Improved Bohr’s inequality for locally univalent harmonic mappings

We prove several improved versions of Bohr’s inequality for the harmonic mappings of the form $f=h+\overline{g}$, where $h$ is bounded by 1 and $\left|g^{\prime }\left(z\right)\right|\le \left|h^{\prime }\left(z\right)\right|$. The improvements are obtained along the lines of an earlier work of Kayumov and Ponnusamy, i.e. (Kayumov and Ponnusamy, 2018) for example a term related to the area of the image of the disk $D(0,r)$ under the mapping $f$ is considered. Our results are sharp. In addition, further improvements of the main results for certain special classes of harmonic mappings are provided.     

Asymmetric epoxidation of some arylalkenyl sulfones using a modified Juliá-Colonna procedure

A modified procedure for performing the Juliá-Colonna epoxidation reaction effects the oxidation of some vinyl sulfones to generate the corresponding epoxides 5-8 in good to excellent optical purity.     

Doubly charged protonated a ions derived from small peptides

Protonated a(2) and a(3) (therefore doubly charged) ions in which both charges lie on the peptide backbone are formed in collision-induced dissociations of [La(III)(peptide)(CH(3)CN)(m)](3+) complexes. Abundant (a(3)+H)(2+) ions are formed from triproline (PPP) and peptides with a proline residue at the N-terminus; these peptides are the most effective in producing ions of the type (a(2)+H)(2+) and (a(3)+H)(2+). A systematic study of the effect of the location of the proline residue and other residues of aliphatic amino acids on the generation of protonated a ions is reported. Density functional theory calculations at B3LYP/6-311++G(d,p) gave the proton affinity of the a(3) ion derived from PPP to be 167.6 kcal mol(-1), 2.6 kcal mol(-1) higher than that of water. The protonated a(2) ions of diglycine and diproline and a(3) ions of triglycine have lower proton affinities and are only observed in lower abundances, possibly due to proton transfer to water in ion-molecule reactions.     

Riesz-Fejér Inequalities for Harmonic Functions

Potential Analysis - In this article, we prove the Riesz - Fejér inequality for complex-valued harmonic functions in the harmonic Hardy space hp for all p >?1. The result is sharp...     

Characterization and the pre-Schwarzian norm estimate for concave univalent functions

Let Co(α) denote the class of concave univalent functions in the unit disk ${\mathbb{D}}$ . Each function ${f\in Co(\alpha)}$ maps the unit disk ${\mathbb{D}}$ onto the complement of an unbounded convex set. In this paper we find the exact disk of variability for the functional ${(1-|z|^2)\left ( f^{\prime\prime}(z)/f^{\prime}(z)\right), f\in Co(\alpha)}$ . In particular, this gives sharp upper and lower estimates for the pre-Schwarzian norm of concave univalent functions. Next we obtain the set of variability of the functional ${(1-|z|^2)\left(f^{\prime\prime}(z)/f^{\prime}(z)\right), f\in Co(\alpha)}$ whenever f′′(0) is fixed. We also give a characterization for concave functions in terms of Hadamard convolution. In addition to sharp coefficient inequalities, we prove that functions in Co(α) belong to the H ^p space for p < 1/α.     

Effect of size of metal ion on MS4N2 chromophore: Synthesis,spectral and single crystal X-ray structural studies on (2,2′-bipyridine)bis(N-cyclohexyl-N-methyldithiocarbamato)M(II) (M = zinc,cadmium)

Isomorphous complexes [Zn(S₂CN(Me)Cy)₂(bipy)] (1) and [Cd(S₂CN(Me)Cy)₂(bipy)] (2) (where Cy(Me)NCS₂⁻ = N-cyclohexyl-N-methyldithiocarbamate anion and bipy = 2,2′-bipyridine) have been synthesised. Their structures and spectroscopic properties have been studied by IR, ¹H and ¹³C NMR spectroscopy and single crystal X-ray analysis. IR spectrum of the complexes show the contribution of thioureide form to the structures. ¹H NMR spectra of the complexes showed the desheilding of methyl protons and H-1 of cyclohexyl group on complexation. The downfield shift of N¹³CS₂, methyl carbon and C-1 of cyclohexyl group carbon signals for 2 (205.5, 36.7 and 64.4 ppm) from the chemical shift value of 1 (204.2, 35.6 and 63.1 ppm) is attributed to the movement of more electron density from dithiocarbamate towards cadmium. Single crystal X-ray structures of 1 and 2 indicate that the central metal atom is in a distorted octahedral environment for both complexes. The presence of added 2,2′-bipyridine ligand in the coordination sphere of M(S₂CN(Me)Cy)₂ increases Zn–S distances and decreases S–Zn–S angles in 1 and slightly increases Cd–S distances in 2. S–Cd–S angles are not affected. This is due to the relatively larger size of the cadmium ion compared to zinc ion which alleviates the strain involved in transformation from tetrahedral to octahedral.     

The extent and effects of peptide sequence scrambling via formation of macrocyclic <Emphasis Type="Italic">b</Emphasis> ions in model proteins

The extent and effects of sequence scrambling in peptide ions during tandem mass spectrometry (MS/MS) have been examined using tryptic peptides from model proteins. Sequencescrambled b ions appeared in about 35% of 43 tryptic peptides examined under MS/MS conditions. In general, these ions had relatively low abundances with averages of 8% and 16%, depending on the instrumentation used. A few tryptic peptides gave abundant scrambled b ions in MS/MS. However, peptide and protein identifications under proteomic conditions with Mascot were not affected, even for these peptides wherein scrambling was prominent. From the 43 tryptic peptides that have been investigated, the conclusion is that sequence scrambling is unlikely to impact negatively on the accuracy of automated peptide and protein identifications in proteomics.     

Dirichlet Problem,Univalency and Schwarz Lemma for Biharmonic Mappings

In this paper, we shall discuss the family of biharmonic mappings for which the maximum principle holds. As a consequence of our study, we present Schwarz Lemma for certain class of biharmonic mappings. Also we discuss the univalency of certain class of biharmonic mappings.