Cauchy products of positive sequences (Q809315)

For sequences \((a_ n)\), \((b_ n)\), the Cauchy product is the sequence \((c_ n)\) defined by \(c_ n=\sum^{n}_{j=1}a_{n-j}b_ j\). Consider the following two properties for positive sequences: \[ (*)\quad \lim_{n}\inf (\inf_{k\geq 1}t^{-1}_{k+n}t_ k)^{1/n}\geq 1;\quad (**)\quad t^ 2_ k\geq t_{k-1}t_{k+1}. \] Motivated by considerations from operator theory the authors show that if two positive sequences satisfy (*) then so does their Cauchy product. A similar result is obtained for logarithmic concavity, condition (**).