engleski

Positivity of sums and integrals for n-convex functions via the Fink identity

We consider the positivity of the sum $\sum_{; ; ; ; i=1}; ; ; ; ^n p_iF(x_i)$ where F is a convex function of higher order as well as analogous results involving the integral $\int_a^bp(x)F(g(x))dx$. We use a representation of the function $F$ 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.

n-convex function ; Fink identity ; Green function ; Čebyšev functional

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