This thesis is closely related to a result of K. L. Tseng, S. R. Hwang, and S. S. Dragomir which is published recently [Comput. Math. Appl. 62 (2011), 401–418]. In that article, by considering four points x, y, y0 , x0 in an interval [a, b] such that x < y ≤ y 0 < x0 and x + x 0 = y + y 0 , some new Hermite-Hadamard-type inequalities were proved. We give some extensions of these new inequalities from a weighted viewpoint. Throughout this thesis, we suppose that f is a convex function on [a, b] and a ≤ x < y ≤ y 0 < x0 ≤ b. In addition, we impose the assumption λx + λ 0x 0 = λy + λ 0 y 0 for some λ, λ0 > 0. When λ = λ 0 = 1, these assumptions reduce to those imposed in paper of Tseng, Hwang and Dragomir