An integral problem of Konyagin and the $(C,L)$-constants of Nikol'skii

An extremal problem concerning functions with small supports posed by Konyagin in connection with number-theoretic applications is considered. It is shown to be related to extremal problems on the best Nikol'skii constants in the inequalities for $C$- and $L$-norms of trigonometric polynomials and entire functions of exponential type. New estimates for constants in these problems are obtained.