Prasanna Sahoo

• Ph.D
• Professor at University of Louisville

The notion of (ψ, γ)-stability was introduced in [V. A. Faiziev, Th. M. Rassias, and P. K. Sahoo, Trans. Amer. Math. Soc., 354, 4455 (2002)]. It was shown that the Cauchy equation f(xy) = f(x) + f(y) is (ψ, γ)-stable both on any Abelian group and on any meta-Abelian group. In [V. A. Faiziev and P. K. Sahoo, Publ. Math. Debrecen, 75, 6 (2009)], it w...

.The notion of (?,?)-stability of the Jensen functional equation was introduced in [Nonlinear Funct. Anal. Appl. 11 (2006), 759-791] and it was shown that the Jensen functional equation is (?,?)-stable on any step-two nilpotent group. In this paper, we prove a more general result by showing that the Jensen functional equation is (?,?)-stable on ste...

Let G be an abelian group, (Formula presented.) be the field of complex numbers, (Formula presented.) be any fixed element and (Formula presented.) be an involution. In this paper, we determine the general solution (Formula presented.) of the functional equation (Formula presented.) for all (Formula presented.).

Let G be a group, $${\mathbb{C}}$$C be the field of complex numbers, z0 be any fixed, nonzero element in the center Z(G) of the group G, and $${\sigma : G \to G}$$σ:G→G be an involution. The main goals of this paper are to study the functional equations $${f(x{\sigma}yz_{0}) - f(xyz_{0}) = 2f(x)f(y)}$$f(xσyz0)-f(xyz0)=2f(x)f(y) and $${f(x{\sigma}yz...

This paper aims to determine the general solution f:F2→S of the equation f(ϕ(x,y,u,v))=f(x,y)f(u,v) for suitable conditions on the function ϕ:F4→F2, where F will denote either R or C, and S is a multiplicative semigroup. Using this result, we determine the general solution of several functional equations studied earlier, namely f(ux+vy,uy+vx)=f(x,y...

Let G be a group and \(\mathbb {C}\) the field of complex numbers. Suppose \(\sigma : G \rightarrow G\) is an involutive endomorphism, that is, \(\sigma \) is an endomorphism of G and it satisfies the condition \(\sigma (\sigma (x)) = x\) for all x in G. In this paper, we find the solutions \(f, g, h, k : G\rightarrow \mathbb {C}\) of the equation...

In this paper, we consider the Ulam(Hyers stability of the functional equations f(ux - vy, uy - vx) = f(x, y)f(u, v), f(ux + vy, uy - vx) = f(x, y)f(u, v), f(ux + vy, uy + vx) = f(x, y)f(u, v), f(ux - vy, uy + vx) = f(x, y)f(u, v) for all x, y, u, v ∈ ℝ, where f: ℝ² → ℝ, which arise from number theory and are connected with the characterizations of...

Let G be a group and C the field of complex numbers. Suppose σ1,σ 2 : G → G are endomorphisms satisfying the condition σi(σi(x)) = x for all x in G and for i = 1, 2. In this paper, we find the central solution f : G → C of the equation f (xy) + f (σi(y)x) =2f (x) + f (y) + f (σ2(y)) for all x,y ∈ G which is a variant of the Drygas functional equati...

In this paper, we prove some stability results concerning the generalized quadratic and quartic type functional equation in the context of non-Archimedean fuzzy normed spaces in the spirit of Hyers-Ulam-Rassias. As applications, we establish some results of approximately generalized quadratic and quartic type mapping in non-Archimedean normed space...

Using the fixed point method, we investigate the generalized Hyers–Ulam stability of the ternary homomorphisms and ternary derivations between fuzzy ternary Banach algebras for the additive functional equation of n-Apollonius type, namely ∑i=1nf(z-xi)=-1n∑1≤i<j≤nf(xi+xj)+nf(z-1n2∑i=1nxi),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackag...

We determine the general solutions f : R2 → R of the functional equation f(ux-vy, uy+v(x+y)) = f(x, y)f(u, v) for all x, y, u, v ∈ R. We also investigate both bounded and unbounded solutions of the functional inequality f(ux-vy, uy+v(x+y))-f(x, y)f(u, v)| ≤ φ(u, v) for all x, y, u, v ∈ R, where φ : R2 → R+ is a given function.

