As an alternative, we also design an iterative algorithm to solve the nonconvex problem by using the convex features of the original cost function w.r.t.
These ideas led to the study of the class of all functions such that is starlike for some fixed function in In this direction, Shanmugam [2] introduced and investigated various subclasses of analytic functions by using the convex hull method [3 5] and the method of differential subordination.
By using Jensen's inequality with the convex function Ψ u) = u p, p ≥ 1, p < 0, and reversing the order of integration, we find that ∫ 0 ℓ 1 x ∫ 0 x g ( y ) d y p d x x ≤ ∫ 0 ℓ 1 x ∫ 0 x g p ( y ) d y d x x = ∫ 0 ℓ g p ( y ) ∫ y ℓ 1 x 2 d x d y = ∫ 0 ℓ g p ( y ) 1 y - 1 ℓ d y = ∫ 0 ℓ g p ( y ) 1 - y ℓ d y y.
Remark 1 Using the Schur-convex function decision theorem, Liu et al. [9] have proved Theorem 3.
Using the discrete Jensen inequality for the convex function, we have the following conclusion: (33).
Using the Hermite-Hadamard inequality for the convex function, we can extend inequality (3.1) on the left and on the right hand side as follows: (34).
From the subdifferentiability of the convex function to scheme (3.1), using the first-order necessary optimality condition, we have (3.7).
For instance, the convex function can be (varphi(x) := |x|^{p}) with (p geq1), namely the convex 'φ-function' used of the general theory of Orlicz spaces, used to generate the (L^{p} -spaces; see, e.g., [4, 5].
We establish an explicit and general decay rate result using some properties of the convex functions.
We prove an explicit and general decay rate result using some properties of the convex functions.
