Characterizing the Logarithm Through Continuity, Monotonicity, and Functional Equations

A universal Cauchy functional equation over the positive reals

5 - EXPONENTIALS AND LOGARITHMS

The functional equation af(xy)+bf(x)f(y)+cf(x+y)+d(f(x)+f(y)) = 0 whose shape contains all the four well‐known forms of Cauchy’s functional equation is solved for solutions which are functions having the positive reals as their domain. This complements an earlier work of Dhombres in 1988 where the same functional equation was solved for solutions whose domains contain zero, which leaves out the logarithmic function. Here not only the logarithmic function is recovered but the analysis is entirely different and is based on solving appropriate difference equations.

CHAPTER 5 - EXPONENTIAL AND LOGARITHM FUNCTIONS

We prove the generalized Hyers–Ulam stability of a mean value type functional equation f ( x ) − g ( y ) = ( x − y ) h ( x + y ) by applying a method originated from fixed point theory.

Considering the serial operations as multiplication and power, we introduced the next step of them in [M.H. Hooshmand, Ultra power and ultra exponential functions, Integral Transforms Spec. Funct. 17 (2006), pp. 549–558], and called it ultra power. In that way, we proved a uniqueness theorem for the (ultra exponential) functional equation f(x)=a f(x−1), x>−1, with the initial condition f(0)=1, and got the new function uxp a for which uxp a (n)=a n (a to the ultra power of n) for every n∈ℕ. For this reason, we call uxp a ultra exponential function. In this paper, we want to introduce another new function Iog a , namely infra logarithm function, that is the dual of uxp a and for a>1 is its inverse. We show that Iog a satisfies the dual of the ultra exponential equation that is the same Abel's exp a -functional equation (f(a x )=f(x)+1), where x runs over ℝ\\ [δ1, δ2] (δ1, δ2 will be introduced). For this reason, sometimes we call it infra logarithm functional equation. There after, we prove that Iog a is the unique solution of the Abel's equation, under some conditions, and show that it together with ultra power part function (another new function) are the essential solutions of it, and then determine the general solution of the Abel's exp a -equation for every 0<a≠1. Also we state another functional equation f(a x )=f((x))+[x]+1, namely co-infra logarithm functional equation, and prove that it is equivalent to the Abel's equation when x runs over . Moreover, we introduce some properties of the infra logarithm function and its relations to the ultra exponential function and their related equations.

An Introduction to Real Analysis

6 - Exponentials and Logarithms

6 - Exponentials and Logarithms

The present work continues the study for the superstability and solution of the Pexider type functional equation , which is the mixed functional equation represented by sum of the sine, cosine, tangent, hyperbolic trigonometric, and exponential functions. The stability of the cosine (d'Alembert) functional equation and the Wilson equation was researched by many authors: Baker [7], Badora [5], Kannappan [14], Kim ([16, 19]), and Fassi, etc [11]. The stability of the sine type equations was researched by Cholewa [10], Kim ([18], [20]). The stability of the difference type equation for the above equation was studied by Kim ([21], [22]). In this paper, we investigate the superstability of the sine functional equation and the Wilson equation from the Pexider type difference functional equation , which is the mixed equation represented by the sine, cosine, tangent, hyperbolic trigonometric functions, and exponential functions. Also, we obtain additionally that the Wilson equation and the cosine functional eqaution in the obtained results can be represented by the composition of a homomorphism. In here, the domain (G; +) of functions is a noncommutative semigroup (or 2-divisible Abelian group), and A is an unital commutative normed algebra with unit 1A. The obtained results can be applied and expanded to the stability for the difference type's functional equation which consists of the (hyperbolic) secant, cosecant, logarithmic functions.

It is not difficult to demonstrate the Hyers–Ulam stability of the logarithmic functional equation \(f(xy)=f(x)+f(y)\ {\rm for\ functions}\ f:(0,\infty)\rightarrow E\), where E is a Banach space. More precisely, if a function \(f:(0,\infty)\rightarrow E\) satisfies the functional inequality \(\| f(xy)-f(x)-f(y)\| \leq \delta\ {\rm for\ some}\ \delta > 0\ {\rm and\ for\ all}\ x,y > 0\), then there exists a unique logarithmic function \(L:(0,\infty)\rightarrow E\) (this means that \(L(xy)=L(x)+L(y)\ {\rm for\ all}\ x,y >0\)) such that \(\| f(x)-L(x)\| \leq \delta\ {\rm for\ any}\ x >0\). In this chapter, we will introduce a new functional equation \(f(x^y)=yf(x)\) which has the logarithmic property in the sense that the logarithmic function \(f(x)={\rm ln}x (x > 0)\) is a solution of the equation. Moreover, the functional equation of Heuvers \(f(x+y)=f(x)+f(y)+f(1/x+1/y)\) will be discussed.

CHAPTER VI - EXPONENTIAL AND LOGARITHMIC FUNCTIONS

To date, the bestmethodsfor estimating the growth of mean values of arithmetic functions rely on the Voronoï summation formula. By noticing a general pattern in the proof of his summation formula, Voronoï postulated that analogous summation formulas for $\sum a(n)f(n)$ can be obtained with ‘nice’ test functions f(n), provided a(n) is an ‘arithmetic function’. These arithmetic functions a(n) are called so because they are expected to appear as coefficients of some L-functions satisfying certain properties. It has been well-known that the functional equation for a general L-function can be used to derive a Voronoï-type summation identity for that L-function. In this article, we show that such a Voronoï-typesummation identity in fact endows the L-function with some structural properties, yielding in particular the functional equation. We do this by considering Dirichlet series satisfying functional equations involving multiple Gamma factors and show that a given arithmetic function appears as a coefficient of such a Dirichlet series if and only if it satisfies the aforementioned summation formulas.

In this paper we derive the explicit, closed-form, recursion-free formulae for the arbitrary-order Fr?chet derivatives of the exponential and logarithmic functions in unital Banach algebras (complex or real). These computations are obtained via the Bochner integrals for the Banach algebra valued functions, with respect to the standard Lebesgue measure. As an application, we utilize our results in the approximation schemes of the solutions to stochastic functional differential equations.

We consider the functional equation \( f(xf(x))=\varphi (f(x)) \) where \( \varphi: J\rightarrow J \) is a given increasing homeomorphism of an open interval \( J \subset (0,\infty) \), and \( f:(0,\infty )\rightarrow J \) is an unknown continuous function. If 1 is a fixed point of \( \varphi \) then the solutions are pointwise complete in the class of continuous monotone functions, i.e., for any point \( (x_0,y_0) \in (0,\infty) \times J \) there is a continuous monotone solution passing through it. On the other hand, if 1 is not fixed by \( \varphi \) then there exist exceptional values y 0 if and only if there is a minimal open \( \varphi \)-invariant interval \( K \subset J \) (i.e., \( \varphi (K)=K \)) containing 1; the exceptional values of y 0 are then just the elements of K. We also show that the continuous solutions cannot cross the line y = p where p is a fixed point of \( \varphi \), unless p = 1.¶We give a characterization of the class of monotone continuous solutions; it depends on a family of monotone functions defined on a compact “initial” interval. We provide a sufficient condition for any continuous solution to be monotone. It implies, among others, that 0 or \( \infty \); is an end-point of J. We conjecture that this condition is also necessary.