What do you mean by "three main options in pairs"? I interpret this as asking which of these would be possible "u" parts of a function for which
some "v" will give an analytic function.
I presume you know that if u(x,y)+ iv(x,y) is analytic if and only if u and v satisfy the "Cauchy-Riemann" equations:
Now, notice what happens if we differentiate both sides of the first equation with respect to x and the second equation with respect to y:
and
That, is, u must satisfy
or
That is, a function, u(x,y) can be part of an analytic if and only if it satisfies that equation.
(Surely whoever gave you this problem expects you to know that.)