Does the presence of non harmonic real parts of complex functions imply that they are not holomorphic? If $f $ is a complex function such that $f(x,y)=u+iv$ and $u$ is not harmonic can we say that $f$ is not holomorphic and therefore not analytic?

Yes, you can (at least in a domain). For every holomorphic function in a domain, its real and imaginary parts are harmonic functions.  See https://en.wikipedia.org/wiki/Holomorphic_function#Properties
