In this paper the problem of stability for comprehensive classes of Analytic functions in T-neigh-borhoods are studied. Also the lower and upper bounds of radius of stability are obtained.

Let X be a Banach space of functions Analytic on a plane domain  W such that for every l in W the functional of evaluation at l is bounded. Assume further that X contains the constants and admits multiplication by the independent variable z, Mz, as a bounded operator. We give sufficient conditions for Mz to be reflexive.

For Analytic functions f in the open unit disc U, a generalization operator Dlf(z) of Sălăgean operator is introduced. Some properties for Dlf(z) are discussed in the present paper.

We shall provide a brief account on new achievements in the study of composition and composition-differentiation operators acting on classical spaces of Analytic functions on the unit disk, as well as on the polydisk and the unit ball. The emphasis is given to the Hardy space, the Bergman space, and the Dirichlet space on the unit disk in complex plane. In the last section, we shall provide some new results regarding the Hilbert-Schmidt operators in the setting of several complex variables.

The purpose of the paper is to derive some interesting implications associated with some differential inequalities, for certain Analytic functions in the open unit disk. Connections to previously well known results are also established.

We investigate on some subclasses of Analytic fuctions defined by subordination. Also, we give estimates of $\sup_{|z|<1}\big(1-|z|^{2}\big)\big|\dfrac{f^{''}(z)}{f^{'}(z)}\big|$, for functions belonging to extended class of starlike functions. For a locally univalent Analytic function $f$ defined on $\Delta =\{z\in \mathbb{C}: |Z|<1\}$, we consider the pre-Schwarzian norm by $\Vert T\Vert=\sup _{|z|<1}\big(1-|z|^{2}\big)\big|\dfrac{f^{''}(z)}{f^{'}(z)}\big|$. In this work, we find the sharp norm estimate for the functions $f$ in the extended classes of starlike functions.