I can empathise with him a bit. I really hated my younger sibling for a while.
Things that really didn't help:
Dm telling me how much he "adored me" (I still don't think he did!)
Dm pushing him to do what I did and then telling me how brilliant he was at it.
Dm giving him my things to "borrow" (they never came back undamaged)
Dm telling me I had to make allowances because he was younger.
Dm dropping everything to do something for/with him (because he was younger so I was expected to understand that he couldn't entertain himself)
Dm telling I couldn't join in because they were doing something together, but always expecting me to allow him to join.
Any games we played allowances were made to extremes for him. I was expected to have the same as the adults. For example trivial pursuits (which I am very bad on) dm would read through the questions until she found one she thought he might know-then he got several guesses and if he still didn't guess got a second question.
Just read through those. I suspect you do at least the first one.
It really does not help.
It comes across as "oh your younger sibling is just perfect so everyone loves them, whereas you should be grateful for any grains of acceptance."
I felt I couldn't "win" on anything we both did-and he always had to do what I did. If I did better, then it was expected as I was older. If he did half as well as me it was taken as a sign of genius.
Anything I got, he had to have because "otherwise he felt left out", even if I'd been told I couldn't have something until a specific age.
Also do remember when he was 6yo you had a 2yo. I bet he had to fend far more on his own at 6yo than she does? I know my eldest would have chosen her clothes dressed herself and made her own breakfast by 3yo. #3 when they started school still struggled with dressing and definitely wouldn't have made their own breakfast-why should they? If I wasn't available, they had 2 willing big sisters to help 
So when he's saying "you do far more for her", he's probably right. Out of necessity, it's probably right!
What would have helped: Time apart. Doing our own things and being accepted that we each had our own talents.
Time spent with you when she's awake too. You spend time after she's in bed. Great... but maybe in his mind you give him the crumbs of your time when she's asleep. You give her the time and ignore him. At 6yo she's old enough to do something on her own while you play with him.
Give him advantages of being older. Maybe he can have a phone, or more pocketmoney, or trust him to do things. Maybe ask him to help choose something - when it doesn't matter, let him make a decision about the new printer you're getting.
Take interest in what he's interested in. Let him explain while you listen to him.
You want them to both feel you value them for who they are. Currently he feels you don't value him for who he is.
The good news is that I do get on with my brother now. It really took until i left home and there was space between us (sorry). But I think that things dm did which she thought would push us together, the sharing of mutual interests etc actually drove us further apart.