On certain inequalities involving the constant \(e\) and their applications
This is a survey of results, mainly by the first author, some improved, with several further related inequalities and limit values offered. Among the latter the following is typical (and relatively less technical). For all \(y>0,\) \[ \Biggl(1+\frac{1}{y} \Biggr)^{1/2} \exp \Biggl[-\frac{1}{12y^2 +12y}\Biggr]< e\Bigl(1+\frac{1}{y}\Bigr)^{-y}< \Biggl(1+\frac{1}{y}\Biggr)^{1/2} \exp \Biggl[-\frac{1}{12y^2 +12y+3}\Biggr]. \] Connections between different inequalities and limits are emphasized. [Note: The inequality \(\sum_{k=1}^n a_k \log a_k \geq \sum_{k=1}^n a_k\log b_k\) if \(\sum_{k=1}^n a_k = \sum_{k=1}^n b_k\) is mentioned on p. 570; the name ``Shannon inequality'' is not].
scientific article; zbMATH DE number 1533185