I am reading a paper where $F(u) = \frac{1}{p+1}|u|^{p+1}$ and $f(u) = |u|^{p-1} u$ and $R= Q_1+Q_2$ where the functions $u,Q_1,Q_2:\mathbb{R}^{n}\to \mathbb{C}.$
The authors claim that by Taylor expansion for $p\geq 2$ we have that, $$|F(Q_1+Q_2) -F(Q_1)-F(Q_2)-f(Q_1)Q_2-f(Q_2)Q_1|\leq K|Q_1Q_2|^{3/2}$$ for some constant $K>0.$ I am not sure how to justify this formally. I tried writing $Q_1 = a+ib$ and $Q_2 = x+iy$ for real valued functions $a,b,x$ and $y.$ Then I write each expression as $$F(Q_1+Q_2) = \frac{1}{p+1}[(a+x)^2+(b+y)^2]^{(p+1)/2}$$ $$f(Q_1)Q_2 = |Q_1|^{p-1}Q_1 Q_2= (a^2+b^2)^{(p-1)/2}[ax-by+i(ay+bx)]$$ and so on, but this is quite tedious and I am guessing there is a quick way to see this since there was no justification provided in the paper. Any hints/comments would be much appreciated.