The function $f\left(z\right)=z^p+{\sum }_{k=1}^{\infty }{a}_{p+k}{z}^{p+k}$ (p ∈ ℕ = 1,2,3,...) analytic in the unit disk E is said to be in the class $K_{n,p}\left(h\right)$ if ($D^{n+p}f)/\left(D^{n+p-1}f\right)\prec h$, where $D^{n+p-1}f=\left(z^p\right)/\left({(1-z)}^{p+n}\right)*f$ and h is convex univalent in E with h(0) = 1. We study the class $K_{n,p}\left(h\right)$ and investigate whether the inclusion relation $K_{n+1,p}\left(h\right)\subseteq K_{n,p}\left(h\right)$ holds for p > 1. Some coefficient estimates for the class are also obtained. The class $A_{n,p}(a,h)$ of functions satisfying the condition $a*\left(D^{n+p}f\right)/\left(D^{n+p-1}f\right)+(1-a)*\left(D^{n+p+1}f\right)/\left(D^{n+p}f\right)\prec h$ is also studied.

The note develops results from [5] where an invariance under the Möbius transform mapping the upper halfplane onto itself of the Weinstein operator $W_k:=\Delta +\frac{k}{x_n}\frac{\partial }{\partial x_n}$ on ${ℝ}^n$ is proved. In this note there is shown that in the cases $k\ne 0$, $k\ne 2$ no other transforms of this kind exist and for case $k=2$, all such transforms are described.

We study asymptotic properties of solutions for a system of second differential equations with $p$-Laplacian. The main purpose is to investigate lower estimates of singular solutions of second order differential equations with $p$-Laplacian ${\left(A\left(t\right){\Phi }_p\left(y^{\text{'}}\right)\right)}^{\text{'}}+B\left(t\right)g\left(y^{\text{'}}\right)+R\left(t\right)f\left(y\right)=e\left(t\right)$. Furthermore, we obtain results for a scalar equation.

We introduce and study the linear symmetric systems associated with the modified Cherednik operators. We prove the well-posedness results for the Cauchy problem for these systems. Eventually we describe the finite propagation speed property of it.

Let $p^{\text{'}}$ and $q^{\text{'}}$ be points in ${ℝ}^n$. Write $p^{\text{'}}\sim q^{\text{'}}$ if $p^{\text{'}}-q^{\text{'}}$ is a multiple of $(1,...,1)$. Two different points $p$ and $q$ in ${ℝ}^n/\sim$ uniquely determine a tropical line $L(p,q)$ passing through them and stable under small perturbations. This line is a balanced unrooted semi-labeled tree on $n$ leaves. It is also a metric graph. If some representatives $p^{\text{'}}$ and $q^{\text{'}}$ of $p$ and $q$ are the first and second columns of some real normal idempotent order $n$ matrix $A$, we prove that the tree $L(p,q)$ is described by a matrix $F$, easily obtained from $A$. We also...