The main target of this paper is to discuss operator Hermite–Hadamard inequality for convex functions, without appealing convexity. Several forms will be presented and some applications including norm mean inequalities shown too.

Abstract. The main result includes features of a Hardy-type inequality and an inequality of either Sobolev or Gagliardo-Nirenberg type. It is inspired by the method of proof of a recent improved Sobolev inequality derived by M. Ledoux which brings out the connection between Sobolev embeddings and heat kernel bounds. Here Ledoux’s technique is applied to the operator L := x · ∇ and the analysis ...

We review some recent results concerning lower eigenvalues estimates for the Dirac operator [6, 7]. We show that Friedrich’s inequality can be improved via certain well-chosen symmetric tensors and provide an application to Sasakian spin manifolds.