论文标题
在赖特功能的对数洞中
On the log-concavity of the Wright function
论文作者
论文摘要
我们研究了Wright函数$ ϕ（-α，β，-x）的半线上的对数concavity，概率设置$α\ in（0,1）$和$β\ ge0中的$。制定了一种自然的猜想，以表达赖特功能的对数洞穴与这种广义熵的存在之间的等效性。对于$β\geqα$解决了该问题，在经典情况下，Mittag-Leffler分布的$β= 1-α$，该分布表现出一定的关键参数$α_*= 0.771667 ... $ $ $在伽马功能上隐含地定义并表征逻辑concavity。我们还证明，概率的赖特函数始终是单峰的，并且仅当$β\geqα$或$α\ le 1/2 $和$β= 0时，它们在多种模态上是单型的。
We investigate the log-concavity on the half-line of the Wright function $ϕ(-α,β,-x),$ in the probabilistic setting $α\in (0,1)$ and $β\ge 0.$ Applications are given to the construction of generalized entropies associated to the corresponding Mittag-Leffler function. A natural conjecture for the equivalence between the log-concavity of the Wright function and the existence of such generalized entropies is formulated. The problem is solved for $β\geqα$ and in the classical case $β= 1-α$ of the Mittag-Leffler distribution, which exhibits a certain critical parameter $α_*= 0.771667...$ defined implicitly on the Gamma function and characterizing the log-concavity. We also prove that the probabilistic Wright functions are always unimodal, and that they are multiplicatively strongly unimodal if and only if $β\geqα$ or $α\le 1/2$ and $β= 0.$
