Understanding a family of Sobolev-type inequalities

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 .

Now the claim is: $$\forall 0 < s < r < \infty$$, $$\forall 0, such that $$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.

Since $$s' < r' \leq r$$, there exists $$\alpha \in (0,1]$$ such that $$$$\|f\|_{r'} \lesssim \|f\|_{s'}^{1-\alpha} \, \|f\|_r^\alpha.\label{1}\tag{1}$$$$ Hence (by the $$S^\theta_{r,s}$$ assumption) $$$$\|f\|_{r'} \lesssim \|f\|_{s'}^{1-\alpha} \, \|f\|_s^{\alpha(1-\theta)}\,\|\nabla f\|_p^{\alpha\theta}. \label{2}\tag{2}$$$$
Now consider the cases (a) $$s' < s \leq r'$$ and (b) $$r' \leq s < r$$. For (a), you obtain a bound on $$\|f\|_s$$ in terms of $$\|f\|_{s'}$$ and $$\|f\|_{r'}$$ just like \eqref{1} which you can insert in \eqref{2} to get $$S^{\theta'}_{r',s'}$$ for some appropriate $$\theta'$$ to be calculated. For (b), the same ansatz gives a bound on $$\|f\|_s$$ in terms of $$\|f\|_{r'}$$ and $$\|f\|_r$$. Using assumption $$S^\theta_{r,s}$$, you get $$S^\tau_{s,r'}$$ for some $$\tau \in [0,1)$$ which can then be inserted into \eqref{1} to yield $$S^{\theta'}_{r',s'}$$.