On the quantities related to multiply monotone functions (Q2901720)

The present work is devoted to the study of the so-called multiply monotone functions. Given \(m\in {\mathbb N},\) the function \(h\in C^{m-1}(0, +\infty)\) is said to be \(m\)-multiply monotone, denoted by \(h\in M_m,\) if and only if the function \((-1)^{m-1}h^{(m-1)}\) to be nonnegative, decreasing, lower convex and has a finite limit at \(+\infty\). For such functions we consider a quantity \(\gamma_m(\rho, h)\) by the following way. Let \(h\in M_{m+1},\) \(0<h(+0)\leq 1\) and \(h(+\infty)=0,\) then we define \(\gamma_m(\rho, h),\) \(\rho\geq 1,\) as supremum over all \(\gamma\in {\mathbb R}\) for which the combination \(\lambda_{\rho, \gamma}(x)=\frac{1-(1+\gamma x)h(x)}{x^{\rho}}\) belongs to \(M_m\). The exact value of \(\gamma_m(\rho, h)\) for the functions \(e^{-x},\;(1+x)^{-\mu}\) and \((1-x)_+^\mu\) are obtained in the paper. More detailed, Theorem 1 states that \(\gamma_m(1, h)=\frac{1}{m+2},\) \(\gamma_m(\rho, h)=1\) as \(\rho\geq 2\) and \(0<\widetilde{\gamma}_{m+1}(\rho)\leq \gamma_m(\rho, h)<1\) whenever \(h=e^{-t}\) and a constant \(\widetilde{\gamma}_{m+1}(\rho)\) depends only on \(m\) and \(\rho\). Theorem 2 states that \(\gamma_m(\rho, h_{\mu})=\mu\) as \(\rho\geq 2,\) \(\gamma_m(1, h_{\mu})=\frac{\mu+m+1}{\mu+2}\) and \(\gamma_m(\rho, h_1)=1\) as \(\rho\geq 1,\) where \(h_{\mu}(t)=(1+t)^{-\mu},\) \(\mu\geq 1\). Moreover, \(\gamma_m(\rho, h_{\mu})<\mu\) as \(\mu>1\) and \(1\leq\rho<2\). Some other results such as estimates of \(\gamma_m(\rho, h)\) for the function \(h(t)=(1-t)^{\mu}_{+},\) \(\mu\geq m+1,\) and a criterion of the belonging of the arbitrary function \(h\) to \(M_m,\) are obtained in the work.