Abstract
We consider the averages An(f) = 1/(n − 1)Pn−1
r=1 f(r/n) and Bn(f) = 1/(n+ 1)Pn
r=0 f(r/n). If f is convex, then An(f) increases with n and Bn(f) decreases.
For the class of functions called superquadratic, a lower bound is given for the successive differences in these sequences, in the form of a convex combination of functional values, in all cases at least f(1/3n). Generalizations are formulated in which r/n is replaced by ar/an and 1/n by 1/cn. Inequalities are derived involving the sum Pn r=1(2r − 1)p
r=1 f(r/n) and Bn(f) = 1/(n+ 1)Pn
r=0 f(r/n). If f is convex, then An(f) increases with n and Bn(f) decreases.
For the class of functions called superquadratic, a lower bound is given for the successive differences in these sequences, in the form of a convex combination of functional values, in all cases at least f(1/3n). Generalizations are formulated in which r/n is replaced by ar/an and 1/n by 1/cn. Inequalities are derived involving the sum Pn r=1(2r − 1)p
| Original language | English |
|---|---|
| Article number | 91 |
| Number of pages | 14 |
| Journal | Journal of Inequalities in Pure and Applied Mathematics |
| Volume | 5 |
| State | Published - 2004 |