A note on knot Floer homology of satellite knots with (1,1)-patterns

We prove that if $P$ is a $(1,1)$-pattern knot, the two inequalities $\dim \widehat{HFK}(P(K))⩾\dim \widehat{HFK}(P(U))$ and $\dim \widehat{HFK}(P(K))⩾\dim \widehat{HFK}(K)$ hold for the unknot $U\subset S^3$ and any companion knot $K\subset S^3$.