Let $f(z)={\sum }_{k\ge 1}{a}_k{z}^{n_k}$ be an entire function of finite order such that $\lim n_k/k=\infty$. Let $M(r)={\max }_{|z|=r}|f(z)|$ and $m(r)={\min }_{|z|=r}|f(z)|$. Is it true that$$\mathrm{limsup}\frac{\log m(r)}{\log M(r)}=1?$$

AMS 30 · solved (literature)

theorem erdos_516 {f : ℂ → ℂ} {n : ℕ → ℕ}
    (hn : HasFabryGaps n) {a : ℕ → ℂ} (ha : ∀ n, a n ≠ 0)
    (hfn : ∀ z, HasSum (fun k => a k * z ^ n k) (f z)) (hf : OfFiniteOrder f) :
    limsup (fun r => ratio r f) atTop = 1