A Liapunov type inequality for Sugeno integrals

The classical Liapunov inequality shows an interesting upper bound for the Lebesgue integral of the product of two functions. This paper proposes a Liapunov type inequality for Sugeno integrals. That is, we show that H-s,H-t,H-r ((s) integral(1)(0) f(x)(s)d mu)(r-t) <= ((s) integral(1)(0) f(x)(t)d mu)(r-s) ((s) integral(1)(0) f(x)(r)d mu)(s-t) holds for some constant H-s,H-t,H-r where 0 < t < s < r, f : [0, 1] -> [0,infinity) is a non-increasing concave function, and mu is the Lebesgue measure on R. We also present two interesting classes of functions for which the classical Liapunov type inequality for Sugeno integrals with H-s,H-t,H-r = 1 holds. Some examples are provided to illustrate the validity of the proposed inequality. (C) 2011 Elsevier Ltd. All rights reserved.