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.