In this paper we prove two results. The first is an extension of the result of G. D. Jones [4]: (A) Every nontrivial solution for $$\left\}\begin{array}{c}{(-1)}^n{u}^{\left(2n\right)}+f(t,u)=0,\text{in}(\alpha ,\infty ),u^{\left(i\right)}\left(\xi \right)=0,i=0,1,\cdots ,n-1,\text{and}\xi \in (\alpha ,\infty ),\hfill \end{array}\right.$$ must be unbounded, provided $f(t,z)z\ge 0$, in $E\times ℝ$ and for every bounded subset $I$, $f(t,z)$ is bounded in $E\times I$. (B) Every bounded solution for ${(-1)}^n{u}^{\left(2n\right)}+f(t,u)=0$, in $ℝ$, must be constant, provided $f(t,z)z\ge 0$ in $ℝ\times ℝ$ and for every bounded subset $I$, $f(t,z)$ is bounded in $ℝ\times I$.

The purpose of this paper is to study the class ${*}_{\infty }\left(ℍ\right)$ of univalent analytic functions defined in the right halfplane ℍ and starlike w.r.t. the boundary point at infinity. An analytic characterization of functions in ${*}_{\infty }\left(ℍ\right)$ is presented.

For n ≥ 1, let denote the class of all analytic functions f in the unit disk Δ of the form $f\left(z\right)=z+{\sum }_{k=2}^{\infty }{a}_k{z}^k$. For Re α < 2 and γ > 0 given, let (γ,α) denote the class of all functions f ∈ satisfying the condition |f’(z) - α f(z)/z + α - 1| ≤ γ, z ∈ Δ. We find sufficient conditions for functions in (γ,α) to be starlike of order β. A generalization of this result along with some convolution results is also obtained.

A function $f:ℝ\to ℝ$ is said to have the $n$-th Laplace derivative on the right at $x$ if $f$ is continuous in a right neighborhood of $x$ and there exist real numbers ${\alpha }_0,...,{\alpha }_{n-1}$ such that $s^{n+1}{\int }_0^{\delta }{e}^{-st}[f(x+t)-{\sum }_{i=0}^{n-1}{\alpha }_i{t}^i/i!]dt$ converges as $s\to +\infty$ for some $\delta >0$. There is a corresponding definition on the left. The function is said to have the $n$-th Laplace derivative at $x$ when these two are equal, the common value is denoted by $f_{\langle n\rangle }\left(x\right)$. In this work we establish the basic properties of this new derivative and show that, by an example, it is more general than...

Let n ≥ 2 and $H_k^{s,s^{\text{'}}}=u\in S^{\text{'}}\left({ℝ}^n\right):\parallel u{\parallel }_{s,s^{\text{'}}}<\infty$, where $\parallel u\parallel {²}_{s,s^{\text{'}}}={\left(2\pi \right)}^{-n}\int {(1+|\xi \left|²\right)}^s(1+|{\xi }^{\text{'}}{\left|²\right)}^{s^{\text{'}}}\left|Fu\left(\xi \right)\right|²d\xi$, $Fu\left(\xi \right)=\int e^{-ix\xi }u\left(x\right)dx$, ${\xi }^{\text{'}}\in {ℝ}^k$, k < n. We prove that for some s,s’ the space $H_k^{s,s^{\text{'}}}$ is a multiplicative algebra.

We consider functions of the type $F\left(z\right)=z{\prod }_{j=1}^n{[f_j\left(z\right)/z]}^{a_j}$, where $a_j$ are real numbers and $f_j$ are ${\beta }_j$-strongly close-to-starlike functions of order ${\alpha }_j$. We look for conditions on the center and radius of the disk (a,r) = z:|z-a| < r, |a| < r ≤ 1 - |a|, ensuring that F((a,r)) is a domain starlike with respect to the origin.

Let $\mathbb{P}:={\left(P_t\right)}_{t>0}$ be a measurable semigroup and $m$ a $\sigma$-finite positive measure on a Lusin space $X$. An $m$-exit law for $\mathbb{P}$ is a family ${\left(f_t\right)}_{t>0}$ of nonnegative measurable functions on $X$ which are finite $m$-a.e. and satisfy for each $s,t>0$ $P_s{f}_t=f_{s+t}$ $m$-a.e. An excessive function $u$ is said to be in $ℛ$ if there exits an $m$-exit law ${\left(f_t\right)}_{t>0}$ for $\mathbb{P}$ such that $u={\int }_0^{\infty }{f}_{t}dt$, $m$-a.e. Let $𝒫$ be the cone of $m$-purely excessive functions with respect to $\mathbb{P}$ and $ℐmV$ be the cone of $m$-potential functions. It is clear that $ℐmV\subseteq ℛ\subseteq 𝒫$. In this...