Increasing the pitch after the first milliseconds so that it sounds correct to the human brain

At your risk.

I have not heard anyone teaching this expect Sean Davies (the disc cutting lathe restauration expert) when he commented that I am singing pitch low the endings of words.

I think all good musicians seem to increase the pitch just after the attack of the note. 

An example is here at 1:30, the hand of the cello player's sliding slightly down while doing vibrato at the same time, meaning, the average frequency is subtly increased till it sounds correct to the listener's brain.

Or listening to Maria Callas: