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 bounded linear operators defined on complex Hilbert space. As applications these we deduce chain new classical numerical which improve existing ones. In particular, prove that A, $$\frac{1}{4} \Vert A^*A+AA^*\Vert +\frac{\mu }{2}\max \{\Vert \Re (A)\Vert ,\Vert \Im \} \le w^2(A) \, w^2( |\Re (A)| +i |\Im (A)|),$$ where $$\mu = \big | (A)+\Im -\Vert (A)-\Im |.$$ This radius, namely, $$\begin{aligned} \frac{1}{4} \frac{1}{2} . \end{aligned}$$