Differential Inequalities and Univalent Functions

Let ${\mathcal M}$ be the class of analytic functions in the unit disk \ID with the normalization $f(0)=f'(0)-1=0$, and satisfying the condition \left |z^2\left (\frac{z}{f(z)}\right )''+ f'(z)\left(\frac{z}{f(z)} \right)^{2}-1\right |\leq 1, \quad z\in \ID. Functions in $\mathcal{M}$ are known to be univalent in \ID. In this paper, it is shown that the harmonic mean of two functions in ${\mathcal M}$ are closed, that is, it belongs again to ${\mathcal M}$. This result also holds for other related classes of normalized univalent functions. A number of new examples of functions in $\mathcal{M}$ are shown to be starlike in \ID. However we conjecture that functions in $\mathcal{M}$ are not necessarily starlike, as apparently supported by other examples.Comment: 10 pages; To appear in Lobachevskii Journal of Mathematic

Univalence and starlikeness of certain transforms defined by convolution of analytic functions

AbstractLet U(λ) denote the class of all analytic functions f in the unit disk Δ of the form f(z)=z+a2z2+⋯ satisfying the condition|f′(z)(zf(z))2−1|⩽λ,z∈Δ. In this paper we find conditions on λ and on c∈C with Rec⩾0≠c such that for each f∈U(λ) satisfying (z/f(z))∗F(1,c;c+1;z)≠0 for all z∈Δ the transformG(z)=Gfc(z)=z(z/f(z))∗F(1,c;c+1;z),z∈Δ, is univalent or starlike. Here F(a,b;c;z) denotes the Gauss hypergeometric function and ∗ denotes the convolution (or Hadamard product) of analytic functions on Δ

On harmonic combination of univalent functions

Let ${\mathcal S}$ be the class of all functions $f$ that are analytic and univalent in the unit disk \ID with the normalization $f(0)=f'(0)-1=0$. Let $\mathcal{U} (\lambda)$ denote the set of all $f\in {\mathcal S}$ satisfying the condition |f'(z)(\frac{z}{f(z)})^{2}-1| <\lambda ~for $z\in \ID$, for some $\lambda \in (0,1]$. In this paper, among other things, we study a "harmonic mean" of two univalent analytic functions. More precisely, we discuss the properties of the class of functions $F$ of the form $$\frac{z}{F(z)}=1/2(\frac{z}{f(z)}+\frac{z}{g(z)}),$$ where $f,g\in \mathcal{S}$ or $f,g\in \mathcal{U}(1)$. In particular, we determine the radius of univalency of $F$, and propose two conjectures concerning the univalency of $F$.Comment: 10 pages. the article is with a journa

On the instability of equilibrium of nonholonomic systems with nonhomogeneous constraints

The first Lyapunov method, extended by V. Kozlov to nonlinear mechanical systems, is applied to the study of the instability of the equilibrium position of a mechanical system moving in the field of potential and dissipative forces. The motion of the system is subject to the action of the ideal linear nonholonomic nonhomogeneous constraints. Five theorems on the instability of the equilibrium position of the above mentioned system are formulated. The theorem formulated in [V. V. Kozlov, On the asymptotic motions of systems with dissipation, J. Appl. Math. Mech. 58 (5) (1994) 787-792], which refers to the instability of the equilibrium position of the holonomic scleronomic mechanical system in the field of potential and dissipative forces, is generalized to the case of nonholonomic systems with linear nonhomogeneous constraints. In other theorems the algebraic criteria of the Kozlov type are transformed into a group of equations required only to have real solutions. The existence of such solutions enables the fulfillment of all conditions related to the initial algebraic criteria. Lastly, a theorem on instability has also been formulated in the case where the matrix of the dissipative function coefficients is singular in the equilibrium position. The results are illustrated by an example

A quality of life assessment and the correlation between generic and disease-specific questionnaires scores in outpatients with chronic liver disease-pilot study

