On the Bohr inequality

The Bohr inequality, first introduced by Harald Bohr in 1914, deals with finding the largest radius $r$, $0<r<1$, such that $\sum_{n=0}^\infty |a_n|r^n \leq 1$ holds whenever $|\sum_{n=0}^\infty a_nz^n|\leq 1$ in the unit disk $\mathbb{D}$ of the complex plane. The exact value of this largest radius, known as the \emph{Bohr radius}, has been established to be $1/3.$ This paper surveys recent advances and generalizations on the Bohr inequality. It discusses the Bohr radius for certain power series in $\mathbb{D},$ as well as for analytic functions from $\mathbb{D}$ into particular domains. These domains include the punctured unit disk, the exterior of the closed unit disk, and concave wedge-domains. The analogous Bohr radius is also studied for harmonic and starlike logharmonic mappings in $\mathbb{D}.$ The Bohr phenomenon which is described in terms of the Euclidean distance is further investigated using the spherical chordal metric and the hyperbolic metric. The exposition concludes with a discussion on the $n$-dimensional Bohr radius

Radical Cyclisation of α-Halo Aluminium Acetals: A Mechanistic Study

International audienceα‐Bromo aluminium acetals are suitable substrates for Ueno–Stork‐like radical cyclisations affording γ‐lactols and acid‐sensitive methylene‐γ‐lactols in high yields. The mechanistic study herein sets the scope and limitation of this reaction. The influence of the halide (or chalcogenide) atom X (X=Cl, Br, I, SPh, SePh) in the precursors α‐haloesters, as well as influence of the solvent and temperature was studied. The structure of the aluminium acetal intermediates resulting from the reduction of the corresponding α‐haloesters has been investigated by low‐temperature 13C‐INEPT diffusion‐ordered NMR spectroscopy (DOSY) experiments and quantum calculations, providing new insights into the structures of these thermally labile intermediates. Oxygen‐bridged dimeric structures with a planar Al2O2 ring are proposed for the least hindered aluminium acetals, while monomeric structures seem to prevail for the most hindered species. A comparison against the radical cyclisation of aluminium acetals derived from allyl and propargyl alcohols with the parent Ueno–Stork has been made at the BHandHLYP/6‐311++G(d,p) level of theory, highlighting mechanistic similarities and differences

New routes to Cu(l)/Cu nanocatalysts for the multicomponent click synthesis of 1,2,3-triazoles

An array of copper and copper–zinc based nanoparticles (NPs) have been fabricated employing a variety of polymeric capping agents. Analysis by TEM, XRPD and XPS suggests that by manipulating reagent, reductant and solvent conditions it is possible to achieve materials that are mono-/narrow disperse with mean particle sizes in the ≤10 nm regime. Oxidative stability in air is achieved for monometallic NPs using poly(methyl methacrylate) (PMMA) anti-agglomerant in conjunction with a variety of reducing conditions. In contrast, those encapsulated by either poly(1-vinylpyrrolidin-2-one) (PVP) or poly(4-vinylpyridine) (PVPy) rapidly show Cu2O formation, with all data suggesting progressive oxidation from Cu to Cu@Cu2O core–shell structure and finally Cu2O. Bimetallic copper–zinc systems, reveal metal segregation and the formation of Cu2O and ZnO. Catalysts have been screened in the synthesis of 1,2,3-triazoles through multicomponent azide–alkyne 1,3-dipolar cycloaddition. Whereas PMMA- and PVPy-coating results in reduced catalytic activity, those protected by PVP are highly active, with quantitative triazole syntheses achieved at room temperature and with catalyst loadings of 0.03 mol% metal for Cu and CuZn systems prepared using NaH2PO2, N2H4 or NaBH4 reductants.This work was generously supported by the Spanish Ministerio de Ciencia e Innovación (MICINN; CTQ2007-65218 and Consolider Ingenio 2010-CSD2007-00006), the Generalitat Valenciana (GV; PROMETEO/2009/039), and FEDER. Y. M. and B. R. K. thank the ISO of the Universidad de Alicante, the UK EPSRC and the University of Cambridge for grants

Multicomponent click synthesis of 1,2,3-triazoles from epoxides in water catalyzed by Copper nanoparticles on activated carbon

Copper nanoparticles on activated carbon have been found to effectively catalyze the multicomponent synthesis of β-hydroxy-1,2,3-triazoles from a variety of epoxides and alkynes in water. The catalyst is easy to prepare, reusable at a low copper loading (0.5 mol %), and exhibits higher catalytic activity than some commercially available copper sources. The regio- and stereochemistry of the reaction has been revised and unequivocally established on the basis of X-ray crystallographic analyses. An NMR experiment has been implemented for the rapid and unmistakable determination of the regiochemistry of the process. Some mechanistic aspects of the reaction have been also undertaken which unveil the participation of copper(I) acetylides.This work was generously supported by the Spanish Ministerio de Ciencia e Innovación (MICINN; CTQ2007-65218 and Consolider Ingenio 2010-CSD2007-00006), the Generalitat Valenciana (GV; PROMETEO/2009/039), and FEDER. Y.M. acknowledges the ISO of the Universidad de Alicante for a grant

A Non-Linear Extremal Problem on the Hardy Space

We relate some classical extremal problems on the Hardy space to norms of truncated Toeplitz operators and complex symmetric operators