Denote by a set of the form and let be the -th Chebyshev constant for . One of the main purposes of the paper is to give a new proof of the estimate , where does not depend on . In addition, another result is given concerning the approximation of by polynomial inverse images of with order . The two theorems above are interrelated and arise from the new approach introduced by the author.
This approach is based on the statement in the so-called inheritance problem, and has the advantage of avoiding the appearance of -intervals, with the technical difficulties that they entail. A section is devoted to give another application of the latter problem in a more general setting.