In this talk we propose explicit a priori error bounds for approaching f(A)b by help of the Arnoldi method. Here A is a large real not necessarily symmetric matrix, and f some function analytic on the field of values or numerical range W (A) = {y∗Ay : ‖y‖ = 1}. An essential tool in our work is the inequality ‖Fn(A)‖ ≤ 2 derived in [1] where Fn is the nth Faber polynomial corresponding to W (A),...

Motivated by the recent work on symmetric analytic functions using concept of Faber polynomials, this article introduces and studies two new subclasses bi-close-to-convex quasi-close-to-convex associated with Janowski functions. By polynomial expansion method, it determines general coefficient bounds for belonging to these classes. It also finds initial coefficients bi-quasi-convex Some known c...

The idea to study infinite matrices whose entries are the coefficients of the powers of a given formal series is rather old and dates back at least to Schur’s posthumous papers on Faber polynomials [39-41]. In 1953, Jabotinsky reconsidered Schur and Shiffer’s [38] work on the subject and developed a systematic study of these matrices [20]. Since then, several applications confirmed that Jabotin...

It has been recently shown that ||Fn(A)|| ≤ 2, where A is a linear continuous operator acting in a Hilbert space, and Fn is the Faber polynomial of degree n corresponding to some convex compact E ⊂ C containing the numerical range of A. Such an inequality is useful in numerical linear algebra, it allows for instance to derive error bounds for Krylov subspace methods. In the present paper we ext...