arXiv subjects

Jinjing Qiao

Publications and source records attributed to Jinjing Qiao.

5 recordsLinked to original sources

arXiv2026

Properties for ($α,β$)-harmonic functions

We investigate properties of ($α,β$)-harmonic functions. First, we discuss the coefficient estimates for ($α,β$)-harmonic functions. In particular, we obtain Heinz's inequality for ($α,β$)-harmonic functions, propose a coefficient bound for normalized univalent ($α,β$)-harmonic functions and prove that this holds for the subclass that consists of starlike functions. Furthermore, by utilizing the relationship between ($α,β$)-harmonic functions and harmonic functions, we obtain Radó's theorem, Koebe type covering theorems and an area theorem. Finally, we show growth estimates and distortion estimates for ($α,β$)-harmonic functions by using the $L^p$ norms of the boundary functions.

math.CV↗

arXiv2021

On Harmonic Entire mappings

In this paper, we investigate properties of harmonic entire mappings. Firstly, we give the characterizations of the order and the type for a harmonic entire mapping $f=h+\overline{g}$, respectively, and also consider the relationship between the order and the type of $f$, $h$, and $g$. Secondly, we investigate the harmonic mappings $f=h+\overline{g}$ such that $f^{(n_p)}=h^{(n_p)}+\overline{g^{(n_p)}}$ are univalent in the unit disk, where $\{n_p\}_{p=1}^{\infty}$ be a strictly increasing sequence of nonnegative integers. In terms of the sequence $\{n_p\}_{p=1}^{\infty}$, we derive several necessary conditions for these mappings to be entire and also establish an upper bound for the order of these mappings.

math.CV↗

arXiv2020

Properties of Normal Harmonic Mappings

In this paper, we present several necessary and sufficient conditions for a harmonic mapping to be normal. Also, we discuss maximum principle and five-point theorem for normal harmonic mappings. Furthermore, we investigate the convergence of sequences for sense-preserving normal harmonic mappings and show that the asymptotic values and angular limits are identical for normal harmonic mappings.

math.CV↗

arXiv2019

Extreme points and support points of families of harmonic Bloch mappings

In this paper, the main aim is to discuss the existence of the extreme points and support points of families of harmonic Bloch mappings and little harmonic Bloch mappings. First, in terms of the Bloch unit-valued set, we prove a necessary condition for a harmonic Bloch mapping (resp. a little harmonic Bloch mapping) to be an extreme point of the unit ball of the normalized harmonic Bloch spaces (resp. the normalized little harmonic Bloch spaces) in the unit disk $\mathbb{D}$. Then we show that a harmonic Bloch mapping $f$ is a support point of the unit ball of the normalized harmonic Bloch spaces in $\mathbb{D}$ if and only if the Bloch unit-valued set of $f$ is not empty. We also give a characterization for the support points of the unit ball of the harmonic Bloch spaces in $\mathbb{D}$.

math.CV↗

arXiv2016

Generalized Zalcman conjecture for convex functions of order $α$

Let $\mathcal S$ denote the class of all functions of the form $f(z)=z+a_2z^2+a_3z^3+\cdots$ which are analytic and univalent in the open unit disk $\ID$ and, for $λ>0$, let $Φ_λ(n,f)=λa_n^2-a_{2n-1}$ denote the generalized Zalcman coefficient functional. Zalcman conjectured that if $f\in \mathcal S$, then $|Φ_1 (n,f)|\leq (n-1)^2$ for $n\ge 3$. The functional of the form $Φ_λ(n,f)$ is indeed related to Fekete-Szegő functional of the $n$-th root transform of the corresponding function in $\mathcal S$. This conjecture has been verified for a certain special geometric subclasses of $\mathcal S$ but the conjecture remains open for $f\in {\mathcal S}$ and for $n > 6$. In the present paper, we prove sharp bounds on $|Φ_λ(n,f)|$ for $f\in \mathcal{F}(α)$ and for all $n\geq 3$, in the case that $λ$ is a positive real parameter, where $ \mathcal{F}(α)$ denotes the family of all functions $f\in {\mathcal S}$ satisfying the condition $${\rm Re } \left( 1+\frac{zf''(z)}{f'(z)}\right) > α~\mbox{ for $z\in \ID$}, $$ where $-1/2\leq α<1$. Thus, the present article proves the generalized Zalcman conjecture for convex functions of order $α$, $α\in [-1/2,1)$.

math.CV↗