A note on functional equations connected with the Cauchy mean value theorem
Aequationes mathematicae - 2018
Tóm tắt
The aim of this paper is to describe the solution (f, g) of the equation
$$\begin{aligned}{}[f(x)-f(y)]g'(\alpha x+(1-\alpha )y)= [g(x)-g(y)]f'(\alpha x+(1-\alpha )y),\ x,y\in I, \end{aligned}$$
where
$$I\subset \mathbb {R}$$
is an open interval,
$$f,g:I\rightarrow \mathbb {R}$$
are differentiable,
$$\alpha $$
is a fixed number from (0, 1).
Từ khóa
Tài liệu tham khảo
Aczél, J.: A mean value property of the derivative of quadratic polynomials-without mean values and derivatives. Math. Mag. 58(1), 42–45 (1985)
Balogh, Z.M., Ibrogimov, O.O., Mityagin, B.S.: Functional equations and the Cauchy mean value theorem. Aequ. Math. 90, 683–697 (2016)
Haruki, S.: A property of quadratic polynomials. Am. Math. Monthly 86(7), 577–579 (1979)
Lundberg, A.: A rational Sûto equation. Aequ. Math. 57, 254–277 (1999)
Lundberg, A.: Sequential derivatives and their application to a Sûto equation. Aequ. Math. 61, 48–59 (2001)
Sahoo, P.K., Riedel, T.: Mean Value Theorems and Functional Equations. World Scientific Publishing Co., Inc., River Edge (1998)