The Schwarz inequality and Jensen’s one are fundamental in a Hilbert space. Regarding a sesquilinear map B(X, Y ) = Y*X as an operatorvalued inner product, we discuss operator versions for the above inequalities and give simple conditions that the equalities hold.

In this investigation, using Opoola differential operator ($D^{m}(\mu,\beta,t)f(z)$), a new integral operator: $I_{t,\beta,\mu}^{m,\sigma}(f_{1},...,f_{n})(z): A^{n}\rightarrow A$  is defined in the unit disk, $U=\left\lbrace z\in C:\left|z\right|<1\right\rbrace$; and we investigated the Univalence conditions of this generalized operator. Finally, a number of corollaries and remarks which show the extension of our results are presented.

A generalized integral operator of order $\alpha$ of a real function $f$  including a parameter set $P$, namely $K_P^\alpha f(t)$ has been introduced by O. P.  Agrawal  (Computers and Mathematics with Applications, 59 (2010) 1852--1864), which is a generalization of some important fractional integrals such as the Riemann-Liouville fractional integral. Using pseudo-analysis, this paper introduces a pseudo-operator integral of order $\alpha$ including a parameter set $P$  in a semiring $([a, b], \oplus, \odot)$, which is  a generalization of  $K_P^\alpha f(t)$. We also discuss some particular cases and we obtain the well-known H\"{o}lder's and Minkowski's  inequalities  for this kind of pseudo-operator integral. The results given in this paper provide a generalization of several inequalities obtained in earlier studies.

In the present paper, taking some advantages offered by the context of finite dimensional Hilbert spaces, we shall give a complete characterizations of certain distinguished classes of operators (self-adjoint, unitary reflection, normal) in terms of operator inequalities. These results extend previous characterizations obtained by the second author.