Entire function with zeroes on real line

Question

Let f be an entire function $\mathbb{C}\to\mathbb{C}$ of order $1$ with zeroes only on real line. I need to prove that this function has following bound: $$\exists a>0\:\:\:|f(z)| \ge e^{-a|Im \:z|}|Re\:f(z)|$$ I have searched over some theorems about order and zeroes, found some approximation and factorisation theorems but not sure how they could help.