The inequality

\(\displaystyle{f{{\left({\frac{{{a}+{b}}}{{{2}}}}\right)}}}\leq{\frac{{{1}}}{{{b}-{a}}}}{\int_{{a}}^{{b}}}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}\)

is a special case of Jensen's inequality.

And since f is convex, we have

\(\displaystyle{f{{\left({x}\right)}}}\leq{f{{\left({a}\right)}}}+{\frac{{{f{{\left({b}\right)}}}-{f{{\left({a}\right)}}}}}{{{b}-{a}}}}\cdot{\frac{{{\left({b}-{a}\right)}^{{2}}}}{{{2}}}}\)

\(\displaystyle={\left({b}-{a}\right)}{\frac{{{f{{\left({a}\right)}}}+{f{{\left({b}\right)}}}}}{{{2}}}}\)

which is equivalent to the second inequality.

\(\displaystyle{f{{\left({\frac{{{a}+{b}}}{{{2}}}}\right)}}}\leq{\frac{{{1}}}{{{b}-{a}}}}{\int_{{a}}^{{b}}}{f{{\left({x}\right)}}}{\left.{d}{x}\right.}\)

is a special case of Jensen's inequality.

And since f is convex, we have

\(\displaystyle{f{{\left({x}\right)}}}\leq{f{{\left({a}\right)}}}+{\frac{{{f{{\left({b}\right)}}}-{f{{\left({a}\right)}}}}}{{{b}-{a}}}}\cdot{\frac{{{\left({b}-{a}\right)}^{{2}}}}{{{2}}}}\)

\(\displaystyle={\left({b}-{a}\right)}{\frac{{{f{{\left({a}\right)}}}+{f{{\left({b}\right)}}}}}{{{2}}}}\)

which is equivalent to the second inequality.