Tsallis relative operator entropy is defined and then its properties are given. Shannon inequality and its reverse one in Hilbert space operators derived by T.Furuta [4] are extended in terms of the parameter of the Tsallis relative operator entropy. Moreover the generalized Tsallis relative operator entropy is introduced and then several operator inequalities are derived.

Tsallis relative operator entropy is defined and then its properties are given. Shannon inequality and its reverse one in Hilbert space operators derived by T.Furuta [5] are extended in terms of the parameter of the Tsallis relative operator entropy. Moreover the generalized Tsallis relative operator entropy is introduced and then several operator inequalities are derived.

In the paper, authors propose the concepts of operator extended s-convex function and operator extended s-convex on the co-ordinates, and establish some new HermiteHadamard type inequalities for operator extended s-convex functions and operator extended s-convex functions on the co-ordinates.

In this paper, we consider the eigenvalue problem of the Schrödinger operator, and obtain some universal inequalities for eigenvalues of the Schrödinger operator in complete simple connected noncompact Riemmannian manifolds admitting special functions which include hyperbolic space.

A new class of harmonic multivalent functions defined by an integral operator is introduced. Coefficient inequalities, extreme points, distortion bounds, inclusion results and closure under an integral operator for this class are obtained .

In this paper, we introduce operator s-convex functions and establish some Hermite-Hadamard type inequalities in which some operator s-convex functions of positive operators in Hilbert spaces are involved.