Exact inequalities for derivatives of functions of low smoothness defined on an axis and a semiaxis - Babenko, V. F.; Kofanov, V. A.; Pichugov, S. A. (Ukrains’kyi Matematychnyi Zhurnal, 2006)
We obtain new exact inequalities of the form$$∥x(k)∥_q ⩽ K∥x∥^{α}_p ∥x(r)∥^{1−α}_s$$for functions defined on the axis $R$ or the semiaxis $R_{+}$ in the case where$$r = 2,\; k = 0,\; p ∈ (0,∞),\; q ∈ (0,∞],\; q > p,\; s=1,$$for functions defined on the axis $R$ in the case where$$r = 2,\; k = 1,\; q ∈ [2,∞),\; p = ∞,\; s= 1,$$and for functions of constant sign on $R$ or $R_{+}$ in the case where$$r = 2,\; k = 0,\; p ∈ (0,∞),\; q ∈ (0,∞],\; q > p,\; s = ∞$$and in the case where$$r = 2,\; k = 1,\; p ∈ (0,∞),\; q = s = ∞.$$