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This paper presents several numerical radii and norm inequalities for Hilbert space operators. These inequalities improve some earlier related inequalities. For an operator $A$, we prove that $$\begin{array}{ccc}\hfill {\omega }^2\left(A\right)\le & \parallel \frac{A^{*}A+A{A}^{*}}{2}-\frac{1}{2R}((1-t)A^{*}A+tA{A}^{*}\hfill & \hfill -{((1-t){\left(A^{*}A\right)}^{1/2}+{\left(A{A}^{*}\right)}^{1/2})}^2)\parallel \end{array}$$ where $R=max\{t,1-t\}$ and $0\le t\le 1$.

We prove an inner product inequality for Hilbert space operators. This inequality will be utilized to present a general numerical radius inequality using convex functions. Applications of the new results include obtaining new forms that generalize and extend some well known results in the literature, with an application to the newly defined generalized numerical radius. We emphasize that the approach followed in this article is different from the approaches used in the literature to obtain such...