If i am given with a hilbert space $H$, Let $$P+Q = I$$ be the projection operators such that their sum is identity operators on $H$. Now i am given with the estimates for $x, y \in H$ $$\frac{1}{2}|\langle x,y\rangle| \leq |\langle Px, y\rangle|$$ and $$\frac{1}{2}|\langle x,y\rangle| \leq |\langle Qx, y\rangle|$$
Now by using above how can i write the following? $$\langle x, y\rangle \leq 2 |\langle Px, y\rangle|^{1-\nu} |\langle Qx, y\rangle|^{\nu}$$ for $\nu \in (0,1)$?
Is it really possible to write above expression or not? Please help
We have that
$$|\langle Px, y\rangle|\ge \frac{1}{2}|\langle x,y\rangle| \implies |\langle Px, y\rangle|^{1-\nu}\ge \frac{1}{2^{1-\nu}}|\langle x,y\rangle|^{1-\nu}$$ and
$$|\langle Qx, y\rangle|\ge \frac{1}{2}|\langle x,y\rangle| \implies |\langle Qx, y\rangle|^{\nu}\ge \frac{1}{2^{\nu}}|\langle x,y\rangle|^{\nu}.$$ Thus
$$2|\langle Px, y\rangle|^{1-\nu}|\langle Qx, y\rangle|^{\nu}\ge2\frac{1}{2^{1-\nu}}|\langle x,y\rangle|^{1-\nu}\frac{1}{2^{\nu}}|\langle x,y\rangle|^{\nu}=|\langle x,y\rangle|\ge \langle x,y\rangle.$$