For all positive integers m,d, b let ρ(m,d) (resp. ρ(m,d, b)) be the maximal symmetric tensor rank of any f ∈ C[x0, . . . , xn] {0} homogeneous of degree d (resp. and with border rank ≤ b). Here we prove that ρ(m,d) ≤ ( m+d m ) −m for all m ≥ 2 and d ≥ 2 (only by 1 better than a far more general result of Landsberg and Teitler), that ρ(m,d, b) ≤ ρ(b − 1, d, b) if 2 ≤ b ≤ m and d ≥ b − 1 and that ρ(m,d, b) ≤ d · ⌈ ( m+d m ) /(m + 1)⌉ if 2 ≤ b ≤ m and 3 ≤ d ≤ b− 2. AMS Subject Classification: 14N05, 14Q05, 15A69