International Journal of Analysis
Volume 2016 (2016), Article ID 7160348, 5 pages
http://dx.doi.org/10.1155/2016/7160348

Research Article

Strong -Convexity and Separation Theorems

Teodoro Lara,¹ Nelson Merentes,² and Kazimierz Nikodem³

¹Departamento de Física y Matemáticas, Universidad de los Andes, N. U. Rafael Rangel, Trujillo, Venezuela
²Universidad Central de Venezuela, Escuela de Matemáticas, Caracas, Venezuela
³Department of Mathematics, University of Bielsko-Biala, Ul. Willowa 2, 43-309 Bielsko-Biala, Poland

Received 9 July 2016; Accepted 16 October 2016

Academic Editor: Baruch Cahlon

Copyright © 2016 Teodoro Lara et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Jensen inequality for strongly -convex functions and a characterization of pairs of functions that can be separated by a strongly -convex function are presented. As a consequence, a stability result of the Hyers-Ulam type is obtained.

1. Introduction

Let be a normed space, let be a convex subset of , and let A function is called strongly convex with modulus (see, e.g., [1, 2]) if for all and Recall also that the usual notion of convex functions corresponds to the case Strongly convex functions, introduced by Polyak [3], play an important role in optimization theory and mathematical economics. Many properties and applications of them can be found in the literature (see, e.g., [2, 4–7] and the references therein).

In [8] Varošanec introduced the notion of -convexity. Let be a given function. A function is said to be -convex iffor all and . This notion unifies and generalizes the known classes of convex functions, -convex functions, Godunova-Levin functions, and -functions, which are obtained by putting in (2) , (where ), (with ), and , respectively. Some properties of them can be found, for example, in [8–13].

Combining the above two ideas we say that a function is strongly -convex with modulus (cf. [14]) iffor all and

In this note we present a Jensen-type inequality for such functions and give a characterization of pairs of functions that can be separated by a strongly -convex one. Separation (or sandwich) theorems, that is, theorems providing conditions under which two given functions can be separated by a function from some special class, play an important role in many fields of mathematics and have various applications. In the literature one can find numerous results of this type (see, e.g., [12, 15–24]).

2. Jensen-Type Inequality

In the whole paper we assume that is a real inner product space (i.e., the norm in is induced by an inner product: ). is a convex nonempty subset of and is a positive constant.

A function is said to be multiplicative if Note that if is multiplicative, then it is nonnegative and either or In what follows we assume that .

The following result is a counterpart of the classical Jensen inequality for strongly -convex functions. It generalizes the Jensen-type inequality for strongly convex functions obtained in [4]. Similar results for -convex functions are proved in [8, 12].

Theorem 1. Let be a multiplicative function such that for all If a function is strongly -convex with modulus , thenfor all , , and with and

Proof. For inequality (5) is trivial and for it follows from the definition of strong -convexity (note that ). Now, assuming (2) holds for some , we will prove it for . By the definition of strong -convexity we getwhereBy the inductive assumption we have Now, using the above inequalities, the multiplicativity of , and the assumption , we obtain To finish the proof it is enough to show that or, equivalently,Sincewe havewhich finishes the proof.

3. Separation by Strongly -Convex Functions

It is proved in [15] that two functions defined on a convex subset of a vector space can be separated by a convex function if and only if for all , , and with .

In this section we present counterparts of that result related to strong -convexity.

Theorem 2. Let be given functions and be a multiplicative function such that for all If there exists a function strongly -convex with modulus such thatthenfor all , , and with and

Proof. By the Jensen inequality for strongly -convex functions (Theorem 1) we have

Theorem 3. Let be given functions and be a multiplicative function such that for all Iffor all , , and with and , then there exists a function strongly -convex with modulus such that

Proof. Fix and define a function byBy (18) the definition is correct and for all . On the other hand, taking in the above definition (and, consequently, , ) and using the fact that , we get for all .
To prove that is strongly -convex with modulus , fix and . Take arbitrary , and , such that , and , . Since , the point is a convex combination of , , andTherefore, by the definition of we havewhere . By the multiplicativity of we haveNote also thatand, similarly,Hence, using the fact that , we getSubstituting (23) and (26) into (22), we obtainNow, taking the infimum in the first term and next in the second term of the right hand side of (27) and using the definition of , we getwhich shows that is strongly -convex with modulus and finishes the proof.

Remark 4. The method used in the proof of Theorem 3 is similar to that in [12, Theorem3]. However, in our case we assume additionally that , . The following example shows that this assumption is essential. Let , and consider , and . Then condition (18) is satisfied with , but there is no function strongly -convex (with any modulus) satisfying . Indeed, if is strongly -convex with , then is nonnegative and some of its values are positive (putting in the definition of strong -convexity, we get , but is not strongly -convex).

