A specific function f(r) involving a ratio of complicated gamma functions depending upon a real variable r(> 0) is handled. Details are explained regarding how this function f(r) appeared naturally for our investigation with regard to its behavior when r belongs to R⁺. We determine explicitly where this function attains its unique minimum. In doing so, quite unexpectedly the customary Cramér-Rao inequality comes into play in order to nail down a valid proof of the required lower bound for f(r) and locating where is that lower bound exactly attained.

中文翻译：

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Cramér-Rao不等式在证明复杂伽玛函数比率可达到的下界上的不寻常应用

处理涉及取决于实际变量r（> 0）的复杂伽马函数比率的特定函数f（r）。详细解释关于这个功能如何˚F（[R ）自然出现了我们的调查就其行为时，[R属于[R ⁺。我们明确确定该函数在何处达到其唯一最小值。这样做时，出乎意料的是，通常的Cramér-Rao不等式开始起作用，以便确定f（r）所需下界的有效证明，并确定下界确切地在哪里获得。