Bull. Malays. Math. Sci. Soc. https://doi.org/10.1007/s40840-020-00971-2 Semi-doubly Stochastic Operators and Majorization of Integrable Functions Farid Bahrami 1 · Seyed Mahmoud Manjegani 1 · Shirin Moein 1 Received: 8 July 2019 / Revised: 25 June 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020 Abstract In this paper, we introduce semi-doubly stochastic (SDS ) operators on L 1 ( X , μ). The Ryff’s theorem extended to sigma-finite measure space using semi-doubly stochastic operators on L 1 ( X , μ). Keywords Majorization · Integrable functions · σ -finite measure space · Semi-doubly stochastic operator Mathematics Subject Classification 47B60 · 60E15 1 Introduction After some conferences on inequalities with special emphasis on majorization in the California institute of technology and the University of Queensland, Australia, respec- tively, in 1998 and 2002, a surge of interest in majorization theory was generated because of its potential applications in a wide range of fields, especially in Quantum physics, as well as other sciences like economics and physics. For more details, see [3,7,9]. In this work, we provide a generalization of majorization based on a class of operators, which will be called semi-doubly stochastic operators, initially moti- Communicated by Fuad Kittaneh. This work is partially supported by a Grant from Isfahan University of Technology. B Seyed Mahmoud Manjegani manjgani@cc.iut.ac.ir Farid Bahrami fbahrami@cc.iut.ac.ir Shirin Moein s.moein@math.iut.ac.ir 1 Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran 123