Zygfryd Kominek
(C-K)-strongly n-convex functions in linear spaces
In tribute to Julian Musielak on his 85th birthday
Abstract. The notion of strongly n-convex functions with a control symmetric n- linear function ϕ in the class of functions acting from one real linear space to another one are introduced. Some connections between such functions and n-convex functions are also given.
2010 Mathematics Subject Classification: Proszę wpisać SUBCLASS.
Key words and phrases: Proszę wpisać słowa kluczowe.
1. Introduction. Let I ⊂ R be an interval, f : I → R be a function and let c be a positive constant. Following R. Ger and K. Nikodem [3] we recall the notions of n-convex functions and strongly n-convex functions with modulus c. For arbitrary points x0, x1, ..., xn∈ I, x0< x1<· · · < xn we define divided differences in the following way:
[x0; f ] := f (x0),
[x0, x1, ..., xm; f ] := [x1, ..., xm; f ]− [x0, ..., xm−1; f ]
xm− x0 , m = 1, ..., n.
Following E. Hopf [5] and T. Popoviciu [9], a function f : I → R is called convex of order of (n − 1) (or shortly (n − 1)-convex) if and only if
(1) [x0, x1, ..., xn; f ]≥ 0
for all x0, x1, ..., xn ∈ I, x0 < x1 <· · · < xn. The notion of n-convex functions was generalized by many authors. Some properties of n-convex functions may be found in the papers [1], [2], [4], [6], [7], [8], [11], [13] and [14]. The notion of strongly convex functions have been introduced by B.T. Polyak [10] and the notion of strongly (n − 1)-convex functions with modulus c > 0 have been introduced by
R. Ger and K. Nikodem [3]. They called f : I → R strongly (n − 1)-convex function if and only if it satisfies the following inequalities
(2) [x0, x1, ..., xn; f ]≥ c
for all x0, x1, ..., xn ∈ I, x0 < x1 <· · · < xn. Some connections with strongly n-convexity may be found in an interesting article [12]. The aim of our paper is to generalizes the definition giving by Ger and Nikodem to the class of functions defined on a subset of a real linear space and with the values in an another one.
2. Results. Let X, Y be real linear spaces and let C (K) be a proper convex cone in X (Y ), 0 6∈ C (0 6∈ K). The relation ≤ giving by the following way
x≤ y ⇐⇒ y − x ∈ C (K) or x = y
define a partial order in X (Y ). We will sometimes write y ≥ x instead of x ≤ y and x < y (x > y) is equivalent to x ≤ y (x ≥ y) and x 6= y. Let D ⊂ X be a convex set and let n be a fixed positive integer. Fix n + 1 collinear points x0, x1, ..., xn
belonging to the set D such that x0 < x1 < ... < xn, and let h ∈ X, h > 0, and nonnegative numbers λi = λi(h), λ0< λ1< ... < λn be so chosen that
xi= x0+ λih, i = 0, 1, ..., n.
For a function f : D → Y we put
(3) U (x0, ..., xn; f ; h) :=
1, λ0, λ20, ... , λn0−1, f (x0) 1, λ1, λ21, ... , λn1−1, f (x1) ...
1, λn, λ2n, ... , λn−1n , f (xn)
Moreover, by V (λ0, λ1, ... , λn)we denotes the Van der Monde’s determinant of the form
V (λ0, λ1, ... , λn) :=
1, λ0, λ20, ... , λn−10 , λn0 1, λ1, λ21, ... , λn1−1, λn1 ...
1, λn, λ2n, ... , λnn−1, λnn
Conditions λ0< λ1< ... < λn imply the equivalence of the following inequality:
U (x0, ..., xn; f ; h)≥ 0,
and U (x0, ..., xn; f ; h)
V (λ0, λ1, ... , λn)≥ 0.
