Bemerkungen zu einem Satz von Fejér

Последовательность {ita_k} _(n) ^k =1/∞ вещественных ч исел называется дважды мо нотонной, еслиa _k -2a _k+1 +a _k+2 ≧0 дляk≧1. В работе доказываютс я следующие утвержде ния, являющиеся обобщени ем двух теорем Фейера:

1. Если {ita_k — дважды моно тонная последовател ьность, то для ¦z¦<1 $$\operatorname{Re} \sum\limits_{\kappa = 1}^\infty {a_\kappa z^\kappa } /\sum\limits_{\kappa = 1}^n {a_\kappa z^\kappa } > 1/2$$ дляи≧ 1.
2. Если О≦β<1 и последова тельность (k+1-2β)ak} дважд ы монотонна, то для ¦z¦<1 $$\operatorname{Re} \sum\limits_{\kappa = 1}^\infty {ka_\kappa z^\kappa } /\sum\limits_{\kappa = 1}^\infty {a_\kappa z^\kappa } > \beta $$ , то есть $$\sum\limits_{\kappa = 1}^\infty {a_\kappa z^\kappa } \varepsilon S_\beta ^\kappa $$ . При помощи 2) получены о бобщения и уточнения теорем из работы [1] о линейных комбинациях некотор ых однолистных функц ий.

     

Schwarz—Pick Inequalities for Derivatives of Arbitrary Order

Abstract. Let Ω and Π be two simply connected domains in the complex plane C which are not equal to the whole plane C and let λ _Ω and λ _Π denote the densities of the Poincare metric in Ω and Π , respectively. For f: Ω → Π analytic in Ω , inequalities of the type $$\frac{{|f^{(n)} (z)|}}{{n!}} \leqslant M_n (z,\Omega ,\Pi )\frac{{(\lambda _\Omega (z))^n }}{{\lambda _\Pi (f(z))}},z \in \Omega$$ are considered where M _n (z,Ω, Π) does not depend on f and represents the smallest value possible at this place. We prove that $$M_n (z,\Delta ,\Pi ) = (1 + |z|)^{n - 1}$$ if Δ is the unit disk and Π is a convex domain. This generalizes a result of St. Ruscheweyh. Furthermore, we show that $$C_n (\Omega ,\Pi ) = sup\left\{ {M_n (z,\Omega ,\Pi )|z \in \Omega } \right\} \leqslant 4^{n - 1}$$ holds for arbitrary simply connected domains whereas the inequality 2 ^n-1 ≤ C _n (Ω,Π) is proved only under some technical restrictions upon Ω and Π .     

Global integral criteria for composition operators

Let D_α denote the Dirichlet-type space of functions analytic on the unit disk U and Q_α the conformal invariant version of this space. Any analytic self-map φ of U induces a composition operator C_φ acting on D_α, respectively, Q_α by C_φf=f°φ, where f∈D_α, respectively, f∈Q_α. The aim of this paper is to characterize boundedness and compactness of such operators in terms of global area integrals of φ.     

Über Polyanion-Polykation-Komplexe mit Polyphosphat

Polyanion-Polycation Complexes with Polyphosphate With two Na polyphosphates of different molar weight and a series of cationic polyelectrolytes (polydimethyldiallylammonium chloride, polyethylenimine, cationically modified polyacryloamides of different charge density) polysalts (symplexes) have been prepared. The precipitates were investigated with regard to stoichiometry of cationic to anionic groups, with regard to swelling in water, and with regard to morphology. Applying special conditions of component concentration, stoichiometric 1:1 symplexes were obtained also with polyphosphate. According to our results, the cohesion in these polyphosphate symplexes is caused mainly by electrostatic forces only, in contrast to symplexes with anionic cellulose derivatives.     

Über holomorphe Funktionen,die einer Wachstumsbeschränkung unterliegen

Ohne Zusammenfassung     

The conformal radius as a function and its gradient image

Let Ω be a domain in with three or more boundary points in andR(w, Ω) the conformal, resp. hyperbolic radius of Ω at the pointw ε Ω/{∞}. We give a unified proof and some generalizations of a number of known theorems that are concerned with the geometry of the surface in the case that the Jacobian of ∇R(w, Ω), the gradient ofR, is nonegative on Ω. We discuss the function ∇R(w, Ω) in some detail, since it plays a central role in our considerations. In particular, we prove that ∇R(w, Ω) is a diffeomorphism of Ω for four different types of domains. This work was supported by a grant of the Deutsche Forschungsgemeinschaft for F. G. Avkhadiev.