Inequalities for cyclic functions (Q5959031)

The cyclic functions are defined as the cyclic partial sums of the exponential function NEWLINE\[NEWLINE \varphi_n(z):=\sum_{\nu=0}^\infty {z^{n\nu}\over (n\nu)!} \qquad (z\in {\mathbb C},\;2\leq n\in{\mathbb N}). NEWLINE\]NEWLINE In the special case \(n=2\), \(\varphi_2\) is the classical cosine hyperbolic function. The authors offer an interesting generalization of the well-known inequality NEWLINE\[NEWLINE {\sinh x\over x} < \cosh x <\left({\sinh x\over x}\right)^3 \qquad (x>0) NEWLINE\]NEWLINE in the following form NEWLINE\[NEWLINE \left({(n-k)! \varphi_n^{(k)}(x)\over x^{n-k}}\right)^\alpha < \varphi_n(x) < \left({(n-k)! \varphi_n^{(k)}(x)\over x^{n-k}}\right)^\beta,NEWLINE\]NEWLINE for all \(1\leq k\leq n-1\) and \(x>0\) with the best possible constants NEWLINE\[NEWLINE \alpha=1 \qquad\text{and} \qquad \beta={2n-k\choose n} .NEWLINE\]