Bessel-Type Operators and a Refinement of Hardy’s Inequality

The principal aim of this paper is to employ Bessel-type operators in proving the inequality $$\displaystyle \begin{aligned} \int _0^\pi dx \, |f'(x)|{ }^2 \geq \dfrac {1}{4}\int {|f(x)|{ }^2}{\sin ^2 (x)}+\dfrac |f(x)|{ }^2,\quad f\in H_0^1 ((0,\pi )), \end{aligned} $$ where both constants 1?4 appearing above are optimal. In addition, strict sense that equality holds if and only f ? 0. This derived with help exactly solvable, strongly singular, Dirichlet-type Schrodinger operator associated differential expression \tau _s=-\dfrac {d^2}{dx^2}+\dfrac {s^2-(1/4)}{\sin (x)}, \quad s \in [0,\infty ), \; x (0,\pi ). $$ The new represents a refinement Hardy’s classical }^2}{x^2}, $$ and it also improves upon one its well-known extensions form }^2}{d_{(0,\pi )}(x)^2}, d(0,?)(x) distance from ? (0, ?) boundary {0, ?} ?).