In the present note we give an elementary proof of the necessary and sufficient condition for a univariate function to belong the class $\mathcal N_\varkappa^+$. This class was introduced mainly to deal with the indefinite version of the Stieltjes moment problem (and corresponding $π$-Hermitian operators), although it is applicable beyond the original scope. The proof relies on asymptotic analysis of the corresponding Hermitian forms. Our result closes a gap in the criterion given by Krein and Langer in their joint paper of 1977. The correct condition was stated by Langer and Winkler in 1998, although they provided no proper reasoning.
Let ${\mathcal A}$ denote the family of all functions $f$ analytic in the unit disk $\ID$ and satisfying the normalization $f(0)=0= f'(0)-1$. Let $\mathcal{S}$ denote the subclass of ${\mathcal A}$ consisting of univalent functions in $\ID$. We consider the subclass $\mathcal{U} $ of $\mathcal{S}$ that is defined by the condition that for its members $f$ the condition $$\left |\left (\frac{z}{f(z)} \right )^{2}f'(z)-1\right | < 1 ~\mbox{ for $z\in \ID$} $$ holds. To theses relations belong striking similarities and on the other hand big differences. We show that some results about $\mathcal{S}…
Let $\mathcal{A}$ denote the class of analytic functions $f$ in the unit disk $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$ normalized by $f(0)=0$, $f'(0)=1$. In the present article, we obtain the sharp estimates of the Schwarzian norm for functions in the classes $\mathcal{G}(β)=\{f\in \mathcal{A}:{\rm Re\,}[1+zf''(z)/f'(z)]<1+β/2\}$, where $β>0$ and $\mathcal{F}(α)=\{f\in \mathcal{A}:{\rm Re\,}[1+zf''(z)/f'(z)]>α\}$, where $-1/2\le α\le 0$. We also establish two-point distortion theorem for functions in the classes $\mathcal{G}(β)$ and $\mathcal{F}(α)$.
In this paper we have introduced two new classes $\mathcal{H}\mathcal{M}(β, λ, k, ν)$ and $\overline{\mathcal{H}\mathcal{M}} (β, λ, k, ν)$ of complex valued harmonic multivalent functions of the form $f = h + \overline g$, satisfying the condition \[ Re \{(1 - λ) \frac{Ω^vf}{z} + λ(1-k) \frac{(Ω^vf)'}{z'} + λk \frac{(Ω^vf)''}{z''} \} > β, (z\in \mathcal{D})\] where $h$ and $g$ are analytic in the unit disk $\mathcal{D} = \{z : |z| < 1\}.$ A sufficient coefficient condition for this function in the class $\mathcal{H}\mathcal{M}(β, λ, k, ν)$ and a necessary and sufficient coefficient condition f…
In this paper we study the relation between the weighted energy class $\mathcal{E}_χ$ introduced by S. Benelkouchi, V. Guedj and A. Zeriahi recently with the classes $\mathcal{E}$ and $\mathcal{N}$ studied by Cegrell. Moreover, we establish a generalized comparison principle for the operator $\text{M}_χ$ and, as an application, we prove a slight version of existence of solutions of Monge-Ampère type equation in the class $\mathcal{E}_χ(H,Ø)$.
In this paper we study the class $\mathcal{E}_{m}(Ω)$ of $m-$subharmonic functions introduced by Lu in \cite{L1}. We prove that the convergence in $m-$capacity implies the convergence of the associated Hessian measure for functions that belong to $\mathcal{E}_{m}(Ω)$. Then we extend those results to the class $\mathcal{E}_{m,χ}(Ω)$ that depends on a given increasing real function $χ$. A complete characterization of those classes using the Hessian measure is given as well as a subextension theorem relative to $\mathcal{E}_{m,χ}(Ω)$.
We give coefficient estimates for a class of close-to-convex harmonic mappings, and discuss the Fekete-Szegő problem of it. We also introduce two classes of polyharmonic mappings $\mathcal{HS}_{p}$ and $\mathcal{HC}_{p}$, consider the starlikeness and convexity of them, and obtain coefficient estimates on them. Finally, we give a necessary condition for a mapping $F$ to be in the class $\mathcal{HC}_{p}$.
