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 \mathbbD{\mathbb{D}}. Each function f ? Co(a){f\in Co(\alpha)} maps the unit disk \mathbbD{\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)( f^￠￠(z)/f^￠(z)), f ? Co(a){(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)(f^￠￠(z)/f^￠(z)), f ? Co(a){(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.     

Schwarzian derivative, integral means, and the affine and linear invariant families of biharmonic mappings

In this paper we discuss the properties of the Schwarzian derivative, integral means and the affine and linear invariant families of biharmonic mappings. First, we introduce the Schwarzian derivative S(F) for biharmonic mappings F = ∣z∣²G + H, and obtain several necessary and sufficient conditions for S(F) to be analytic. Second, we introduce the subordination of biharmonic mappings and obtain inequalities for integral means of subordinate biharmonic mappings. Finally, we introduce the affine and linear invariant families of biharmonic mappings and prove several estimates related to the Jacobian of functions in these invariant families.     

Vanadium(V)-substituted Keggin-type heteropolyoxotungstophosphates as electron transfer and antimicrobial agents: oxidation of glutathione and sensitization of MRSA towards β-lactam antibiotics

Glutathione (GSH) undergoes facile electron transfer with vanadium(V)-substituted Keggin-type heteropolyoxometalates, [ \textPV^\textV \textW _{1 1} \textO _{4 0} ] ^{4 -} [ {\text{PV}}^{\text{V}} {\text{W}}_{ 1 1} {\text{O}}_{ 4 0} ]^{{ 4 { - }}} (HPA1) and [ \textPV^\textV \textV^\textV \textW _{1 0} \textO _{4 0} ] ^{5 -} [ {\text{PV}}^{\text{V}} {\text{V}}^{\text{V}} {\text{W}}_{ 1 0} {\text{O}}_{ 4 0} ]^{{ 5 { - }}} (HPA2). The kinetics of these reactions have been investigated in phthalate buffers spectrophotometrically at 25 °C in aqueous medium. One mole of HPA1 consumes one mole of GSH and the product is the one-electron reduced heteropoly blue, [ \textPV^\textIV \textW _{1 1} \textO ₄₀ ] ^5- [ {\text{PV}}^{\text{IV}} {\text{W}}_{ 1 1} {\text{O}}_{ 40} ]^{ 5- } . But in the GSH-HPA2 reaction, one mole of HPA2 consumes two moles of GSH and gives the two-electron reduced heteropoly blue [ \textPV^\textIV \textV^\textIV \textW ₁₀ \textO ₄₀ ] ^7- [ {\text{PV}}^{\text{IV}} {\text{V}}^{\text{IV}} {\text{W}}_{ 10} {\text{O}}_{ 40} ]^{ 7- } . Both reactions show overall third-order kinetics. At constant pH, the order with respect to both [HPA] species is one and order with respect to [GSH] is two. At constant [GSH], the rate shows inverse dependence on [H⁺], suggesting participation of the deprotonated thiol group of GSH in the reaction. A suitable mechanism has been proposed and a rate law for the title reaction is derived. The antimicrobial activities of HPA1, HPA2 and [ \textPV^\textV \textV^\textV \textV^\textV \textW ₉ \textO _{4 0} ] ^{6 -} [ {\text{PV}}^{\text{V}} {\text{V}}^{\text{V}} {\text{V}}^{\text{V}} {\text{W}}_{ 9} {\text{O}}_{ 4 0} ]^{{ 6 { - }}} (HPA3) against MRSA were tested in vitro in combination with vancomycin and penicillin G. The HPAs sensitize MRSA towards penicillin G.     

Application of gamma irradiation to reduce microbial contamination in herbal cosmetic products

This study addresses the decontamination of herbal powder cosmetics by gamma irradiation to reduce the total microbial colony count in facial herbal powder, herbal rose brush on and talcum. Pre-irradiated samples showed total colony counts of 3.00×10⁴, 2.70×10⁴ and 1.00×10³ CFU/g. At 3rd day after application, irradiation reduced the total colony counts to 1.90×10², 6.00×10² and 1.20×10² CFU/g. Moreover, the total colony counts of the three samples were found to be less than 100 CFU/g after 3 months storage. The non-uniformity of ΔE^? revealed that time affected the color of brush on and talcum, which differed from their original color; however, irradiation affected the colors of the brush on only (P<0.05), by reducing its brightness and increasing redness and yellowness of the products. Paired preference tests were conducted in facial herbal powder and herbal rose brush on. The results showed no significant preferences between the non irradiated and irradiated of the two products at P max=75%, α=0.05, β=0.10. This concludes that the irradiation does not affect the preference of the products, and it can be an alternative technology to reduce microbial decontaminations in herbal products.     

