Assuming that by "mix" you mean "multiply", then there will be many different components of the mixed sine waves - eight to be precise. If you only want the lowest of these components, they'll be well below the audio range of any of the input signals (since you said they're close in frequency) you should be able to get away with computing the parity (use three dual-input XOR gates) and LP filter that.
Parity is multiplication in the digital realm.
Clifford Heath