Notes that if ˜h = ρ h for a ρ > 0 then λi(h) = ρ λi(˜h)and U (x0, ..., xn; f ; h) = ρn(n−1)2 U (x0, ..., xn; f ; ˜h)
as well as
V (λ0(h), λ1(h), ..., λn(h)) = ρn(n+1)2 V (λ0(˜h), λ1(˜h), ..., λn(˜h)) Consequently
(4) U (x0, ..., xn; f ; h)
V (λ0(h), λ1(h), ... , λn(h)) = 1 ρn
U (x0, ..., xn; f ; ˜h) V (λ0(˜h), λ1(˜h), ... , λn(˜h)). Let us put
(5) [x0, x1, ..., xn; f ; h]?:= U (x0, ..., xn; f ; h) V (λ0, λ1, ... , λn). Every function f : D → Y fulfilling condition
(6) [x0, x1, ..., xn; f ; h]?≥ 0,
for arbitrary system of collinear points x0, x1, ..., xn∈ D such that there exist h > 0 and nonnegative numbers λ0 < λ1, ..., λn fulfilling conditions xi = x0+ λih, i = 0, ..., n, is called (C-K)-convex function of (n − 1) order.
Lemma 2.1 Let D, x0, x1, ..., xn, h, λi, i = 0, 1, ..., n, have the same meaning as above. For each function f : D → Y we have
(7) [x0, x1, ..., xn; f ; h]?= [x1, ..., xn; f ; h]?− [x0, x1, ..., xn−1; f ; h]?
λn− λ0 .
Proof By virtue of (5) and (3) [x0, ..., xn−1; f ; h]?= U (xV (λ0,...,xn−1;f ;h)
0,λ1,...,λn−1)
=Pn−1
i=0(−1)n−1−if (xi)V (λ0,...,λV (λi−1,λi+1,...,λn−1)
0,...,λn−1)
=Pn−1
i=0(−1)n−1−if (xi)(λn−λ0)...(λV (λn0−λ,...,λi−1i−1)(λ,λn−λi+1i+1,...,λ)...(λn)n−λn−1)(λn−λV (λ0)...(λ0,...,λn−λn)n−1)
=Pn−1
i=0(−1)n−1−if (xi)V (λ0,...,λV (λi−10,...,λ,λi+1n),...,λn)(λn− λi).
Analogously, taking into account equalities xi = x1+ (λi− λ1)h, i = 1, ...., n, we obtain
[x1, ..., xn; f ; h]?=Pn
i=1(−1)n−if (xi)V (λ1−λ1,λV (λ2−λ1,...,λi−1−λ1,λi+1−λ1,...,λn−λ1)
1−λ1,λ2−λ1,...,λn−λ1)
=Pn
i=1(−1)n−if (xi)V (λ1,...,λV (λi−1,λi+1,...,λn)
1,...,λn)
=Pn
i=1(−1)n−if (xi)(λ1−λ0)...(λV (λ0i−1,...,λ−λi−10)(λ,λi+1i+1−λ,...,λ0)...(λn) n−λ0)(λ1−λV (λ0)...(λ0,...,λnn−λ) 0)
=Pn
i=1(−1)n−if (xi)V (λ0,...,λV (λi−10,...,λ,λi+1n),...,λn)(λi− λ0).
Thus, according to (3),
[x1, ..., xn; f ; h]?− [x0, ..., xn−1; f ; h]? λn− λ0
= 1
λn− λ0[
n−1
X
i=1
(−1)n−if (xi)V (λ0, ..., λi−1, λi+1, ..., λn)
V (λ0, ..., λn) (λi− λ0) +f (xn)V (λ0, ..., λn−1)
V (λ0, ..., λn) (λn− λ0) + (−1)nf (x0)V (λ1, ..., λn)
V (λ0, ..., λn)(λn− λ0) +
nX−1 i=1
(−1)n−if (xi)V (λ0, ..., λi−1, λi+1, ..., λn)
V (λ0, ..., λn) (λn− λi)]
= Xn i=0
(−1)n−if (xi)V (λ0, ..., λi−1, λi+1, ..., λn) V (λ0, ..., λn)
= [x0, ..., xn; f ; h]?.
