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Already during the existence of first Yugoslavia, there occurred the secession of Boka Kotorska from Dalmatia, i. e. Croatia. During the entire World War II, the expression "Montenegro and Boka Kotorska" was in use, but towards the end of WW II "Boka Kotorska" was simply dropped from the syntagma. This was accompanied by all sorts of pressures on Croatian populace there. There are numerous cases of imprisonments and murders of Croats from Boka at the end of war and during the immediate years after. The fate of Oslav Smodlaka, a photographer from Tivat, best illustrates the difficulties that Croats experienced there. Smodlaka was recognized by historiography as a Croat assassinated at the end of war. Actually, he was killed in 1949. Simply because he refused to relocate, he was declared state enemy, tortured to the point of turning gray in one single night, and finally killed. He left a spouse and an eleven-month daughter. The fate of O. Smodlaka served as a disciplining measure for the rest of Croats in Boka, who lived under a permanent fear and oppression. (SOI : CSP: S. 385)

We consider positivity of sum $\sum_{i=1}^np_if(x_i)$ involving convex functions of higher order. Analogous for integral $\int_a^bp(x)f(g(x))dx$ is also given. Representation of a function $f$ via the Fink identity and the Green function leads us to identities for which we obtain conditions for positivity of the mentioned sum and integral. We obtain bounds for integral remainders which occur in those identities as well as corresponding mean value theorems.

In this article, we derive the Popoviciu-type inequalities by using the weighted version of the extension of Montgomery's identity and Green functions. Some results for n-convex functions at a point are also obtained. Besides that, some Ostrowski-type inequalities are also presented, which are interrelated with the obtained inequalities.
Mathematics Subject Classification (2010): 26A51, 26D15, 26D20.
Received 26 September 2021; Accepted 13 May 2022