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The theory of analytic functions and more specific p-valent functions, is one of the most fascinating topics in one complex variable. There are many remarkable theorems dealing with extremal problems for a class of p-valent functions on the unit disk U. Recently, many researchers have shown great interests in the study of differential operator. The objective of this paper is to define a new generalized derivative operator of p-valent analytic functions of fractional power in the open unit disk U denoted by Dm,bλ1λ2 p,αf(z). This operator generalized some well-known operators studied earlier, we mention some of them in the present paper. Motivated by the generalized derivative operator Dm,bλ1λ2 p,αf(z) we introduce and investigate two new subclasses Sm,bλ1,λ2,p,α(μ,ν) and TSm,bλ1,λ2,p,α(μ,ν), which are subclasses of starlike p-valent analytic functions of fractional power with positive coefficients and starlike p-valent analytic functions of fractional power with negative coefficients, respectively. In addition, a sufficient condition for functions f ∈ Σp,α to be in the class Sm,bλ1,λ2,p,α(μ,ν) and a necessary and sufficient condition for functions f ∈ Tp,α will be obtained. Some corollaries are also pointed out. Moreover, we determine the extreme points of functions belong to the class TSm,bλ1,λ2,p,α(μ,ν).

In our paper we provide some direct extentions of our recent sharp results on traces in the analytic function spaces, which we proved earlier in case of the unit ball in C^n, to the case of the bounded strongly pseudoconvex domains with a smooth boundary. To be more precise we consider the analytic Bloch space in the strongly pseudoconvex domains with a smooth boundary, mixed norm spaces and so-called the new Herz type spaces of analytic functions in the domains of the same type. The Bloch spaces, for various complicated domains, were studied by many authors, but the various Herz type spaces are introduced in this paper, as far as we know, for the first time. The role of so-called r-lattices and their new properties are essential for our proofs. These techniques based on the lattices in the strongly pseudoconvex domains were invented and heavily used in the recent papers of Abate and coauthors. The arguments in the proofs in the case of the unit ball and the strongly pseudoconvex domains have some similarity.

This paper contains an overview of recent results of Area-Nevanlinna classes in higher dimension. We here consider various aspects of this new interesting research area of analytic function theory in higher dimension (integral operations, embedding theorems, Taylor coefficients). Previously in one dimension all these results were known. New open interesting Problems in this new research area will be also discussed and indicated.

The purpose of the note is to obtain equivalent quasinorm, sharp estimates for the quasinorm of Hardy’s and new Bergman’s analytic classes of in the polydisk. We extend some classical onedimensional assertions to the case of several complex variables. Our results more precisely provide direct new extention of some known one variable theorems concerning area integral to the case of simplest product domains namely the unit polydisk in Cn. Let further D be a bounded or unbounded domain in Cn. For example, tubular domain over symmetic cone or bounded pseudoconvex domain with smooth boundary. Our results can be probably extended to the case of products of such type complicated domains, namely even to D×…×D. This can be probably done based on some approaches we suggested and used in this paper. On the other hand our results in simpler case namely in the unit polydisk may also have various interesting applications in complex function theory in the unit polydisk. We finnaly provide similar type sharp. results in some new Bergman spaces in bounded strongly pseudoconvex domains

Hypergeometric functions are of special interests among the complex analysts especially in looking at the properties and criteria of univalent. Hypergeometric functions have been around since 1900’s and have special applications according to their own needs. Recently, we had an opportunity to study on q-hypergeometric functions and quite interesting to see the behavior of the functions in the complex plane. There are many different versions by addition of parameters and choosing suitable variables in order to impose new set of q-hypergeometric functions. The aim of this paper is to study and introduce a new convolution operator of q-hypergeometric typed. Further, we consider certain subclasses of starlike functions of complex order. We derive some geometric properties like, coefficient bounds, distortion results, extreme points and the Fekete-Szego inequality for these subclasses.

In this paper we introduce and study the class$\ \mathcal{VR}_{\delta ,\eta }(n,\lambda ,\alpha )\ $of analytic functions with varying arguments of coefficients. We obtain coefficients inequalities, distortion theorems involving fractional calculus, radii of close to convexity, starlikeness and convexity and square root transformation for functions in the class $% \mathcal{VR}_{\delta ,\eta }(n,\lambda ,\alpha )$.\ Finally, integral convolution for functions in this class are considered.