Let ϕ : X × ... × X → Y be an n-linear function i.e., linear with respect to each variable for arbitrary fixed other (n − 1) variables. We say that ϕ is symmetric if and only if for every points x1, ..., xn ∈ X and every permutation (σ1, ..., σn)of (1, ..., n) the following equality ϕ(x1, ..., xn) = ϕ(xσ1, ..., xσn) holds true. We say that an n-linear function ϕ : X × ... × X → Y is positive definite if and only if ϕ(h, ..., h
| {z }
n
) > 0for every h > 0.
In the proof of the following Lemma 2 we will write xk instead of (x, ..., x
| {z }
k
) and hn−k instead of (h, ..., h
| {z }
n−k
), k = 0, 1, ..., n. Let us put
ϕ?(x) := ϕ(xn), x∈ X.
Lemma 2.2 Let X, x0, x1, ..., xn, h, λi, i = 0, 1, ..., n, n≥ 1, have the same meaning as above. If ϕ : Xn→ Y is a symmetric n-linear functions then
(8) [x0, ..., xn; ϕ?; h]?= ϕ?(h), h∈ X.
Proof Let us fix n ≥ 1. According to [7, p.393], (5), (3) (with ϕ?instead of f) we
get
[x0, x1, ..., xn; ϕ?; h]?= U (xV (λ0,x01,λ,...,x1,...,λn;ϕn?);h)
=V (λ0,λ11,...,λn)Pn
i=0(−1)n−iϕ?(xi)V (λ0, ..., λi−1, λi+1, ..., λn)
=V (λ0,λ11,...,λn)Pn
i=0(−1)n−iPn k=0
n k
ϕ(xk0, hn−k)λn−ki V (λ0, ..., λi−1, λi+1, ..., λn)
=V (λ0,λ11,...,λn)Pn k=0
n k
ϕ(xk0, hn−k)Pn
i=0(−1)n−iλni−kV (λ0, ..., λi−1, λi+1, ..., λn)
=V (λ0,λ11,...,λn)Pn k=0
n k
ϕ(xk0, hn−k)
1, λ0, λ20, ... , λn0−1, λn0−k 1, λ1, λ21, ... , λn1−1, λn1−k ...
1, λn, λ2n, ... , λn−1n , λn−kn
= ϕ(hn) = ϕ?(h).
Now, we introduce the notion of strongly n-convexity of a function.
Definition 2.3 A function f : D → Y is called (C-K)-strongly (n − 1)-convex with a control positive definite symmetric n-linear function ϕ : X × ... × X → Y if and only if for every system of collinear points x0, ..., xn∈ D satisfying conditions xi= x0+ λih, i = 0, 1, ..., n, h > 0, λ0< λ1< ... < λn, the following inequality
[x0, ..., xn; f ; h]?≥ ϕ?(h).
holds true.
Remark 2.4 If x0, ..., xn is a system of collinear points, xi = x0 + λih, i = 0, 1, ..., n, h > 0, λ0 < λ1 < ... < λn, and ˜h = ρh, ρ > 0, then the inequali- ties
[x0, ..., xn; f ; h]?≥ ϕ?(h) and
[x0, ..., xn; f ; ˜h]?≥ ϕ?(˜h).
are equivalent.
Proof It follows from (5) and (4) that
[x0, ..., xn; f ; ˜h]?= ρn[x0, ..., xn; f ; h]?
Now our assertion follows from the equality ϕ?(˜h) = ρnϕ?(h).
The following theorem shows some connection between the properties of n- convexity and strongly n-convexity.
Theorem 2.5 Let X, Y and D have the same meaning as above. Then the function f : D → Y is (C-K)-strongly (n − 1)-convex with a positive definite symmetric n- linear control function ϕ if and only if the function g giving by the formula g(x) :=
f (x)− ϕ?(x), x∈ D, is (C-K)-convex of (n − 1) order.
Proof It is a simple consequence of the above definitions and of the linearity of
the operator [x0, ..., xn; · ; h]?.
The case X = Y = R, C = K = (0, ∞)
Proposition 2.6 Assume X = Y = R, C = K = [0, ∞), D = I. Then for every system of points x0, x1, ..., xn∈ I, x0< x1<· · · < xn we have
[x0, x1, ..., xn; f ] = 1
hn[x0, x1, ..., xn; f ; h]? with a suitable h > 0.
