Coefficient estimates for bi-univalent Ma-Minda starlike and convex functions

Estimates on the initial coefficients are obtained for normalized analytic functions $f$ in the open unit disk with $f$ and its inverse $g=f^{-1}$ satisfying the conditions that $zf'(z)/f(z)$ and $zg'(z)/g(z)$ are both subordinate to a starlike univalent function whose range is symmetric with respect to the real axis. Several related classes of functions are also considered, and connections to earlier known results are made

Coefficient Conditions for Starlikeness of Nonnegative Order

Sufficient conditions on a sequence {ak} of nonnegative numbers are obtained that ensures f(z)= E°°k=1 akzk is starlike of nonnegative order in the unit disk. A result of Vietoris on trigonometric sums is extended in this pursuit. Conditions for close to convexity and convexity in the direction of the imaginary axis are also established. These results are applied to investigate the starlikeness of functions involving the Gaussian hypergeometric functions

A Third-Order Differential Equation and Starlikeness of a Double Integral Operator

Functions f(z)= z+E°2 anzn that are analytic in the unit disk and satisfy the differential equation f'(z) + azf"(z)+yz2f"(z) = g(z) are considered, where g is subordinated to a normalized convex univalent function h. These functions f are given by a double integral operator of the form f(z) = (10(10G(ztμsν�t−μs−νds dt with G" subordinated to h. The best dominant to all solutions of the differential equation is obtained. Starlikeness properties and various sharp estimates of these solutions are investigated for particular cases of the convex function h

Dibut­yl{N′-[1-(5-chloro-2-oxidophenyl-κO)ethyl­idene]-3-hy­droxy-2-naphtho­hydrazidato-κ2 N′,O 2}tin(IV)

The five-coordinate SnIV atoms in the two crystallographically independent mol­ecules of the title compound, [Sn(C4H9)2(C19H13ClN2O3)], are in distorted cis-C2NO2Sn trigonal-bipyramidal coordination environments. The tridentate dianion of the Schiff base, N′-[1-(5-chloro-2-oxidophen­yl)ethyl­idene]-3-hy­droxy-2-naphtho­hydrazide, displays inter­molecular O—H⋯N hydrogen bonding, which stabilizes the overall compound

[N′-(5-Chloro-2-oxidobenzyl­idene-κO)-3-hydr­oxy-2-naphthohydrazidato-κ2 N′,O 2]diphenyl­tin(IV)

The SnIV atom in the title compound, [Sn(C6H5)2(C18H11ClN2O3)], is O,N,O′-chelated by the deprotonated Schiff base ligand and further bonded by two phenyl rings in a distorted cis-C2SnNO2 trigonal-bipyramidal geometry [C—Sn—C = 125.7 (2)°]. The two phenyl rings are oriented at a dihedral angle of 55.2 (3)°. Intra­molecular O—H⋯N hydrogen bonding is present in the crystal structure

The resin-embedded cornea prepared via rapid processing protocol : a good histomorphometric target for clinical investigation in opthalmology and optometry.

This study illustrates and quantifies the changes on corneal tissue between the paraffin-embedded and resin-embedded blocks and thus, selects a better target in investigational ophthalmology and optometry via light microscopy. Corneas of two cynomolgus monkeys (Macaca fascicularis) were used in this study. The formalin-fixed cornea was prepared in paraffin block via the conventional tissue processing protocol (4-day protocol) and stained with haematoxylin and eosin. The glutaraldehyde-fixed cornea was prepared in resin block via the rapid and modified tissue processing procedure (1.2-day protocol) and stained with toluidine blue. The paraffin-embedded sample exhibits various undesired tissue damage and artifact such as thinner epithelium (due to the substantial volumic extraction from the tissue), thicker stroma layer (due to the separation of lamellae and the presence of voids) and the distorted endothelium. In contrast, the resin-embedded corneal tissue has demonstrated satisfactory corneal ultrastructural preservation. The rapid and modified tissue processing method for preparing the resin-embedded is particularly beneficial to accelerate the microscopic evaluation in ophthalmology and optometry

Bis[4-(dimethyl­amino)pyridinium] tetra­bromidobis(4-chloro­phen­yl)stannate(IV)–4-bromo­chloro­benzene (1/1)

In the title compound, (C7H11N2)2[SnBr4(C6H4Cl)2]·C6H4BrCl, the SnIV atom in the tetra­bromidobis(4-chloro­phen­yl)stannate(IV) anion lies on a centre of inversion. The distances between the 4-(dimethyl­amino)pyridinium N atom and the Br atoms of the anion are 3.450 (2) and 3.452 (2) Å, suggesting weak hydrogen bonding. The 4-bromo­chloro­benzene solvent mol­ecule, which is a bromination by-product from the reaction, is disordered about a twofold rotation axis with approximately equal occupancy

Bis[4-(dimethyl­amino)pyridinium] tetra­bromidobis(4-methyl­phen­yl)stannate(IV)

In the title compound, (C7H11N2)2[SnBr4(C7H7)2], the tetra­bromidobis(4-methyl­phen­yl)stannate(IV) anion possesses a centre of inversion located at the SnIV atom. In the crystal structure, two inversion-related cations are linked to the anion via weak N—H⋯Br hydrogen bonds