Inequalities of Hölder and Minkowski type for pseudo-integrals with respect to interval-valued ⊕-measures
Introduction
In mathematical analysis, inequalities between integrals, such as Hölder and Minkowski inequalities, are important tools for studying spaces. Let be a space with a nonnegative measure m. From Hölder inequality follows that if and , where and , then . Minkowski inequality implies that if , where , then .
One generalization of mathematical analysis is pseudo-analysis. It is obtained by the replacement of the field of real numbers with a real interval and by the replacement of the operations of addition and multiplication with some new operations, so-called pseudo-addition and pseudo-multiplication. In pseudo-analysis field, instead of (additive) measures, non-additive monotone set-functions with additional property, so-called pseudo-additivity are considered. Those set-functions are called pseudo-additive measures. Based on pseudo-operations and pseudo-additive measures, pseudo-integrals are obtained [18], [19], [21]. There are various relations and applications of pseudo-analysis to purely mathematical, as well as applied mathematical sciences. For example, pseudo-analysis has been used in solving non-linear partial differential equations [20] and game theory [13], [14]. The inequalities of Hölder and Minkowski type in pseudo-analysis surrounding have been studied in [2], [3].
The investigation of upper and lower probabilities is performed in [6]. For a measurable space , two set functions and are considered. Both and map ∅ to 0 and Ω to 1, has the property of super-additivity, has the property of sub-additivity and , for all . Set-valued functions and are called lower and upper probability functions, respectively.
Upper and lower probabilities can also be introduced using multi-valued mappings. Let be a probability space, a measurable space and a multi-valued mapping. For and sets and , upper probability induced by multi-valued mapping Γ on is defined by and lower probability is , for every . For the class of the measurable selections of multi-valued mapping Γ and the class of probability distributions induced on by measurable selections, in the paper [17], authors investigate the conditions under which the equality holds, i.e. can be represented as the interval. Also, it is shown when and .
Non-additive interval-valued measures and integrals of real-valued functions with respect to this type of measures are studied in [7], [10], [12].
One generalization of pseudo-additive measures can be constructed using the family of pseudo-additive measures, such that is a dense linear order with endpoints, where ⪯ is a total order [7]. The obtained pseudo-additive measure is an interval-valued measure, i.e. it has the form and it has the property of pseudo-additivity. This type of measure, the pseudo-integral of a real-valued function with respect to it, and the inequalities of Jensen and Chebyshev type for this type of measures are investigated in [7].
Choquet integral with respect to an interval-valued capacity is considered in [12]. Some convergence theorems for interval-valued capacity functionals and interval-valued probability measures are proven. Choquet integral of a bounded real-valued continuous function with respect to the interval-valued upper semi-continuous capacity has the similar form as the pseudo-integral of a real-valued function with respect to the interval-valued pseudo-additive measure considered in the present paper.
So-called p-additive set-valued measures and the set-valued fuzzy measure defined by set-valued fuzzy integrals are studied in [10]. The fuzzy integral used in this paper is Sugeno's integral. The concept of the set-valued g-measure is introduced and Radon–Nikodym Theorem is shown.
Our first results on Hölder and Minkowski type inequalities based on g-integrals and some elementary examples were presented at the 12th IEEE International Symposium on Intelligent Systems and Informatics, Subotica, Serbia. The paper [15] deals with Chebyshev, Hölder and Minkowski inequalities based only on g-integrals (as just one special case of pseudo-integrals) with respect to the interval-valued pseudo-additive measure.
The rest of the paper is organized as follows. Section 2 contains necessary facts on pseudo-operations, semirings, pseudo-additive measures and pseudo-integrals, as well as certain facts from [18], [19], [21]. This section also contains the definition of the interval-valued pseudo-additive measure and the construction of the pseudo-integral of a real-valued function with respect to it [7], [15]. It is shown that, by using a sequence of probability measures and a strictly monotone function g, it is possible to construct a nontrivial interval-valued pseudo-additive measure. In the third section, the notion of pseudo-power is extended to the pseudo-power of a set. Since the research is focused on closed intervals and g-semiring, some basic properties of the pseudo-power of intervals are proven and illustrative examples are provided. In the fourth section the main results of this paper, namely Hölder and Minkowski type of inequalities for the pseudo-integral of a real-valued function with respect to the interval-valued pseudo-additive measure, are proven. In the special case, when the interval-valued measure , the results coincide with the results from [3]. This section also contains certain illustrative examples, for Hölder and Minkowski type of inequalities, with respect to the interval-valued pseudo-additive measure constructed in one of the examples from Section 2. Concluding remarks are presented in the final section.
Section snippets
Preliminary notions
This section contains some basic notions, such as pseudo-operations and a semiring, that are the core of the presented results. Pseudo-operations are extended on the space of nonempty subsets of . This section also contains the definition of a ⊕-measure and an interval-valued ⊕-measure. The notion of the pseudo-integral of a real-valued function with respect to the ⊕-measure, as well as the pseudo-integral of a real-valued function with respect to the interval-valued ⊕-measure, is given
Pseudo-power on sets
For , the pseudo-power can be extended to the pseudo-power of a set in the following manner.
Definition 3.1 For and we define
If pseudo-multiplication ⊙ is an idempotent operation ( or ) then and .
In the work that follows, we will only consider the non-idempotent pseudo-multiplication given by a generator g.
Theorem 3.2 For and , where g is a generating function, the following holds ,
The inequalities of Hölder and Minkowski type based on the interval-valued measure
This section contains the main results of this paper, i.e. Hölder and Minkowski type inequalities based on the interval-valued ⊕-measure (6).
Let be a semiring such that pseudo-multiplication is given by some generator g. Let be measurable pseudo-integrable functions, i.e., pseudo-integrals and exist as a finite value in the sense of the observed semiring for all .
Conclusion
Some of the recent research by a number of authors is focused on inequalities for the integrals based on non-additive measures (see [1], [2], [3], [4], [5], [11], [22], [23], etc.). Since integrals with respect to ⊕-measure where or can be obtained as limits of families of g-integrals [16], the focus of this research was on g-semirings and the corresponding pseudo-integral of real-valued function with respect to interval-valued ⊕-measure. For the future work we propose Hölder and
Acknowledgements
The authors are partially supported by the Ministry of Science and Technological Development of the Republic of Serbia through grants III41103, TR36001, TR32035, OI174009 and TR36030.
References (23)
- et al.
Chebyshev type inequalities for pseudo-integrals
Nonlinear Anal.
(2010) - et al.
General Minkowski type inequalities for Sugeno integrals
Fuzzy Sets Syst.
(2010) - et al.
Hölder and Minkowski type inequalities for pseudo-integral
Appl. Math. Comput.
(2011) - et al.
Berwald type inequality for Sugeno integral
Appl. Math. Comput.
(2010) - et al.
A Cauchy–Schwarz type inequality for fuzzy integrals
Nonlinear Anal.
(2010) - et al.
An approach to pseudo-integration of set-valued functions
Inf. Sci.
(2011) A Liapunov type inequality for Sugeno integrals
Nonlinear Anal.
(2011)A note on convergence properties of interval-valued capacity functionals and Choquet integrals
Inf. Sci.
(2012)- et al.
Idempotent integral as limit of g-integrals
Fuzzy Sets Syst.
(1999) - et al.
Approximations of upper and lower probabilities by measurable selections
Inf. Sci.
(2010)