Generalized Hilbert operator and Fejér-Riesz type inequalities on the polydisc

Let f be a holomorphic function on the unit polydisc n, with Taylor expansion f(z)∑|k|=0∞akzk≡∑k1+⋯+kn=0∞ak1,…,knz1k1⋯znknwhere k = (k1, ⋯, kn  ) ∈ ℤ+n. The authors define generalized Hilbert operator on n by ℋγ,n(f)(z)=∑|k|=0 i1,⋯,in≥0∞ai1,⋯,in∏j=1nΓ(γj+kj+1)Γ(kj+ij+1)Γ(kj+1)Γ(kj+ij+γj+2)zkwhere γ ∈ ℂn, such that ℜγjℜγj > − 1, 2, ⋯, n. An upper bound for the norm of the operator on Hardy spaces ℍp(n) is found. The authors also present a Fejér-Riesz type inequality on the weighted Bergman space on p and find an invariant space for the generalized Hilbert operator.