this is a (3/4" cutting) LSU Gold that I've rooted over a month ago and it has not grow sins then but looks healthy, what is the problem with this cutting ?
It's use to full sun behind my window

LSU Gold likes room to grow initially, pot it up and add 1 tblspoon of Epsom salts.  After the initial growth it will settle in and flourish

Magnesium Sulfate (Epsom Salt) it will promote leaf growth.

Thanks for the information about LSU Gold and needing room - I have had a cutting that has been doing the exact same thing for about 4 months without any signs of good root development. I am going to pot it up and see if that helps it develop more.

I would fertilize it.

if anyone has any spare LSU Gold cuttings or plants will be appreciated

chances are it wasn't fertilizer that killed it, but too much water that came with fertilizer, if you used liquid fertilizer. usually best bet when everything looks good is to leave it alone. no water, no fertilizer, nothing. leave it be and occationally check to see if roots are still good. when there is enough root, up pot to 1 gal even if the top looks the just the same.

Roots are always the main indicator as to when to pot up to 1 gallon in my opinion.

if you have good strong roots, they should be ready for the 1 gal. top growth is nice to see, but the main thing we are after is the root. once the weather start to warm up, the top will come.

can anyone post any pictures of what's good roots for the next stage please?

For me, it should look something like this all around the cup.  This one rooted extremely well, but I've transferred with not so thick roots too.  You'll find that not all cuttings have nice thick roots.  Some have very thin roots but still circle the cup, which is good enough for me too.

up until winter rooting what i look for was about 4-5 roots coming through peat pot. but something like frank pix above is what you want.