Let G be a group and (Formula presented.) the field of complex numbers. Suppose (Formula presented.) is an involution on G. In this paper, we determine the general solution (Formula presented.) of the functional equation(Formula presented.)for all (Formula presented.). In Chung et al. (J Korean Math Soc 38:37–47, 2001), the solution of the above eq...

In this paper, a network tomography (NT) approach is proposed to study network performance in ad-hoc networks. An analysis of network performance is presented in a dynamic MANET. An expectation-maximization (EM) algorithm is used in NT to estimate the network performance parameter in accordance with network performance observations. Over the dynami...

Using the direct method and the fixed point method, we prove the Hyers-Ulam stability of the following additive-cubic-quartic (ACQ) functional equation 11[f(x + 2y) + f(x - 2y)] = 44[f(x + y) + f(x - y)] + 12f(3y) - 48f(2y) + 60f(y) - 66f(x) in matrix Banach spaces. Furthermore, using the fixed point method, we also prove the Hyers-Ulam stability o...

Let $G$ be a commutative group and $\mathbb{C}$ the field of complex numbers, $\mathbb{R}^{+}$ the set of positive real numbers and $f,g,h,k:G\times \mathbb{R}^{+}\rightarrow \mathbb{C}$ . In this paper, we first consider the Levi-Civitá functional inequality $$\begin{eqnarray}\displaystyle |f(x+y,t+s)-g(x,t)h(y,s)-k(y,s)|\leq {\rm\Phi}(t,...

In this note we provide the solution to a problem posed by the first author in a previous paper. In particular, we prove a result relating the number of nonzero coefficients of a certain functional equation to the order of any derivation satisfying that equation.

This book is both a tutorial and a textbook. It is based on over 15 years of lectures in senior level calculus based courses in probability theory and mathematical statistics at the University of Louisville, USA. This book presents an introduction to probability and mathematical statistics and it is intended for students already having some mathema...

Using the fixed point method, we prove some results concerning the stability of the functional equation (FORMULA) where f is defined on a vector space and taking values in a fuzzy Banach space, which is said to be a functional equation related to a characterization of inner product spaces.

Let S be a nonunital commutative semigroup, an involution, and the set of complex numbers. In this paper, first we determine the general solutions of Wilson’s generalizations of d’Alembert’s functional equations and on nonunital commutative semigroups, and then using the solutions of these equations we solve a number of other functional equations...

In this paper, we study the Hyers–Ulam stability of a simple Levi–Civitá functional equation f(x+y)=f(x)h(y)+f(y) and its pexiderization f(x+y)= g(x) h(y)+k(y) on non-unital commutative semigroups by investigating the functional inequalities |f(x+y)−f(x)h(y)−f(y)|≤𝜖 and |f(x+y)−g(x)h(y)−k(y)|≤𝜖, respectively. We also study the bounded solutions of...

This work aims to study of the stability of two generalizations of the functional equa- tion f(pr,qs)+f(ps,qr )= f(p,q) f(r,s), namely (i) f(pr,qs)+g(ps,qr )= h(p,q)h(r,s) ,a nd (ii) f(pr,qs)+g(ps,qr )= h(p,q)k(r,s) for all p,q,r,s ∈ G ,w hereG is a commutative semi- group. Thus this work is a continuation of our earlier works (15 )a nd (16), and t...

Let S be a commutative semigroup, \({\mathbb{C}}\) the set of complex numbers, \({\mathbb{R}^+}\) the set of nonnegative real numbers, \({f, g : S \to \mathbb{C}\, \, {\rm and} \, \, \sigma : S \to S}\) an involution. In this article, we consider the stability of the Wilson’s functional equations with involution, namely \({f(x + y) + f(x + \sigma y...

Let double-struck R sign be the set of real numbers, double-struck R sign+={x∈double-struck R sign|x>0}, ε∈double-struck R sign+, and f,g,h:double-struck R sign+→d As classical and L∞ versions of the Hyers-Ulam stability of the logarithmic type functional equation in a restricted domain, we consider the following inequalities: |f(x+y)-g(xy)-h((1/x)...

This work aims to determine all functions \({f, g, h, k : G \to \mathbb{C}}\) that satisfy the functional equation f(pr, qs) + g(ps, qr) = h(p,q) + k(r,s) for all p, q, r, s ∈G, where G is a group and \({\mathbb{C}}\) is the field of complex numbers.

