If you are on a compact domain  $\Omega$in $\mathbb{R}^n$ you need to impose some elliptic boundary conditions  or  it is difficult to make predictions.   Assume for simlicity that $f$ satisfies the Dirichlet boundary conditions, i.e., it vanishes on the boundary of $\Omega$.    Then

$$\lambda_3\Vert f\Vert_{L^\infty}=\sup_{x\in\Omega} |D_3f(x)| \leq  C\lambda^{\frac{3}{2}} \Vert f\Vert_{L^2}\leq  C'\lambda^{\frac{3}{2}} \Vert f\Vert_{L^\infty}$$ 

This gives you

$$\lambda_3\leq C'  \lambda^{\frac{3}{2}}. $$


**Edit.** In my earlier post I misquoted the result I used. I have fixed the errors.    For details see my source X. Bin: *Derivatives of the spectral function  and Sobolev norms of eigenfunctions on a closed Riemann manifold*, Annal. Glob. Annalysis, vol. 26(2004), 231-252, Theorem 1.2.