On Stolarsky inequality for Sugeno and Choquet integrals

Keywords: Fuzzy measure Sugeno integral Choquet integral Stolarsky's inequality a b s t r a c t Recently Flores-Franulič, Román-Flores and Chalco-Cano proved the Stolarsky type inequality for Sugeno integral with respect to the Lebesgue measure k. The present paper is devoted to generalize this result by relaxing some of its requirements. Moreover, Stolar-sky inequality for Choquet integral is added, too. Non-additive measures and corresponding integrals can be used for modelling problems in non-additive environment. Since Sugeno [23] initiated research on fuzzy measure and fuzzy integral (known as Sugeno integral), this area has been widely developed and a wide variety of topics have been investigated (see, e.g., [3,7,19,21,25] and references therein). Integral inequalities are an important aspect of the classical mathematical analysis [4,22]. Recently, Román-Flores and his collaborators generalized several classical integral inequalities to Sugeno integral (cf. [6,8]). Flores-Franulič and Román-Flo-res [6] provided a Chebyshev type inequality for Lebesgue measure-based Sugeno integral of continuous and strictly monotone functions. This inequality was generalized to arbitrary fuzzy measure-based Sugeno integral of monotone functions by Ouyang et al. [15]. Later, Mesiar, Ouyang and Li further generalized this inequality to a rather general form [12,16–18]. Jensen inequality was generalized in [20]. Some other inequalities are proved in [1,2]. In [8] Flores-Franulič et al. proved a Stolarsky type inequality for Lebesgue measure-based Sugeno integral and a continuous and strictly monotone function f : ½0; 1Š ! ½0; 1Š. In this contribution, we generalize this inequality to fuzzy measure-based Sugeno integral and a general monotone function f. After recalling some basic concepts and known results in the next section, Section 3 presents our main results, as generalization of Stolarsky inequality for Sugeno integral obtained in [8], including illustrative examples. In Section 4, Stolarsky theorem for Choquet integral is shown. Finally, in Section 5, some concluding remarks are added.