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Your last inequality implies that $v(x,y):=u(x,y)-u(x,-y)\geq 0,\; y>0$, and similarly $v(x,y)\leq 0,\; y<0$. Since $v$ is harmonic, this easily implies that $v(x,y)=cy$ for some real constant $c$. Now $w(x,y):=u(x,y)-cy/2$ will satisfy $w(x,y)=w(x,-y)$ which happens for a harmonic function if and only $w(x,0)=0$. This implies $u(x,0)=0$ and we obtain a characterization: these are exactly those entire functions which are real on the real line. (And your original inequality is in fact always equality).