2 fixed typo.

An upper bound for the maximum cut mean value Alberto Bertoni, Paola Campadelli and Roberto Posenato

Their bound is the same as yours; more precisely, for a random graph with $n$ vertices and $x n$ edges, for sufficiently large $x,$ they claim the size of max cut divided by $x n$ is bounded above by

$$\frac12 + \frac1{\sqrt{x}} + \frac12 \frac{\log x}{x},$$

so I assume this is tight.

EDIT

A matching lower bound is provided in MR2060633 (2005c:68088) Coppersmith, Don(1-IBM); Gamarnik, David(1-IBM); Hajiaghayi, Mohammad Taghi(1-MIT); Sorkin, Gregory B.(1-IBM) Random MAX SAT, random MAX CUT, and their phase transitions. (English summary) Random Structures Algorithms 24 (2004), no. 4, 502–545. 68Q25 (60C05 68T20 68W40 82B26 82B44)

See Theorem 20.

1

Their bound is the same as yours; more precisely, for a random graph with $n$ vertices and $x n$ edges, for sufficiently large $x,$ they claim the size of max cut divided by $x n$ is bounded above by
$$\frac12 + \frac1{\sqrt{x}} + \frac12 \frac{\log x}{x},$$