In this paper we determine the general solution of the equation [ f (x) + f (y)][ f (u) + f (v)] = f (xu − yv) + f (xv + yu) for all x, y, u,v without assuming any regularity condition on the unknown function f . This functional equation arises from a property of complex numbers and the solution of this functional equation is determined using an el...

More than 33 years ago M. Kuczma and R. Ger posed the problem of solving the alternative Cauchy functional equation ${f(xy) - f(x) - f(y) \in \{ 0, 1\}}$ where ${f : S \to \mathbb{R}, S}$ is a group or a semigroup. In the case when the Cauchy functional equation is stable on S, a method for the construction of the solutions is known (see Forti...

In 1997, Bailey and Bannister showed that a + b > c + h holds for all triangles with < arctan(22/7) where a, b, and c are the sides of the triangle, h is the altitude to side c, and is the angle opposite c. In this paper, we show that a + b > c + h holds approximately 92% of the time for all triangles with ɣ < π/2.

Let G be a group and H an abelian group. Let J *(G, H) be the set of solutions f: G → H of the Jensen functional equation f(xy) + f(xy-1) = 2f(x) satisfying the condition f(xyz) - f(xzy) = f(yz) - f(zy) for all x, y, z ∈ G. Let Q*(G, H) be the set of solutions f: G → H of the quadratic equation f(xy) + f(xy-1) = 2f(x) + 2f(y) satisfying the Kannapp...

Let S be a commutative semigroup, σ:S→S an endomorphism of order 2, G a 2-cancellative abelian group, and n a positive integer. One of the goals of this paper is to determine the general solutions of the functional equations f 1 (x+y)+f 2 (x+σy)=f 3 (x) and also f 1 (x+y)+f 2 (x+σy)=f 3 (x)+f 4 (y) for all x,y∈S n , where f 1 ,f 2 ,f 3 ,f 4 :S n →G...

Let S be a semigroup and X a Banach space. The functional equation φ(xyz)+φ(x)+φ(y)+φ(z)=φ(xy)+φ(yz)+φ(xz) is said to be stable for the pair (X, S) if and only if f:S→X satisfying f(xyz)+f(x)+f(y)+f(z)-f(xy)-f(yz)-f(xz)≤δfor some positive real number δ and all x,y,z∈S, there is a solution φ: S →X such that f-φ is bounded. In this paper, among other...

The present work aims to find the general solution f 1, f 2, f 3 : G 2 → H and f : G → H of the Sincov type functional equation \({f}_{1}(x,y) + {f}_{2}(y,z) + {f}_{3}(z,x) = f(x + y + z)\) for all x, y, z ∈ G without any regularity assumption. Here G and H are additive abelian groups, and the division by 2 is uniquely defined in H.

The present work aims to study the stability of the following three functional equations: (i) f(pr,qs)+f(ps,qr)=f(p,q)f(r,s), (ii) f(pr,qs)+f(ps,qr)=f(p,q)g(r,s), and (iii) f(pr,qs)+f(ps,qr)=g(p,q)f(r,s) for all p,q,r,s∈(0,1). The first functional equation arises in the characterization of symmetrically compositive sum form distance measures.

Front propagation models represent an important category of image segmentation techniques in the current literature. These models are normally formulated in a continuous level sets framework and optimized using gradient descent methods. Such formulations result in very slow algorithms that get easily stuck in local solutions and are highly sensitiv...

This work aims to determine the general solution f : F(2) -> K of the functional equation f (phi(x, y, u, v)) = f (x, y)f (u, v) for suitable conditions on the function phi : F(4) -> F(2), where F will denote either R or C, and K is an abelian group. Using this result, we determine the solution f : C(2) -> C* of the functional equation f (ux - vy,...

Introduction to Functional Equations grew out of a set of class notes from an introductory graduate level course at the University of Louisville. This introductory text communicates an elementary exposition of valued functional equations where the unknown functions take on real or complex values. In order to make the presentation as manageable as p...

We give a survey of results related the various quasi-mean value theorems for symmetrically differentiable functions and present some new results. The symmetric derivative of a real function is discussed and its elementary properties are pointed out. Some results leading to the quasi-Lagrange mean value theorem for symmetrically differentiable func...

The present work continues the study of the stability of the functional equations of the type f(pr,qs)+f(ps,qr)=f(p,q)f(r,s), namely, (i) f(pr,qs)+f(ps,qr)=g(p,q)g(r,s), and (ii) f(pr,qs)+f(ps,qr)=g(p,q)h(r,s) for all p,q,r,s∈G, where G is an abelian group. These functional equations arise in the characterization of symmetrically compositive sum fo...

