Making use of an extended fractional differintegral Operator (introduced recently by Patel and Mishra), we introduce a new subclass of multivalent analytic functions and investigate certain interesting properties of the subclass.

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.

Using the Riemann-Liouville fractional differintegral Operator, the Lie theory is reformulated. The fractional Poisson bracket over the fractional phase space as 3N state vector is defined to be the fractional Lie derivative. Its properties are outlined and proved. A theorem for the canonicity of the transformation using the exponential Operator is proved. The conservation of its generator is proved in a corollary. A Theorem for the inverse fractional canonical mapping is proved. The composite mappings of two successive transformations is defined. The fractional Lie Operator and its properties are introduced. Some useful lemmas on this Operator are proved. Lie transform depending on a parameter over the fractional phase space is presented and its relations are proved. Two theorems that proved the transformation F= EW Z is completely canonical and is a solution of the Hamiltonian system (30) are given. Recurrence relations are obtained.

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.