Are derived hashes weakening the root? Given a root hash

root = H(plaintext)


and two (or more) derived hashes

h1 = H(salt1 + root)
h2 = H(salt2 + root)


would the knowledge of h1 and h2 alone increase the chances of finding the root hash or even the plaintext?
I'm using SHA256.
To the best of our knowledge, SHA256 does not leak any additional information from related hashes.

On the other hand, the state of "our knowledge" might not be that comprehensive; this security property of SHA256 cannot be derived from the base security assumptions of a hash function (preimage resistance, second preimage resistance and collision resistance), and as far as I know, no one has studied it.  While it may be (and likely is) secure, no one has really studied it, and so there there might be a semiobvious leakage that no one happened to find yet.

Now, there is an obvious variation on this which has been studied: HMAC.  HMAC is a Message Authentication Code, that is, it has a security assumption that if you give someone (who doesn't know the key) a large number of MAC outputs based on known texts, they still cannot derive the key, or any information about MAC outputs for texts they haven't seen.  That is precisely the security assumption that you are looking for; and so HMAC has been analyzed against the exact problem you're looking for.  HMAC uses a hash function (for example, SHA256) as an underlying primitive, and it does have a small additional cost over the underlying hash; if you can do some precomputation, this additional cost is just one more hash compression evaluation.

As for how you would use HMAC; well, HMAC has two inputs, a "key" and a "message".  You would make your "root" the key and the "salt" the message, and use the output of the MAC as your derived hash.