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.

The study of classical Littlewood-Paley OPERATORs has an intrinsic interest for their essential role in harmonic analysis due to their applications in PDEs and other fields. One of the Littlewood-Paley OPERATORs is g λ *  OPERATOR which its p,p  strong boundedness depends on the parameter λ . For example, Fefferman showed strong boundedness of classical g λ *  for 1max 1, 2 p .  In this work, We consider the Laplace-Bessel differential OPERATOR and correspondingly we define the relevant Littlewood-Paley OPERATOR g B,λ *  to investigate both L p,ν - boundedness of g B,λ *  for 2≤P<∞  and λ>1+ 2v n and its unboundedness for 0

In this paper by using a nice criterion, we show that the perturbation of identity OPERATORs by some multiples of the standard backward shift is hypercyclic. This gives a new proof for Salas Theorem in ( [10 ], Theorem 3.3).