Wirtinger integral inequalities for pseudo-integrals and pseudo-additive measure

The main purpose of this paper is to show Wirtinger type inequalities for the pseudo-integral. We are concerned with pseudo-integrals based on the following three canonical cases: in the first case, the real semiring with pseudo-operation is generated by a strictly monotone continuous function g; in the second case, the pseudo-operations include a pseudo-multiplication and a power arithmetic addition; in the last case, ⊕-measures are interval-valued. Examples are given to illustrate these equalities.

Pseudo-analysis is chosen as the research background of this paper, because it presents a contemporary mathematical theory which has been successfully applied in many practical fields. The last decade has shown an increasing research activity on pseudo-analysis [13][14][15][16][17]. As a method to promote the classical mathematical analysis, pseudo-analysis extends the concept of traditional operation to pseudo-operation including pseudo-addition and pseudo-multiplication. Later on, the researchers present pseudo-integrals [16] based on the important theory of pseudo-analysis, namely pseudo-operations and interval-valued measure. The pseudo-integral is now emerging as one of the hottest mathematical subjects, many scholars have studied its application and promotion on generalizations of integral inequalities [18][19][20][21][22].
Since the Wirtinger inequality is one of the most important inequalities, this paper studies three Wirtinger type integral inequalities in pseudo-analysis environment. In the first case, we consider a Wirtinger type integral inequality for an applied pseudointegration equipped with a monotonic continuous mapping g. In the second case, we study a Wirtinger type integral inequality for the pseudo-integration adopting a semiring ([0.1], sup, ) to design the theory. In the last case, we show a Wirtinger type integral inequality with respect to interval-valued ⊕-measures. Moreover, several examples are provided for validation.
The paper is organized as follows: Sect. 2 contains some of preliminaries. Section 3 provides generalizations of two Wirtinger type inequalities to pseudo-integrals on a gsemiring. Section 4 proves a Wirtinger integral inequality for the pseudo-integral of a realvalued function with respect to the pseudo-additive measure. The conclusion is shown in Sect. 5.

Preliminaries
In this section, we review some basic notions about pseudo-operations and pseudointegrals, the relevant literature includes [15,23].
An interval [c, d] is a closed subset of [-∞, +∞]. The complete order in [c, d] is expressed as which can be the usual order of the real line or can be another order. Also, the total order is closely connected to the choice of pseudo-addition. Namely, for the semirings of the first and the third class, i.e., for the semirings with idempotent pseudo-addition, total order is induced by the following For the semiring of the second class given by a generator g, total order is given by , endowed with a pseudoaddition ⊕ and a pseudo-multiplication , is a semiring (see [16,24]). The equation We consider the semiring ([c, d], ⊕, ) in two situations. In the first situation, let g : [c, d] → [0, ∞] be a monotone and continuous mapping and let x ⊕ y = g -1 (g(x) + g(y)), x y = g -1 (g(x) · g(y)) and x (n) = g -1 (g n (x)).
The pseudo-integral for a function p : see for details [17]. The second situation is when where the function ψ means sup-measure m; see [14].
is a continuous density. Then, for any pseudo-addition ⊕ with a generator g there exists a family ) be a semiring, when is generated with g, i.e., we have x y = g -1 (g(x)g(y)) for every x, y ∈ (0, ∞). Let m be the same as in Theorem Pseudo-operations on nonempty subsets C and D of [c, d] adopts the methods similar to the pseudo-addition and the pseudo-multiplication [14]. If C and D stand for two arbitrary nonempty subsets of [c, d] and β ∈ [c, d] + , then The present paper focuses on the interval operation. Let I be the class of all closed subintervals of [c, d] + , i.e., It can be shown (see [13]) that for C, For where C ⊆ X, the properties can be summarized as follows: In Eq. (4), the set-functionμ p M is an interval-valued ⊕-measure.
Proof By (5), we obtain Since g -1 is an increasing function, one has Hence, we have U(s) ds (2) dt.
The proof is therefore complete.
This ends the proof.

Inequalities of Wirtinger type for pseudo-integrals with respect to interval-valued ⊕-measures
This section contains the further results of this paper, i.e., Wirtinger type inequalities based on the interval-valued ⊕-measure [19].
Proof By Theorem 3, we have On the other hand, Since the interval [ can be written in the form Based on Wirtinger inequality for pseudo-integrals (6), one has Since α ∈ [a, b] + and is a positively non-decreasing function, we have Similarly, for β ∈ [a, b] + and -measure μ h , is a positively non-decreasing function, and then, by Inequality (6), we have we have x y and dμ M . Therefore, we have completed the first part of the proof. Furthermore, due to the property of the pseudo-convexity of the subinterval of [a, b] + , can be written in the form Since ⊕ is a non-decreasing function, and is a positively non-decreasing function, one has This yields Hence, the following form holds: This completes the proof.
Remark 9 The proof for a decreasing generator g is similar. In this case, since the total order is opposite to the usual order on the real line, Inequality (7)