Zygmund’s inequality for entire functions of exponential type

Let f∈Lp(R)f∈Lp(R) be an entire function of exponential type σσ whose conjugate indicator diagram lies on interval [0,σi]. We prove that ∫−∞∞|f′(x)|pdx≤cpσp∫−∞∞|Ref(x)|pdx(p≥1) where cp≔πΓ(1+p2)Γ(1+p2) is a best possible constant.