The (ψ,γ)-stability of the Cauchy functional equation is investigated on some noncommutative groups. It is shown that if γ is invariant with respect to inner automorphisms of a step-two solvable group G, then the Cauchy equation f(xy)=f(x)+f(y) is (ψ,γ)-stable on G. If ψ satisfies the condition lim n→∞ ψ(n 2 ) n=0, then the Cauchy equation is (ψ,γ)...

This paper aims to study the Hyers-Ulam stability of a Sincov type [cf. D. Sintzow, Arch. d. Math. u. Phys. (3) 6, 216–217 (1903; JFM 34.0421.03)] functional equation f(x,y)+f(y,z)+f(x,z)=ℓ(x+y+z) when the domain of the functions is an abelian group and the range is a Banach space.

The present work aims to determine the solution f : R2 → R of the equation f (u x - v y, u y - v x) = f (x, y) + f (u, v) + f (x, y) f (u, v) for all x, y, u, v ∈ R without any regularity assumption. The solution of the functional equation f (u x + v y, u y - v x) = f (x, y) + f (u, v) + f (x, y) f (u, v) is also determined. The methods of solution...

In this paper the stability of the quadratic equation is considered on arbitrary groups. Since the quadratic equation is stable on Abelian groups, this paper examines the stability of the quadratic equation on noncommutative groups. It is shown that the quadratic equation is stable on $n$-Abelian groups when $n$ is a positive integer. The stability...

In this paper, the (ψ, γ)–stability of the quadratic functional equa-tion is considered on arbitrary groups. It is proved that every group can be embedded into a group in which the quadratic equation is (ψ, γ)-stable. Further, it is shown that the quadratic functional equation is (ψ, γ)-stable on all abelian groups and some non-abelian groups such...

In this paper, some new results related to the integral mean value theorem are proven.

Let $S$ be a semigroup and $X$ a Banach space. The functional equation $\phi (xyz)+ \phi (x) + \phi (y) + \phi (z) = \phi (xy) + \phi (yz) + \phi (xz)$ is said to be stable for the pair $(X, S)$ if and only if $f: S\to X$ satisfying $\| f(xyz)+f(x) + f(y) + f(z) - f(xy)- f(yz)-f(xz)\| \leq \delta $ for some positive real number $\delta$ and all $x,...

The problem of the Hyers-Ulam stability of the Hermite-Hadamard inequality posed by Zs. P´ ales is solved. It is shown that for continuous functions f : I → R neither the inequality f (x+y 2 ) 1 y−x y x f (t)dt + � nor 1 y−x y x f (t)dt f (x)+f (y) 2 + � implies the c� − convexity of f (with any c > 0) .H owever, iff is continuous and satisfies bot...

We establish the generalized stability of double centralizers associated with the Cauchy, Jensen, and Trif functional equations in the framework of Banach algebras. We also investigate the superstability of double centralizers of Banach algebras strongly without order.

In this paper, we study the stability of the system of functional equations $f(xy)+f(xy^{-1})=2f(x)+f(y)+f(y^{-1})$ and $f(yx)+f(y^{-1}x)=2f(x)+f(y)+f(y^{-1})$ on groups. Here $f$ is a real-valued function that takes values on a group. Among others we proved the following results: 1) the system, in general, is not stable on an arbitrary group; 2) t...

Drygas (1987) introduced the functional equation f(xy) + f(xy-1) = 2f(x) + f(y) + f(y-1) in connection with the characterization of quasi-inner-product spaces. In this paper, we study the system of functional equations f(xy) + f(xy-1) = 2f(x) + f(y) + f(y-1) and f(yx) + f(y-1x) = 2f(x) + f(y) + f(y-1) on groups. Here f is a real-valued function tha...

In this paper, we introduce a Graph Cut Based Level Set (GCBLS) formulation that incorporates graph cuts to optimize the curve evolution energy function presented earlier by Chan and Vese. We present a discrete form of the level set energy function, prove that it is graph-representable, and minimize it using graph cuts. The major advantages of this...

The present work aims to find the solutions f,g,h,l,m:ℝ 2 →ℝ of the functional equation f(ux-vy;uy-vx)=g(x,y)+h(u,v)+l(x,y)m(u,v) for all x,y,u,v∈ℝ without any regularity assumptions on the unknown functions. This equation is a generalization of a functional equation which arises from the characterization of the determinant of symmetric matrices.

