A lower bound for the L2[—1; 1]-norm of the logarithmic derivative of polynomials with zeros on the unit circle Текст научной статьи по специальности «Математика»

Probl. Anal. Issues Anal. Vol. 8 (26), No 2, 2019, pp. 67-72 67

DOI: 10.15393/j3.art.2019.6030

UDC 517.538.5

M. A. Komarov

A LOWER BOUND FOR THE L2[-1,1]-NORM OF THE LOGARITHMIC DERIVATIVE OF POLYNOMIALS WITH ZEROS ON THE UNIT CIRCLE

Abstract. Let C be the unit circle {z : |z| = 1} and Qn(z) be an arbitrary C-polynomial (i. e., all its zeros zi,..., zn £ C). We prove that the norm of the logarithmic derivative Q'n/Qn in the complex space L2[—1,1] is greater than 1/8.

Key words: logarithmic derivative, C-polynomial, simplest fraction, norm, unit circle

2010 Mathematical Subject Classification: 41A20, 41A29

1. The main result. The problem of whether the logarithmic derivatives of C-polynomials (see the Abstract), i.e., the rational functions of the form

n1

£ ^ (|Zil = ■■■ = |Znl = ^ k= 1

are dense in the complex space L2[-1,1], was raised by Nasyrov in 2014 during the talk of Borodin at the conference "Complex Analysis and its Applications" in Petrozavodsk (see [1, §4]). (Sums J^11/(z — zk) are also known as the simplest fractions or simple partial fractions.)

We find that the Nasyrov's question has the negative answer, namely,

-i

n 1

L—

k=ix - Zk

2x1/2 1

dx I > -8

for any z1,... ,zn on the unit circle.

The related result for the area integral was obtained by Newman [2]:

|z|<1

n 1

£iz - Zk

n

dxdy > ig (N = ••• = |zn| = 1),

(g) Petrozavodsk State University, 2019

i

where x + iy = z. Using the techniques in [2], Chui and Shen [3, §5] derived the order of approximation by sums of the form Y^i 1/(z — zk), zk E C, in the Bers spaces Aq(D), q > 2 (D is the interior of C). To prove (1), we use some ideas in [2], too.

2. Proof of (1). Let |zi| = ■ ■ ■ = |zn| = 1. Obviously,

^^ 1 = 1 / ^^ zk + z

k=1z — zk = 2 A k=i zk— z n

P(Zk; z) := Re ^ = 1 1 + ^ 0, |z| ^ 1, zk — z 1 — 2Re(zzk) + | z |2

and for any h > 0 and every k = 1,..., n the set of points z for which P(zk; z) ^ h, i.e.,

(h + 1)|z|2 — 2hRe(zzk) + h — 1 ^ 0, is the disk |z — zkh/(h + 1)| ^ 1/(h + 1) in the closed unit disk. Put

cn =1 — 6(5n)-2.

Case 1: There is a pole z* E {z1,... ,zn} such that |Re z*| ^ cn. We can assume that cn ^ Re z* ^ 1. Then there is the segment [an,pn],

4 5n — 4

0 < an < Pn ^ 1, fin an > - , ,

5n + 4 n n 5n + 4

such that

P(z*; x) ^ 5n/4 for an ^ x ^ Pn.

Indeed, the intersection of the disk {z : P(z*; z) ^ 5n/4} with the real axis is the segment [x1,x2] C (0,1], where x1 and x2 > x1 are the (real) roots of the equation

(5n + 4)x2 — 2(5n)x Re z * + 5n — 4 = 0

with the discriminant

d ^ 4(5n)2{1 — 12(5n)-2 + 36(5n)-4} — 4{(5n)2 — 16} > 16.

It is clear that x1x2 = (5n — 4)/(5n + 4) and x2 — x1 = \fd/(5n + 4).

Now, using the identity (3) and putting Zn = (zi,..., zn}, we get

1 o i

p n i 2 [ f n x2

Z(Zn):= / £ dx> / E P(zk; x)

k=1 k ^ k=1

dx > M > P(zk; x) - n ) 4xr2 >

x - zfc -1 k=1 -1

in

> /(p (* N )2 > & - an > ^ 4 > > _1_

> J (P(z ; x) n) 4x2 > 43 «n^n > 43 5n - 4 " 24 > 64.

an

Case 2: |Re zk| < cn for any k = 1,..., n. We have

P (zk; x) < 1/3 for x e An := [-1, - Yn] U [Yn, 1], k = 1,...,n,

Yn := 1 - (9/5)(5n)-2, since the product of the roots of every polynomial

q(x) = q(zk; x) = 4x2 - 2x Re zk - 2

is negative (= -1/2), while the values of q at the points ±1 and ±Yn are positive, so that q(x) > 0 for x e An. Thus

2 1

dx „ f n2 2 1

/(" - E p (zk; x^4s2 >21

An k=1 Yn

/(Zn) > / (n -> ^(zk; x)) — > 2 I — dx = — > -

and the proof is complete. Note that /(Zn) > n/80 in Case 1.

