Rev. Anal. Num´er. Th´eor. Approx., vol. 40 (2011) no. 1, pp. 47–51 ictp.acad.ro/jnaat
h-STRONGLY E-CONVEX FUNCTIONS
DANIELA MARIAN∗
Abstract. Starting from strongly E-convex functions introduced by E. A.
Youness, and T. Emam, from h-convex functions introduced by S. Varoˇsanec and from the more general concept ofh-convex functions introduced by A. H´azy we define and study h-strongly E-convex functions. We study some properties of them.
MSC 2000. 26B25.
Keywords. Strongly E-convex sets, strongly E-convex functions, h-convex functions,h-stronglyE-convex functions.
1. PRELIMINARY NOTIONS AND RESULTS
The concepts of E-convex sets andE-convex functions were introduced by Youness in [8]. Subsequently, Chen introduced a new concept of semi-E-convex functions in [2]. Based upon these approaches, in [9] Youness and Emam intro- duced the concepts of stronglyE-convex sets and stronglyE-convex functions.
We firstly recall the definitions of convex sets, convex functions,E-convex sets andE-convex functions then of stronglyE-convex sets and stronglyE-convex functions and finally the definitions of h-convex functions, in the sense of Varoˇsanec [7] and H´azy [4].
Definition 1. A set A ⊂ Rn is called convex if λx+ (1−λ)y ∈ A, for every pair of points x, y∈A and every λ∈[0,1].
Definition 2. A function f : Rn → R is called convex on a convex set A⊂Rn if for every pair of points x, y∈A and every λ∈[0,1], the following inequality is satisfied:
f(λx+ (1−λ)y)≤λf(x) + (1−λ)f(y).
We consider a function E:Rn→Rn.
Definition3. [8]A setA⊂Rn is calledE-convex ifλE(x)+(1−λ)E(y)∈ A, for every pair of points x, y∈A and every λ∈[0,1].
∗Department of Mathematics, Faculty of Automation and Computer Science, Technical University of Cluj-Napoca, Constantin Daicoviciu, no. 15, 400020 Cluj-Napoca, Romania, e-mail: [email protected].
Definition 4. [8]A function f : Rn → R is called E-convex on an E- convex set A⊂Rn if for every pair of pointsx, y∈Aand every λ∈[0,1], the following inequality is satisfied:
f(λE(x) + (1−λ)E(y))≤λf(E(x)) + (1−λ)f(E(y)).
Definition 5. [9]A setA⊂Rn is called strongly E-convex if λ(αx+E(x)) + (1−λ) (αy+E(y))∈A, for every pair of pointsx, y∈A, α∈[0,1] and λ∈[0,1].
Definition 6. [9]A function f :Rn→ R is called strongly E-convex on a strongly E-convex set A ⊂Rn if for every pair of points x, y ∈ A, α ∈ [0,1]
and λ∈[0,1], the following inequality is satisfied:
f(λ(αx+E(x)) + (1−λ) (αy+E(y)))≤λf(E(x)) + (1−λ)f(E(y)).
In the following lines we recall the definition of h-convex functions intro- duced in [7] by S. Varoˇsanec.
We consider I and J intervals in R, (0,1) ⊆ J and the real non-negative functions h:J →R, f :I →R, h6= 0.
Definition 7. [7]The function f :I →R is calledh-convex on I or is said to belong to the class SX(h, I) if for every pair of points x, y ∈I and every λ∈(0,1), the following inequality is satisfied:
f(λx+ (1−λ)y)≤h(λ)f(x) +h(1−λ)f(y).
In [1] Bombardelli and Varoˇsanec omitted the assumption that f and h are non-negative. We recall now the definitions of h-convex functions introduced in [4] by A. H´azy.
Let X be a real (complex) linear space andA⊂X nonempty, convex, open.
Leth: [0,1]→R, f :A→R.
Definition 8. [4]The function f : A → R is called h-convex on A if for every pair of points x, y ∈ A and every λ ∈ [0,1], the following inequality is satisfied:
f(λx+ (1−λ)y)≤h(λ)f(x) +h(1−λ)f(y).
2. PROPERTIES OFh-STRONGLYE-CONVEX FUNCTIONS
Starting from strongly E-convex functions and fromh-convex functions in the sense of H´azy we define and study h-strongly E-convex functions.
In the following lines we consider a map E : Rn → Rn and a strongly E- convex setA⊂Rn.We also consider the functionsh: [0,1]→R, f :A→R.
Definition 9. A function f :A → R is called h-strongly E-convex on A if for every pair of points x, y ∈ A, α ∈ [0,1] and λ ∈ [0,1], the following inequality is satisfied:
(1)
f(λ(αx+E(x))+(1−λ) (αy+E(y)))≤h(λ)f(E(x))+h(1−λ)f(E(y)).
Theorem10. Iff :A→Rish-strongly E-convex onAand h(0) = 0then (2) f(αx+E(x))≤h(1)f(E(x)).
Proof. We putλ= 1 in (1) and we obtain (2).
Theorem 11. If the functions fi : A → R, i = 1,2, . . . , k are h-strongly E-convex on A, then, for ai ≥ 0, i = 1,2, . . . , k the function F : A → R, F(x) =
k
P
i=1
aifi(x) ish-strongly E-convex on A.
