In chapter~12, (p.99) of this note it is written that
"In the case $p \geq 2$, the inequality $$|b|^p \geq |a|^p + p<|a|^{p−2}a, b − a> + C(p)|b − a|^p$$ holds with a constant $C(p) > 0$. The case $1 < p < 2$ requires a modification of the last term."
My question is what is exactly the inequality for the case $1 < p < 2$, what about the case $p=1$? Thank you in advance