Schur Convexity of Functional Bonferroni Mean

In this paper, we introduce a functional extension of the classical Bonferroni mean. Using tools from majorization theory together with differential criteria for Schur convexity, we establish sufficient conditions under which the functional Bonferroni mean and the functional generalized Bonferroni harmonic mean are Schur convex, Schur concave, or Schur harmonically convex. As applications, we derive separation inequalities between these two means, obtain several new integral inequalities, and prove norm inequalities for functional Bonferroni means.