Asymptotic Relations between Maximums of Absolute Values and Maximums of Real Parts of Entire Functions

According to the classical Wiman–Valiron theorem, the maximum of the absolute value and the maximum of the real part of an entire function are asymptotically equal at infinity outside an exceptional set of a finite logarithmic measure. In this paper, we study the following problem concerning the exceptional set: how does the ratio of the maximum of the absolute value and the maximum of the real part of an entire function depend on its Taylor coefficients? In particular, our results imply that the maximum of the absolute value can increase arbitrarily fast with respect to the maximum of the real part or the Nevanlinna characteristic.