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Register a new userGeometric properties of the shifted hypergeometric functions
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We will provide sufficient conditions for the shifted hypergeometric function $z_2F_1(a,b;c;z)$ to be a member of a specific subclass of starlike functions in terms of the complex parameters $a,b$ and $c.$ For example, we study starlikeness of order $alpha,$ $lambda$-spirallikeness of order $alpha$ and strong starlikeness of order $alpha.$ In particular, those properties lead to univalence of the shifted hypergeometric functions on the unit disk.
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In this paper our aim is to find the radii of starlikeness and convexity of Bessel function derivatives for three different kind of normalization. The key tools in the proof of our main results are the Mittag-Leffler expansion for nth derivative of Bessel function and properties of real zeros of it. In addition, by using the Euler-Rayleigh inequalities we obtain some tight lower and upper bounds for the radii of starlikeness and convexity of order zero for the normalized nth derivative of Bessel function. The main results of the paper are natural extensions of some known results on classical Bessel functions of the first kind.
Let $f$ be the infinitesimal generator of a one-parameter semigroup $left{ F_{t}right} _{tge0}$ of holomorphic self-mappings of the open unit disk $Delta$. In this paper we study properties of the family $R$ of resolvents $(I+rf)^{-1}:DeltatoDelta~ (rge0)$ in the spirit of geometric function theory. We discovered, in particular, that $R$ forms an inverse Lowner chain of hyperbolically convex functions. Moreover, each element of $R$ satisfies the Noshiro-Warschawski condition and is a starlike function of order at least $frac12$,. This, in turn, implies that each element of $R$ is also a holomorphic generator. We mention also quasiconformal extension of an element of $R.$ Finally we study the existence of repelling fixed points of this family.
For $ngeq 4$ (even), the function $varphi_{nmathcal{L}}(z)=1+nz/(n+1)+z^n/(n+1)$ maps the unit disk $mathbb{D}$ onto a domain bounded by an epicycloid with $n-1$ cusps. In this paper, the class $mathcal{S}^*_{nmathcal{L}} = mathcal{S}^*(varphi_{nmathcal{L}})$ is studied and various inclusion relations are established with other subclasses of starlike functions. The bounds on initial coefficients is also computed. Various radii problems are also solved for the class $mathcal{S}^*_{nmathcal{L}}$.
Kalantaris Geometric Modulus Principle describes the local behavior of the modulus of a polynomial. Specifically, if $p(z) = a_0 + sum_{j=k}^n a_jleft(z-z_0right)^j,;a_0a_ka_n eq 0$, then the complex plane near $z = z_0$ comprises $2k$ sectors of angle $frac{pi}{k}$, alternating between arguments of ascent (angles $theta$ where $|p(z_0 + te^{itheta})| > |p(z_0)|$ for small $t$) and arguments of descent (where the opposite inequality holds). In this paper, we generalize the Geometric Modulus Principle to holomorphic and harmonic functions. As in Kalantaris original paper, we use these extensions to give succinct, elegant new proofs of some classical theorems from analysis.
The purpose of this paper is twofold. One is to enrich from a geometrical point of view the theory of octonionic slice regular functions. We first prove a boundary Schwarz lemma for slice regular self-mappings of the open unit ball of the octonionic space. As applications, we obtain two Landau-Toeplitz type theorems for slice regular functions with respect to regular diameter and slice diameter respectively, together with a Cauchy type estimate. Along with these results, we introduce some new and useful ideas, which also allow to prove the minimum principle and one version of the open mapping theorem. Another is to strengthen a version of boundary Schwarz lemma first proved in cite{WR} for quaternionic slice regular functions, with a completely new approach. Our quaternionic boundary Schwarz lemma with optimal estimate improves considerably a well-known Osserman type estimate and provides additionally all the extremal functions.
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