## A note on Hadamard’s inequality.(English)Zbl 0880.26019

Let $$f$$ be a real-valued convex function on $$[a,b]$$ and let $F(t)= {1\over 2(b-a)} \int^b_a \Biggl(f\Biggl({1+ t\over 2} a+ {1-t\over 2} x\Biggr)+ f\Biggl({1+ t\over 2} b+{1- t\over 2} x\Biggr)\Biggr)dx,\quad t\in [0,1].$ The authors prove that $$F$$ is convex and increasing on $$[0,1]$$, and ${1\over b-a} \int^b_a f(x)dx\leq F(t)\leq {1\over 2} (f(a)+ f(b)).$

### MSC:

 26D15 Inequalities for sums, series and integrals 26A51 Convexity of real functions in one variable, generalizations