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Strongly convex functions of higher order
Nonlinear Analysis 74 (2011) 661–665
Contents lists available at ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Strongly convex functions of higher order Roman Ger a , Kazimierz Nikodem b,∗ a
Institute of Mathematics, Silesian University, ul. Bankowa 14, 40-007 Katowice, Poland
b
Department of Mathematics and Computer Science, University of Bielsko-Biała, ul. Willowa 2, 43-309 Bielsko-Biała, Poland
article
info
Article history: Received 25 June 2010 Accepted 8 September 2010 MSC: primary 26A51 secondary 39B62
abstract The notion of strongly n-convex functions with modulus c > 0 is introduced and investigated. Relationships between such functions and n-convex functions in the sense of Popoviciu as well as generalized convex functions in the sense of Beckenbach are given. Characterizations by derivatives are presented. Some results on strongly Jensen n-convex functions are also given. © 2010 Elsevier Ltd. All rights reserved.
Keywords: Strongly convex functions Convex functions of higher order Generalized convex functions
1. Introduction Let I ⊂ R be an interval and c be a positive constant. A function f : I → R is called strongly convex with modulus c if f (tx + (1 − t )y) ≤ tf (x) + (1 − t )f (y) − ct (1 − t )(x − y)2 ,
(1)
for all x, y ∈ I and t ∈ [0, 1]. Strongly convex functions have been introduced by Polyak [1] and they play an important role in optimization theory. Many properties of these functions can be found, among others, in [2–6]. In the classical theory of convex functions (i.e. functions satisfying (1) with c = 0) their natural generalization are convex functions of higher order. Let us recall the definition. Let n ∈ N and x0 , . . . , xn be distinct points in I. Denote by [x0 , . . . , xn ; f ] the divided difference of f at x0 , . . . , xn defined by the recurrence
[x0 ; f ] = f (x0 ),
[x1 , . . . , xn ; f ] − [x0 , . . . , xn−1 ; f ] , n ∈ N. xn − x0 Following Hopf [7] and Popoviciu [8] a function f : I → R is called convex of order n (or n-convex) if [x0 , . . . , xn ; f ] =
[ x 0 , . . . , x n +1 ; f ] ≥ 0 for all x0 < · · · < xn+1 in I. It is well known (and easy to verify) that 1-convex functions are ordinary convex functions. Many results on n-convex functions one can found, among others, in [8,9,4,10–13]. In this paper we introduce the notion of strongly n-convex functions and investigate properties of this class of functions. Let c be a positive constant and n ∈ N. We say that a function f : I → R is strongly convex of order n with modulus c (or strongly n-convex with modulus c) if
[ x 0 , . . . , x n +1 ; f ] ≥ c , ∗
Corresponding author. E-mail addresses: [email protected] (R. Ger), [email protected] (K. Nikodem).
0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.09.021
(2)
662
R. Ger, K. Nikodem / Nonlinear Analysis 74 (2011) 661–665
for all x0 < · · · < xn+1 in I. Note that for n = 1 condition (2) is equivalent to f ( x1 ) f (x2 ) f (x0 ) + + ≥ c, (x0 − x1 )(x0 − x2 ) (x1 − x0 )(x1 − x2 ) (x2 − x0 )(x2 − x1 ) or f (x1 ) ≤
x2 − x1 x2 − x0
Hence, putting t =
f ( x0 ) +
x 2 −x 1 x 2 −x 0
x1 − x0 x2 − x0
f (x2 ) − c (x2 − x1 )(x1 − x0 ),
and, consequently, 1 − t =
x 1 −x 0 x 2 −x 0
x0 < x1 < x2 .
and x1 = tx0 + (1 − t )x2 , we get
f (tx0 + (1 − t )x2 ) ≤ tf (x0 ) + (1 − t )f (x2 ) − ct (1 − t )(x2 − x0 )2 for all x0 , x2 ∈ I and t ∈ (0, 1), which means that f is strongly convex with modulus c. 2. Characterization by n-convexity We start with a theorem which gives a relationship between strongly n-convex and n-convex functions. It plays a crucial role in our further investigations. For n = 1 such result can be found in [3, Prop. 1.1.2]. Theorem 1. Let I ⊂ R be an interval, n ∈ N and c > 0. A function f : I → R is strongly n-convex with modulus c if and only if the function g (x) = f (x) − cxn+1 , x ∈ I, is n-convex. The proof of this theorem is based on the following simple facts whose proofs are straightforward. Lemma 2. For each distinct x0 , . . . , xn ∈ R the operator [x0 , . . . , xn ; ·] is linear. Lemma 3. [x0 , . . . , xn ; xn ] = 1 for each n ∈ N and distinct x0 , . . . , xn ∈ R. Proof of Theorem 1. If f is strongly n-convex with modulus c and g (x) = f (x) − cxn+1 , then, by Lemmas 2 and 3, we get
[x0 , . . . , xn+1 ; g ] = [x0 , . . . , xn+1 ; f ] − [x0 , . . . , xn+1 ; cxn+1 ] ≥ c − c = 0, which means that g is n-convex. Conversely, if g is n-convex then for f (x) = g (x) + cxn+1 we have
[x0 , . . . , xn+1 ; f ] = [x0 , . . . , xn+1 ; g ] + [x0 , . . . , xn+1 ; cxn+1 ] ≥ 0 + c = c , which proves that f is strongly n-convex with modulus c.
