Bulletin of the
Korean Mathematical Society BKMS

A mean-value result, saying that the difference quotient of a differentiable function in a real interval is a mean value of its derivatives at the endpoints of the interval, leads to the functional equation \[ \frac{f(x)-F(y)}{x-y}=M\left( g(x),G(y)\right) ,\text{ \ \ \ \ }x\neq y, \] where $M$ is a given mean and $f,F,g,G$ are the unknown functions. Solving this equation for the arithmetic, geometric and harmonic means, we obtain, respectively, characterizations of square polynomials, homographic and square-root functions. A new criterion of the monotonicity of a real function is presented.

Keywords: mean-value theorem, classical means, monotonic functions, quadratic function, homographic function, square root function, functional equation

MSC numbers: Primary 39B22, 26A24, 26A48