Holomorphic means differentiable in a neighborhood of a point. The functions (a) and (c) are not differentiable in a "region" about a point.
As for (b) don't confuse a "complex polynomial" with a "real polynomial" with complex coefficients. Now, the complex polynomial is entire but your function is not of that form but rather just two real functions with an "i" coefficient in front of the second one.
And the function is complex analytic iff h(x,y) is the complex congugate of g(x,y).