Inequalities of Different Metrics for Differentiable Periodic Functions

• Date: 2015
• Main Authors: Kofanov, V. A., Кофанов, В. А.
• Format: Article
• Language: Russian; English
• Published: Institute of Mathematics, NAS of Ukraine 2015
• Online Access: https://umj.imath.kiev.ua/index.php/umj/article/view/1974
• Journal Title: Ukrains’kyi Matematychnyi Zhurnal

We prove the following sharp inequality of different metrics: $$\begin{array}{cc}\hfill {\left\Vert x\right\Vert}_q\le {\left\Vert {\varphi}_r\right\Vert}_q{\left(\frac{{\left\Vert x\right\Vert}_p}{{\left\Vert {\varphi}_r\right\Vert}_p}\right)}^{\frac{r+1/q}{r+1/p}}{\left\Vert {x}^{(r)}\right\Vert}_...