Proof. Since the functions fi : A → R, i = 1,2, . . . , k are h-strongly E- convex on A, then, for each x, y ∈ A, every α ∈ [0,1] and λ ∈ [0,1], we have
F(λ(αx+E(x)) + (1−λ) (αy+E(y)))
=
k
X
i=1
aifi(λ(αx+E(x)) + (1−λ) (αy+E(y)))
≤h(λ)
k
X
i=1
aifi(E(x)) +h(1−λ)
k
X
i=1
aifi(E(y))
=h(λ)F(E(x)) +h(1−λ)F(E(y)).
Hence the function F is h-strongly E-convex on A.
We consider a strongly E-convex setA ⊂Rn,a function f :Rn→ R, and a function ϕ:R→R linear and nondecreasing.
Theorem12. If the functionf :Rn→Rish-stronglyE-convex on A then the composite function ϕ◦f is h-strongly E-convex on A.
Proof. Since f ish-strongly E-convex on A, for eachx, y∈A,α∈[0,1] and λ∈[0,1], we havef(λ(αx+E(x)) + (1−λ) (αy+E(y)))≤h(λ)f(E(x)) + h(1−λ)f(E(y)) and hence
(ϕ◦f) (λ(αx+E(x)) + (1−λ) (αy+E(y)))
≤ϕ[h(λ)f(E(x)) +h(1−λ)f(E(y)))]
=h(λ) (ϕ◦f) (E(x)) +h(1−λ) (ϕ◦f) (E(y)),
which implies that ϕ◦f is h-strongly E-convex on A.
We denote E(x) byEx for simplicity.
Theorem13. If the function f :Rn→Ris non-negative and differentiable h-strongly E-convex on a strongly E-convex set A and h is a non-negative function with the property h(λ)≤λ for everyλ∈[0,1]then
(3) (Ex−Ey)∇(f ◦E) (y)≤(f ◦E) (x)−(f◦E) (y) for everyx, y∈A.
Proof. Since f ish-stronglyE-convex on A,
f(λ(αx+Ex) + (1−λ) (αy+Ey))≤h(λ) (f ◦E) (x) +h(1−λ) (f◦E) (y) for eachx, y∈A, λ∈[0,1] andα∈[0,1]. Since h(x)≤x for everyx∈[0,1]
we have
f((αy+Ey) +λ[(αx+Ex)−(αy+Ey)])
≤λ(f ◦E) (x) + (1−λ) (f ◦E) (y)
= (f ◦E) (y) +λ[(f◦E) (x)−(f◦E) (y)]
and hence
f(αy+Ey) +λ[(αx+Ex)−(αy+Ey)])∇f(αy+Ey) +O λ2
≤(f◦E) (y) +λ[(f◦E) (x)−(f◦E) (y)]
By takingα→0,we get
f(Ey) +λ(Ex−Ey)∇f(Ey)) +O λ2
≤(f ◦E) (y) +λ[(f ◦E) (x)−(f ◦E) (y)]. Dividing byλ >0 and taking λ→0,we obtain
(Ex−Ey)∇(f ◦E) (y)≤(f ◦E) (x)−(f◦E) (y),
for each x, y∈A.
The following theorem provides a characterization of h-strongly E-convex functions with respect to the E-monotonicity of the gradient of map, similar with that obtain fromE-convex functions, by Soleimani-Damaneh in [3].
Definition 14. Let f :Rn →Rbe differentiable. The map ∇f :Rn →Rn is called E-monotone if
(∇f(E(x))− ∇f(E(y))) (E(x)−E(y))≥0, for everyx, y∈Rn.
Theorem15. If the function f :Rn→Ris non-negative and differentiable h-strongly E-convex on a strongly E-convex set A and h is a non-negative function with the property h(λ)≤λ for everyλ∈[0,1]then
(4) (∇f(E(x))− ∇f(E(y))) (E(x)−E(y))≥0 for everyx, y∈A.
Proof. Since f ish-stronglyE-convex on A, from theorem (13) we have (Ex−Ey)∇(f◦E) (y)≤(f◦E) (x)−(f◦E) (y)
and
(Ey−Ex)∇(f◦E) (x)≤(f ◦E) (y)−(f◦E) (x), for everyx, y∈A.Adding these two inequalities we obtain
(∇f(E(x))− ∇f(E(y))) (E(x)−E(y))≥0
for everyx, y∈A.
Theorem 16. Let the functions gi :Rn →R, i= 1,2, . . . , m be h-strongly E-convex on Rn. We consider the set
M ={x∈Rn|gi(x)≤0, i= 1,2, . . . m}.
If E(M) ⊆ M and the function h is positively then the set M is strongly E-convex.
Proof. Since the functions gi : Rn → R, i = 1,2, . . . , m are h-strongly E- convex onRn then, for everyx, y∈M,α∈[0,1] andλ∈[0,1] we have
gi(λ(αx+Ex) + (1−λ) (αy+Ey))
≤h(λ) (gi◦E) (x) +h(1−λ) (gi◦E) (y)≤0,
and henceλ(αx+Ex) + (1−λ) (αy+Ey)∈M.
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Received by the editors: January 12, 2011.