We consider three classes of functions defined using the class $\mathcal{P}$ of all analytic functions $p(z)=1+cz+\dotsb$ on the open unit disk having positive real part and study several radius problems for these classes. The first class consists of all normalized analytic functions $f$ with $f/g\in\mathcal{P}$ and $g/(zp)\in\mathcal{P}$ for some normalized analytic function $g$ and $p\in \mathcal{P}$. The second class is defined by replacing the condition $f/g\in\mathcal{P}$ by $|(f/g)-1|<1$ while the other class consists of normalized analytic functions $f$ with $f/(zp)\in\mathcal{P}$ for s…
Let $\mathcal{A}$ denote the class of analytic functions in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$ satisfying $f(0)=0$ and $f'(0)=1$. Let $\mathcal{U}$ be the class of functions $f\in\mathcal{A}$ satisfying $$\left|f'(z)\left(\frac{z}{f(z)}\right)^2-1 \right|< 1 \quad\mbox{ for } z\in\mathbb{D},$$ and $\mathscr{G}$ denote the class of functions $f\in \mathcal{A}$ satisfying $${\rm Re\,}\left(1+\frac{zf''(z)}{f'(z)}\right)>-\frac{1}{2} \quad\mbox{ for } z\in\mathbb{D}.$$
In Geometric function theory, occasionally attempts have been made to solve a particular problem for the Ma-Minda classes, $\mathcal{S}^*(ψ)$ and $\mathcal{C}(ψ)$ of univalent starlike and convex functions, respectively. Recently, a popular radius problem generally known as Bohr's phenomenon has been studied in various settings, however little is known about Rogosinski radius. In this article, for a fixed $f\in \mathcal{S}^*(ψ)$ or $\mathcal{C}(ψ),$ the class of analytic subordinants $S_{f}(ψ):= \{g : g\prec f \} $ is studied for the Bohr-Rogosinski phenomenon in a general setting. It's applic…
In the present paper we introduce and studied two subclasses of multivalent functions denoted by $\mathcal{M}^λ_{p,n}(γ;β)$ and $\mathcal{N}^λ_{p,n}(μ,η;δ)$. Further, by giving specific values of the parameters of our main results, we will find some connection between these two classes, and moreover, several consequences of main results are also discussed.
We introduce the class of analytic functions $$\mathcal{F}(ψ):= \left\{f\in \mathcal{A}: \left(\frac{zf'(z)}{f(z)}-1\right) \prec ψ(z),\; ψ(0)=0 \right\},$$ where $ψ$ is univalent and establish the growth theorem with some geometric conditions on $ψ$ and obtain the Koebe domain with some related sharp inequalities. Note that functions in this class may not be univalent. As an application, we obtain the growth theorem for the complete range of $α$ and $β$ for the functions in the classes $\mathcal{BS}(α):= \{f\in \mathcal{A} : ({zf'(z)}/{f(z)})-1 \prec {z}/{(1-αz^2)},\; α\in [0,1) \}$ and $\mat…
Let $\mathcal C^M$ denote a Denjoy-Carleman class of $\mathcal C^\infty$ functions (for a given logarithmically-convex sequence $M = (M_n)$). We construct: (1) a function in $\mathcal C^M((-1,1))$ which is nowhere in any smaller class; (2) a function on $\mathbb R$ which is formally $\mathcal C^M$ at every point, but not in $\mathcal C^M(\mathbb R)$; (3) (under the assumption of quasianalyticity) a smooth function on $\mathbb R^p$ ($p \geq 2$) which is $\mathcal C^M$ on every $\mathcal C^M$ curve, but not in $\mathcal C^M(\mathbb R^p)$.
Let $\mathcal{D}_v$ denote the Dirichlet type space in the unit disc induced by a radial weight $v$ for which $\widehat{v}(r)=\int_r^1 v(s)\,ds$ satisfies the doubling property $\int_r^1 v(s)\,ds\le C \int_{\frac{1+r}{2}}^1 v(s)\,ds.$ In this paper, we characterize the Schatten classes $S_p(\mathcal{D}_v)$ of the generalized Hilbert operators \begin{equation*} \mathcal{H}_g(f)(z)=\int_0^1f(t)g'(tz)\,dt \end{equation*} acting on $\mathcal{D}_v$, where $v$ satisfies the Muckenhoupt-type conditions $$ \sup_{0<r<1}\left(\int_r^1 \frac{\widehat{v}(s)}{(1-s)^2} \,ds\right)^{1/2} \left(\int_0^r \frac…
