Invariant Muirhead Mean inequality and some Schur convexity properties

Résumé

The aim of this paper is to define the invariant Muirhead mean, denoted by $^i{M_h}(a,b)$ and to discuss the properties of Schur, Schur geometric and Schur harmonic convexities of this mean. Also, an inequality involving the invariant Muirhead mean is constructed by applying the Newton Raphson formula. These inspections contribute to the theoretical development of generalized symmetric means in Majorization theory and Functional inequalities.