Entire function under a horizontal inverse-square-root bound
= Entire function under a horizontal inverse-square-root bound
If an entire function satisfies $|h(z)|\leq|\operatorname{Re}z|^{-1/2}$ away from the imaginary axis, then $h=0$. Cauchy's formula on $|z|=R$ bounds every Taylor coefficient by a constant times
$$
R^{-n-1/2}\int_0^{2\pi}|\cos\theta|^{-1/2}\,d\theta,
$$
which tends to zero as $R\to\infty$.