CONVEXITY AND CONCAVITY IN SUBMAJORIZATION INEQUALITIES FOR τ-MEASURABLE OPERATORS - Dauitbek Достілек (Journal of Mathematics, Mechanics and Computer Science, 2025)
We proved the following result. Consider a semi-finite von Neumann algebra equipped with a trace, and let there be several τ-measurable operators together with a nonnegative function defined on the nonnegative real line. Suppose also that we are given positive weights whose total sum equals one. If the function obtained by applying f to the square root of its argument is convex and if f vanishes at zero, then the weighted sum of the values of f applied to the absolute values of the operators is at least as large as a certain expression involving f evaluated at both the average of all the operators and their pairwise differences. If the same function of the square root is concave, the inequality reverses: the mentioned expression becomes no smaller than the weighted sum of the transformed absolute values. This theorem yields a significant generalization of Clarkson-type inequalities in the noncommutative setting and extends the result previously established by Alrimawi, Hirzallah, and Kittaneh.