Several upper estimates for the numerical radius of Hilbert space operators are given. Among many other inequalities, it is shown that \begin{align*}{{mega }^{2}}eft( A \right)e \frac{1}{4}eft\| {{eft| A \right|}^{2}}+{{eft| {{A}^{*}} \right|}^{2}} \right\| +\frac{1}{2}mega eft( {{A}^{2}} \right)-\frac{1}{2}\underset{eft\| x \right\|=1} {\mathop{\underset{xı \mathscr H}{\mathop{ıf }}\,}}\,{{eft( qrt{eftangle {{eft| A \right|}^{2}}x,x \right\rangle } -qrt{eftangle {{eft| {{A}^{*}} \right|}^{2}}x,x \right\rangle } \right)}^{2}}.\end{align*}