Ukrainian Mathematical Journal, Vol. 55, No. 5, 2003 COMPARISON OF EXACT CONSTANTS IN INEQUALITIES FOR DERIVATIVES OF FUNCTIONS DEFINED ON THE REAL AXIS AND A CIRCLE V. F. Babenko, V. A. Kofanov, and S. A. Pichugov UDC 517.5 We investigate the relationship between the constants K ( ) R and K ( ) T , where KG ( ) = K Gqps kr , (; , ,; ) α : = sup , ( ) ( ) ( ) ( ) ( ) () ( ) x L G x k L G L G r L G ps r r q p s x x x ∈ ≠ - 0 1 α α is the exact constant in the Kolmogorov inequality, R is the real axis, T is a unit circle, L G ps r , ( ) is the set of functions x L G p ∈ ( ) such that x L G r s () ( ) ∈ , qps , , [, ] ∈ ∞ 1 , kr , ∈ N , k < r, α= - + - + - r k q s r p s 1 1 1 1 / / / / if G = R , and α= - - + - + -       min , / / / / / 1 1 1 1 1 kr r k q s r p s if G = T. We prove that if r k q s r p s kr - + - + - = - 1 1 1 1 1 / / / / / , then K K ( ) ( ) R T = , but if r k q s r p s - + - + - 1 1 1 1 / / / / < 1 - kr / , then K K ( ) ( ) R T ≤ ; moreover, the last inequality can be an equality as well as a strict inequality. As a corollary, we obtain new exact Kolmogorov-type inequalities on the real axis. 1. Introduction Let G denote either the real axis R or a unit circle T realized as the segment [ 0 , 2π ] with identified endpoints, and let Lp ( G ) , 1 ≤ p ≤ ∞ , be the space of measurable functions x : G → R with the finite Lp-norm x L G p ( ) = x p : = xt dt p xt p p G p t G () , sup () . / ∫         ≤ <∞ =∞        ∈ 1 1 if vrai if In what follows, we write ⋅ p instead of x L G p ( ) if this does not lead to misunderstanding. For r ∈ N and s ∈ [ 1 , ∞ ], we denote by L G s r ( ) the set of functions x : G → R such that x r ( ) -1 ( ) () x x 0 = is locally abso- lutely continuous and x r () ∈ L G s ( ) and assume that L G ps r , ( ) = L G L G p s r ( ) ( ) I . Note that L s r ( ) T ⊂ L p ( ) T for any p ∈ [ 1 , ∞ ] . In numerous problems of analysis, an important role is played by inequalities for the norms of intermediate derivatives of functions x ∈ L G ps r , ( ) of the form Dnepropetrovsk National University, Dnepropetrovsk. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 55, No. 5, pp. 579–589, May, 2003. Original article submitted December 11, 2001. 0041–5995/03/5505–0699 $25.00 © 2003 Plenum Publishing Corporation 699