Inequalities Between the Norms of a Function and Its Derivatives

Abstract

The paper is devoted to the problem of finding the maximum of the norm \( \|x\|_q \) with the constraints \( \|x\|_p = \eta \), \( \|\dot{x}\|_r = \sigma \), \( x(0) = a \), \( a, \sigma, \eta > 0 \), for functions \( x \in L_p(\mathbb{R}_-) \) with derivatives \( \dot{x} \in L_r(\mathbb{R}_-) \), \( 0 < p \leq q < \infty \), \( r > 1 \). The arguments employed are based on the standard machinery of the calculus of variations.