If a function $f=u+iv$ is holomorphic $\rightarrow$ Cauchy Riemann eqns hold.
$u_x=v_y$ and $u_y=-v_x$ This means that $u_{xx}=v_{yx}$ and $u_{yy}=-v_{xy}$
At the same time $v_{yy}=u_{xy}$ and $v_{xx}=-u_{yx}$.
This implies: $$u_{xx}+u{yy}+v_{xx}+v_{yy}=0$$ Hence, holomorphic $\Rightarrow$ harmonic. This is useful because harmonic functions attain maximums and minimums at the boundary and I can use max/min modulus principles.
Is this correct? If $g$ is holomorphic
$g(0)=0$ implies that $|g(0)|=0 \Rightarrow g(z)=0$
Next, All harmonic functions are holomorphic, which is harder to show. Is this true?