In this paper, we present a thresholding technique based on two-dimensional Tsallis–Havrda–Charvát entropy. The effectiveness of the proposed method is demonstrated by using examples from the real-world and synthetic images.

In this paper, we establish the conditional Hyers-Ulam-Rassias stability of the generalized Jensen functional equation $r f \left (\frac{sx+ty}{r}) = s g(x) + t h(y)$ on various restricted domains such as inside balls, outside balls, and punctured spaces. In addition, we prove the orthogonal stability of this equation and study orthogonally general...

In this paper, we determine the general solution of the functional equation f 1(2x + y) + f 2(2x − y) = f 3(x + y) + f 4(x − y) + f 5(x) without assuming any regularity condition on the unknown functions f 1, f 2, f 3, f 4, f 5 : ℝ → ℝ. The general solution of this equation is obtained by finding the general solution of the functional equations f(2...

In this paper, we give a new characterization of the Stolarsky means for two positive numbers. Our result generalizes a result due to Liu [5].

Chmielinski has proved in the paper [4] the superstability of the generalized orthogonality equation \<f (x), f (y)>\ = \<x, y>\. In this paper, we will extend the result of Chmielinski by proving a theorem: Let D. be a suitable subset of R(n). If a function f : D(n) --> R(n) satisfies the inequality \\<f(x), f (y)>\ - \<x, y>\\ less than or equal...

In this paper we establish the stability of Jensen’s functional equation on some classes of groups. We prove that Jensen equation is stable on noncommutative groups such as metabelian groups and T (2, K), where K is an arbitrary commutative field with characteristic different from two. We also prove that any group A can be embedded into some group...

For any cardinal number M we construct examples of amal-gamated products and HNN extensions of groups such that the dimen-sion of the space of second bounded cohomologies is at least M. Also we describe the space of pseudocharacters of the group GL(2, F2[z]).

In this paper, we present a new thresholding technique based on two-dimensional Renyi's entropy. The two-dimensional Renyi's entropy was obtained from the two-dimensional histogram which was determined by using the gray value of the pixels and the local average gray value of the pixels. This new method extends a method due to Sahoo et al. (Pattern...

Let \mathbbK \mathbb{K} be a field of real or complex numbers and \mathbbK0 \mathbb{K}_0 denote the set of nonzero elements of \mathbbK \mathbb{K} . Let \mathbbG \mathbb{G} be an abelian group. In this paper, we solve the functional equation f 1 (x + y) + f 2 (x - y) = f 3 (x) + f 4 (y) + g(xy) by modifying the domain of the unkn...

In this paper, we determine the general solu- tion of the functional equation f(x + 2y) + f(x 2y) + 6f(x) = 4(f(x + y) + f(x y)) for all x, y 2 R without assuming any reg- ularity conditions on the unknown function f. The method used for solving this functional equation is elementary but exploits an important result due to M. Hosszu (2). The soluti...

The functional equation arises in the theory of conditionally specified distributions. In this paper, we investigate the Hyers-Ulam stability of this functional equation.

In this paper, we determine the general solution of the quartic equation f (x+2y)+f (x−2y)+6f (x) = 4[f (x+y)+f (x−y)+ 6f (y)] for all x, y ∈ R without assuming any regularity conditions on the unknown function f . The method used for solving this quartic functional equation is elementary but exploits an important result due to M. Hosszú [3]. The s...

In this paper, we show that Flett's points are stable in the sense of Hyers and Ulam.

Our main goal is to determine the solution of the functional equation f(x+t, y+t) + f(x-t, y) + f(x, y-t) = f(x-t, y-t) + f(x, y+t) + f(x+t, y)f(x+t, y+t) + f(x-t, y) + f(x, y-t) = f(x-t, y-t) + f(x, y+t) + f(x+t, y) where f is a complex valued function defined on the abelian group \mathbbZ Å\mathbbZ\mathbb{Z} \oplus \mathbb{Z}. This functional e...

In this paper, we study three functional equations that arise from the representation of the Bose-Einstein Entropy.

H. Drygas introduced the functional equation f(x+y)+f(x¡y )=2 f ( x)+f(y)+ f(¡y) in connection with quasi-inner-product spaces. In this paper, we will prove the Hyers{ Ulam stability of the functional equation of Drygas by investigating the stability problem of the functional equation f(x + y )+ f ( x ¡ y )=2 f ( x )+ g(2y). Besides the stability o...

