Some Newton's like inequalities with applications (Q2006211)

\textit{M. A. Noor} et al. [J. Adv. Math. Stud. 9, No. 1, 7--16 (2016; Zbl 1345.26037)] established some equation to derive Newton-type inequalities for various functions. An arithmetrically-harmonically convex function was defined and Hermite-Hadamard type inequalities for arithmetrically-harmonically convex functions defined on convex subsets in linear spaces were given by \textit{S. S. Dragomir} [Stud. Univ. Babeş-Bolyai, Math. 61, No. 4, 489--502 (2016; Zbl 1399.26048)]. \textit{M. Kadakal} and \textit{I. Iscan} [``Inequalities of Hermite-Hadamard and Bullen type for arithmetic-harmonically convex functions'', Univers. J. Math. Appl. 2, No. 3, 152--158 (2019)] obtained some new generalized inequalities of the Hermite-Hadamard and Bullen type for functions whose certain power of second derivatives in modulus are arithmetrically-harmonically convex. The authors obtain an auxiliary identity by the generalization method given by \textit{I. Iscan} et al., [``Some new inequalities on generalization of Hermite-Hadamard and Bullen type inequalities, application to trapezoidal and midpoint formula'', Kragujevac J. Math. 45, No. 4, 647--665 (2021)] to obtain some new integral inequalities of Newton type for functions whose first derivative in absolute value at certain power are arithmetrically-harmonically convex. Some special cases and applications for special means are given. The results are useful for researchers on integral inequalities of Newton type.