论文标题
系数猜想的基本反例
An elementary counterexample to a coefficient conjecture
论文作者
论文摘要
在本文中，我们考虑了函数家族$ f $ meromorthic在单位磁盘$ \ id = \ {z：\，| z | | <1 \} $在点$ z = p $，taylor扩展\ [f（z）= z+\ sum_ {k = 2}^{\ infty} a_kz^k，\ quad | z | <p，\]并满足条件\ [\ [\ [\左） | \ left（\ frac {z} {f（z）} \ right）-z \ left（\ frac {z} {z} {f（z）} \ right）' - 1 \ right |<λ，\，\，\，\ forall z \ in \ in \ id，\ in \ id，\ in \ in \ in \ in \ in \ in \ in \ in \ in \ in \ in \ in \ in \ in \ in \ in \ in \ in \ in \ rive。我们用$ \ Mathcal {U} _M（λ）$表示此类，我们将证明该类函数的表示定理。结果，我们获得了$ | a_2 | $的估计值的简单证明，并获得了$ f \ in \ Mathcal {u} _m（λ）$的laurent系列的初始系数的不等式。在\ cite {pw2}中，有人指出，对于$ f \ in \ mathcal {u} _m（λ）$不等式\ [| a _n | \，\ leq \ \ leq \，\ frac {1} \ Quad N \ GEQ 2 \]有效。我们为案例$ n = 3 $的猜想提供了反例。
In this article, we consider the family of functions $f$ meromorphic in the unit disk $\ID=\{z :\,|z| < 1\}$ with a pole at the point $z=p$, a Taylor expansion \[f(z)= z+\sum_{k=2}^{\infty} a_kz^k, \quad |z|<p, \] and satisfying the condition \[\left |\left(\frac{z}{f(z)}\right)-z\left(\frac{z}{f(z)}\right)'-1\right |<λ,\, \forall z\in\ID, \] for some $λ$, $0<λ< 1$. We denote this class by $\mathcal{U}_m(λ)$ and we shall prove a representation theorem for the functions in this class. As consequences, we get a simple proof for the estimates of $|a_2|$ and obtain inequalities for the initial coefficients of the Laurent series of $f\in \mathcal{U}_m(λ)$ at its pole. In \cite{PW2} it had been conjectured that for $f\in \mathcal{U}_m(λ)$ the inequalities \[|a_n|\,\leq\,\frac{1}{p^{n-1}}\sum_{k=0}^{n-1}(λp^2)^k, \quad n\geq 2 \] are valid. We provide a counterexample to this conjecture for the case $n=3$.
