A functional inequality for entire functions generalizing the sine functional equation

1974 ◽  
Vol 10 (2-3) ◽  
pp. 318-318
Author(s):  
Lawrence Etigson
Keyword(s):  
Functional Equation ◽  
Entire Functions ◽  
Functional Inequality ◽  
Sine Functional Equation

On an Exponential Functional Inequality and its Distributional Version

2015 ◽  
Vol 58 (1) ◽  
pp. 30-43 ◽  
Author(s):  
Jaeyoung Chung
Keyword(s):  
Functional Equation ◽  
Tensor Product ◽  
Functional Inequality ◽  
Schwartz Distribution ◽  
Exponential Functional ◽  
The Stability

AbstractLet G be a group and 𝕂 = ℂ or ℝ. In this article, as a generalization of the result of Albert and Baker, we investigate the behavior of bounded and unbounded functions f : G → 𝕂 satisfying the inequalityWhere ϕ: Gn-1 → [0,∞]. Also as a a distributional version of the above inequality we consider the stability of the functional equationwhere u is a Schwartz distribution or Gelfand hyperfunction, o and ⊗ are the pullback and tensor product of distributions, respectively, and S(x1, ..., xn) = x1 + · · · + xn.

scholarly journals A Class of functional equations which have entire solutions

1988 ◽  
Vol 38 (3) ◽  
pp. 351-356 ◽  
Author(s):  
Peter L. Walker
Keyword(s):  
Functional Equation ◽  
Entire Function ◽  
Wide Class ◽  
Functional Equations ◽  
Entire Functions ◽  
Inverse Function ◽  
Entire Solution ◽  
Entire Solutions ◽  
Image Position

We consider the Abelian functional equationwhere φ is a given entire function and g is to be found. The inverse function f = g−1 (if one exists) must satisfyWe show that for a wide class of entire functions, which includes φ(z) = ez − 1, the latter equation has a non-constant entire solution.

A sine functional equation

1970 ◽  
Vol 4 (1-2) ◽  
pp. 56-62 ◽  
Author(s):  
J. A. Baker
Keyword(s):  
Functional Equation ◽  
Sine Functional Equation

ON THE SUPERSTABILITY OF THE PEXIDER TYPE SINE FUNCTIONAL EQUATION

2012 ◽  
Vol 25 (1) ◽  
pp. 1-18
Author(s):  
Gwang Hui Kim
Keyword(s):  
Functional Equation ◽  
Sine Functional Equation

scholarly journals On the Superstability of Lobačevskiǐ’s Functional Equations with Involution

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Jaeyoung Chung ◽  
Bogeun Lee ◽  
Misuk Ha
Keyword(s):  
Functional Equation ◽  
Euclidean Space ◽  
Functional Equations ◽  
Direct Consequence ◽  
Functional Inequality ◽  
Commutative Group ◽  
Bounded Functions

LetGbe a uniquely2-divisible commutative group and letf,g:G→Candσ:G→Gbe an involution. In this paper, generalizing the superstability of Lobačevskiǐ’s functional equation, we considerf(x+σy)/22-g(x)f(y)≤ψ(x)orψ(y)for allx,y∈G, whereψ:G→R+. As a direct consequence, we find a weaker condition for the functionsfsatisfying the Lobačevskiǐ functional inequality to be unbounded, which refines the result of Găvrută and shows the behaviors of bounded functions satisfying the inequality. We also give various examples with explicit involutions on Euclidean space.

scholarly journals The Cosine-Sine Functional Equation on Semigroups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Bruce Ebanks
Keyword(s):  
Functional Equation ◽  
General Solution ◽  
Direct Solution ◽  
Continuous Solutions ◽  
Topological Semigroups ◽  
Primary Object ◽  
Object Of Study ◽  
Special Case ◽  
Sine Functional Equation

Abstract The primary object of study is the “cosine-sine” functional equation f(xy) = f(x)g(y)+g(x)f(y)+h(x)h(y) for unknown functions f, g, h : S → ℂ, where S is a semigroup. The name refers to the fact that it contains both the sine and cosine addition laws. This equation has been solved on groups and on semigroups generated by their squares. Here we find the solutions on a larger class of semigroups and discuss the obstacles to finding a general solution for all semigroups. Examples are given to illustrate both the results and the obstacles. We also discuss the special case f(xy) = f(x)g(y) + g(x)f(y) − g(x)g(y) separately, since it has an independent direct solution on a general semigroup. We give the continuous solutions on topological semigroups for both equations.

On a Sine Functional Equation

1963 ◽  
Vol 70 (3) ◽  
pp. 306 ◽  
Author(s):  
Sanford L. Segal
Keyword(s):  
Functional Equation ◽  
Sine Functional Equation

HEAT KERNEL METHOD FOR THE LEVI-CIVITÁ EQUATION IN DISTRIBUTIONS AND HYPERFUNCTIONS

2015 ◽  
Vol 92 (1) ◽  
pp. 77-93
Author(s):  
JAEYOUNG CHUNG ◽  
PRASANNA K. SAHOO
Keyword(s):  
Functional Equation ◽  
Stability Problem ◽  
Kernel Method ◽  
Functional Inequality ◽  
Complex Numbers ◽  
Ulam Stability ◽  
Real Numbers ◽  
Positive Real ◽  
Image Position ◽  
Ulam Stability Problem

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$$\begin{eqnarray}\displaystyle |f(x+y,t+s)-g(x,t)h(y,s)-k(y,s)|\leq {\rm\Phi}(t,s),\quad x,y\in G,t,s>0, & & \displaystyle \nonumber\end{eqnarray}$$where${\rm\Phi}:\mathbb{R}^{+}\times \mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$is a symmetric decreasing function in the sense that${\rm\Phi}(t_{2},s_{2})\leq {\rm\Phi}(t_{1},s_{1})$for all$0<t_{1}\leq t_{2}$and$0<s_{1}\leq s_{2}$. As an application, we solve the Hyers–Ulam stability problem of the Levi-Civitá functional equation$$\begin{eqnarray}\displaystyle u\circ S-v\otimes w-k\circ {\rm\Pi}\in {\mathcal{D}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})\quad [\text{respectively}\;{\mathcal{A}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})] & & \displaystyle \nonumber\end{eqnarray}$$in the space of Gelfand hyperfunctions, where$u,v,w,k$are Gelfand hyperfunctions,$S(x,y)=x+y,{\rm\Pi}(x,y)=y,x,y\in \mathbb{R}^{n}$, and$\circ$,$\otimes$,${\mathcal{D}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})$and${\mathcal{A}}_{L^{\infty }}^{\prime }(\mathbb{R}^{2n})$denote pullback, tensor product and the spaces of bounded distributions and bounded hyperfunctions, respectively.

scholarly journals Stability of the Cosine–Sine Functional Equation on Amenable Groups

2020 ◽  
pp. 355-378
Author(s):  
Ajebbar Omar ◽  
Elqorachi Elhoucien
Keyword(s):  
Functional Equation ◽  
Amenable Groups ◽  
Sine Functional Equation

scholarly journals On two variable functional inequality and related functional equation

2009 ◽  
pp. 799-804
Author(s):  
Mirosław Adamek
Keyword(s):  
Functional Equation ◽  
Functional Inequality
Sign in / Sign up

Export Citation Format

Share Document