Reduction of the Kolmogorov inequality for a non negative part of the second derivative on the real line to the inequality for convex functions on an interval

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 $1/r+1/p=2/q$.