Comparison of Exact Constants in Inequalities for Derivatives of Functions Defined on the Real Axis and a Circle

Abstract

We investigate the relationship between the constants K(R) and K(T), where $K\left( G \right) = K_{k,r} \left( {G;q,p,s;\alpha } \right): = \mathop {\mathop {\sup }\limits_{x \in L_{p,s}^r \left( G \right)} }\limits_{x^{(r)} \ne 0} \frac{{\left\| {x^{\left( k \right)} } \right\|_{L_q \left( G \right)} }}{{\left\| x \right\|_{L_q \left( G \right)}^\alpha \left\| {x^{\left( r \right)} } \right\|_{L_s \left( G \right)}^{1 - \alpha } }}$ is the exact constant in the Kolmogorov inequality, R is the real axis, T is a unit circle,

$$L_{p,s}^r (G)$$

is the set of functions x ∈ L _p(G) such that x ^(r) ∈ L _s(G), q, p, s ∈ [1, ∞], k, r ∈ N, k < r, We prove that if

$$\frac{{r - k + {1 \mathord{\left/{\vphantom {1 q}} \right.\kern-\nulldelimiterspace} q} - {1 \mathord{\left/{\vphantom {1 s}} \right.\kern-\nulldelimiterspace} s}}}{{r + {1 \mathord{\left/{\vphantom {1 q}} \right.\kern-\nulldelimiterspace} q} - {1 \mathord{\left/{\vphantom {1 s}} \right.\kern-\nulldelimiterspace} s}}} = 1 - k/r$$

thenK(R) = K(T),but if

$$\begin{gathered}\frac{{r - k + {1 \mathord{\left/{\vphantom {1 q}} \right.\kern-\nulldelimiterspace} q} - {1 \mathord{\left/{\vphantom {1 s}} \right.\kern-\nulldelimiterspace} s}}}{{r + {1 \mathord{\left/{\vphantom {1 q}} \right.\kern-\nulldelimiterspace} q} - {1 \mathord{\left/{\vphantom {1 s}} \right.\kern-\nulldelimiterspace} s}}} < \hfill \\1 - k/r \hfill \\ \end{gathered} $$

thenK(R) ≤ K(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.