In this paper, we introduce the concept of (ψ, γ)-pseudoadditive mappings from a semigroup into a Banach space, and we provide a generalized solution of Ulam's problem for approximately additive mappings.

By using an idea of Heuvers, Moak and Boursaw [1], we will prove a Hyers-Ulam-Rassias stability (or a general Hyers-Ulam stability) of the functional equation (1), which is closely related to the square root spiral.

Our main goal is to determine the general solution of the functional equation f(x+t,y+t)+f(x-t,y)+f(x,y-t)=f(x-t,y-t)+f(x,y+t)+f(x+t,y) where f is a complex valued function defined on the Abelian group ℤ n ⊕ℤ n . This functional equation is connected to a problem in spatial filtering of digital images and also arises while characterizing quadratic...

For a class of extensions of free products of groups, a description is given of the space of the real-valued functions ϕ defined on the group G and satisfying the conditions (1) the set {ϕ(xy) − ϕ(x) − ϕ(y) ∣ x, y ∈ G} is bounded; and (2) ϕ(xn) = nϕ(x) for any x ∈ G and any n ∈ (the set of integers).Let G be an arbitrary group and let S be its subs...

In this note, we examine the stability of the Pompeiu functional equation f (x + y + xy )= f (x )+ f (y )+ f (x)f (y).

In this paper, we introduce the concept of ( ;){pseudoadditive mappings from a semigroup into a Banach space, and we provide a generalized solution of Ulam's problem for approximately additive mappings.

In this paper, we investigate the Hyers-Ulam stability of a generalized Hosszu functional equation, namely f(x + y - xy ) + g(xy) = h(x) + k(y), where f, g, h, k are functions of a real variable with values in a Banach space.

Let \( f: G \times G \to {\Bbb C} \), where G denotes a 2-divisible abelian group and \( {\Bbb C} \) the set of complex numbers. The general solution of the functional equation¶¶f(x + t,y + t) + f(x - t,y) + f(x,y - t) = f(x - t,y - t) + f(x,y + t) + f(x + t,y)¶ for all x,y,t in G is determined. It is shown that the solution of this functional equa...

In 1839, De Morgan gave a mathematical justification of Gompertz's law of mortality through a composite functional equation, f(x+y)+f(x+z)=f(x+h(y,z)). A slightly more general version of this equation was studied in 1905 by M. Chini. Both solved their equations in the class of differentiable functions on the real line. Here we solve the equation f(...

In this paper, we will prove the Hyers-Ulam stability of the quadratic functional equation of Pexider type, f 1(x + y) + f 2(x - y) = f 3(x) + f 4(y).

We present the general solution of the functional equation f(xy,xy) + f(xy(sup)-1,x) + f(x,xy(sup)-1) = f(xy(sup)-1,xy(sup)-1) + f(xy,x) + f(x,xy). Furthermore, we also prove the Hyers-Ulam stability of the above functional equation.

We study the generalized Hyers-Ulam stability of the functional equation f[x1,x2,x3]=h(x1+x2+x3).

We prove some generalizations of Flett's mean value theorem for a class of Gateaux differentiable functions f:X→Y, where X and Y are topological vector spaces.

Flett’s mean value theorem states that if f is differentiable on [a,b] and f ' (a)=f ' (b), then there is an η∈(a,b) such that f(η)-f(a)=f ' (η)(η-a). In this article we prove an extension of this theorem and some of its variants. In particular, we show that the conclusions continue to hold for approximately differentiable functions.

In this paper, we prove the stability results of a mean value type functional equation, namely f (x) − g(y) = (x − y)h(x + y) which arises from the mean value theorem. We also prove that the above functional equation is superstable for the class of functions f, g, h : Z → Z (the set of integers).

We will prove the Hyers-Ulam stability of the Davison functional equation f(xy) + f(x+y) = f(xy+x) + f(y) for a class of functions from a field (or a commutative algebra) of characteristic different from 2 and 3 into a Banach space.

In all measures of information, the entropy f of a single event with probability p plays a fundamental role. The entropy function f, which is also known as the self information satisfies axioms of nonnegativity, additivity and maximality, that is $$ \begin{gathered} \left( a \right)\;f\left( p \right) \geqslant 0,\quad p \in \left] {0,1} \right[, \...