ON A FUNCTIONAL EQUATION RELATED TO A GENERALIZATION OF FLETT’S MEAN VALUE THEOREM

In this paper, we characterize all the functions that attain their Flett mean value at a particular point between the endpoints of the interval under consideration. These functions turn out to be cubic polynomials and thus, we also characterize these.


Introduction.
In [5], Sahoo and Riedel gave a generalization of Flett's mean value theorem [2] as follows. It is easy to see that if f (b) = f (a), then this reduces to Flett's mean value theorem. Aczél [1] and Haruki [3] used the Lagrange mean value theorem to ask the question of which functions attained their mean value at a prescribed point c ∈ (a, b), in particular, at the midpoint c = (a + b)/2. The answer is that only quadratic polynomials have the property that the mean value on any interval is attained at the midpoint of that interval. A natural question to ask is this same question for the above mean value theorem. It turns out that quadratic polynomials satisfy (1.1) for any c, but, more interestingly, cubic polynomials satisfy it for c = (a + 3b)/4. Thus, the main question becomes whether cubic polynomials are the only functions having this property.
Following the approach in [1], we pexiderize (1.1) to obtain and now setting c = (a + 3b)/4 yields More generally, setting c = sa + tb with s + t = 1 and 0 < s, t < 1, we obtain The question we answer, in this paper, is: What are the functions f ,h that satisfy the functional equations (1.4) and (1.5) for all a, b ∈ R? In solving this functional equation, we do not assume any regularity conditions on f or h.

Solution of the functional equation.
The main work in solving this functional equation is to reduce (1.4) and (1.5) to a form where we can apply the following result by Székelyhidi [6,Thm. 9.5] and Wilson [7].
then h is a generalized polynomial of degree at most n.
Thus, we are able to prove our main result (Theorem 2.2).

Theorem 2.2. The real valued functions f and h are solutions of the functional equation (1.5) if and only if
Proof. It is easy to check that the functions f and h, given above, do satisfy the functional equation (1.5).
To show that these are the only solutions, we start by rewriting (1.5) using s + t = 1 as follows: Now, we replace a by a − tu in (2.5) and get Similarly, using a = a − 2tu in (2.5), we get Interchanging u with −u in (2.7) gives (2.8) Collecting the terms of h that have the same argument, we obtain Writing x = a + 2tu and dividing (2.11) by (2 − t) yields Thus, since t ≠ 0 is fixed, (2.12) is of the form of equation (2.1) and hence, h(x) is a generalized polynomial of degree at most 2, where β is a symmetric, biadditive function and α is an additive function and C is an arbitrary real constant. Setting a = 0 in (2.5), we get and substituting from (2.13), we obtain To prove the continuity of f and h, let us substitute the solutions given in (2.15) into (2.5). We see that both the left-and the right-hand side of (2.5) are polynomial functions in a and u. The equality of the two sides implies, therefore, the equality of terms which are of the same degree with respect to a and u. First, comparing the terms of degree 1 with respect to each variable, we get In particular, α(a)/a does not depend on a and, therefore, α(a) = 2Ba for some constant B. Now, let us compare the terms of degree 2 with respect to a and those of degree 1 with respect to u. We get (2.24) Inserting (2.24) into (1.5), we get, after simplifying, whence, it follows that A = 0 provided t ≠ 3/4. Note that, for t = 3/4, we have 3A(1 − (1/2t)) = A and the assertion follows from (2.24).

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management. For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy. Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made. Until now, computational methods and models are central to the analysis of economic and financial decisions. However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models. In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches. For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset. Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results. The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making. In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning • Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation