Bounding Zolotarev Numbers Using Faber Rational Functions

Faber Polynomials for Rational Mapping Functions

Faber Polynomials Corresponding to Rational Exterior Mapping Functions

Faber polynomials corresponding to rational exterior mapping functions of degree (m,m − 1) are studied. It is shown that these polynomials always satisfy an (m + 1)-term recurrence. For the special case m = 2, it is shown that the Faber polynomials can be expressed in terms of the classical Chebyshev polynomials of the first kind. In this case, explicit formulas for the Faber polynomials are de...

Estimates for Derivatives of Rational Functions and the Fourth Zolotarev Problem

An estimate is obtained for the derivatives of real rational functions that map a compact set on the real line to another set of the same kind. Many well-known inequalities (due to Bernstein, Bernstein–Szegő, V. S. Videnskĭı, V. N. Rusak, and M. Baran–V. Totik) are particular cases of this estimate. It is shown that the estimate is sharp. With the help of the solution of the fourth Zolotarev pr...

Optimal Ray Sequences of Rational Functions Connected with the Zolotarev Problem

where r is a rational function of degree n. We consider, more generally, the infimum Zmn of such ratios taken over the class of all rational functions r with numerator degree m and denominator degree n. For any "ray sequence" of integers (m, n); that is, m/n -+ A, m + n -+ 00, we show that z~m+n) approaches a limit L(l) that can be described in terms of the solution to a certain minimum energy ...

Erratum: Optimal Ray Sequences of Rational Functions Connected with the Zolotarev Problem

Proposition 1. Suppose the disjoint compacta E1' £2 are regular and), = p/q is rational. Set -r = )./(). + 1) and construct the Leja-Bagby points for the condenser (£1, E~) and corresponding rationals rn(Z) E Rnp,nq as in [1, p. 259], /fthe associated sequence {O"n}~l of signed measures defined in (7.9) of [1] coffi'erges in the weakstar sense to the signed measure jL, then jL = J1,*(-r), ~\!he...