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Register a new userGeometric properties of the nonlinear resolvent of holomorphic generators
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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.
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We prove a theorem on separation of boundary null points for generators of continuous semigroups of holomorphic self-mappings of the unit disk in the complex plane. Our construction demonstrates the existence and importance of a particular role of the binary operation $circ$ given by $1 / f circ g = 1/f + 1/g$ on generators.
We present a rigidity property of holomorphic generators on the open unit ball $mathbb{B}$ of a Hilbert space $H$. Namely, if $finHol (mathbb{B},H)$ is the generator of a one-parameter continuous semigroup ${F_t}_{tgeq 0}$ on $mathbb{B}$ such that for some boundary point $tauin partialmathbb{B}$, the admissible limit $K$-$limlimits_{ztotau}frac{f(x)}{|x-tau|^{3}}=0$, then $f$ vanishes identically on $mathbb{B}$.
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.
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.
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.
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