Generalization of the Lyapunov type inequality for pseudo-integrals

We prove two kinds of Lyapunov type inequalities for pseudo-integrals. One discusses pseudo-integrals where pseudo-operations are given by a monotone and continuous function g. The other one focuses on the pseudo-integrals based on a semiring 0; 1 ½ Š; sup; ð Þ , where the pseudo-multiplication is generated. Some examples are given to illustrate the validity of these inequalities. As a generalization of classical analysis, Pseudo-analysis [19,24,31] based on a semiring a; b ½ Š; È; ð Þ , in which pseudo-addition È and pseudo-multiplication are given by a monotone and continuous function generator g, has been investigated. In this structures, many concepts such as È-measure (pseudo-additive measure), pseudo-integral, pseudo-convolu-tion, pseudo-Laplace transform, etc. have been proposed. The integral inequalities are good mathematical tools both in theory and application. Different integral inequalities including Chebyshev, Jensen, Hölder and Minkowski inequalities are widely used in various fields of mathematics, such as probability theory, differential equations, decision-making under risk and information sciences. Recently, some classical inequalities have been generalized for fuzzy integrals. Román-Flores et al. [4,13–15,26–30] investigated several kinds of inequalities for Sugeno integral including Geometric inequality, Jensen type inequality, Chebyshev type inequality, Hardy type inequality, Convolution type inequality, Markov type inequality and General Barnes–Goduno-va–Levin type inequality. Girotto and Holzer [16,17] illustrated a characterization of comonotonicity property by a Cheby-shev type inequality for Sugeno integral and Choquet integral. Mesiar and Ouyang [1–3,20–23] proposed Chebyshev, Minkowski and Berwald inequalities for Sugeno integral. Caballero and Sadarangani [8–11] presented Chebyshev, Cauchy–Schwarz, Fritz Carlson and sandor inequalities for Sugeno integral. In addition, some famous inequalities have also been generalized for pseudo-integral. Agahi et al. [5,6] studied Chebyshev, Hölder and Minkowski inequalities for pseudo-integrals. Pap and Šrboja [25] discussed generalization of the Jensen type inequalities for pseudo-integrals. Daraby [12] obtained generalization of the Stolarsky type inequality for pseudo-integrals. The purpose of this paper is to prove a generalization of the Lyapunov type inequality for pseudo-integrals. We discuss two kinds of Lyapunov type inequalities for pseudo-integrals. The first discusses pseudo-integrals where pseudo-operations