The impact of Stieltjes' work on continued fractions and orthogonal polynomials

Stieltjes' work on continued fractions and the orthogonal polynomials related to continued fraction expansions is summarized and an attempt is made to describe the influence of Stieltjes' ideas and work in research done after his death, with an emphasis on the theory of orthogonal polynomials

Small values of polynomials: Cartan, Pólya and others

Let be a monic polynomial of degree , and , . A classic lemma of Cartan asserts that the lemniscate can be covered by balls , whose diameters satisfy For , this shows that has an area at most . P&#243;lya showed in this case that the sharp estimate is . We discuss some of the ramifications of these estimates, as well as some of their close cousins, for example when is normalized to have norm 1 on some circle, and Remez' inequality.</p