Some Hadamard–Fejér Type Inequalities for LR-Convex Interval-Valued Functions

The purpose of this study is to introduce the new class of Hermite–Hadamard inequality for LR-convex interval-valued functions known as LR-interval Hermite–Hadamard inequality, by means of pseudo-order relation ( ≤p ). This order relation is defined on interval space. We have proved that if the interval-valued function is LR-convex then the inclusion relation “ ⊆ ” coincident to pseudo-order relation “ ≤p ” under some suitable conditions. Moreover, the interval Hermite–Hadamard–Fejér inequality is also derived for LR-convex interval-valued functions. These inequalities also generalize some new and known results. Useful examples that verify the applicability of the theory developed in this study are presented. The concepts and techniques of this paper may be a starting point for further research in this area.

Inclusion Properties of The Homogeneous Herz-Morrey Spaces With Variable Exponent

In this paper, we have discussed about the inclusion properties of the homogeneous Herz-Morrey spaces with variable exponent and the weak homogeneous spaces with variable exponent. We also studied the inclusion relation between those spaces.

Some Integral Inequalities for Generalized Convex Fuzzy-Interval-Valued Functions via Fuzzy Riemann Integrals

In this study, we introduce the new concept of $$h$$ h -convex fuzzy-interval-valued functions. Under the new concept, we present new versions of Hermite–Hadamard inequalities (H–H inequalities) are called fuzzy-interval Hermite–Hadamard type inequalities for $$h$$ h -convex fuzzy-interval-valued functions ($$h$$ h -convex FIVF) by means of fuzzy order relation. This fuzzy order relation is defined level wise through Kulisch–Miranker order relation defined on fuzzy-interval space. Fuzzy order relation and inclusion relation are two different concepts. With the help of fuzzy order relation, we also present some H–H type inequalities for the product of $$h$$ h -convex FIVFs. Moreover, we have also established strong relationship between Hermite–Hadamard–Fej´er (H–H–Fej´er) type inequality and $$h$$ h -convex FIVF. There are also some special cases presented that can be considered applications. There are useful examples provided to demonstrate the applicability of the concepts proposed in this study. This paper's thoughts and methodologies could serve as a springboard for more research in this field.

Improved cosine and cotangent function-based similarity measures for q-rung orthopair fuzzy sets and TOPSIS method

Despite the importance of cosine and cotangent function- based similarity measures, the literature has not provided a satisfactory formulation for the case of q-rung orthopair fuzzy set (qROFS). This paper criticizes the existing attempts in terms of respect of the basic axioms of a similarity measure and strict inclusion relation. In addition, the maximum operator-based similarity measures are criticized. Then, new improved, axiomatically supported cosine and cotangent function-based similarity measures for qROFSs are proposed. Additional properties of the new similarity measures are discussed to guarantee their good performance. Two algorithmic procedures of TOPSIS method that based on fixed and relative ideal solutions are discussed. The numerical examples are provided to support the findings

Some new Jensen, Schur and Hermite-Hadamard inequalities for log convex fuzzy interval-valued functions

<abstract> <p>The inclusion relation and the order relation are two distinct ideas in interval analysis. Convexity and nonconvexity create a significant link with different sorts of inequalities under the inclusion relation. For many classes of convex and nonconvex functions, many works have been devoted to constructing and refining classical inequalities. However, it is generally known that log-convex functions play a significant role in convex theory since they allow us to deduce more precise inequalities than convex functions. Because the idea of log convexity is so important, we used fuzzy order relation $\left(\preceq \right)$ to establish various discrete Jensen and Schur, and Hermite-Hadamard (H-H) integral inequality for log convex fuzzy interval-valued functions (L-convex F-I-V-Fs). Some nontrivial instances are also offered to bolster our findings. Furthermore, we show that our conclusions include as special instances some of the well-known inequalities for L-convex F-I-V-Fs and their variant forms. Furthermore, we show that our conclusions include as special instances some of the well-known inequalities for L-convex F-I-V-Fs and their variant forms. These results and different approaches may open new directions for fuzzy optimization problems, modeling, and interval-valued functions.</p> </abstract>

Some fuzzy-interval integral inequalities for harmonically convex fuzzy-interval-valued functions

<abstract> <p>It is well-known fact that fuzzy interval-valued functions (F-I-V-Fs) are generalizations of interval-valued functions (I-V-Fs), and inclusion relation and fuzzy order relation on interval space and fuzzy space are two different concepts. Therefore, by using fuzzy order relation (FOR), we derive inequalities of Hermite-Hadamard (<italic>H</italic>·<italic>H</italic>) and Hermite-Hadamard Fejér (<italic>H</italic>·<italic>H</italic> Fejér) like for harmonically convex fuzzy interval-valued functions by applying fuzzy Riemann integrals. Moreover, we establish the relation between fuzzy integral inequalities and fuzzy products of harmonically convex fuzzy interval-valued functions. The outcomes of this study are generalizations of many known results which can be viewed as an application of a defined new version of inequalities.</p> </abstract>