Functions and differences whose roots have the same real part

Let $f : \mathbb{C} → \mathbb{C}$ be an entire function of order $≤ 1$. Assume that all the roots of $f$ have the same real part, abbreviated by $f ∈ SRP.$ For $λ ∈ \mathbb{R}\setminus\{0\},$ define $\Delta_λf(z) := f(z +λ)− f(z).$ We investigate the situation when $\Delta_λf ∈ SRP.$