Inverse Hölder inequalities in one and several dimensions

We study certain functionals and obtain an inverse Hölder inequality for n functions f₁^a1,…,f_n^an (f_k concave, 1 dimension).We also prove a multidimensional inverse Hölder inequality for n functions f₁,…,f_n, where $\text{?}{}^2\text{f}{}_{\mathrm{k}}\text{?x}{}_{\mathrm{i}}{}^2\text{? 0, i = 1,…, d, k = 1,…, n}$.Finally we give an inverse Minkowski inequality for concave functions.