Proof Let h > 0, λ0 < λ1 <, ..., < λn, be such chosen that xi = x0+ λih, i = 0, 1, ..., n. Then
[x0; f ] = f (x0) = 1
h0f (x0) = 1
h0[x0; f ; h]?.
Assuming [x0, x1, ..., xn−1; f ] = hn1−1[x0, x1, ..., xn−1; f ; h]?, by virtue of the defini- tion od n-th divided difference and (7) we get
[x0, x1, ..., xn; f ] = [x1,...,xn;f ]x−[xn−x00,...,xn−1;f ]
= hn1−1
[x1,...,xn;f ;h]?−[x0,...,xn−1;f ;h]? (λn−λ0)h
= h1n[x0, x1, ..., xn; f ; h]?.
Remark 2.7 Definitions of the n-convexity of f giving by conditions (1) and (6) are equivalent.
Immediately from Proposition 1 easily follows the following remark.
Remark 2.8 For every system of points x0, x1, ..., xn ∈ I, x0 < x1 <· · · < xn and a constant c > 0 we have
[x0, x1, ..., xn; f ]≥ c if and only if
[x0, x1, ..., ; f ; h]?≥ chn with an h > 0.
Since in our case every positive definite symmetric n-linear function ϕ : Rn → R has the form
ϕ(s1, ..., sn) = cs1· · · sn, s1, ..., sn ∈ R, thus the conditions
[x0, x1, ..., xn; f ]≥ c and
[x0, x1, ..., xn; f ; h]?≥ ϕ(h, ..., h) are equivalent. Hence we obtain the following remark
Remark 2.9 The definitions of strongly n-convexity with modulus c > 0 given in (2) and of the strongly n-convexity given in Definition 2.3 are equivalent.
References
[1] R. Ger, Convex functions of higher orders in Euclidean spaces, Ann. Polon.Math.25 (1972), 293-302.
[2] R. Ger, n-Convex Functions in Linear Spaces, Aequationes Math.10 (2/3) (1974), 172-176.
[3] R. Ger and K. Nikodem, Strongly convex functions of higher order, Nonlinear analysis74 (2011), 661-665.
[4] A. Gilanyi and Zs. Páles, Onconvex functions of higher order, Math. Inequal. Appl.11 (2008), 271-282.
[5] E. Hopf, Über die Zusammenhan¨ge zwischen gevissen höheren Differenzenquotientienten reeller Funktionen einer reellen Variables und deren Differenzierbarkeitseigenschaften , Friedrich-Wilhelms-Universität, Berlin, 1926.
[6] Z. Kominek, and M. Kuczma, p-convex functions in linear spaces, Ann. Polon. Math.53 (1991), 91-108.
[7] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities. Cau- chy’s Equation and Jensen’s Inequality, Silesian University and PWN, Warszawa-Kraków- Katowice.
[8] A. Pinkus and D. Wulbert,Extending n-convex functions, Studia Math.171/2 (2005), 125- 152.
[9] T. Popoviciu, Les fonctions convexes, Hermann et Cie, Paris, 1944.
[10] B. T. Polyak, Existence theorems and converegence of minimmizing of minimizing sequence in extremum problems with restriction, Sovciet Math, Dokl.7 (1966), 72-75.
[11] A. W. Roberts and D. E. Varberg, Comvex Functions, Academic Press, New York-London, 1973.
[12] T. Szostok, On ω-Convex Functions, Function Spaces IX, Banach Center Publications, vol.
92, Institute of Math. Polish Acad. of Sciences, W-wa, 2011, 351-359.
[13] Sz. Wąsowicz, Some properties of generalized higher-order convexity, Publ. Math. Debrecen 68/1-2 (2006), 171-182.
[14] Sz. Wąsowicz, Support-type properties of convex functions of higher order and Hadamard- type inequalities, J. Math. Anal. Appl.332/2 (2007), 1229-1241.
Zygfryd Kominek
Institute of Mathematics, Silesian University Bankowa 14, PL-40-007 Katowice, Poland E-mail: zkominek@ux2.math.us.edu.pl
(Received: 11.09.2013)