3. Characterization via derivatives It is known that a function f : I → R defined on an open interval I is n-convex with n > 1 if and only if it is of the class C n−1 in I and its (n − 1)-th derivative f (n−1) is convex (see [9, Thm. 15.8.4]). Moreover, if f is of the class C n in I then it is n-convex if and only if f (n) is increasing in I, and also if f is of the class C n+1 in I then it is n-convex if and only if f (n+1) is nonnegative in I (see [9, Thm. 15.8.5, Thm. 15.8.6]). In this section we will present analogous results for strongly n-convex functions. Theorem 4. Let I ⊂ R be an open interval, c > 0 and n > 1. A function f : I → R is strongly n-convex with modulus c if and only if it is of the class C n−1 in I and its (n − 1)-th derivative f (n−1) is strongly convex with modulus 2c (n + 1)!. Proof. (⇒) Assume that f is strongly n-convex with modulus c. By Theorem 1 f can be represented in the form f (x) = g (x) + cxn+1 , x ∈ I, where g is an n-convex function. Hence c f (n−1) (x) = g (n−1) (x) + (n + 1)!x2 , 2
x ∈ I.
Since g (n−1) is convex, this representation means that f (n−1) is strongly convex with modulus 2c (n + 1)!.
(⇐) By the assumption and Theorem 1 f (n−1) is of the form f (n−1) (x) = g (x) + 2c (n + 1)!x2 , x ∈ I, with a convex function g. Integrating both sides n − 1 times, we obtain f (x) = G(x) + cxn+1 ,
x ∈ I,
where G is an n-convex function. Thus, by Theorem 1, f is strongly n-convex with modules c.
The next theorem shows that f is strongly n-convex if and only if its n-th derivative is strongly increasing in some sense.
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Theorem 5. Let I ⊂ R be an open interval and f : I → R be of the class C n in I. Then f is strongly n-convex with modulus c if and only if f (n) satisfies the condition
(f (n) (x) − f (n) (y))(x − y) ≥ c (n + 1)!(x − y)2 ,
x, y ∈ I .
(3)
Proof. (⇒) By Theorem 1 f is of the form f (x) = g (x) + cxn+1 , x ∈ I, with an n-convex g. Hence f (n) (x) = g (n) (x) + c (n + 1)!x, Since g
(n)
(g
x ∈ I.
is increasing, we have
(n)
(x) − g (n) (y))(x − y) ≥ 0,
x, y ∈ I .
Thus, for all x, y ∈ I,
(f (n) (x) − f (n) (y))(x − y) = (g (n) (x) − g (n) (y))(x − y) + c (n + 1)!(x − y)2 ≥ c (n + 1)!(x − y)2 . (⇐) Assume (3) and put g (x) = f (x) − cxn+1 , x ∈ I. Then
(g (n) (x) − g (n) (y))(x − y) = (f (n) (x) − f (n) (y))(x − y) − c (n + 1)!(x − y)2 ≥ 0, which means that g is n-convex. Thus, by Theorem 1 again, f is strongly n-convex with modulus c.
Theorem 6. Let I ⊂ R be an open interval and f : I → R be of the class C n+1 in I. Then f is strongly n-convex with modulus c if and only if f (n+1) ≥ c (n + 1)!, x ∈ I. Proof. (⇒) Since f (x) = g (x) + cxn+1 , x ∈ I, with an n-convex g, we have f (n+1) (x) = g (n+1) (x) + c (n + 1)! ≥ c (n + 1)!, (⇐) Put g (x) = f (x) − cx g
(n)
(x) = f
(n)
n+1
x ∈ I.
, x ∈ I. Then
(x) − c (n + 1)! ≥ 0,
x ∈ I,
which means that g is n-convex. Hence f is strongly n-convex with modulus c.
