Let \(X\) be a nonempty set, \(F: X \times X \to \mathbb{C}\) a function, \(c\ge 0\) a fixed constant and \(H\) be an inner product. The author studies a connection between \[ |F(a,x) F(x,b) - F(a,b)|\le c, \quad a,b,x\in X,\tag{1} \] which is related to the Sincov functional equation (see [D. Gronau, ESAIM, Proc. Surv. 46, 43–46 (2014; Zbl 1330.01049)]), and \[ |F(u,v)|\le c, \quad u,v\in X.\tag{2} \] The inequality (1) generalizes \[ \left|\frac{2\langle a, x\rangle}{\|a\| \cdot \|x\|} \cdot \frac{2\langle x, b\rangle}{\|x\| \cdot \|b\|} - \frac{2\langle a, b\rangle}{\|a\| \cdot \|b\|} \right| \le 2, \quad a,b,x \in H\setminus \{0\}, \] which is the same as the Richard inequality (see [M. L. Buzano, Univ. Politec. Torino, Rend. Sem. Mat. 31(1971–72/1972–73), 405–409 (1974; Zbl 0285.46016)]), and the inequality (2) generalizes \[ \left| \frac{2\langle u, v\rangle}{\|u\| \cdot \|v\|} \right|\le 2, \quad u,v \in H\setminus \{0\}, \] which is equivalent to the Cauchy-Schwarz inequality (see [S. S. Dragomir, Linear Multilinear Algebra 65, No. 3, 514–525 (2017; Zbl 1370.47003)]).
The results are useful for researchers on inequalities, but the references are mixed up.