An Application of the Schur Algorithm to Variability Regions of Certain Analytic Functions-I

Abstract

Let ? be a convex domain in the complex plane C with ? ? C, and P be a conformal map of the unit disk D onto ?. Let F? be the class of analytic functions g in D with g(D) ? ?. Also, let H1?(D) be the well known closed unit ball of the Banach space H?(D) of bounded analytic functions ? in D, with norm ? ?? ?= sup z?D| ?(z) |. Let C(n)={(c0,c1,�,cn)?Cn+1:there exists??H1?(D)satisfying?(z)=c0+c1z+?+cnzn+?forz?D}. For each fixed z? D, j= - 1 , 0 , 1 , 2 , � and c= (c, c1, � , cn) ? C(n) , we use the Schur algorithm to determine the region of variability V?j(z0,c)={?0z0zj(g(z)-g(0))dz:g?F?with(P-1?g)(z)=c0+c1z+?+cnzn+?}. We also show that for z? D\ { 0 } and c?IntC(n), V?j(z0,c) is a convex closed Jordan domain, which we determine by giving a parametric representation of the boundary curve ?V?j(z0,c). � 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH, DE part of Springer Nature.