 Solution

For $g\in K$, convexity gives $h+t(g-h)\in K$ for $0\le t\le 1$. Thus $F(t)=\Vert h-f+t(g-h){\Vert }_p^p$ has a right minimum at $t=0$ and $F^{\prime }(0)\ge 0$. The derivative of a power of the Lp norm isConsequentlyThe printed hint has coefficient $2$ where the general coefficient is $p$. This does not affect the desired sign, since $p>0$. The complex conjugation is important for complex-valued functions; for real functions it can be omitted.