Remarks on free mutual information and orbital free entropy

The present notes provide a proof of i* (ℂP + ℂ(I - P); ℂQ + ℂ(I - Q)) = – χorb(P,Q) for any pair of projections P,Q with τ(P) = τ(Q) = 1/2. The proof includes new extra observations, such as a subordination result in terms of Loewner equations. A study of the general case is also given.

Remarks on free mutual information and orbital free entropy

The present notes provide a proof ofi* (ℂP + ℂ(I - P); ℂQ + ℂ(I - Q))= – χorb(P,Q) foranypair of projectionsP,Qwithτ(P) = τ(Q) = 1/2. The proof includes new extra observations, such as a subordination result in terms of Loewner equations. A study of the general case is also given.

On a subordination result for analytic functions defined by convolution

In this paper we discuss some subordination results for a subclass of functions analytic in the unit disk U

Corrigendum of the paper: some results on certain classes of multivalently analytic functions based on differential subordination involving a convolution structure math. Slovaca 60 (2010), 471–484 (subordination result and generalizations)

This note gives the correct form of a result (Theorem 2, p. 478) published recently in [Math. Slovaca 60 (2010), 471–484].

Certain Conditions for Starlikeness of Analytic Functions of Koebe Type

For , , we consider the of normalized analytic convex functions defined in the open unit disc . In this paper, we investigate the class , that is, , with is Koebe type, that is, . The subordination result for the aforementioned class will be given. Further, by making use of Jack's Lemma as well as several differential and other inequalities, the authors derived sufficient conditions for starlikeness of the class of -fold symmetric analytic functions of Koebe type. Relevant connections of the results presented here with those given in the earlier works are also indicated.

Subordination Properties for Certain Analytic Functions

The purpose of the present paper is to derive a subordination result for functions in the class of normalized analytic functions in the open unit disk . A number of interesting applications of the subordination result are also considered.