Radii of covering disks for locally univalent harmonic mappings

Monatshefte für Mathematik - For a univalent smooth mapping f of the unit disk $${\mathbb D}$$ of complex plane onto the manifold $$f({\mathbb D})$$ , let $$d_f(z_0)$$ be the radius of the...     

Bounds for the product of modified Bessel functions

In this note our aim is to present some monotonicity properties of the product of modified Bessel functions of the first and second kind. Certain bounds for the product of modified Bessel functions of the first and second kind are also obtained. These bounds improve and extend known bounds for the product of modified Bessel functions of the first and second kind of order zero. A new Turán type inequality is also given for the product of modified Bessel functions, and some open problems are stated, which may be of interest for further research.     

Effect of gravity modulation on linear,weakly-nonlinear and local-nonlinear stability analyses of stationary double-diffusive convection in a dielectric liquid

Meccanica - The paper deals with the study of effect of gravity modulation on double-diffusive convection in a dielectric liquid for the cases of rigid-rigid and free-free boundaries. Using a...     

Kinetics of the outer-sphere oxidation of thiourea by heteropoly-α<Subscript>2</Subscript>-17-tungsto-1-vanadodiphosphate anion

Kinetics of the oxidation of thiourea (tu) by heteropoly-α₂-17-tungsto-1-vanadodiphosphate anion, α₂-[P₂V^VW₁₇O₆₂]^7?, have been studied spectrophotometrically in aqueous acidic medium at 25 °C. At low pH (2.4–3.0), the neutral form of tu is the only reactive species. At higher pH (4.2–4.9), both neutral and deprotonated forms of tu participate in the reaction. The observed mixed-order kinetics suggest two parallel reactions: one in which the order in [tu] is unity, and a second in which it is two. In both cases, the order in [α₂-[P₂V^VW₁₇O₆₂]^7?] is unity. Based on the kinetic studies, a mechanism is proposed, in which a second-order proton-coupled electron transfer involving NH₂CSNH₂ and α₂-[P₂V^VW₁₇O₆₂]^7? proceeds through a sequential electron transfer, followed by proton transfer such that the reaction is an “activation-controlled” outer-sphere electron transfer process. By applying the Marcus equation, the self-exchange rate constants for the couples \({\text{NH}}_{2} {\text{CSNH}}_{2}^{ \cdot + }\)/NH₂CSNH₂ and α₂-[P₂V^VW₁₇O₆₂]^7?/α₂-[P₂V^IVW₁₇O₆₂]^8? were evaluated.     

Cellulase Production by Pink Pigmented Facultative Methylotrophic Strains (PPFMs)

Pink pigmented facultative methylotrophs (PPFM) isolated from water samples of Cooum and Adyar rivers in Chennai and soil samples of forests located in various districts of Tamil Nadu, India were screened for cellulase production using carboxymethylcellulose agar (CMC agar) medium. The strains showed wide variations in the production of clearing zones around the colonies on CMC agar medium flooded with Congo red. CMCase and filter paper assays were used to quantitatively measure the cellulase activity of 13 PPFM strains. Among the strains, Methylobacterium gregans, MNW 60, MHW 109, MSF 34, and MSF 40 showed cellulolytic activity ranging from 0.73 to 1.16 U mL⁻¹ with wide temperature (35–65°C) and pH (5 to 8) tolerance. SDS-PAGE analysis of the crude enzyme of PPFM strain MNW 60 exhibited several protein bands, and zymogram analysis revealed two dimeric cellulase bands with molecular mass of ~92 and 42 kDa. Scanning electron microscopic studies revealed significant morphological differences between the cells grown in normal and CMC amended medium. The strain MNW 60 was identified as Methylobacterium sp. based on biochemical, physiological, and morphological analyses, and the methylotrophic nature was authenticated by the presence of mxaF gene, encoding methanol dehydrogenase as a key indicator enzyme of methylotrophs, with 99% similarity to Methylobacterium lusitanum. With the 16S ribosomal RNA sequence showing 97% similarity to M. lusitanum strain MP2, this can be proposed as a novel taxon of the genus Methylobacterium. The study forms the first detailed report on the extracellular cellulase production by pink pigmented Methylobacterium sp., and it is expected that this might be the basis for further studies on cellulase production by PPFMs to explore the molecular mechanism, strain improvement, and large-scale cellulase production for its application.