3. Refinements of (1). It is of interest to find an order of inf(/(Zn) : Zn C C}. We find this order in two cases. Let C+ (C+) be the intersection of the unit circle with the upper (right) closed half-plane.

Proposition. There is an absolute constant 0 < c ^ n/2 such that

/(Zn) ^ cn2 (4)

for all n in N and any Zn in C+ or C+. This bound is sharp in order n.

Proof. First let Zn C C+. By the theorem of Govorov and Lapenko [4, Theorem 2] with r =1 and 5 = (2/3)(35e)-1, we have

E

x - zk k=1 k

2 1 2

> n • - •- for x G A C [—1,1], mes A > -;

3 35e L ' J' 3'

therefore /(Zn) > c'n2, C := (2/3)3(35e)-2

1

If Zn C C+, i.e., all Rezk * 0, then we have q(zk; x) * 0 for any x £ [—1, — 1^V/2] and every k (see Case 2, § 2), so that

-i/V2

I (Zn) > J (n — n/3)2 4dX2 = c"n2, C" ,

-i

C > cC. Thus, (4) holds with c = cC.

Finally, the sharpness of (4) in order n as well as the estimate c ^ n/2 follow from the example of the C-polynomial (z — i)n:

i i

J |n/(x — i)|2dx = 2n2 J dx/(x2 + 1) = nn2/2. -i o ^

It seems that in the general case, Zn C C, the following result is true.

Conjecture. There is an absolute constant 0 < c ^ ln2 such that

I (Zn) * cn (5)

for all n in N and any Zn in C. This bound is sharp in order n. To prove the sharpness of (5) we use the C-polynomial zn — i: i „ i

nxn i

, ti-(i/n) f , , s

dx = 2n t 1 dt = nln2 + t-i-(i/n) ln(t2 + 1)dt

-i o

(the last integral is less than 1, because ln(t2 + 1) < t2 for 0 < t ^ 1).

4. A lower bound for the L2-norm in the case of the unit disk.

It follows from (2) and the Schwarz inequality that

/ r r i 2 \ i/2 /—

[JJ £ — dxdy) * f (N = - = lz.l = D.

|z|<i k=i

Using the techniques in the proof of (1), we can derive a more sharp bound. Indeed, the consideration of two cases (cf. §2):

1) there is a pole z* £ {zi,... ,zn} such that cn ^ Rez* ^ 1,

2) —1 ^ Re zk < cn for all k = 1,..., n

xn i

shows that for any = {zi,..., zn} C C

E

fc=i

x - zfc

>

2 • 82

But all points zke ^ G R, also belong to C, so that i

I:=

E

fc=i

1

re^ — zk

rdr >

1

2 • 82

for any ^ G R.

Finally, for any zi,... ,zn on the unit circle, we have

2n

|z|<1

E

fc=1

1

z — zfc

dxdy I

\ 1/2

I(Zra; <p)d^j

i/2

>

n

8

Acknowledgment. The author is grateful to the referees for their useful comments and suggestions.

This work was supported by RFBR project 18-31-00312 mol_a.

References

[1] Borodin P. A., Approximation by simple partial fractions with constraints on the poles. II. Sb. Math., 2016, vol. 207, no. 3, pp. 331-341.

DOI: https://doi .org/10.1070/SM8500.

[2] Newman D. J., A lower bound for an area integral. Amer. Math. Monthly, 1972, vol. 79, no. 9, pp. 1015-1016. DOI: https://doi.org/10.2307/ 2318074.

[3] Chui C. K., Shen X.-C., Order of approximation by electrostatic fields due to electrons. Constr. Approx., 1985, vol. 1, no. 1, pp. 121-135.

DOI: https://doi .org/10.1007/BF01890026.

[4] Govorov N. V., Lapenko Yu. P., Lower bounds for the modulus of the logarithmic derivative of a polynomial. Math. Notes, 1978, vol. 23, no. 4, pp. 288-292. DOI: https://doi.org/10.1007/BF01786958.

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Received February 28, 2019. In revised form, May 20, 2019. Accepted May 20, 2019. Published online May 22, 2019.

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