Improved Hardy-Rellich inequalities

<p style='text-indent:20px;'>We investigate Hardy-Rellich inequalities for perturbed Laplacians. In particular, we show that a non-trivial angular perturbation of the free operator typically improves inequality, and may also provide an estimate which does not hold in case. The main examples are related to introduction magnetic field: this is manifestation diamagnetic phenomenon, has been observed by Laptev Weidl [<xref ref-type="bibr" rid="b21">21</xref>] Hardy later Evans Lewis rid="b9">9</xref>] Rellich inequality; however, best our knowledge, so called inequality yet investigated regards. After showing optimal prove constant attained any function domain estimate.</p>