Inequalities among quasi-arithmetic means for continuous field of operators

In this paper we study inequalities among quasi- arithmetic means for a continuous field of selfadjoint operators, a field of positive linear mappings and continuous strictly monotone functions which induce means. We present inequalities with operator convexity and without operator convexity of appropriate functions. Also, we present a general formulation of converse inequalities in each of these cases. Furthermore, we obtain refined inequalities without operator convexity. As applications, we obtain inequalities among power means.