Remark 7. For the special case n = 1 the results presented above can be found in [4, p. 268]. 4. Strongly Jensen n-convex functions A function f : I → R is said to be strongly Jensen convex with modulus c > 0 (cf. [14]) if it satisfies the condition (1) with t = 12 , that is
f
x+y
2
≤
f (x) + f (y) 2
c
− (x − y)2 , 4
x, y ∈ I .
In this section we extend this definition to strongly Jensen n-convex functions. Let △nh be the difference operator of n-th order with increment h > 0 defined by the recurrence:
△0h f (x) = f (x), △nh f (x) = △nh−1 f (x + h) − △nh−1 f (x), n ∈ N. A function f : I → R is said to be n-convex in the sense of Jensen (or Jensen n-convex) if △nh+1 f (x) ≥ 0 for all x ∈ I and h > 0 such that x + (n + 1)h ∈ I (cf. e.g. [9,4,15–17]). We say that a function f : I → R is strongly n-convex with modulus c > 0 in the sense of Jensen (or strongly Jensen n-convex with modulus c) if
△hn+1 f (x) ≥ c (n + 1)!hn+1 for all x ∈ I and h > 0 such that x + (n + 1)h ∈ I. Note that for n = 1 condition (4) reduces to △2h f (x) ≥ 2ch2
(4)
or f (x + 2h) − 2f (x + h) + f (x) ≥ 2ch2 . Putting u = x and v = x + 2h, we obtain
f
u+v 2
≤
f (u) + f (v) 2
c
− (u − v)2 , 4
u, v ∈ I ,
which means that f is strongly Jensen convex with modulus c.
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Remark 8. Every function f : I → R strongly n-convex with modulus c is strongly Jensen n-convex with modulus c. It follows from the fact that if points x0 < · · · < xn+1 are equally spaced, that is xi = x0 + ih, i = 1, . . . , n + 1, with some h > 0, then
[x0 , . . . , xn+1 ; f ] =
△nh+1 f (x0 ) (n + 1)!hn+1
(5)
(see [9, Lem. 15.2.5]). If f is strongly n-convex with modulus c, then [x0 , . . . , xn+1 ; f ] ≥ c for all x0 < · · · < xn+1 in I. In particular, for equally spaced points we get
△nh+1 f (x0 ) = [x0 , . . . , xn+1 ; f ](n + 1)!hn+1 ≥ c (n + 1)!hn+1 , which means that f is strongly Jensen n-convex with modulus c. The next result is analogous to Theorem 1 and gives a relationship between strongly Jensen n-convex functions and Jensen n-convex functions. Theorem 9. Let I ⊂ R be an interval, n ∈ N and c > 0. A function f : I → R is strongly Jensen n-convex with modulus c if and only if the function g (x) = f (x) − cxn+1 , x ∈ I, is Jensen n-convex. The proof of this theorem is based on the following simple facts. Lemma 10 ([9, Lem. 15.1.1]). The operator △nh is linear. Lemma 11. △nh xn = n!hn , for every n ∈ N, x ∈ R and h > 0. Proof of Theorem 9. (⇒) Using the strong Jensen n-convexity of f and Lemmas 10 and 11 we get
△nh+1 g (x) = △nh+1 f (x) − △nh+1 xn+1 ≥ c (n + 1)!hn+1 − c (n + 1)!hn+1 = 0, which shows that g is Jensen n-convex. (⇐) By the Jensen n-convexity of g we have
△nh+1 f (x) = △hn+1 g (x) + △nh+1 xn+1 ≥ c (n + 1)!hn+1 , which proves that f is strongly Jensen n-convex with modulus c.
