Inequalities for derivatives of functions in the spaces Lp
The following sharp inequality for local norms of functions x L^r_∞,∞(R) is proved: 1b-a_a^b|x’(t)|^qdt ≤ 1π_0^π|φ_r-1(t)|^q dt (||x||_L_∞(R) ||φ_r||_∞ )^r-1rq||x^(r)||^q_∞r,r N, where φ_r is the perfect Euler spline, takes place on intervals [a, b] of monotonicity of the function x for q ≥ 1 or for any q > 0 in the cases of r = 2 and r = 3. As a corollary, well-known A. A. Ligun’s inequality for functions x L^r_∞ of the form ||x^(k)||_q ≤ ||φ_r-k||_q||φ_r||_∞^1-k/r ||x||^1-k/r_∞||x^(r)||^k/r_∞,k,r N,k < r, 1 ≤ q < ∞, is proved for q [0,1) in the cases of r = 2 and r = 3., number=10, journal=Ukrains’kyi Matematychnyi Zhurnal, author=Kofanov, V. A., year=2008, month=Oct., pages=1338 – 1349