Norm inequalities

Question

I know very famous Holder's and Minkowski's inequalities. Holder's inequality says that $$\|fg\|_1 \leq \|f\|_p\|g\|_q$$ where $\frac{1}{p}+\frac{1}{q}=1$. Whereas Minkowski's inequality states that $$\|f+g\|_p \leq \|f\|_p + \|g\|_p.$$ My question is, are there any known inequalities for product of functions which are of the form: $$\|fg\|_p \leq C \|f^a\|^b_p \|g^c\|^d_p.$$ Where C,a,b,c,d are any non negative constants.If there exist any, refrences or names will be helpful. If there are not any such inequalities, is there are particular reason for that?

Answer 1

From Hölder inequality, we have for any $q,r$ with $1/q + 1/r = 1$ :
$$\|fg\|_p =\|fg^{p}\|_1^{1/p} \leq \left(\|f^p\|_q \|g^p\|_r\right)^{1/p} = \|f\|_{pq}\|g\|_{pr} = \|f^q\|_p^{1/q}\|f^r\|_p^{1/r}$$

Answer 2

Inequalities of this type can be easily deduced from Hölder's inequality: $$ \|fg\|_p=\||f|^p|g|^p\|_1^{1/p}\leq\||f|^p\|_r^{1/p}\||g|^p\|_s^{1/p}=\||f|^r\|_p^{1/r}\||g|^s\|_{p}^{1/s} $$ whenever $\frac 1 r+\frac 1 s=1$