Jun Wang

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• PhD
• Professor (Associate) at Fudan University

By using asymptotic method, we verify the existence on the slowly growing solutions to second order difference equations discussed by Ishizaki-Yanagihara’s Wiman-Valiron method and Ishizaki-Wen’s binomial series method. The classical problem on finding conditions on the polynomial coefficients Pj(z) (j = 0, 1, 2) and F(z) to guarantee that all nont...

We study the question of when two functions L_1,L_2 in the extended Selberg class are identical in terms of the zeros of L_i-h(i=1,2). Here, the meromorphic function h is called moving target. With the assumption on the growth order of h, we prove that L_1\equiv L_2 if L_1-h and L_2-h have the same zeros counting multiplicities. Moreover, we also c...

We show a necessary and sufficient condition on the existence of finite order entire solutions of linear differential equations $$ f^{(n)}+a_{n-1}f^{(n-1)}+\cdots+a_1f'+a_0f=0,\eqno(+) $$ where $a_i$ are exponential sums for $i=0,\ldots,n-1$ with all positive (or all negative) rational frequencies and constant coefficients. Moreover, under the cond...

In this paper we connect the finiteness property and the periodicity in the study of the generalized Yang’s conjecture and its variations, which involve the inverse question of whether f(z) is still periodic when some differential polynomial in f is periodic. The finiteness property can be dated back to Weierstrass in the characterization of additi...

Let $f(z)=\sum\limits_{j=0}^{\infty} a_j z^j$ be a transcendental entire function and let $f_\omega(z)=\sum\limits_{j=0}^{\infty}\chi_j(\omega) a_j z^j$ be a random entire function, where $\chi_j(\omega)$ are independent and identically distributed random variables defined on a probability space $(\Omega, \mathcal{F}, \mu)$ . In this paper, we firs...

We construct exponential maps for which the singular value tends to infinity under iterates while the maps are ergodic. This is in contrast with a result of Lyubich from 1987 which tells that $e^z$ is not ergodic.

The growth of meromorphic solutions of linear difference equations containing Askey—Wilson divided difference operators is estimated. The ϕ-order is used as a general growth indicator, which covers the growth spectrum between the logarithmic order ρlog (f) and the classical order ρ(f) of a meromorphic function f.

We prove new results on upper and lower limits of real-valued functions by means of ψ-densities introduced by P. D. Barry in 1962. This allows us to improve several existing results on the growth of non-decreasing and unbounded real-valued functions in sets of positive density. The ψ-densities allow us to introduce a new concept of a limit for real...

The φ-order was introduced in 2009 for meromorphic functions in the unit disc, and was used as a growth indicator for solutions of linear differential equations. In this paper, the properties of meromorphic functions in the complex plane are investigated in terms of the φ-order, which measures the growth of functions between the classical order and...

This paper has two purposes. One is to establish a version of Nevanlinna theory based on the historic so-called Jackson difference operator \({D_q}f(z) = {{f(qz) - f(z)} \over {qz - z}}\) for meromorphic functions of zero order in the complex plane ℂ. We give the logarithmic difference lemma, the second fundamental theorem, the defect relation, Pic...

For entire or meromorphic function f, a value θ ∈ [0, 2π) is called a Julia limiting direction if there is an unbounded sequence {zn} in the Julia set satisfying \(\mathop {\lim }\limits_{n \to \infty } \;\arg {z_n} = \theta \). Our main result is on the entire solution f of P(z, f) + F(z)fs = 0, where P(z, f) is a differential polynomial of f with...

Meromorphic solutions of non‐linear differential equations of the form f n + P ( z , f ) = h are investigated, where n ≥ 2 is an integer, h is a meromorphic function, and P ( z , f ) is differential polynomial in f and its derivatives with small functions as its coefficients. In the existing literature this equation has been studied in the case whe...

The growth of meromorphic solutions of linear difference equations containing Askey-Wilson divided difference operators is estimated. The $\varphi$-order is used as a general growth indicator, which covers the growth spectrum between the logarithmic order $\rho_{\log}(f)$ and the classical order $\rho(f)$ of a meromorphic function $f$.

Let $f_\omega(z)=\sum\limits_{j=0}^{\infty}\chi_j(\omega) a_j z^j$ be a random entire function, where $\chi_j(\omega)$ are independent and identically distributed random variables defined on a probability space $(\Omega, \mathcal{F}, \mu)$. In this paper, we first define a family of random entire functions, which includes Gaussian, Rademacher, Stei...