Introduction. Chronic liver diseases (CLD) are an important cause of morbidity and mortality in general population. The aim of this study was to analyze potential differences between patients with CLD and healthy control group, and to estimate the severity of CLD by using simple questionnaires: general health questionnaire (GHQ-12) and chronic liver disease questionnaire (CLDQ). Methods. A cross-sectional pilot study was performed in Zemun Clinical Hospital during years 2014 and 2015. Sixty participants were divided into 4 groups (15 per group): chronic alcoholic hepatitis, other chronic hepatitis, liver cirrhosis, and healthy control group. Entire study population chose one of four offered answers of structured questionnaires GHQ-12 and CLDQ, based on which mean model of end-stage liver disease (MELD) and Child-Turcotte-Pugh (CTP) scores were calculated. Results. Mean GHQ12 and CLDQ scores were 10.5 and 5.21 +/- 1.11 respectively. Regarding certain CLDQ domain scores, a significant difference between alcoholic and non-alcoholic hepatitis groups in the worry domain was observed. Mean MELD score was 7.42 +/- 2.89 and did not differ between chronic hepatitis groups, while mean CTP score was 5.73 +/- 0.88. A statistically significant correlation was observed between GHQ12 and CLDQ scores (rho = -0.404, p LT 0.01), but not between subjective and objective scores. Conclusions. Mean GHQ12 and CLDQ scores pointed out to general psychological no-distress condition of the studied participants, as well as scarcely expressed CLD-specific complaints. Mean MELD and CTP scores indicated stable chronic liver diseases, with low three-month mortality rates in the cases of chronic hepatitis, as well as determination to Child A group in the case of liver cirrhosis

Isothermal sintering of BZT ceramics

Starting mixtures of BaCO3 •ZnO and Ti02 were mechanically activated for 0,5, 10,20.40 and 80 minutes in a planetary ball mill. The powders obtained were sintered isothermally to temperatures between 1000 and 1300 °C. The phase composition of powders and sintered samples were followed by X-ray analyses. Also, the changes in microstructures were detected using SEM

Identification and molecular characterization of Chryseobacterium vrystaatense ST1 isolated from oligomineral water of southeast Serbia

The isolation and molecular characterization of bacterial strains isolated from water sources in the Vlasina Mountain in southeast Serbia, confirmed the presence of a new species Chryseobacterium vrystaatense ST1. This Gram- negative species showed an extremely low level of biochemical reactivity in biochemical tests. The gene for 16S rRNA was amplified by PCR using universal primers and sequenced. Comparison of 16S rRNA gene sequence and phenotypic features indicated that the isolate ST belonged to Chryseobacterium vrystaatense. A BLAST search of sequenced 1088 nucleotides of the 16S rRNA gene with all sequences deposited in the NCBI collection showed the highest similarity (98%) with the strain Chryseobacterium vrystaatense sp. nov., designated as strain R-23533. The very high homology of these two strains allowed classification of our strain at the species level, but some differences indicate, and indirectly confirm, that the isolate ST is an authentic representative. On the basis of these results, we could conclude that Chryseobacterium vrystaatense ST was for first time isolated in Serbia, which is particularly important when one bears in mind that there are only three sequences of this species deposited in the NCBI collection

Transport of silver nanoparticles from nanocomposite Ag/alginate hydrogels under conditions mimicking tissue implantation

The aim of this work was to assess phenomena occurring during AgNP transport from nanocomposite Ag/alginate hydrogels under conditions relevant for potential biomedical applications as antimicrobial soft tissue implants. First, we have studied AgNP migration from the nanocomposite to the adjacent alginate hydrogel mimicking soft tissue next to the implant. AgNP deposition was carried out by the initial burst release lasting for similar to 24 h yielding large aggregates on hydrogel surfaces and smaller clusters (similar to 400 nm in size) inside. However, the overall released content was low (0.67%) indicating high nanocomposite stability. In the next experimental series, release of AgNPs, 10-30 nm in size, from Ag/alginate microbeads in water was investigated under static conditions as well as under continuous perfusion mimicking vascularized tissues. Mathematical modeling has revealed AgNP release by diffusion under static conditions with the diffusion coefficient within the Ag/alginate hydrogel of 6.9x10(-19) m(2) s(-1). Conversely, continuous perfusion induced increased AgNP release by convection with the interstitial fluid velocity estimated as 4.6 nm s(-1). Overall, the obtained results indicated the influence of hydrodynamic conditions at the implantation site on silver release and potential implant functionality, which should be investigated at the experimentation beginning using appropriate in vitro systems

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