With EC secp256k1 is there a way of transforming a function of the private key to a function of the public key? A key pair has a private key $D_A$ and a public key $Q_A$.

$D_A$ is an integer less than the curve's $n$.

Is there any (boolean) function of the private key $f(D_A)$ which can be transformed into a function of the public key $f'(Q_A)$?

i.e. are there any relationships between private keys which can be calculated knowing only the public keys?
We aren't aware of any sich relations. In fact, one assumes that the group of elliptic curve points behave like a Generic Group.
A generic group is a group, where the encoding of the elements are chosen as a random values.