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Advances in Fuzzy Systems
Volume 2019, Article ID 5080723, 10 pages
https://doi.org/10.1155/2019/5080723
Research Article

Hölder Type Inequalities for Sugeno Integrals under Usual Multiplication Operations

1Department of Mathematics, Myongji University, Yongin, Kyunggido 449-728, Republic of Korea
2BangMok College of Basic Studies, Myongji University, Yongin, Kyunggido 449-728, Republic of Korea

Correspondence should be addressed to Jae Duck Kim; rk.ca.ujm@mikdj

Received 5 November 2018; Accepted 18 December 2018; Published 3 January 2019

Academic Editor: Michal Baczynski

Copyright © 2019 Dug Hun Hong and Jae Duck Kim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The classical Hölder inequality shows an interesting upper bound for Lebesgue integral of the product of two functions. This paper proposes Hölder type inequalities and reverse Hölder type inequalities for Sugeno integrals under usual multiplication operations for nonincreasing concave or convex functions. One of the interesting results is that the inequality, where and is the Lebesgue measure on holds if and are nonincreasing and concave functions. As a special case, we consider Cauchy-Schwarz type inequalities for Sugeno integrals involving nonincreasing concave or convex functions. Some examples are provided to illustrate the validity of the proposed inequalities.