An internal polya inequality for ℂ-convex domains in ℂ n

Official URL: https://dx.doi.org/10.1134/S0001434619030052

Let K ⊂ ℂ be a polynomially convex compact set, f be a function analytic in a domain ℂ¯ \ K with Taylor expansion f(z)=∑k=0∞ak/zk+1 at ∞, and Hi(f):=det(ak+l)k,l=0i be the related Hankel determinants. The classical Polya theorem [11] says that lim supi→∞|Hi(f)|1/i2≤d(K), where d(K) is the transfinite diameter of K. The main result of this paper is a multivariate analog of the Polya inequality for a weighted Hankel-type determinant constructed from the Taylor series of a function analytic on a ℂ-convex (= strictly linearly convex) domain in ℂ n .