It is known that Jensen n-convex functions need not be continuous (and hence they need not be n-convex). However, for continuous functions the concepts of n-convexity and Jensen n-convexity are equivalent. There are also many theorems giving relatively weak conditions under which Jensen n-convex functions are continuous (cf. e.g. [9, Chpt.15], [4,15,16] and the references therein). Similar results hold for strongly Jensen n-convex functions. We present here, as an example, a counterpart of the classical theorem of Ciesielski [15] (cf. also [16]). Theorem 12. Let I be an open interval and n ∈ N. If a function f : I → R is strongly Jensen n-convex with modulus c > 0 and bounded on a set A ⊂ I having positive Lebesgue measure (or of the second category and with the Baire property), then f is continuous on I and strongly n-convex with modulus c. Proof. By Theorem 9 f is of the form f (x) = g (x) + cxn+1 , x ∈ I, where g is Jensen n-convex. If f is bounded on A, then g is also bounded on A (without loss of generality we may assume that A is bounded). Hence, by the theorem of Ciesielski, g is continuous and n-convex. Consequently, f is continuous and strongly n-convex with modulus c. 5. Connections with generalized convexity The convexity of a function f : I → R means that for any two distinct points on the graph of f the segment joining these points lies above the corresponding part of the graph. In 1937 Beckenbach [18] generalized this concept by replacing the segments by graphs of continuous functions belonging to a two-parameter family F of functions. Next, Tornheim [19] extended this idea by taking n-parameter families. The so obtained generalized convex functions have many properties known for classical convex (n-convex) functions (see e.g. [18,20,19,4,21]). In this section we will show that strong nconvexity is equivalent to generalized convexity with respect to a certain n-parameter family. Let n ≥ 2. A family F of continuous real functions defined on an interval I ⊂ R is called an n-parameter family if for any n points (x1 , y1 ), . . . , (xn , yn ) ∈ I × R with x1 < · · · < xn there exists exactly one ϕ ∈ F such that
ϕ(xi ) = yi for i = 1, . . . , n. The unique function ϕ ∈ F determined by the points (x1 , y1 ), . . . , (xn , yn ) will be denoted by ϕ(x1 ,y1 ),...,(xn ,yn ) . A function f : I → R is said to be convex with respect to the n-parameter family F (briefly, F -convex) if for any x1 < · · · < xn in I f (x) ≤ ϕ(x1 ,f (x1 )),...,(xn ,f (xn )) (x),
x ∈ [xn−1 , xn ].
R. Ger, K. Nikodem / Nonlinear Analysis 74 (2011) 661–665
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It is well known that if
Fn = {an xn + · · · + a1 x + a0 : a0 , . . . , an ∈ R}, i.e. Fn is the set of all polynomials of degree at most n, then Fn is an (n + 1)-parameter family and the generalized convexity with respect to Fn coincides with n-convexity (cf. [4,19,11]). In a similar way we can characterize the strong n-convexity. Let c > 0 be a fixed number and
Fn,c = {cxn+1 + an xn + · · · + a1 x + a0 : a0 , . . . , an ∈ R}. Clearly, Fn,c is also an (n + 1)-parameter family and the following theorem holds. Theorem 13. A function f : I → R is strongly n-convex with modulus c if and only if f is Fn,c -convex. Proof. Fix arbitrarily points x1 , . . . , xn+1 in I. Let ϕ be the unique polynomial in Fn,c determined by ϕ(xi ) = f (xi ), i = 1, . . . , n + 1. Then ψ defined by
ψ(x) = ϕ(x) − cxn+1 ,
x ∈ I,
belongs to Fn and is uniquely determined by ψ(xi ) = f (xi ) − cxni +1 , i = 1, . . . , n + 1. Clearly, f (x) ≥ ϕ(x),
x ∈ [xn , xn+1 ]
if and only if f (x) − cxn+1 ≥ ψ(x),
x ∈ [xn , xn+1 ].
This means that f is Fn,c -convex if and only if f (x) − cxn+1 is Fn -convex. Since the Fn -convexity is equivalent to the nconvexity, we obtain, by Theorem 1, that Fn,c -convexity is equivalent to the strong n-convexity with modulus c. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
