Holder-type inequality or not (optimal transport in Riemannian manifolds)?

Question

I'm reading a paper about optimal transport on Riemannian manifolds, and I came into a part where the author talks about "distortion coefficients". Here there's an inequality that I don't know how to prove it (I tried but I get contradictions): \begin{align*} (a+b)^{\frac{N-1}{N}}(c+d)^{\frac{1}{N}}\geq a^{\frac{N-1}{N}}c^{\frac{1}{N}}+b^{\frac{N-1}{N}}d^{\frac{1}{N}}\qquad\mbox{with}\quad a,b,c,d>0. \end{align*} He states that he used the Holder inequality with $p=\frac{N}{N-1}$ and $q=N$.