4, _W_α is univalent in the disk {z ∈ C : |z| ≤ α}. Let us take the integral on the right hand side of (4) along the level curve

{u ∈ C : arg(Wα(u)) = arg(Wα(z)), 0 < |u| < |z|}.

On this curve |W_α(_{u)| is an increasing function of |u|. Therefore we have} _z 0 [_W_α(_u)]m_{du =} _W_α_(z) 0 W m α _WdW α α(u)

= exp_{i(m + 1) arg(W}_α(_z))

|Wα(z)| 0 |Wα| md|Wα| W_α(_u) = [Wα(z)] m+1 (_{m + 1)W}_α(_z)− exp i(m + 1) arg(Wα(z)) · _|W_α_(z)| 0 |Wα| m 1 W_α(_u) − 1 W_α(_z) d|Wα| = [Wα(z)] m+1 (_{m + 1)W}_α(_z)+O(|Wα(z)| m+1₎ = [Wα(z)] m+1 (_{m + 1)W}_α(_z)(1 +O(1)) as m → ∞.

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5. Main result

For any open disc _{D ⊂ C, denote by ν}_mn(_{D) the number of non-trivial zeros of} a polynomial _Q_mn in _{D. We say that the non-trivial zeros of the sequence Q}_mn have density _{d(D) in D if there exists the limit}

d(D) = lim

m→∞

νmn(D)

m .

Theorem. The setZ coincides with the curve C_α. Furthermore, for an arbitrary disc D ⊂ C the non-trivial zeros of the sequence Qmn have in D the density

d(D) = 1

2_πlengthWα(D ∩ Cα).

Proof. According to Lemma 3 we have Z ⊆ {z ∈ C : |z| ≤ α}. Let K be a compact subset of {z ∈ C : |z| ≤ α} lying entirely either inside of C_α or outside of_C_α. Then |W_α(_{z)| < 1 for z ∈ K in the former case and |W}_α(_{z)| > 1 for z ∈ K} in the latter. Therefore formula (5) implies that there exists the uniform limit

lim

m→∞Qmn(z) =

1 on any compact set lying inside of the curve_C_α_,

∞ on any compact set lying outside of the curve Cα.

This proves Z ⊆ C_α.

Lemma 6 also shows that the following limit exists:

Q(z) := lim

m→∞

1

m log|Qmn(z)| = log bα+ log

+_|W

α(z)| |z| ≤ α.

The function _{Q(z) is subharmonic in the disk {z ∈ C : |z| < α} and harmonic} in{z ∈ C : |z| ≤ α} \ C_α. Therefore its Riesz’ measure is supported by_C_α with the density ρ(z) = 1 2_π ∂ ∂nlog|Wα(z)| = 1 2_π ∂ ∂sarg(Wα(z)), z ∈ Cα

with respect to the Lebesgue measure on_C_α. Hence

d(D) = 1 2π D∩Cα ∂ ∂sarg(Wα(z)) |dz| = 1 2πlengthWα(D ∩ Cα).

Remark 1. Since zeros of _P_mn are images of non-trivial zeros of _Q_mn under the bilinear transformation _{T (z) := z/(1 − z), we can readily derive from the} theorem above that the set of all accumulation points of the set

mn

{z ∈ C : Pmn(z) = 0}

(under condition (1)) coincides with

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where _K_α is the closed disc with diameter [−α/(1 + α), α/(1 − α)] ⊂ R. Note also that Lemma 2 shows that, even without condition (1), all zeros of _P_mn, 0_{< m < n − 1, are contained in the open half-plane {z ∈ C : Re z > −1/2}.}

Remark 2. It can be shown that the function _W_α is univalent in the region_G_α lying on the left of

∂Gα = z ∈ C : arg(z) + 1 α − 1 arg(1− z) = 0, Im(z) = 0 ∪ {α}.

This curve goes through the point _{z = α and is located in}

{z ∈ C : Re z < α} ∪ {α}

for 0_{< α < 1/2, in}

{z ∈ C : Re z > α} ∪ {α}

for 1_{/2 < α < 1, and coincides with the half plane}

{z ∈ C : Re z = α}

for_{α = 1/2. The asymptotic formulas (4) and (5) remain in force on any compact} set in _G_α. -1.5 -1 -0.5 0.5 1 1.5 -1.5 -1 -0.5 0.5 1 1.5 α = 0.3 -1.5 -1 -0.5 0.5 1 1.5 -1.5 -1 -0.5 0.5 1 1.5 α = 0.5 -1.5 -1 -0.5 0.5 1 1.5 -1.5 -1 -0.5 0.5 1 1.5 α = 0.7

Figure 2. _{The curve} _∂G_α_.

Remark 3. After this work was completed, A. Klyachko informed the author that he has obtained a similar result by a diﬀerent method but has not published it.

Acknowledgement. I express my deep gratitude to Alexander Iljinskii and Natalya Zheltukhina for careful reading of the manuscript and valuable remarks. My greatest thanks to the copy editor for creating Figures 1 and 2 and many other improvements to the ﬁnal text.

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References

1. P. B. Borwein and W. Chen, Incomplete rational approximation in the complex plane, Constructive Approximation 11 (1995), 85–106.

2. A. Edrei, E. B. Saﬀ and R. S. Varga, Zeros of Sections of Power Series, Lecture Notes in Math. (Springer) 1002, 1983.

3. I. V. Ostrovskii, On zero distribution of sections and tails of power series, Isr. Math. Conf. Proc. 15 (2001), 297–310.

4. G. Szeg˝o, ¨_{Uber eine Eigenschaft der Exponentialreihe, Sitzungsber. Berl. Math. Ges. 23} (1924), 50–64; see also: G. Szeg¨o, Collected Papers, Vol.1, Birkh¨auser, Boston, 1982, 645–662.

Iossif V. Ostrovskii E-mail: iossif@fen.bilkent.edu.tr

Address: Department of Mathematics, Bilkent University, 06800 Bilkent, Ankara, Turkey; In-stitute for Low Temperature Physics and Engineering, 47 Lenin ave, 61103 Kharkov, Ukraine.