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) ^th power of the Riemann zeta function with the 2h^th power of its derivative on the critical line in the range 1 k 2, 0 h 1. Unconditionally, we extend these upper bounds to all 0 h k 2, and obtain lower bounds for all 0 h k+1/2. Assuming the Riemann hypothesis, we give sharp bounds for all 0 h k. We also prove upper bounds of the conjectured order for more general joint moments of zeta with its higher derivatives.