## A mapping in connection to Hadamard’s inequalities.(English)Zbl 0747.26015

The following generalization of the well-known Hadamard inequalities for convex functions is given: Let $$f:[a,b]\to R$$ be a convex function on the interval $$[a,b]$$ from $$R$$. Let the function $$H:[0,1]\to R$$ be defined by $H(t):={1\over b-a}\int^ b_ af\left(tx+(1-t){a+b\over 2}\right)dx.$ Then (i) $$H$$ is convex on $$[0,1]$$, (ii) we have $$\inf_{t\in[0,1]}H(t)=H(0)=f\left({a+b\over 2}\right)$$, $$\sup_{t\in [0,1]}H(t)=H(1)={1\over b-a}\int^ b_ af(x)dx$$, (iii) $$H$$ is monotonously increasing on $$[0,1]$$.

### MSC:

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

### Keywords:

logarithmic mean; Hadamard inequalities; convex functions