B.T. Polyak, Existence theorems and convergence of minimizing sequences in extremum problems with restrictions, Sov. Math. Dokl. 7 (1966) 72–75. E.S. Polovinkin, Strongly convex analysis, Sb. Math. 187 (2) (1996) 103–130. J.-B. Hiriart-Urruty, C. Lemaréchal, Fundamentals of Convex Analysis, Springer-Verlag, Berlin, Heidelberg, 2001. A.W. Roberts, D.E. Varberg, Convex Functions, Academic Press, New York, London, 1973. N. Merentes, K. Nikodem, Some remarks on strongly convex functions, Aequationes Math. (in press). K. Nikodem, Zs. Páles, Characterizations of inner product spaces by strongly convex functions, Banach J. Math. Anal. (in press). E. Hopf, Über die Zusammenhänge zwischen gewissen höheren Differenzenquotienten reeller Funktionen einer reellen Variablen und deren Differenzierbarkeitseigenschaften, Friedrich-Wilhelms-Universität, Berlin, 1926. T. Popoviciu, Les Fonctions Convexes, Hermann et Cie, Paris, 1944. M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities. Cauchy’s Equation and Jensen’s Inequality, PWN—Uniwersytet Ślaski, ¸ Warszawa, Kraków, Katowice, 1985, second ed., Birkhäuser, Basel, Boston, Berlin, 2009. A. Gilányi, Zs. Páles, On convex functions of higher order, Math. Inequal. Appl. 11 (2008) 271–282. A. Pinkus, D. Wulbert, Extending n-convex functions, Studia Math. 171 (2) (2005) 125–152. Sz. Wasowicz, ¸ Some properties of generalized higher-order convexity, Publ. Math. Debrecen 68 (1–2) (2006) 171–182. Sz. Wasowicz, ¸ Support-type properties of convex functions of higher order and Hadamard-type inequalities, J. Math. Anal. Appl. 332 (2) (2007) 1229–1241. A. Azócar, J. Giménez, K. Nikodem, J.L. Sánchez, On strongly midconvex functions, Opuscula Math. (in press). Z. Ciesielski, Some properties of convex functions of higher order, Ann. Polon. Math. 7 (1959) 1–7. R. Ger, Convex functions of higher order in Euclidean spaces, Ann. Polon. Math. 25 (1972) 293–302. R. Ger, n-convex functions in linear spaces, Aequationes Math. 10 (1974) 172–176. E.F. Beckenbach, Generalized convex functions, Bull. Amer. Math. Soc. 43 (1937) 363–371. L. Tornheim, On n-parameter families of functions and associated convex functions, Trans. Amer. Math. Soc. 69 (1950) 457–467. M. Bessenyei, Zs. Páles, Hadamard-type inequalities for generalized convex functions, Math. Inequal. Appl. 6 (3) (2003) 379–392. K. Nikodem, Zs. Páles, Generalized convexity and separation theorems, J. Convex Anal. 14 (2) (2007) 239–247.
Contents lists available at ScienceDirect
Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Strongly convex functions of higher order Roman Ger a , Kazimierz Nikodem b,∗ a
Institute of Mathematics, Silesian University, ul. Bankowa 14, 40-007 Katowice, Poland
b
Department of Mathematics and Computer Science, University of Bielsko-Biała, ul. Willowa 2, 43-309 Bielsko-Biała, Poland
article
info
Article history: Received 25 June 2010 Accepted 8 September 2010 MSC: primary 26A51 secondary 39B62
abstract The notion of strongly n-convex functions with modulus c > 0 is introduced and investigated. Relationships between such functions and n-convex functions in the sense of Popoviciu as well as generalized convex functions in the sense of Beckenbach are given. Characterizations by derivatives are presented. Some results on strongly Jensen n-convex functions are also given. © 2010 Elsevier Ltd. All rights reserved.
Keywords: Strongly convex functions Convex functions of higher order Generalized convex functions
1. Introduction Let I ⊂ R be an interval and c be a positive constant. A function f : I → R is called strongly convex with modulus c if f (tx + (1 − t )y) ≤ tf (x) + (1 − t )f (y) − ct (1 − t )(x − y)2 ,
(1)
for all x, y ∈ I and t ∈ [0, 1]. Strongly convex functions have been introduced by Polyak [1] and they play an important role in optimization theory. Many properties of these functions can be found, among others, in [2–6]. In the classical theory of convex functions (i.e. functions satisfying (1) with c = 0) their natural generalization are convex functions of higher order. Let us recall the definition. Let n ∈ N and x0 , . . . , xn be distinct points in I. Denote by [x0 , . . . , xn ; f ] the divided difference of f at x0 , . . . , xn defined by the recurrence
[x0 ; f ] = f (x0 ),
[x1 , . . . , xn ; f ] − [x0 , . . . , xn−1 ; f ] , n ∈ N. xn − x0 Following Hopf [7] and Popoviciu [8] a function f : I → R is called convex of order n (or n-convex) if [x0 , . . . , xn ; f ] =
[ x 0 , . . . , x n +1 ; f ] ≥ 0 for all x0 < · · · < xn+1 in I. It is well known (and easy to verify) that 1-convex functions are ordinary convex functions. Many results on n-convex functions one can found, among others, in [8,9,4,10–13]. In this paper we introduce the notion of strongly n-convex functions and investigate properties of this class of functions. Let c be a positive constant and n ∈ N. We say that a function f : I → R is strongly convex of order n with modulus c (or strongly n-convex with modulus c) if
[ x 0 , . . . , x n +1 ; f ] ≥ c , ∗
Corresponding author. E-mail addresses: [email protected] (R. Ger), [email protected] (K. Nikodem).
