Stability of multiplicative inverse functional equations in three variables (Q5891748)

The article deals with the Ulam stability property for the following functional equations NEWLINE\[NEWLINEr\bigg(\frac{x + y + z}3\bigg) - r(x + y + z) - \frac{2r\big(\frac{x+y}2\big)r\big(\frac{y+z}2\big)r\big(\frac{x+z}2\big)}{r\big(\frac{x+y}2\big)r\big(\frac{y+z}2\big) + r\big(\frac{y+z}2\big)r\big(\frac{x+z}2\big) + r\big(\frac{x+z}2\big)r\big(\frac{x+y}2\big)} = 0\tag{1}NEWLINE\]NEWLINE and NEWLINE\[NEWLINEr\bigg(\frac{x + y + z}3\bigg) + r(x + y + z) - \frac{4r\big(\frac{x+y}2\big)r\big(\frac{y+z}2\big)r\big(\frac{x+z}2\big)}{r\big(\frac{x+y}2\big)r\big(\frac{y+z}2\big) + r\big(\frac{y+z}2\big)r\big(\frac{x+z}2\big) + r\big(\frac{x+z}2\big)r\big(\frac{x+y}2\big)} = 0.\tag{2}NEWLINE\]NEWLINE Both equations are considered in the spaces of non-zero real numbers (?). The author describes the conditions for a function \(\psi(x,y,z)\) under which the inequality \(|I| \leq \psi(x,y,z)\) (\(I\) is the left hand part of (1) or (2)) implies that \(r\) is represented in the form \(r(x) = A(x) + \omega(x)\), where \(A(x)\) is a solution correspondingly to (1) or (2) (these solutions are called multiplicative inverse functions, among them \(f(x) = x^{-1}\)) and \(\omega(x)\) is small in some sense. Some partial cases are considered.