As a consequence of Theorem 3 we obtain the following Hyers-Ulam-type stability result for strongly -convex functions.

Let be a positive constant. We say that a function is -strongly -convex with modulus if for all , , and with and .

Corollary 5. Let be a multiplicative function such that for all If a function is -strongly -convex with modulus , then there exists a function strongly -convex with modulus such that

Proof. Define , . Then and satisfy (18). Therefore, by Theorem 3, there exists , strongly -convex with modulus , such that , on

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This research has been partially supported by Central Bank of Venezuela.

References

J.-B. Hiriart-Urruty and C. Lemaréchal, Fundamentals of Convex Analysis, Springer, Berlin, Germany, 2001.

A. W. Roberts and D. E. Varberg, Convex Functions, Academic Press, New York, NY, USA, 1973.

B. T. Polyak, “Existence theorems and convergence of minimizing sequences in extremum problems with restrictions,” Soviet Mathematics Doklady, vol. 7, pp. 72–75, 1966.

N. Merentes and K. Nikodem, “Remarks on strongly convex functions,” Aequationes Mathematicae, vol. 80, no. 1-2, pp. 193–199, 2010.

K. Nikodem, “Strongly convex functions and related classes of functions,” in Handbook of Functional Equations: Functional Inequalities, Th. M. Rassias, Ed., vol. 95 of Springer Optimizations and Its Applications, pp. 365–405, Springer, Berlin, Germany, 2015.

K. Nikodem and Zs. Páles, “Characterizations of inner product spaces by strongly convex functions,” Banach Journal of Mathematical Analysis, vol. 5, no. 1, pp. 83–87, 2011.

J.-P. Vial, “Strong convexity of sets and functions,” Journal of Mathematical Economics, vol. 9, no. 1-2, pp. 187–205, 1982.

S. Varošanec, “On h-convexity,” Journal of Mathematical Analysis and Applications, vol. 326, no. 1, pp. 303–311, 2007.

M. Bombardelli and S. Varošanec, “Properties of h-convex functions related to the Hermite-Hadamard-Fejér inequalities,” Computers & Mathematics with Applications, vol. 58, no. 9, pp. 1869–1877, 2009.

S. S. Dragomir, J. Pečarić, and L. E. Persson, “Some inequalities of Hadamard type,” Soochow Journal of Mathematics, vol. 21, no. 3, pp. 335–341, 1995.

A. Házy, “Bernstein-Doetsch type results for h-convex functions,” Mathematical Inequalities & Applications, vol. 14, no. 3, pp. 499–508, 2011.

A. Olbryś, “On separation by h-convex functions,” Tatra Mountains Mathematical Publications, vol. 62, pp. 105–111, 2015.

A. Olbryś, “Representation theorems for h-convexity,” Journal of Mathematical Analysis and Applications, vol. 426, no. 2, pp. 986–994, 2015.

H. Angulo, J. Giménez, A.-M. Moros, and K. Nikodem, “On strongly h-convex functions,” Annals of Functional Analysis, vol. 2, no. 2, pp. 85–91, 2011.

K. Baron, J. Matkowski, and K. Nikodem, “A sandwich with convexity,” Mathematica Pannonica, vol. 5, no. 1, pp. 139–144, 1994.

M. Bessenyei and P. Szokol, “Convex separation by regular pairs,” Journal of Geometry, vol. 104, no. 1, pp. 45–56, 2013.

M. Bessenyei and P. Szokol, “Separation by convex interpolation families,” Journal of Convex Analysis, vol. 20, no. 4, pp. 937–946, 2013.

W. Forg-Rob, K. Nikodem, and Z. Páles, “Separation by monotonic functions,” Mathematica Pannonica, vol. 7, no. 2, pp. 191–196, 1996.

N. Merentes and K. Nikodem, “Strong convexity and separation theorems,” Aequationes Mathematicae, vol. 90, no. 1, pp. 47–55, 2016.

K. Nikodem and Z. Páles, “Generalized convexity and separation theorems,” Journal of Convex Analysis, vol. 14, no. 2, pp. 239–247, 2007.

K. Nikodem, Z. Páles, and S. Wąsowicz, “Abstract separation theorems of Rodé type and their applications,” Annales Polonici Mathematici, vol. 72, no. 3, pp. 207–217, 1999.

K. Nikodem and S. Wąsowicz, “A sandwich theorem and Hyers-Ulam stability of affine functions,” Aequationes Mathematicae, vol. 49, no. 1-2, pp. 160–164, 1995.

Zs. Páles, “Separation by approximately convex functions,” Grazer Mathematische Berichte, vol. 344, pp. 43–50, 2001.

S. Wąsowicz, “Polynomial selections and separation by polynomials,” Studia Mathematica, vol. 120, no. 1, pp. 75–82, 1996.