On Generalizations of Hölder's and Minkowski's Inequalities

We present the generalizations of Hölder's inequality and Minkowski's inequality along with the generalizations of Aczel's, Popoviciu's, Lyapunov's and Bellman's inequalities. Some applications for the metric spaces, normed spaces, Banach spaces, sequence spaces and integral inequalities are further specified. It is shown that $({mathbb{R}}^n,d)$ and $left(l_p,d_{m,p}right)$ are complete metric spaces and $({mathbb{R}}^n,{left|xright|}_m)$ and $left(l_p,{left|xright|}_{m,p}right)$ are $frac{1}{m}-$Banach spaces. Also, it is deduced that $left(b^{r,s}_{p,1},{left|xright|}_{r,s,m}right)$ is a $frac{1}{m}-$normed space.