On Cauchy–Schwarz type inequalities and applications to numerical radius inequalities

In this work, a refinement of the Cauchy–Schwarz inequality in inner product space is proved. A more general Kato’s or so called mixed Schwarz established. Refinements some famous numerical radius inequalities are also pointed out. As shown these refinements generalize and refine recent old results obtained literature. Among others, it proved that if $$T\in \mathscr {B}\left( {H}\right) $$ , then $$\begin{aligned} \omega ^{2}\left( T\right)&\le \frac{1}{12} \left\| \left| T \right| +\left| {T^* } \right\| ^2 + \frac{1}{3} \left( T\right) \\&\le \frac{1}{6} ^2+ \end{aligned}$$ which refines by Kittaneh Moradi [10].