arXiv subjects
Publications and source records attributed to Jianren Long.
All the finite order entire solutions of \begin{equation*} f^n(z)+q(z)e^{Q(z)}f^{(k)}(z+c)=P(z) \end{equation*} are given, where $ q(z) $, $ Q(z), P(z) $ are polynomials, $ k $ and $ n \geq 2 $ are integers, and $ c \in \mathbb{C} \setminus \{0\} $.This solves an open problem of Heittokangas-Ishizaki-Tohge-Wen [Bull. Lond. Math. Soc. 55, 1-77 (2023)].
Let \(0<q<p<\infty\), \(\Omega\) be a bounded \(\bbC\)-convex domains in \(\bbC^n\). We establish several equivalent characterizations for the boundedness of Carleson embedding \(J_\mu:A_\alpha^p\hookrightarrow L^q(\mu)\) on \(\Omega\) with sharp \(\cB\) condition. Furthermore, we prove that the boundedness of \(J_\mu\) is equivalent to its compactness.
For $\mu$ is a positive Borel measure on $\mathbb{D}$, The $r$ summing Carleson embdedding $J_\mu: A_w^p\to L^q(\mu)$ are characterized in this paper, some conditions which ensure that the Carleson embedding for $J_\mu: A_w^p\to L^q(\mu)$ is $r$ summing are obtained for all $0<p,q<\infty.$
Let $f$ be a transcendental entire function with hyper-order strictly less than 1 and having a Borel exceptional small function. If $f$ and $\Delta^n f$, or $f'$ and $f(z+1)$, share a function CM, then the exact form of $f$ is determined, which improves the previous results given by L\"u et al. [Results Math. 74, article number 30 (2019)] and Liu et al. [Bull. Korean Math. Soc. 51, 1453-1467 (2014)]. As an application, the relationship between $f$ and $\Delta^n f$ is established under the condition that they share a finite set, which partially resolves Liu's question raised in [J. Math. Anal. Appl. 359, 384-393 (2009)]. Furthermore, several examples are presented to demonstrate these results.
The meromorphic solutions $f$ with $\rho_2(f)<1$ of the non-linear difference equation \begin{align*} f^n(z)+P_d(z,f)=p_1e^{{\lambda_1}z}+p_2e^{{\lambda_2}z}+p_3e^{{\lambda_3}z}, \end{align*} are characterized in terms of exponential functions using Nevanlinna theory, under certain conditions on $\lambda_j$ for $j=1,2,3$. Here, $n>2$, $P_d(z,f)$ is a difference polynomial in $f$ of degree $\le n-1$, and $\lambda_j,~p_j\not=0$ for $~j=1,2,3$. These results improve upon those previously obtained by Chen et al.[Bull. Korean Math. Soc. 61, 745-762 (2024)]. Some examples are provided to illustrate these results. Additionally, if $P_d(z,f)$ is a differential-difference polynomial, then under the supplementary condition $N(r,f)=S(r,f)$, by applying the same proof method, these conclusions still hold.
On the one hand, the fractional order derivative characterization of the Besov-Morrey type space $B_{p}^{K}(s)$ is established by $K$-Carleson measures, and it was also shown that $f \in B_{p}^{K}(s_1) \Leftrightarrow f^{\left(\frac{s_2 - s_1}{p}\right)} \in B_{p}^{K}(s_2)$, which extended the results of Sun et al. on the fractional derivative of Morrey type space. On the other hand, some sufficient conditions for the growth of solutions to linear complex differential equations have been obtained by using $n$th derivative criterion.
The two weights inequality for Hankel operators $$\|H_f^\omega (\cdot)\|_{L_\eta^q}\leq C \|\cdot\|_{A_v^p},$$ induced by some radial weights under the regular assumptions is considered, the boundedness and compactness of Hankel operators $H^\omega_f$ is characterized for $1<p,q<\infty$ and $\omega\neq v\neq \eta\in \mathcal{R}$.
The existence of the meromorphic solutions to Fermat type delay-differential equation \begin{equation} f^n(z)+a(f^{(l)}(z+c))^m=p_1(z)e^{a_1z^k}+p_2(z)e^{a_2z^k}, \nonumber \end{equation} is derived by using Nevanlinna theory under certain conditions, where $k\ge1$, $m,$ $n$ and $l$ are integers, $p_i$ are nonzero entire functions of order less than $k$, $c$, $a$ and $a_i$ are constants, $i=1,2$. These results not only improve the previous results from Zhu et al. [J. Contemp. Math. Anal. 59(2024), 209-219], Qi et al. [Mediterr. J. Math. 21(2024), article no. 122], but also completely solve two conjectures posed by Gao et al. [Mediterr. J. Math. 20(2023), article no. 167]. Some examples are given to illustrate these results.
The paired Hayman's conjecture of different types are considered. More accurately speaking, the zeros of a pair of $f^nL(z,g)-a_1(z)$ and $g^mL(z,f)-a_2(z)$ are characterized using different methods from those previously employed, where $f$ and $g$ are both transcendental entire functions, $L(z,f)$ and $L(z,g)$ are non-zero linear delay-differential polynomials, $\min\{n,m\}\ge 2$, $a_1,a_2$ are non-zero small functions with relative to $f$ and $g$, or to $f^n(z)L(z,g)$ and $g^m(z)L(z,f)$, respectively. These results give answers to three open questions raised by Gao, Liu[Bull. Korean Math. Soc. 59 (2022)] and Liu, Liu[J. Math. Anal. Appl. 543 (2025)].
We study the question posed by G. Gundersen and C. C. Yang, in which the following two types of binomial differential equations are investigated, $$ a(z)f'f''-b(z)(f)^{2}=c(z)e^{2d(z)},~~a(z)ff'-b(z)(f'')^{2}=c(z)e^{2d(z)}, $$ where $a(z)$, $b(z)$ and $c(z)$ are polynomials such that $a(z)b(z)c(z)\not\equiv 0$, $d(z)$ is non-constant polynomial. The explicit forms of entire solutions of the above binomial differential equations are obtained by using the Nevanlinna theory, which gives partial solutions to the question of G. Gundersen and C. C. Yang. In addition, some examples are given to illustrate these results.
Let $(z_k)$ be a sequence of distinct points in the unit disc $\mathbb{D}$ without limit points there. We are looking for a function $a(z)$ analytic in $\mathbb{D}$ and such that possesses a solution having zeros precisely at the points $z_k$, and the resulting function $a(z)$ has `minimal' growth. We focus on the case of non-separated sequences $(z_k)$ in terms of the pseudohyperbolic distance when the coefficient $a(z)$ is of zero order, but $\sup_{z\in \mathbb{D}} (1-|z|)^p |a(z)|=+\infty$ for any $p>0$. We established a new estimate for the maximum modulus of $a(z)$ in terms of the functions $n_z(t)=\sum_{|z_k-z|\le t} 1 $ and $N_z(r)=\int_0^r \frac{(n_z(t)-1)^+}{t}dt.$ The estimate is sharp in some sense. The main result relies on a new interpolation theorem.