Редукция неравенства Колмогорова для положительной срезки второй производной на оси к неравенству для выпуклых функций на отрезке

In this paper we delve into connection between sharp constants in the inequalities
$$\|y'\|_{L_q(\mathbb{R})}\le K_+ \sqrt{\|y\|_{L_r(\mathbb{R})}\|y''_+\|_{L_p(\mathbb{R})} },$$

$$\|u'\|_{L_q(0,1)}\le \overline{K} \sqrt{\|u\|_{L_r(0,1)} \|u''\|_{L_p(0,1)}},$$
where the second one is considered for convex functions $u(x)$, $x\in[0,1]$ with an absolutely continuous derivative that vanishes at the point $x=0$. We prove that $K_+=\overline{K}$ under conditions $1 \le q,r,p<\infty$ and $1/r+1/p=2/q$.