I cant remember having seen this relation before is it valid, and is the below derivation valid?
$Var(\lambda X+(1-\lambda)Y)= \lambda^2 Var(X)+(1-\lambda)^2Var(Y)+2\lambda(1-\lambda)Cor(X,Y)\sqrt{Var(X)Var(Y)}\leq\lambda^2 Var(X)+(1-\lambda)^2Var(Y)+2\lambda(1-\lambda)\sqrt{Var(X)Var(Y)}=(\lambda\sqrt{ Var(X)}+(1-\lambda)\sqrt{Var(Y)})^2\leq \lambda Var(X)+(1-\lambda) Var(Y)$
Using in the first inequality that $Cor(X,Y)\leq 1$ and in the second inequality that $(\cdot)^2$ is convex.