Let me introduce my recent result with V.N. Huy published in Applied Mathematics Letters last year.
I want to start by recalling the following results due to Nenad Ujevic:
Let be an open interval such that and let be a twice differentiable function such that is bounded and integrable. Then we have
Let be an open interval such that and let be a twice differentiable function such that . Then we have
In the above mentioned results, constants in the first and in the second result are sharp in sense that these cannot be replaced by smaller ones. This leads us to strengthen them by enlarging the number of knots (2 knots in both results) and replacing the norms in the first and in the second.
Before stating our main result, let us introduce the following notation
Let and . For each , we assume such that
We are in a position to state our main result.
Let be an open interval such that and let be a -th differentiable function such that . Then we have
As can be seen the above result is not sharp. It will be very interesting if we can derive a sharp estimate. Note that, the results due to Nenad Ujevic can be rewritten as the following