What perennials can compete with a tree's root system?

There is a section of my yard, under the spread of a large Swamp Maple, where the tree's dense, matte-like roots extend toward the surface, thus consuming most of the moisture and air. The tree's canopy is high so the area receives full sun. I have a sprinkler system there so lack of water is not the main problem.

If you look closely at the picture, you can see that the grass dries out the closer you get to the tree. Along the horizontal section of the fence, I've planted Double Knock-Out roses. The three nearest the tree are barren, followed on the right by a partial bloomer, and to the right of it a full bloomer.

Along the vertical section of the fence, I've planted Paeonia lactiflora 'Karl Rosenfield', which are in their second year. While none have bloomed yet, the two nearest the tree are stunted.

I’m thinking about replacing the three roses and three peonies closest to the tree with something that will thrive under those conditions. What perennials can compete with the tree’s root system? I would prefer a colorful rebloomer. Could Coneflower compete?

I have a Rose of Sharon that does fine near a big maple and a few others at the edge of my 'woods'. I planted 2 more today along a wooded edge. The soil is rocky, I only water the one under the maple which is next to an old lilac.
I started pruning them last spring to maintain shape and height. They flower even better now, over a month so far.
Andy P

Asters do fine for me 5' from the base of a 20' Juniper, hostas @ 4' distance, my neighbor has hostas right at the base of their 40' maple, Golden Rod might work if it gets ~6hrs sun. Generally drought resistant perennials.

I was going to say the same thing, drought resistant perennials.

None (perennials) that you will won't have problems with in the near future. Maples are surface root growers so even if you changed the grade (i.e. a raised bed) the roots will make their way back to the top. At least with a raised bed you can defy the Maple roots for a couple years.

Find a great flowering ground cover like Plumbago to cover most of the area and get a few hardy drought resistant perennials (natives are good), as everyone has recommended, for some interest.

If you do not create a raised bed or planter box you will be on the losing end of the battle the moment you start digging the hole.

A trick I learned for planting under trees is to make your planting holes twice as big and line them with a 1 inch layer of newspaper. Soak the newspaper really well before adding soil and planting. The newspaper acts like a barrier to the tree roots and by the time it degrades, your plants have large enough roots to compete for water.