Injectivity of sections of convex harmonic mappings and convolution theorems

In the article the authors consider the class ${\mathcal H}_0$ of sense-preserving harmonic functions $f=h+\overline{g}$ defined in the unit disk $|z|<1$ and normalized so that $h(0)=0=h'(0)-1$ and $g(0)=0=g'(0)$, where $h$ and $g$ are analytic in the unit disk. In the first part of the article we present two classes $\mathcal{P}_H^0(\alpha)$ and $\mathcal{G}_H^0(\beta)$ of functions from ${\mathcal H}_0$ and show that if $f\in \mathcal{P}_H^0(\alpha)$ and $F\in\mathcal{G}_H^0(\beta)$, then the harmonic convolution is a univalent and close-to-convex harmonic function in the unit disk provided certain conditions for parameters $\alpha$ and $\beta$ are satisfied. In the second part we study the harmonic sections (partial sums) $$s_{n, n}(f)(z)=s_n(h)(z)+\overline{s_n(g)(z)},$$ where $f=h+\overline{g}\in {\mathcal H}_0$, $s_n(h)$ and $s_n(g)$ denote the $n$-th partial sums of $h$ and $g$, respectively. We prove, among others, that if $f=h+\overline{g}\in{\mathcal H}_0$ is a univalent harmonic convex mapping, then $s_{n, n}(f)$ is univalent and close-to-convex in the disk $|z|< 1/4$ for $n\geq 2$, and $s_{n, n}(f)$ is also convex in the disk $|z|< 1/4$ for $n\geq2$ and $n\neq 3$. Moreover, we show that the section $s_{3,3}(f)$ of $f\in {\mathcal C}_H^0$ is not convex in the disk $|z|<1/4$ but is shown to be convex in a smaller disk.Comment: 16 pages, 3 figures; To appear in Czechoslovak Mathematical Journa

Steam sauna and mother roasting in Lao PDR: practices and chemical constituents of essential oils of plant species used in postpartum recovery

<p>Abstract</p> <p>Background</p> <p>Fundamental in traditional postpartum recovery in Lao PDR is the use of hotbeds, mother roasting, steam sauna and steam baths. During these treatments medicinal plants play a crucial role, but little has been published about how the treatments are carried out precisely, which species are used, the medicinal properties of these species, and the medicinal efficacy of their chemical constituents.</p> <p>Methods</p> <p>Sixty-five interviews, in 15 rural villages, with women of 4 different ethnic groups were conducted to survey confinement rituals, and postpartum plant use and salience. Essential oils from the main species used were extracted using steam distillation and the main chemical constituents characterized using gas chromatography-mass spectrometry (GC-MS).</p> <p>Results</p> <p>A total of 10 different species were used by three or more of the ethnic groups included in this study. All species were used in steam sauna and bath, but only 3 species were used in hotbed and mother roasting. Essential oils of <it>Amomum villosum, Amomum microcarpum </it>and <it>Blumea balsamifera </it>were found to contain significant amounts of the following terpenes: β-pinene, camphor, bornyl acetate, borneol, linalool, D-limonene, fenchone, terpinen-4-ol and α-terpinene.</p> <p>Conclusions</p> <p>Many of these terpenes have documented antimicrobial and analgesic properties, and some have also synergistic interactions with other terpenes. The mode of application in hotbed and mother roasting differs from the documented mechanisms of action of these terpenes. Plants in these two practices are likely to serve mainly hygienic purposes, by segregating the mother from infection sources such as beds, mats, stools, cloth and towels. Steam sauna medicinal plant use through inhalation of essential oils vapors can possibly have medicinal efficacy, but is unlikely to alleviate the ailments commonly encountered during postpartum convalescence. Steam sauna medicinal plant use through dermal condensation of essential oils, and steam bath cleansing of the perineal area is possibly a pragmatic use of the reported medicinal plants, as terpene constituents have documented antimicrobial, analgesic and anti-inflammatory properties.</p