0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.09.021
(2)
662
R. Ger, K. Nikodem / Nonlinear Analysis 74 (2011) 661–665
for all x0 < · · · < xn+1 in I. Note that for n = 1 condition (2) is equivalent to f ( x1 ) f (x2 ) f (x0 ) + + ≥ c, (x0 − x1 )(x0 − x2 ) (x1 − x0 )(x1 − x2 ) (x2 − x0 )(x2 − x1 ) or f (x1 ) ≤
x2 − x1 x2 − x0
Hence, putting t =
f ( x0 ) +
x 2 −x 1 x 2 −x 0
x1 − x0 x2 − x0
f (x2 ) − c (x2 − x1 )(x1 − x0 ),
and, consequently, 1 − t =
x 1 −x 0 x 2 −x 0
x0 < x1 < x2 .
and x1 = tx0 + (1 − t )x2 , we get
f (tx0 + (1 − t )x2 ) ≤ tf (x0 ) + (1 − t )f (x2 ) − ct (1 − t )(x2 − x0 )2 for all x0 , x2 ∈ I and t ∈ (0, 1), which means that f is strongly convex with modulus c. 2. Characterization by n-convexity We start with a theorem which gives a relationship between strongly n-convex and n-convex functions. It plays a crucial role in our further investigations. For n = 1 such result can be found in [3, Prop. 1.1.2]. Theorem 1. Let I ⊂ R be an interval, n ∈ N and c > 0. A function f : I → R is strongly n-convex with modulus c if and only if the function g (x) = f (x) − cxn+1 , x ∈ I, is n-convex. The proof of this theorem is based on the following simple facts whose proofs are straightforward. Lemma 2. For each distinct x0 , . . . , xn ∈ R the operator [x0 , . . . , xn ; ·] is linear. Lemma 3. [x0 , . . . , xn ; xn ] = 1 for each n ∈ N and distinct x0 , . . . , xn ∈ R. Proof of Theorem 1. If f is strongly n-convex with modulus c and g (x) = f (x) − cxn+1 , then, by Lemmas 2 and 3, we get
[x0 , . . . , xn+1 ; g ] = [x0 , . . . , xn+1 ; f ] − [x0 , . . . , xn+1 ; cxn+1 ] ≥ c − c = 0, which means that g is n-convex. Conversely, if g is n-convex then for f (x) = g (x) + cxn+1 we have
[x0 , . . . , xn+1 ; f ] = [x0 , . . . , xn+1 ; g ] + [x0 , . . . , xn+1 ; cxn+1 ] ≥ 0 + c = c , which proves that f is strongly n-convex with modulus c.
3. Characterization via derivatives It is known that a function f : I → R defined on an open interval I is n-convex with n > 1 if and only if it is of the class C n−1 in I and its (n − 1)-th derivative f (n−1) is convex (see [9, Thm. 15.8.4]). Moreover, if f is of the class C n in I then it is n-convex if and only if f (n) is increasing in I, and also if f is of the class C n+1 in I then it is n-convex if and only if f (n+1) is nonnegative in I (see [9, Thm. 15.8.5, Thm. 15.8.6]). In this section we will present analogous results for strongly n-convex functions. Theorem 4. Let I ⊂ R be an open interval, c > 0 and n > 1. A function f : I → R is strongly n-convex with modulus c if and only if it is of the class C n−1 in I and its (n − 1)-th derivative f (n−1) is strongly convex with modulus 2c (n + 1)!. Proof. (⇒) Assume that f is strongly n-convex with modulus c. By Theorem 1 f can be represented in the form f (x) = g (x) + cxn+1 , x ∈ I, where g is an n-convex function. Hence c f (n−1) (x) = g (n−1) (x) + (n + 1)!x2 , 2
x ∈ I.
Since g (n−1) is convex, this representation means that f (n−1) is strongly convex with modulus 2c (n + 1)!.
(⇐) By the assumption and Theorem 1 f (n−1) is of the form f (n−1) (x) = g (x) + 2c (n + 1)!x2 , x ∈ I, with a convex function g. Integrating both sides n − 1 times, we obtain f (x) = G(x) + cxn+1 ,
x ∈ I,
where G is an n-convex function. Thus, by Theorem 1, f is strongly n-convex with modules c.
The next theorem shows that f is strongly n-convex if and only if its n-th derivative is strongly increasing in some sense.
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Theorem 5. Let I ⊂ R be an open interval and f : I → R be of the class C n in I. Then f is strongly n-convex with modulus c if and only if f (n) satisfies the condition
(f (n) (x) − f (n) (y))(x − y) ≥ c (n + 1)!(x − y)2 ,
x, y ∈ I .
(3)
Proof. (⇒) By Theorem 1 f is of the form f (x) = g (x) + cxn+1 , x ∈ I, with an n-convex g. Hence f (n) (x) = g (n) (x) + c (n + 1)!x, Since g
(n)
(g
x ∈ I.
is increasing, we have
(n)
(x) − g (n) (y))(x − y) ≥ 0,
x, y ∈ I .
