Return to Revisions

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 $w(x,y)=w(x,-y)$ This implies $(\partial w/\partial y)(0,y)=0$ and by Cauchy-Riemann, the conjugate function $w^*$ to $w$ is constant on the real line. Since $(u+iu^*)(z)=w+iw^*(z)+ciz/2$, we obtain a characterization: these are exactly those entire functions whose imaginary part is of the form $a+bx$ on the real line, where $a$ and $b$ are constants, $b$ is real. (And your original inequality is in fact always equality). Or in other words, the general form of such $f$ is $f=g+a+biz$ where $g$ is real on the real line.