Calculus of functions of a single variable (extrema where the derivative is 0 and/or at the boundary of domains) is applied in order to prove improvements of the Hölder, Lyapunov, Minkowski and power means inequality. We reproduce here the first: if \(0<a_ 1\leq...\leq a_ n,\) \(0<b_ 1\leq...\leq b_ n,\) \(p>0,\) \(q>0,\) \((1/p)+(1/q)=1,\) then \[ 0\leq (a^ p_ 1+a^ p_ n)^{1/p}(b^ q_ 1+b^ q_ n)^{1/q}-a_ 1b_ 1-a_ nb_ n\leq (\sum^{n}_{k=1}a^ p_ k)^{1/p}(\sum^{n}_{k=1}b^ q_ k)^{1/q}-\sum^{n}_{k=1}a_ kb_ k\leq \]
\[ \leq \min [na^ p_ np^{-1}(b^ q_ n-b^ q_ 1)^{p-1}q^{1-p}a_ 1^{1-p}(b_ n-b_ 1)^{1-p}-na_ 1b_ 1b_ n(b_ n^{q-1}-b_ 1^{q-1})(b^ q_ n-b^ q_ 1)^{-1}, \]
\[ nb^ q_ nq^{-1}(a^ p_ n-a^ p_ 1)^{q-1}p^{1-q}b_ 1^{1-q}(a_ n-a_ 1)^{1-q}-na_ 1b_ 1a_ n(a_ n^{p-1}-a_ 1^{p-1})(a^ p_ n-a^ p_ 1)^{-1}]. \] The others are lengthier.