Thus, for all x, y ∈ I,
(f (n) (x) − f (n) (y))(x − y) = (g (n) (x) − g (n) (y))(x − y) + c (n + 1)!(x − y)2 ≥ c (n + 1)!(x − y)2 . (⇐) Assume (3) and put g (x) = f (x) − cxn+1 , x ∈ I. Then
(g (n) (x) − g (n) (y))(x − y) = (f (n) (x) − f (n) (y))(x − y) − c (n + 1)!(x − y)2 ≥ 0, which means that g is n-convex. Thus, by Theorem 1 again, f is strongly n-convex with modulus c.
Theorem 6. Let I ⊂ R be an open interval and f : I → R be of the class C n+1 in I. Then f is strongly n-convex with modulus c if and only if f (n+1) ≥ c (n + 1)!, x ∈ I. Proof. (⇒) Since f (x) = g (x) + cxn+1 , x ∈ I, with an n-convex g, we have f (n+1) (x) = g (n+1) (x) + c (n + 1)! ≥ c (n + 1)!, (⇐) Put g (x) = f (x) − cx g
(n)
(x) = f
(n)
n+1
x ∈ I.
, x ∈ I. Then
(x) − c (n + 1)! ≥ 0,
x ∈ I,
which means that g is n-convex. Hence f is strongly n-convex with modulus c.
Remark 7. For the special case n = 1 the results presented above can be found in [4, p. 268]. 4. Strongly Jensen n-convex functions A function f : I → R is said to be strongly Jensen convex with modulus c > 0 (cf. [14]) if it satisfies the condition (1) with t = 12 , that is
f
x+y
2
≤
f (x) + f (y) 2
c
− (x − y)2 , 4
x, y ∈ I .
In this section we extend this definition to strongly Jensen n-convex functions. Let △nh be the difference operator of n-th order with increment h > 0 defined by the recurrence:
△0h f (x) = f (x), △nh f (x) = △nh−1 f (x + h) − △nh−1 f (x), n ∈ N. A function f : I → R is said to be n-convex in the sense of Jensen (or Jensen n-convex) if △nh+1 f (x) ≥ 0 for all x ∈ I and h > 0 such that x + (n + 1)h ∈ I (cf. e.g. [9,4,15–17]). We say that a function f : I → R is strongly n-convex with modulus c > 0 in the sense of Jensen (or strongly Jensen n-convex with modulus c) if
△hn+1 f (x) ≥ c (n + 1)!hn+1 for all x ∈ I and h > 0 such that x + (n + 1)h ∈ I. Note that for n = 1 condition (4) reduces to △2h f (x) ≥ 2ch2
(4)
or f (x + 2h) − 2f (x + h) + f (x) ≥ 2ch2 . Putting u = x and v = x + 2h, we obtain
f
u+v 2
≤
f (u) + f (v) 2
c
− (u − v)2 , 4
u, v ∈ I ,
which means that f is strongly Jensen convex with modulus c.
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Remark 8. Every function f : I → R strongly n-convex with modulus c is strongly Jensen n-convex with modulus c. It follows from the fact that if points x0 < · · · < xn+1 are equally spaced, that is xi = x0 + ih, i = 1, . . . , n + 1, with some h > 0, then
[x0 , . . . , xn+1 ; f ] =
△nh+1 f (x0 ) (n + 1)!hn+1
(5)
(see [9, Lem. 15.2.5]). If f is strongly n-convex with modulus c, then [x0 , . . . , xn+1 ; f ] ≥ c for all x0 < · · · < xn+1 in I. In particular, for equally spaced points we get
△nh+1 f (x0 ) = [x0 , . . . , xn+1 ; f ](n + 1)!hn+1 ≥ c (n + 1)!hn+1 , which means that f is strongly Jensen n-convex with modulus c. The next result is analogous to Theorem 1 and gives a relationship between strongly Jensen n-convex functions and Jensen n-convex functions. Theorem 9. Let I ⊂ R be an interval, n ∈ N and c > 0. A function f : I → R is strongly Jensen n-convex with modulus c if and only if the function g (x) = f (x) − cxn+1 , x ∈ I, is Jensen n-convex. The proof of this theorem is based on the following simple facts. Lemma 10 ([9, Lem. 15.1.1]). The operator △nh is linear. Lemma 11. △nh xn = n!hn , for every n ∈ N, x ∈ R and h > 0. Proof of Theorem 9. (⇒) Using the strong Jensen n-convexity of f and Lemmas 10 and 11 we get
△nh+1 g (x) = △nh+1 f (x) − △nh+1 xn+1 ≥ c (n + 1)!hn+1 − c (n + 1)!hn+1 = 0, which shows that g is Jensen n-convex. (⇐) By the Jensen n-convexity of g we have
△nh+1 f (x) = △hn+1 g (x) + △nh+1 xn+1 ≥ c (n + 1)!hn+1 , which proves that f is strongly Jensen n-convex with modulus c.
