On page 3 of the paper "Ismail Kombe and Murad Ozaydin, Improved Hardy and Rellich Inequalities On Riemannian Manifolds, Transactions of the American Mathematical Society, Volume 361, Number 12, December 2009, Pages 6191–6203." There are the inequalities: $$ |a+b|^p-|a|^p\geq c(p)|b|^p+p|a|^{p-2}a\cdot b, \qquad (1)$$ where $a,b \in \mathbb{R^{n}}$, $c(p)>0$ a constant, and $p>2$. If $p<2$, then we have $$ |a+b|^p-|a|^p\geq c(p)\dfrac{|b|^2}{\left(|a|+|b|\right)^{2-p}}+p|a|^{p-2}a \cdot b \qquad (2)$$.
I am looking for an upper bound for $ |a+b|^p-|a|^p$, $p>1$ better than the trivial bound $(2^{p-1}-1)|a|^p+2^{p-1}|b|^p$ that follows from the triangle inequality.
A typical argument is exploiting Gateaux-differentiability of the Euclidean norm and the fundamental theorem of calculus to write:
$$ |a+b|^p-|a|^p=p\int_{0}^{1}|a+tb|^{p-2}(a+bt)\cdot b \;dt $$.
Then I need to bound the integrand on the RHS from above. I don't know how to continue. In fact any reference for the proof of (1) or (2) is appreciated. Thanks a lot.