  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.
