I am interpreting all your constraints as not being strict. if you need them to be strict, then accomplish that by making the RHS be a small positive number. (or subtract small number times identity matrix from LHS for senidefinite constraints, or lambda_min(LHS) >= small number).
Declare p1 and p2 and alpha to be CVX variables. Either declare them as nonnegative, or explicitly include p1 >= 0 and p2 >= 0. Or you can use [p1 0;0 p2] == semidefinite(2), which is equivalent.
For your last constraint, you want the negative of the LHS to be semidefinite, so negative of LHS == senidefinite(3). You can optionally add minimize(alpha).