It is known that Jensen n-convex functions need not be continuous (and hence they need not be n-convex). However, for continuous functions the concepts of n-convexity and Jensen n-convexity are equivalent. There are also many theorems giving relatively weak conditions under which Jensen n-convex functions are continuous (cf. e.g. [9, Chpt.15], [4,15,16] and the references therein). Similar results hold for strongly Jensen n-convex functions. We present here, as an example, a counterpart of the classical theorem of Ciesielski [15] (cf. also [16]). Theorem 12. Let I be an open interval and n ∈ N. If a function f : I → R is strongly Jensen n-convex with modulus c > 0 and bounded on a set A ⊂ I having positive Lebesgue measure (or of the second category and with the Baire property), then f is continuous on I and strongly n-convex with modulus c. Proof. By Theorem 9 f is of the form f (x) = g (x) + cxn+1 , x ∈ I, where g is Jensen n-convex. If f is bounded on A, then g is also bounded on A (without loss of generality we may assume that A is bounded). Hence, by the theorem of Ciesielski, g is continuous and n-convex. Consequently, f is continuous and strongly n-convex with modulus c. 5. Connections with generalized convexity The convexity of a function f : I → R means that for any two distinct points on the graph of f the segment joining these points lies above the corresponding part of the graph. In 1937 Beckenbach [18] generalized this concept by replacing the segments by graphs of continuous functions belonging to a two-parameter family F of functions. Next, Tornheim [19] extended this idea by taking n-parameter families. The so obtained generalized convex functions have many properties known for classical convex (n-convex) functions (see e.g. [18,20,19,4,21]). In this section we will show that strong nconvexity is equivalent to generalized convexity with respect to a certain n-parameter family. Let n ≥ 2. A family F of continuous real functions defined on an interval I ⊂ R is called an n-parameter family if for any n points (x1 , y1 ), . . . , (xn , yn ) ∈ I × R with x1 < · · · < xn there exists exactly one ϕ ∈ F such that
ϕ(xi ) = yi for i = 1, . . . , n. The unique function ϕ ∈ F determined by the points (x1 , y1 ), . . . , (xn , yn ) will be denoted by ϕ(x1 ,y1 ),...,(xn ,yn ) . A function f : I → R is said to be convex with respect to the n-parameter family F (briefly, F -convex) if for any x1 < · · · < xn in I f (x) ≤ ϕ(x1 ,f (x1 )),...,(xn ,f (xn )) (x),
x ∈ [xn−1 , xn ].
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It is well known that if
Fn = {an xn + · · · + a1 x + a0 : a0 , . . . , an ∈ R}, i.e. Fn is the set of all polynomials of degree at most n, then Fn is an (n + 1)-parameter family and the generalized convexity with respect to Fn coincides with n-convexity (cf. [4,19,11]). In a similar way we can characterize the strong n-convexity. Let c > 0 be a fixed number and
Fn,c = {cxn+1 + an xn + · · · + a1 x + a0 : a0 , . . . , an ∈ R}. Clearly, Fn,c is also an (n + 1)-parameter family and the following theorem holds. Theorem 13. A function f : I → R is strongly n-convex with modulus c if and only if f is Fn,c -convex. Proof. Fix arbitrarily points x1 , . . . , xn+1 in I. Let ϕ be the unique polynomial in Fn,c determined by ϕ(xi ) = f (xi ), i = 1, . . . , n + 1. Then ψ defined by
ψ(x) = ϕ(x) − cxn+1 ,
x ∈ I,
belongs to Fn and is uniquely determined by ψ(xi ) = f (xi ) − cxni +1 , i = 1, . . . , n + 1. Clearly, f (x) ≥ ϕ(x),
x ∈ [xn , xn+1 ]
if and only if f (x) − cxn+1 ≥ ψ(x),
x ∈ [xn , xn+1 ].
This means that f is Fn,c -convex if and only if f (x) − cxn+1 is Fn -convex. Since the Fn -convexity is equivalent to the nconvexity, we obtain, by Theorem 1, that Fn,c -convexity is equivalent to the strong n-convexity with modulus c. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21]
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