We introduce some new refinements of numerical radius inequalities for Hilbert space invertible operators. More precisely, we prove that if $T\in {\mathcal{B}}({\mathcal{H}})$ is an invertible operator, then $\Vert T\Vert \leq \sqrt{2}\unicode[STIX]{x1D714}(T)$ .

The main purpose of this paper is to discuss operator Jensen inequality for convex functions, without appealing to operator convexity. Several variants of this inequality will be presented, and some applications will be shown too.

We give an alternative lower bound for the numerical radii of Hilbert space operators. As a by-product, we find conditions such that ω0RS0=∥R∥+∥S∥2 $$\begin{array}{} \displaystyle \omega\left(\left[\begin{array}{cc} 0 & R \\ S & 0 \end{array}\right]\right)=\frac{\Vert R \Vert +\Vert S\Vert }{2} \end{array}$$where R, S ∈ 𝔹(𝓗).

We prove several numerical radius inequalities for products of two Hilbert space operators. Some of our inequalities improve well-known ones. More precisely, we prove that, if A,B ? B(H) such that A is self-adjoint with ?1 = min ?i ? ?(A) (the spectrum of A) and ?2 = max ?i ? ?(A). Then ?(AB) ?||A||?(B) + (||A|| - |?1 + ?2|/2)DB where DB = inf ??C ||B - ?I||. In particular, if A > 0 and ?(A) ? [k||A||,||A||], then ?(AB) ? (2 - k)||A|| ?(B).

In this paper, we present several numerical radius inequalities for Hilbert space operators. More precisely, we prove if T , U ∈ B H $ T,U\in\mathbb{B}\left(\mathcal{H}\right) $ such that U is unitary, then ω ⁢ ( T ⁢ U ± U * ⁢ T ) ≤ 2 ⁢ ω ⁢ ( T 2 ) + ∥ T ± T * ∥ 2 . $$\displaystyle\omega(TU\pm U^{*}T)\leq 2\sqrt{\omega(T^{2})+\|T\pm T^{*}\|^{2}}. $$ Also, we have compared our results with some known outcomes.