Euclidean operator radius inequalities of a pair of bounded linear operators and their applications

We obtain several sharp lower and upper bounds for the Euclidean operator radius of a pair of bounded linear operators defined on a complex Hilbert space. As applications of these bounds we deduce a chain of new bounds for the classical numerical radius of a bounded linear operator which improve on the existing ones. In particular, we prove that for a bounded linear operator $A,$ $$\frac{1}{4}\mathrm{\parallel }{A}^{\ast }A+A{A}^{\ast }\mathrm{\parallel }+\frac{\mu }{2}\max \{\mathrm{\parallel }\mathrm{\Re }(A)\mathrm{\parallel },\mathrm{\parallel }\mathrm{\Im }(A)\mathrm{\parallel }\}\le w^2(A)\le w^2(\mathrm{\mid }\mathrm{\Re }(A)\mathrm{\mid }+i\mathrm{\mid }\mathrm{\Im }(A)\mathrm{\mid }),$$ where $\mu =\mid \mathrm{\parallel }\mathrm{\Re }(A)+\mathrm{\Im }(A)\mathrm{\parallel }-\mathrm{\parallel }\mathrm{\Re }(A)-\mathrm{\Im }(A)\mathrm{\parallel }\mid \mathrm{.}$ This improve the existing upper and lower bounds of the numerical radius, namely, $$\frac{1}{4}\mathrm{\parallel }{A}^{\ast }A+A{A}^{\ast }\mathrm{\parallel }\le w^2(A)\le \frac{1}{2}\mathrm{\parallel }{A}^{\ast }A+A{A}^{\ast }\mathrm{\parallel }\mathrm{.}$$