The modulus of smoothness of a Banach space $X$ is
$\rho_X(\tau) = \sup \{ \frac{1}{2}(\left\lVert x+y \right\rVert + \left\lVert x-y \right\rVert) - 1 : \left\lVert x \right\rVert = 1, \left\lVert y \right\rVert = \tau \}$,
and the modulus of convexity is
$\delta_X(\varepsilon) = \inf \{ 1 - \frac{1}{2} \left\lVert x+y \right\rVert : \left\lVert x \right\rVert = \left\lVert y \right\rVert = 1, \left\lVert x-y \right\rVert > \varepsilon\}$.
$X$ is said to have a modulus of smoothness (convexity) of power type $p$ if there is a constant $C > 0$ such that $\rho_X(\tau) \leq C\tau^p$ ($\delta_X(\varepsilon) \geq C\varepsilon^p$).
Lindenstrauss and Tzafriri proved, in Classical Banach Spaces II (Springer-Verlag), that $L^p$, $1 < p < \infty$ has a modulus of convexity of power type max$\{2, p \}$, and then a duality result between the moduli of smoothness and convexity is used to obtain a power type result on the modulus of smoothness, which states that it is of type min$\{2, p\}$.
I am looking for any direct proofs of the power type result for the modulus of smoothness for $L^p$, without using a duality argument.