  Studying the shortness of longest cycles in maximal planar graphs, we improve
the upper bound on the shortness exponent of the class of $\frac{5}{4}$-tough
maximal planar graphs presented by Harant and Owens [Discrete Math. 147 (1995),
301--305]. In addition, we present two generalizations of a similar result of
Tk\'{a}\v{c} who considered $1$-tough maximal planar graphs [Discrete Math. 154
(1996), 321--328]; we remark that one of these generalizations gives a tight
upper bound. We fix a problematic argument used in the first paper.
