On the order of convexity for the shifted hypergeometric functions

Ratios of the Gauss hypergeometric functions with parameters shifted by integers: part I

We consider the ratio of two Gauss hypergeometric functions with real parameters shifted by arbitrary integers. We find a formula for the jump of this ratio over the branch cut in terms of a real hypergeometric polynomial, the beta density and the absolute value of the Gauss hypergeometric function. This allows us to construct explicit integral representations for such ratio when the asymptotic behaviour at unity is mild and the denominator does not vanish. Multiplying the ratio by the aforementioned polynomial, we obtain a function that belongs to a generalized Nevanlinna class for arbitrary …

Mapping properties of the zero-balanced hypergeometric functions

In the present paper, the order of convexity of z\Gauss(a,b;c;z) is first given under some conditions on the positive real parameters a, b and c. Then we show that the image domains of the unit disc \D under some shifted zero-balanced hypergeometric functions z\Gauss(a,b;a+b;z) are convex and bounded by two horizontal lines which solves the problem raised by Ponnusamy and Vuorinen in \cite{PonVuor:2001

涉及超几何函数的亚纯单叶函数的一个子类

On a Subclass of Meromorphic Univalent Functions Involving Hypergeometric Function

本文的主要目的是介绍。涉及的亚纯单叶函数类超几何性质的函数.根据系数不等式、增长定理、偏差定理、星形半径和凸性，得到了超几何性质的一些有趣的几何性质.

基本超几何函数的几何性质

Geometric properties of basic hypergeometric functions

本文考虑Heine引入的基本超几何函数。我们研究了具有移位参数的基本超几何函数的某些比值的映射性质，并表明它们将解析域映射到在虚轴方向上凸的域上。为了研究这些映射的性质，得到了一些关于基本超几何函数的有用恒等式。此外，我们还得到了基本超几何函数属于$q$ -近于凸函数族的条件.

On geometric properties of ratio of two hypergeometric functions

R. Küstner proved in his 2002 paper that the function $w_{a,b,c}(z)=$ $F(a+1,b;c;z)/F(a,b;c;z)$ maps the unit disk $|z|<1$ onto a domain convex in the direction of the imaginary axis under some condition on the real parameters $a,b,c.$ Here $F(a,b;c;z)$ stands for the Gaussian hypergeometric function. In this paper, we study the order of convexity of $w_{a,b,c}.$ In particular, we partially solve the problem raised by the afore-mentioned paper by Küstner.

Hausdorff 矩序列与超几何函数

Hausdorff moment sequences and hypergeometric functions

本文完整刻画了系数为 Hausdorff 矩序列的超几何函数，并利用 Phragmèn-Lindelöf 型定理给出充分条件，进而得到平移超几何函数全星形的充要条件。

On Meijer's $G$ function $G^{m,n}_{p,p}$ for $m+n=p$

The paper is devoted to the piece-wise analytic case of Meijer's $G$ function $G^{m,n}_{p,p}$. While the problem of its analytic continuation was solved in principle by Meijer and Braaksma we show that in the ''balanced'' case $m+n=p$ the formulas take a particularly simple form. We derive explicit expressions for the values of these analytic continuations on the banks of the branch cuts. It is further demonstrated that particular cases of this type of $G$ function having integer parameter differences satisfy identities similar to the Miller-Paris transformations for the generalized hypergeome…