The $\varphi$-order was introduced in 2009 for meromorphic functions in the unit disc, and was used as a growth indicator for solutions of linear differential equations. In this paper, the properties of meromorphic functions in the complex plane are investigated in terms of the $\varphi$-order, which measures the growth of functions between the cla...

Let f be a meromorphic function in the complex plane. A value θ ∈ [0, 2π) is called a Julia limiting direction of f if there is an unbounded sequence {zn} in the Julia set J(f) satisfying limn→∞ arg zn = θ (mod 2π). We denote by L(f) the set of all Julia limiting directions of f. Our main result is that, for any non-empty compact set E ⊆ [0, 2π) an...

In this paper, we investigate the Bohr radius for K-quasiregular sense-preserving harmonic mappings \(f=h+{\overline{g}}\) in the unit disk \({\mathbb {D}}\) such that the translated analytic part \(h(z)-h(0)\) is quasi-subordinate to some analytic function. The main aim of this article is to extend and to establish sharp versions of four recent th...

Motivated by earlier results from the complex function theory, the growth of non-decreasing and unbounded real-valued functions is studied in sets of positive linear/logarithmic density. The results improve several existing results and they are of interest from the real analysis and the complex analysis points of view.

In this paper, we investigate the Bohr radius for $K$-quasiregular sense-preserving harmonic mappings $f=h+\overline{g}$ in the unit disk $\mathbb{D}$ such that the translated analytic part $h(z)-h(0)$ is quasi-subordinate to some analytic function. The main aim of this article is to extend and to establish sharp versions of four recent theorems by...

Meromorphic solutions of non-linear differential equations of the form $f^n+P(z,f)=h$ are investigated, where $n\geq 2$ is an integer, $h$ is a meromorphic function, and $P(z,f)$ is differential polynomial in $f$ and its derivatives with small functions as its coefficients. In the existing literature this equation has been studied in the case when...

The authors study a family of transcendental entire functions which lie outside the Eremenko-Lyubich class in general and are of infinity growth order. Most importantly, the authors show that the intersection of Julia set and escaping set of these entire functions has full Hausdor. dimension. As a by-product of the result, the authors also obtain t...

We investigate transcendental entire solutions of complex differential equations f″+A(z)f=H(z), where the entire function A(z) has a growth property similar to the exponential functions, and H(z) is an entire function of order less than that of A(z). We first prove that the lower order of the entire solution to the equation is infinity. By using ou...

This paper establishes a version of Nevanlinna theory based on Jackson difference operator $D_{q}f(z)=\frac{f(qz)-f(z)}{qz-z}$ for meromorphic functions of zero order in the complex plane $\mathbb{C}$. We give the logarithmic difference lemma, the second fundamental theorem, the defect relation, Picard theorem and five-value theorem in sense of Jac...

In this paper, we study the large scaled geometric structure of Julia sets of entire and meromorphic functions. Roughly speaking, the structure gives us some asymptotic information about the Julia set near the essential singularity. We will show that one part of this structure is determined by the transcendental directions coming from function theo...

In this paper, we study the large scaled geometric structure of Julia sets of entire and meromorphic functions. Roughly speaking, the structure gives us some asymptotic information about the Julia set near the essential singularity. We will show that part of this structure are determined by the transcendental directions coming from function theoret...

In this paper, we mainly investigate the dynamical properties of entire solutions of complex differential equations. With some conditions on coefficients, we prove that the set of common limiting directions of Julia sets of solutions, their derivatives and their primitives must have a definite range of measure.

Let f be a meromorphic function in the complex plane, \(p_j\) polynomials for (\(j=0, 1, 2, \ldots , n\)) and R(z, f) an irreducible rational function in f with small meromorphic functions relative to f as coefficients. Let n be a positive integer and I, J two index sets in \(\mathbb {Z}^n\). In this paper, we systematically study the growth order...

In this paper, we mainly investigate entire solutions of complex differential equations with coefficients involving exponential functions, and obtain the dynamical properties of the solutions, their derivatives and primitives. With some conditions on coefficients, for the solutions, their derivatives and their primitives, we consider the common lim...

This paper is devoted to studying the radial oscillation of solutions of linear differential equations , where , and are transcendental entire functions. With some added conditions on coefficients, we show that there exists some close relation between the Borel directions of solutions and that of , and also obtain some estimates of growth of soluti...

