I am reading *Aspects of Sobolev-Type Inequalities* by professor Laurent Saloff-Coste, where I found a claim on page 66 claiming the following:

Denote the following inequality as $S_{r,s}^{\theta}$: $\forall f \in C^{\infty}_{0}(M)$, $||f||_r \leq (C||\nabla f||_p)^{\theta}||f||_s^{1-\theta}$, where M is a complete non-compact Riemannian manifold and the parameters satisfy $0<\theta\leq 1$ and $0 < s <r\leq\infty$.

Now the claim is: $\forall 0 < s < r < \infty$, $\forall 0<s'<r'<\infty$, such that $r' \leq r$ and $s'\leq s$, and $\frac{1}{r} = \frac{1-\theta}{s}$ and $\frac{1}{r'} = \frac{1-\theta'}{s'}$, we have that $S_{r,s}^{\theta}$ implies $S_{r',s'}^{\theta'}$ by Holder inequality.

My question is how can we prove this claim. This should be elementary but from simple applications of Holder inequality I can only see the claim when $r'\leq r$ and $s' = s$. How can I go to the case where $s'\leq s$. The direction of inequality in $S_{r,s}^{\theta}$ seems to go against Holder's inequality.

Thank you very much for the help.