New refinements of some classical inequalities via Young’s inequality

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 \(T,S \in {\mathcal {B(H)}}\) $T,S\in \mathcal{B}(\mathcal{H})$ , \(\alpha \in (0,1)\) $\alpha \in (0,1)$ and \(x, y \in {\mathcal {H}}\) $x,y\in \mathcal{H}$ , we prove that \(\begin{aligned}{} & {} \left( 1+ L(\alpha )\log ^2\left( \frac{|\langle TS x, y\rangle | }{r(S)\Vert f(|T|) x\Vert \left\| g\left( \left| T^*\right| \right) y\right\| }\right) \right) |\langle TSx, y\rangle | \\{} & {} \quad \le r(S)\Vert f(|T|) x\Vert \left\| g\left( \left| T^*\right| \right) y\right\| , \end{aligned}\) $$\begin{array}{cc}\hfill & \left(1,+,L,(\alpha ),{log}^2,\left(\frac{|⟨TSx,y⟩|}{r(S)\Vert f(|T|)x\Vert ∥g,\left(\left|T^{\ast }\right|\right),y∥}\right)\right)|⟨TSx,y⟩|\hfill \\ \hfill & \le r(S)\Vert f(|T|)x\Vert ∥g,\left(\left|T^{\ast }\right|\right),y∥,\hfill \end{array}$$ 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: \(\begin{aligned} \left| \langle TSx,y \rangle \right| \le r(S)\Vert f(|T|) x\Vert \left\| g\left( \left| T^*\right| \right) y\right\| , \end{aligned}\) $$\begin{array}{c}\hfill \left|⟨,T,S,x,,,y,⟩\right|\le r(S)\Vert f(|T|)x\Vert ∥g,\left(\left|T^{\ast }\right|\right),y∥,\end{array}$$ where \(|T| S=S^*|T|\) $|T|S=S^{\ast }|T|$ and f, g are non-negative continuous functions defined on \([0, \infty )\) $[0,\infty )$ satisfying that \(f(t) g(t)=t\,(t \ge 0)\) $f(t)g(t)=t(t\ge 0)$ .