Integral inequalities related to the uniform measure on \(\mathclose {[}0,1\mathclose {]}\)

By using semigroup techniques, we state and prove a new family of sharp integral inequalities for the Sturm–Liouville operator acting on the Hilbert space \({\mathbb {L}}^{2}([0,1],\textrm{d}x)\) ${\mathbb{L}}^2([0,1],\text{d}x)$ with respect to the uniform measure.