Here I have this shape with 2 concave angles and I want to sharpen them without affecting the surrounding loops. I dont want to use crease or edge split because it create a 100% sharp edge which I dont want.
It seems you are using modifiers: Solidify and Subsurf.
The problem here, is that you don't have any real edge to apply crease.
First you should apply the Solidify Modifier, then you could get the edge in the red circle you drawn.
The crease amount will influence the shape, from completly smooth to completly sharp.
Alternatively, you could select the edge and apply a Bevel.
Select the edge, W, Bevel
In edit mode use edge crease.
In the image above an additional extrusion was created. Then certain vertices had the edge crease applied. By moving the mouse you can the mean crease was set to 1.00 as indicated by the gold arrow.
Also consider using Mark Sharp
In the images above I am using a Subdivision Surface modifier to be clear. I feel comfortable in assuming you do as well. So the smoothing is a balance between Subdivision Surface Modifier, Edge Crease, Mark Sharp, Extra Proximity Loops, and Edge Split.