On Certain Multivalent Functions

Let 𝒮*(p,α) be the class of functions f(z) which are analytic and p-valently starlike of order α in the open unit disk E. The object of the present paper is to derive an interesting condition for f(z) to be in the class 𝒮*(p,α)

Pinpoint-fluorinated polycyclic aromatic hydrocarbons (F-PAHs): Syntheses of difluorinated subfamily and their properties

Difluorinated polycyclic aromatic hydrocarbons (PAHs) containing three to five benzene rings were systematically synthesized by the Pd(II)-catalyzed Friedel–Crafts-type cyclization of 1,1,2-trifluoro- and 1,1-difluoro-1-alkenes and the In(III)-catalyzed tandem cyclization of bis(1,1-difluoroallene)s. Using an array of the difluorinated PAHs that were obtained and previously reported monofluorinated PAHs, the physical properties of the pinpoint-fluorinated PAHs were investigated. (i) The 19F NMR signals of the bay-region fluorine atoms were shifted downfield by ca. 8–14 ppm for vic-difluorinated PAHs and ca. 11–19 ppm for non-vic-difluorinated and monofluorinated PAHs. (ii) The introduction of fluorine into PAH molecules increased their solubilities in organic solvents, which was best exemplified by the high solubilities of 6,7-difluoropicene (5.4 wt%) and 6-fluoropicene (5.3 wt%) in THF. (iii) The HOMO–LUMO energy gaps of the pinpoint-fluorinated PAHs were smaller than that of the corresponding fluorine-free PAH (i.e., picene) by 0.02–0.26 eV, and the HOMO and LUMO energy levels were lowered by 0.10–0.22 eV and 0.12–0.41 eV, respectively