On Birman's sequence of Hardy-Rellich-type inequalities
In 1961, Birman proved a sequence of inequalities $\{I_{n}\},$ for $n\in\mathbb{N},$ valid for functions in $C_0^{n}((0,\infty))\subset L^{2}((0,\infty)).$ In particular, $I_{1}$ is the classical (integral) Hardy inequality and $I_{2}$ is the well-known Rellich inequality. In this paper, we give a proof of this sequence of inequalities valid on a certain Hilbert space $H_{n}([0,\infty))$ of functions defined on $[0,\infty).$ Moreover, $f\in H_{n}([0,\infty))$ implies $f^{\prime}\in H_{n-1}([0,\infty));$ as a consequence of this inclusion, we see that the classical Hardy inequality implies each of the inequalities in Birman's sequence. We also show that for any finite $b>0,$ these inequalities hold on the standard Sobolev space $H_0^{n}((0,b))$. Furthermore, in all cases, the Birman constants $[(2n-1)!!]^{2}/2^{2n}$ in these inequalities are sharp and the only function that gives equality in any of these inequalities is the trivial function in $L^{2}((0,\infty))$ (resp., $L^2((0,b))$). We also show that these Birman constants are related to the norm of a generalized continuous Ces\`aro averaging operator whose spectral properties we determine in detail.