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In this paper, the authors define a new identity for differentiable functions. By using of this identity, authors obtain new estimates on generalization of Hadamard and Simpson type inequalities for quasi-geometrically convex functions.

In this work, by using an integral identity together with both the Hölder integral inequality and the power-mean integral inequality we establish several new inequalities for three-times differentiable arithmetic-harmonically-convex function. Then, using this inequalities, we obtain some new inequalities connected with means.

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In this paper, we obtain refinements of the left-sided Hermite-Hadamard inequality for functions whose first derivatives in absolute value are trigonometrically -function.