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Register a new userOn the generalization of Forellis theorem
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The purpose of this paper is to present a solution to perhaps the final remaining case in the line of study concerning the generalization of Forellis theorem on the complex analyticity of the functions that are: (1) $mathcal{C}^infty$ smooth at a point, and (2) holomorphic along the complex integral curves generated by a contracting holomorphic vector field with an isolated zero at the same point.
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In this paper we first prove a version of $L^{2}$ existence theorem for line bundles equipped a singular Hermitian metrics. Aa an application, we establish a vanishing theorem which generalizes the classical Nadel vanishing theorem.
Clunie and Hayman proved that if the spherical derivative of an entire function has order of growth sigma then the function itself has order at most sigma+1. We extend this result to holomorphic curves in projective space of dimension n omitting n hyperplanes in general position.
A motivation comes from {em M. Ismail and et al.: A generalization of starlike functions, Complex Variables Theory Appl., 14 (1990), 77--84} to study a generalization of close-to-convex functions by means of a $q$-analog of a difference operator acting on analytic functions in the unit disk $mathbb{D}={zin mathbb{C}:,|z|<1}$. We use the terminology {em $q$-close-to-convex functions} for the $q$-analog of close-to-convex functions. The $q$-theory has wide applications in special functions and quantum physics which makes the study interesting and pertinent in this field. In this paper, we obtain some interesting results concerning conditions on the coefficients of power series of functions analytic in the unit disk which ensure that they generate functions in the $q$-close-to-convex family. As a result we find certain dilogarithm functions that are contained in this family. Secondly, we also study the famous Bieberbach conjecture problem on coefficients of analytic $q$-close-to-convex functions. This produces several power series of analytic functions convergent to basic hypergeometric functions.
This note is devoted to two classical theorems: the open mapping theorem for analytic functions (OMT) and the fundamental theorem of algebra (FTA). We present a new proof of the first theorem, and then derive the second one by a simple topological argument. The proof is elementary in nature and does not use any kind of integration (neither complex nor real). In addition, it is also independent of the fact that the roots of an analytic function are isolated. The proof is based on either the Banach or Brouwer fixed point theorems. In particular, this shows that one can obtain a proof of the FTA (albeit indirect) which is based on the Brouwer fixed point theorem, an aim which was not reached in the past and later the possibility to achieve it was questioned. We close this note with a simple generalization of the FTA. A short review of certain issues related to the OMT and the FTA is also included.
The aim of this research paper is to obtain explicit expressions of (i) $ {}_1F_1 left[begin{array}{c} alpha 2alpha + i end{array} ; x right]. {}_1F_1left[ begin{array}{c} beta 2beta + j end{array} ; x right]$ (ii) ${}_1F_1 left[ begin{array}{c} alpha 2alpha - i end{array} ; x right] . {}_1F_1 left[ begin{array}{c} beta 2beta - j end{array} ; x right]$ (iii) ${}_1F_1 left[ begin{array}{c} alpha 2alpha + i end{array} ; x right] . {}_1F_1 left[begin{array}{c} beta 2beta - j end{array} ; x right]$ in the most general form for any $i,j=0,1,2,ldots$ For $i=j=0$, we recover well known and useful identity due to Bailey. The results are derived with the help of a well known Baileys formula involving products of generalized hypergeometric series and generalization of Kummers second transformation formulas available in the literature. A few interesting new as well as known special cases have also been given.
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