  In this paper, we use a new method to obtain the necessary and sufficient
condition guaranteeing the validity of the Minkowski-H\"{o}lder type inequality
for the generalized upper Sugeno integral in the case of functions belonging to
a wider class than the comonotone functions. As a by-product, we show that the
Minkowski type inequality for seminormed fuzzy integral presented by Daraby and
Ghadimi in General Minkowski type and related inequalities for seminormed fuzzy
integrals, Sahand Communications in Mathematical Analysis 1 (2014) 9--20 is not
true. Next, we study the Minkowski-H\"{o}lder inequality for the lower Sugeno
integral and the class of $\mu$-subadditive functions introduced in On
Chebyshev type inequalities for generalized Sugeno integrals, Fuzzy Sets and
Systems 244 (2014) 51--62. The results are applied to derive new metrics on the
space of measurable functions in the setting of nonadditive measure theory. We
also give a partial answer to the open problem 2.22 posed by
Borzov\'a-Moln\'arov\'a and et al in The smallest semicopula-based universal
integrals I: Properties and characterizations, Fuzzy Sets and Systems 271
(2015) 1--17.
