Claims

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1954_12_01_erdos_macintyre: Erdős and Macintyre prove that an entire lacunary series whose reciprocal gaps have a convergent sum has limsup m(r)/M(r) = 1, which gives the question's equality on that subclass; refereed in Proc. Edinburgh Math. Soc.

1963_12_01_fuchs: Fuchs proves that an entire function of finite order whose exponents satisfy n_k/k to infinity has log m(r) above (1-eps) log M(r) outside a set of logarithmic density zero; refereed in Illinois J. Math.

1965_06_01_kovari: Kővári proves that an entire function of any order whose exponents exceed k (log k)^(2+c) for some c > 0 has limsup log m(r)/log M(r) = 1, covering the finite-order functions among them; refereed in Michigan Math. J.

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