bound of two random variable’s expectation

Question

$$E[X]E[Y] \leq E[XY] + \sqrt{V[X]V[Y]}$$ with $E$ is expectation and $V$ is variance.

Is this true?

Answer 1

Cauchy Schwarz gives $E[ (X-EX) (EY-Y) ] \le \sqrt{ \operatorname{var} X \operatorname{var} Y }$.

Expanding $EX E Y - E[XY] \le \sqrt{ \operatorname{var} X \operatorname{var} Y }$.