On a functional equation arising from Proth identity (Q2820731)
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scientific article; zbMATH DE number 6626033
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On a functional equation arising from Proth identity |
scientific article; zbMATH DE number 6626033 |
Statements
9 September 2016
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exponential-type functional equation
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general solution
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multiplicative function
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Proth identity
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bounded solution
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functional inequality
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unbounded solutions
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On a functional equation arising from Proth identity (English)
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The authors determine the general solutions \(f:\mathbb{R}^2 \to \mathbb{R}\) of the functional equation \(f(ux-uy, uy+v(x+y))=f(x,y)f(u,v)\) for each \(x, y, u, v \in \mathbb{R}\) by a simple but different method from \textit{E. A. Chávez} and \textit{P. K. Sahoo} [Appl. Math. Lett. 24, No. 3, 344--347 (2011; Zbl 1211.39012)]. Then, by finding a condition for the solution \(f\) of the following functional inequality to be bounded, they investigate both the bounded, and unbounded solutions of the functional inequality \(f(ux-uy, uy+v(x+y))-f(x,y)f(u,v)\leq \phi(u, v)\) for each \(x, y, u, v \in \mathbb{R}\) and for some given \(\phi :\mathbb{R}^2 \to \mathbb{R}_+.\)
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