In this paper, we study the dynamics of a correlated continuous time random walk with time averaged waiting time. The mean square displacement (MSD) shows this process is subdiffusive and generalized Einstein relation holds. We also get the asymptotic behavior of the probability density function (PDF) of this process is stretched Gaussian. At last,...

Devaney and Krych showed that for $0<\lambda<1/e$ the Julia set of $\lambda e^z$ consists of pairwise disjoint curves, called hairs, which connect finite points, called the endpoints of the hairs, with $\infty$. McMullen showed that the Julia set has Hausdorff dimension $2$ and Karpi\'nska showed that the set of hairs without endpoints has Hausdorf...

We find all non-rational meromorphic solutions of the equation $ww"-(w')^2=\alpha(z)w+\beta(z)w'+\gamma(z)$, where $\alpha$, $\beta$ and $\gamma$ are rational functions of $z$. In so doing we answer a question of Hayman by showing that all such solutions have finite order. Apart from special choices of the coefficient functions, the general solutio...

We consider the uniform asymptotics of polynomials orthogonal on [ 0,8) with respect to the exponential weight w(x) = x(alpha)e- (Q( x)), where alpha > - 1 and Q( x) is a polynomial with positive leading coefficient. In this paper, we have obtained two types of asymptotic expansions in terms of Laguerre polynomials and elementary functions for z in...

In this paper, we study the anomalous diffusion of a particle in an external force field whose motion is governed by nonrenewal continuous time random walks with correlated memorized waiting times, which involves Reimann-Liouville fractional derivative or Reimann-Liouville fractional integral. We show that the mean squared displacement of the test...

We study the asymptotic behavior of Laguerre polynomials Ln(αn)(z) as n→∞n→∞, where αn/nαn/n has a finite positive limit or the limit is +∞+∞. Applying the Deift–Zhou nonlinear steepest descent method for Riemann–Hilbert problems, we derive the uniform asymptotics of such polynomials, which improves on the results of Bosbach and Gawronski (1998). I...

This paper is devoted to studying the dynamical properties of solutions of f '' +A(z)f = 0, where A(z) is an entire function of finite order. With some added conditions on A(z), we prove that all the Fatou components of E(z) are simply-connected provided that E = f(1)f(2) and f(1), f(2) are two linearly independent solutions of such equations. (c)...

This paper is devoted to studying the limit directions of Julia sets of solutions of f(n)+An-1(z)f(n-1)+⋯+A0(z)f=0, where n(≥2) is an integer and Aj(z)(j=0, 1, . ., n-1) are entire functions of finite lower order. With some additional conditions on coefficients, we know that every non-trivial solution f(z) of such equations has infinite lower order...

The concept of logarithmic order is used to investigate the growth of solutions of the linear differential equations f(k) + Ak-1 +...A1(z)f' + A0(z)f= 0, f(k) + A k-1 +...A1(z)f' + A0(z)f= F(z),where A 0 ≠ 0, A1 ..., Ak-1 and F ≠ 0 are transcendental entire functions with orders zero.

In this paper, we study the uniform asymptotics of the Meixner-Pollaczek polynomials with varying parameter as , where A > 0 is a constant. Two asymptotic expansions are obtained, which hold uniformly for z in two overlapping regions which together cover the whole complex plane. One involves parabolic cylinder functions, and the other is in terms o...

In statistical physics, anomalous diffusion plays an important role, whose applications have been found in many areas. In this paper, we introduce a composite-diffusive fractional Brownian motion X α,H (t)=X H (S α (t)), 0<α,H<1, driven by anomalous diffusions as a model of asset prices and discuss the corresponding fractional Fokker-Planck equatio...

This paper is devoted to studying the dynamical properties of solutions of f(n)+A(z)f=0f(n)+A(z)f=0, where n(⩾2)n(⩾2) is an integer, and A(z)A(z) is a transcendental entire function of finite order. We find the lower bound on the radial distribution of Julia sets of E(z)E(z) provided that E=f1f2⋯fnE=f1f2⋯fn and {f1,f2,…,fn}{f1,f2,…,fn} is a solutio...

In this paper, we study the problem of continuous time option pricing with transaction costs by using the homogeneous subdiffusive fractional Brownian motion (HFBM) Z(t)=X(Sα(t))Z(t)=X(Sα(t)), 0<α<10<α<1, here dX(τ)=μX(τ)(dτ)2H+σX(τ)dBH(τ)dX(τ)=μX(τ)(dτ)2H+σX(τ)dBH(τ), as a model of asset prices, which captures the subdiffusive characteristic of fi...

In this Note, we study the uniform asymptotics of the Meixner–Pollaczek polynomials Pn(λn)(z;ϕ) with varying parameter λn=(n+12)A as n→∞, where A>0 is a constant. Uniform asymptotic expansions in terms of parabolic cylinder functions and elementary functions are obtained for z in two overlapping regions which together cover the whole complex plane.

This paper is devoted to studying the growth property and the pole distribution of meromorphic solutions f of some complex difference equations with all coefficients being rational functions or of growth S(r,f). We find the lower bound of the lower order, or the relation between lower order and the convergence exponent of poles of meromorphic solut...

A model for option pricing of a (γ,2H)-fractional Black–Merton–Scholes equation driven by the dynamics of a stock price S(t) satisfying (dS)2H=μS2H(dt)2H+σS2HdBH(t), where BH(t) is a fractional Brownian motion with Hurst exponent H∈(0,1), is established. We obtain the explicit option pricing formulas for the European call option and put option for...

This paper is devoted to studying the relationship between an entire function and its derivative when they share one small function. We generalize some previous results of Gundersen and Yang [G. Gundersen, L.Z. Yang, Entire functions that share one value with one or two of their derivatives, J. Math. Anal. Appl. 223 (1998) 85–95], Chang and Zhu [J....

A model for option pricing of a two parameter (γ,α)-fractional Black-Scholes-Merton differential equation is established based on the stock price modeled by (dS t ) α =μ(S t ) α (dt) α +σ(S t ) α dW α (t), where α>0, μ,σ are constants and dW α (t)=ε(dt) α/2 , fractional Wiener process, ε obeys standard normal distribution. We solve the bi-fractiona...

In this paper, we prove a theorem on the growth of a solution of a linear differential equation. From this we obtain some uniqueness theorems concerning that a nonconstant entire function and its derivatives sharing a small entire function. The results in this paper improve many known results. Some examples are provided to show that the results in...

This paper is devoted to studying growth of solutions of linear differential equations of type where and are entire functions of finite order.

We investigate the solutions and the first passage time for anomalous diffusion processes governed by the fractional nonlinear diffusion equation with diffusion coefficient separable in time and space, D(t,x)=D(t)|x|−θ, subject to absorbing boundary condition and the conventional initial condition p(x,0)=δ(x−x0). We obtain explicit analytical expre...

This paper is devoted to studying the growth of solutions of equations of type f″+h(z)eazf′+Q(z)f=H(z) where h(z), Q(z) and H(z) are entire functions of order at most one. We prove four theorems of such type, improving previous results due to Gundersen and Chen.

This paper is devoted to studying the uniqueness problem of entire functions sharing a small function with their linear differential polynomials.

We investigate the solutions and the first passage time for anomalous diffusion processes governed by the fractional nonlinear diffusion equation with a space- and time-dependent diffusion coefficient subject to absorbing boundaries and the initial condition. We obtain explicit analytical expression for the probability distribution, the first passa...

This work is devoted to investigate explicit solutions of the time-fractional diffusion equations with external forces by considering various diffusion coefficients and an absorbent rate. Besides, the 2nth moment related to such an equation is also discussed. Consequently, the diffusion type can be determined from the mean-square displacement. In a...

This paper is devoted to studying the growth problem, the zeros and fixed points distribution of the solutions of linear differential equations f″+e −z f′+Q(z)f = F(z), where Q(z) ≡ h(z)e cz and c ∊ ℝ.

This paper investigates solutions of some non-homogeneous linear differential equations, which have non-homogeneous term as the small function of solution. Using the similar method, we can generalize the result of G. Gundersen and L. Z. Yang.

In this paper, we investigate the relationship between an entire function and its linear differential polynomial when they share one small function by applying the value distribution theory and complex oscillation theory. As a consequence of the main result, we generalize the study of Gundersen and Yang and the proof is different from theirs.

In this paper, we investigate the relationship between an entire function of small growth $f$ and its linear differential polynomial $L(f)$ when they share one value by applying value distribution theory and complex oscillation theory. As consequences of the main result we can get the precise form of $f$.

This article studies the problem on the fixed points and hyper-order of differential polynomials generated by solutions of two type of second order differential equations. Because of the control of differential equation, we can obtain some precise estimates of their hyper-order and fixed points.

The uniqueness of meromorphic functions with common range sets and deficient values are studied. This result is related to a question of Gross.