Sharp bounds for joint moments of the Riemann zeta function

In previous work, the first author obtained conjecturally sharp upper bounds for the joint moments of the $(2k-2h{)}^{\text{th}}$ power of the Riemann zeta function with the $2h^{\text{th}}$ power of its derivative on the critical line in the range $1\le k\le 2$, $0\le h\le 1$. Unconditionally, we extend these upper bounds to all $0\le h\le k\le 2$, and obtain lower bounds for all $0\le h\le k+1\mathrm{/}2$. Assuming the Riemann hypothesis, we give sharp bounds for all $0\le h\le k$. We also prove upper bounds of the conjectured order for more general joint moments of zeta with its higher derivatives.