Distribution of the logarithmic derivative of a rational function on the line
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DISTRIBUTION OF THE LOGARITHMIC DERIVATIVE OF A RATIONAL FUNCTION ON THE LINE M. A. KOMAROV Vladimir State University, Gor kogo str. 87, 600000 Vladimir, Russia e-mail: [email protected] (Received April 29, 2020; revised July 7, 2020; accepted July 13, 2020)
Abstract. We prove that for an arbitrary real rational function r of degree n, a measure of the set {x ∈ R : |r (x)/r(x)| ≥ n} is at most 2πΘ (Θ ≈ 1.347 is the weak (1, 1)-norm of the Hilbert transform), and this bound is extremal. A problem of rational approximations on the whole real line is also considered.
1. Main result Let · 1 be the norm in the space L = L1 (R). By the classical Kolmogorov inequality, for any real function f ∈ L we have (1)
m{x : |Hf (x)| ≥ δ} ≤ f 1Θ/δ
(δ > 0),
where m is the Lebesgue measure on the real line, Hf is the Hilbert transform 1 ∞ f (t) dt, −∞ < x < +∞. Hf (x) = v.p. π −∞ t − x The smallest possible value of a constant in (1), (2)
Θ=
2 π2
π 0
−1 θ = 1.347 . . . , ln cot dθ 2
was found by Davis [5] (see also the paper of Baernstein [2]). Using this result, we shall obtain an extremal estimate of the distribution function of the logarithmic derivative of real rational functions on the real line. Key words and phrases: logarithmic derivative of a rational function, Kolmogorov inequality for the Hilbert transform, uniform approximation on the line, Lipschitz condition. Mathematics Subject Classification: 26C15, 42A50, 41A20, 41A25.
0236-5294/$20.00 © 2020 Akade ´miai Kiado ´, Budapest, Hungary
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M. A. KOMAROV
For an arbitrary rational function r of degree n ≥ 1 we set (δ > 0). M (r, δ) = m x : r (x)/(nr(x)) ≥ δ Theorem 1. If r is a real rational function of an arbitrary degree n = 1, 2, . . ., then for any δ > 0 we have (3)
M (r, δ) ≤ 2πΘ/δ,
where Θ is defined by (2). The constant 2πΘ cannot be replaced by a smaller number in the sense that for any ε ∈ (0, 1) there are an even number n ˜=n ˜ (ε) and an n ˜ th-degree real rational function r˜ such that
(4)
M (˜ r, δ) ≥ 2πΘ · (1 − ε)/δ.
Estimates of the quantities M (r, δ) are well known for the case of polynomials. By the fundamental Boole’s result [3] (see also [14, p. 234]), we have M (P, δ) = 2/δ for any δ > 0 and any polynomial P , all of whose zeros are real. Later on, Macintyre and Fuchs [18] proved that if P belongs to P c , the class of all complex polynomials, then M (P, δ) ≤ 2e/δ; the sharp constant was obtained by Govorov and Grushevskii [12]: (5)
sup M (P, δ) = 4/δ
P ∈P c
(it follows from (5) that M (r, δ) ≤ 16/δ for an arbitrary complex rational function r [15], but the constant 16 may not be the best). Developing an approach of [12], the author in [15] established an analogous result for the class P of all real polynomials: (6)
sup M (P, δ) = π/δ.
P ∈P
For regular arcs Γ ⊂ C, an estimate of a measure of the set {z ∈ Γ : |P (z)/P (z)| ≥ δ}, P ∈ P c , was constructed by Pekarskii [23]. Govorov and Lapenko [13] obtained an interesting converse estimate: if all the zeros of an nth-degree polynomial
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