On the Equality of Bajraktarević Means to Quasi-Arithmetic Means

This paper offers a solution of the functional equation (tf(x)+(1-t)f(y) )φ (tx+(1-t)y) =tf(x)φ (x)+(1-t)f(y)φ (y) (x,y∈ I), where t∈ ]0,1[ , φ :I→ℝ is strictly monotone, and f:I→ℝ is an arbitrary unknown function. As an immediate application, we shed new light on the equality problem of Bajraktarević means with quasi-arithmetic means.