This article deals with some ergodic properties for general sequences in the closed convex hull of the orbit of some (not necessarily power-bounded) Operators in Banach spaces. A regularity condition more general than that of ergodicity is used to obtain some versions of the Esterle–Katznelson–Tzafriri theorem. Also, the ergodicity of the backward iterates of a sequence is proved under appropriate assumptions as, for example, its peripheral boundedness on the unit circle. The applications concern uniformly Kreiss-bounded Operators, and other ergodic results are obtained for the binomial means and some Operator means related to the Cesaro means.

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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 paper, we establish further improvements  of the Young inequality and its reverse. Then, we assert Operator versions corresponding them. Moreover, an application including positive linear mappings is given. For example, if $A,B\in {\mathbb B}({\mathscr H})$ are two invertible positive Operators such that $0\begin{align*}& \Phi ^{2} \bigg(A \nabla _{\nu} B+ rMm \left( A^{-1}+A^{-1} \sharp_{\mu} B^{-1} -2 \left(A^{-1} \sharp_{\frac{\mu}{2}} B^{-1} \right)\right)\\& \qquad +\left(\frac{\nu}{\mu} \right) Mm \bigg(A^{-1}\nabla_{\mu} B^{-1} -A^{-1} \sharp_{\mu} B^{-1}\bigg)\bigg) \\& \quad \leq \left( \frac{K(h)}{ K\left( \sqrt{{h^{'}}^{\mu}},2 \right)^{r^{'}}} \right) ^{2} \Phi^{2} (A \sharp_{\nu} B),\end{align*}where $r=\min\{\nu,1-\nu\}$, $K(h)=\frac{(1+h)^{2}}{4h}$,  $h=\frac{M}{m}$, $h^{'}=\frac{M^{'}}{m^{'}}$ and $r^{'}=\min\{2r,1-2r\}$. The results of this paper generalize the results of recent years.

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.