Inequalities for linear functionals and numerical radii on $\mathbf{C}^*$-algebras

Let $\mathcal{A}$ be a unital $\mathbf{C}^*$-algebra with unit $e$. We develop several inequalities for a positive linear functional $f$ on $\mathcal{A}$ and obtain several bounds for the numerical radius $v(a)$ of an element $a\in \mathcal{A}$. Among other inequalities, we show that if $a_k, b_k, x_k\in \mathcal{A}$, $r\in \mathbb{N}$ and $f(e)=1$, then \begin{eqnarray*} \left| f \left( \sum_{k=1}^n a_k^*x_kb_k\right)\right|^{r} &\leq& \frac{n^{r-1}}{\sqrt{2}} \left| f\left( \sum_{k=1}^n \big( (b_k^*|x_k| b_k)^{r}+ i (a_k^*|x_k^*|a_k)^{r} \big) \right) \right| \quad (i=\sqrt{-1}), \end{eqnarray*} \begin{eqnarray*} \left| f\left( \sum_{k=1}^n a_k\right)\right|^{2r} &\leq& \frac{n^{2r-1}}{2} f \left( \sum_{k=1}^n Re(|a_k|^r|a_k^*|^r) + \frac12 \sum_{k=1}^n (|a_k|^{2r}+ |a_k^*|^{2r} ) \right). \end{eqnarray*} We find several equivalent conditions for $v(a)=\frac{\|a\|}{2}$ and $v^2(a)={\frac{1}{4}\|a^*a+aa^*\|}$. We prove that $v^2(a)={\frac{1}{4}\|a^*a+aa^*\|}$ (resp., $v(a)=\frac{\|a\|}{2}$) if and only if $\mathbb{S}_{\frac12{ \| a^*a+aa^*\|}^{1/2}} \subseteq V(a) \subseteq \mathbb{D}_{\frac12 {\| a^*a+aa^*\|}^{1/2}}$ (resp., $\mathbb{S}_{\frac12 \| a\|} \subseteq V(a) \subseteq \mathbb{D}_{\frac12 \| a\|}$), where $V(a)$ is the numerical range of $a$ and $\mathbb{D}_k$ (resp., $\mathbb{S}_k$) denotes the circular disk (resp., semi-circular disk) with center at the origin and radius $k$. We also study inequalities for the $(\alpha,\beta)$-normal elements in $\mathcal{A}.$