New refinements of some classical inequalities via Young’s inequality

Abstract

The main objective of this paper is to use a new refinement of Young’s inequality to obtain two new scalar inequalities. As an application, we derive several new improvements of some well-known inequalities, which include the generalized mixed Schwarz inequality, numerical radius inequalities, Jensen inequalities and others. For example, for every $T,S \in {\mathcal {B(H)}}$, $\alpha \in (0,1)$ and $x, y \in {\mathcal {H}}$, we prove that

$$\begin{aligned}{} & {} \left( 1+ L(\alpha )\log ^2\left( \frac{|\langle TS x, y\rangle | }{r(S)\Vert f(|T|) x\Vert \left\| g\left( \left| T^*\right| \right) y\right\| }\right) \right) |\langle TSx, y\rangle | \\{} & {} \quad \le r(S)\Vert f(|T|) x\Vert \left\| g\left( \left| T^*\right| \right) y\right\| , \end{aligned}$$

where L is a positive 1-periodic function and r(S) is the spectral radius of S, which gives an improvement of the well-known generalized mixed Schwarz inequality:

$$\begin{aligned} \left| \langle TSx,y \rangle \right| \le r(S)\Vert f(|T|) x\Vert \left\| g\left( \left| T^*\right| \right) y\right\| , \end{aligned}$$

where $|T| S=S^*|T|$ and f, g are non-negative continuous functions defined on $[0, \infty )$ satisfying that $f(t) g(t)=t\,(t \ge 0)$.