the only real answers that can be given is something similar to this, (canonwise)
Inuyasha wanted to be acknowledged, as a demon, thus went after the shikon so that he could gain power from it. Sesshomaru wanted Tetsusaiga, and it just so happens that Inuyasha has it, in the beginning, it's almost as if, Sesshomaru thought, that Inuyasha wouldnt have remembered him, this speaks to the fact that they must not have been around one another nearly at all.
B/c of the fact that the 3rd movie is not canon, and some of the anime was filler, you can only really go by the manga. All we know is that Sesshomaru has an abhorrence for weakness, not necessarily hanyou. But he does seem to view Inuyasha as weak and unworthy, and thus does not like him.
But we can logically assume that whom ever Inuyasha needed/wanted acknowledgment from wasn't necessarily Sesshomaru. While fanfiction, loves to paint it so that Sesshomaru's a racist against hanyou, more often then not, Sesshomaru does acknowledge Inuyhasa, as 'Little brother.' it's usually in a mocking way, but he does realize and state often that inuyasha is his little brother.
Inuyasha doesn't seem to care for Sesshomaru, or what he thinks, says or does, but it's never shown just WHY inuyasha hates Sesshomaru. Considering he wanted to become a full demon when he went after the Tama when Kikyo had it, i would assume it's because he envied his older brother for what he had and was.
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