Walter Bergweiler
Wiman-Valiron theory and results of Macintyre about "flat regions" describe the asymptotic behavior of entire functions in certain disks around points of maximum modulus. We estimate the size of these disks for Macintyre's theory from above and below.
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This note corrects Example 3.2 in Two-Variable Wiman-Valiron Theory and PDEs by the authors which appeared in Ann. Acad. Sci. Fenn Math. (35) (2010), 571-580.
Suppose that a transcendental meromorphic function in the plane has finitely many critical values, while its multiple points have bounded multiplicities, and its inverse function has finitely many transcendental singularities. Using the Wiman-Valiron method it is shown that the Schwarzian derivative does not have a direct transcendental singularity over infinity, and does not have infinity as a Borel exceptional value.
Analogues of the key results of Wiman-Valiron theory are proved for a class of functions meromorphic in the unit disc, based on an approach developed by Bergweiler, Rippon and Stallard for the plane setting. The results give local approximations for the function and its logarithmic derivative and, in the case of positive order of growth, for higher order logarithmic derivatives as well.
A note on the Wiman-Valiron inequality
本文回顾 Wiman-Valiron 不等式的已知结果，并给出一个此前文献中似乎被忽视的一般形式，最后提出一个公开问题。
The paper gives a precise asymptotic relation between higher order logarithmic difference and logarithmic derivatives for meromorphic functions with order strictly less then one. This allows us to formulate a useful Wiman-Valiron type estimate for logarithmic difference of meromorphic functions of small order. We then apply this estimate to prove a classical analogue of Valiron about entire solutions to linear differential equations with polynomials coefficients for linear difference equations.
We establish a Wiman-Valiron theory for a polynomial series based on the Wilson operator $\mathcal{D}_\mathrm{W}$. For an entire function $f$ of order smaller than $\frac13$, this theory includes (i) an estimate which shows that $f$ behaves locally like a polynomial consisting of the terms near the maximal term in its Wilson series expansion, and (ii) an estimate of $\mathcal{D}_\mathrm{W}^n f$ compared to $f$. We then apply this theory in studying the growth of entire solutions to difference equations involving the Wilson operator.
We establish a Wiman-Valiron theory of a polynomial series based on the Askey-Wilson operator $\mathcal{D}_q$, where $q\in(0,1)$. For an entire function $f$ of log-order smaller than $2$, this theory includes (i) an estimate which shows that $f$ behaves locally like a polynomial consisting of the terms near the maximal term of its Askey-Wilson series expansion, and (ii) an estimate of $\mathcal{D}_q^n f$ compared to $f$. We then apply this theory in studying the growth of entire solutions to difference equations involving the Askey-Wilson operator.
Estimation for the logarithmic partial derivative and Wiman-Valiron theory in several complex variables
本文旨在将Gundersen和Cao的结果推广到高维，估计多复变量亚纯函数的对数偏导数，发展Wiman-Valiron理论，并研究一类复偏微分方程解的增长率。
We show that the Hausdorff dimension of the set of points of bounded orbit in the Julia set of a meromorphic map with a simply connected direct tract and a certain restriction on the singular values is strictly greater than one. This result is obtained by proving new results related to Wiman-Valiron theory.
Let $f(z)=\sum_{n=0}^{+\infty} a_nz^n$\ $(z\in\mathbb{C})$\ be an analytic function in the unit disk and $f_t$ be an analytic function of the form $f_t(z)=\sum_{n=0}^{+\infty} a_ne^{iθ_nt}z^n,$ where $t\in\mathbb{R},$ $θ_n\in\mathbb{N},$ and $h$ be a positive continuous function on $(0, 1)$ increasing to $+\infty$ and such that $ \int_{r_0}^1h(r)dr=+\infty, r_0\in(0,1).$\ If the sequence $(θ_n)_{n\geq0}$ satisfies the inequality $$ \varlimsup_{n\to+\infty}\frac1{\ln n}\ln\frac{θ_n}{θ_{n+1}-θ_n}\leqδ\in[0,1/2), $$ then for all analytic functions $f_t$ almost surely for $t$ there exists a set $E…
Wiman-Valiron inequalities in the unit disk outside sets of finite logarithmic measure
本文肯定回答了Grosse-Erdmann关于单位圆盘内Wiman-Valiron不等式的两个问题，在有限对数测度例外集外建立迭代对数不等式，并给出Rosenbloom复合估计的圆盘类比，关键利用Khinchin族方差界，且指数1/2不可改进。
Wiman-Valiron method for fractional derivatives and sharp growth estimates of $α$-analytic solutions for linear fractional differential equations
本文通过扩展Wiman-Valiron理论，研究了分数微分方程中α-解析解的精确增长估计，并推广了Kochubei的结果。
This is a survey on Valiron's Theorem about the convergence properties of orbits of analytic self-maps of the disk of hyperbolic type and related questions in one and several variables.
Rate of growth of random analytic functions, with an application to linear dynamics
本文建立随机整函数与单位圆盘随机解析函数的 Wiman–Valiron 型不等式，改进已有结果，并用于刻画任意混沌加权后移算子频繁超循环函数在例外集外的增长率。
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