- Turkish Journal of Mathematics
- Vol: 43 Issue: 2
- Positivity of sums and integrals for $n$-convex functions via the Fink identity
Positivity of sums and integrals for $n$-convex functions via the Fink identity
Authors : Asif R Khan, Josip Pecaric, Marjan Praljak, Sanja Varosanec
Pages : 579-594
View : 8 | Download : 5
Publication Date : 9999-12-31
Article Type : Makaleler
Abstract :We consider the positivity of the sum $\sum_{i=1}^n\rho_i\digamma(\xi_i)$, where $\digamma$ is a convex function of higher order, as well as analogous results involving the integral $\int_{a_0}^{b_0}\rho(\xi)\digamma(g(\xi))d\xi$. We use a representation of the function $\digamma$ via the Fink identity and the Green function that leads us to identities from which we obtain conditions for positivity of the above-mentioned sum and integral. We also obtain bounds for the integral remainders which occur in these identities, as well as corresponding mean value results.Keywords : $n$-convex functions, Fink identity, Green function, v{C}ebyv{s}ev functional