MathDB
Technical problem phi(x)+psi(x)=x functions

Source: Miklós Schweitzer 2017, problem 6

January 13, 2018
algebraFunctional inequalityconvex functionfunctioninterval

Problem Statement

Let II and JJ be intervals. Let φ,ψ:I→R\varphi,\psi:I\to\mathbb{R} be strictly increasing continuous functions and let Φ,Ψ:J→R\Phi,\Psi:J\to\mathbb{R} be continuous functions. Suppose that φ(x)+ψ(x)=x\varphi(x)+\psi(x)=x and Φ(u)+Ψ(u)=u\Phi(u)+\Psi(u)=u holds for all x∈Ix\in I and u∈Ju\in J. Show that if f:I→Jf:I\to J is a continuous solution of the functional inequality f(φ(x)+ψ(y))≤Φ(f(x))+Ψ(f(y))(x,y∈I),f\big(\varphi(x)+\psi(y)\big)\le \Phi\big(f(x)\big)+\Psi\big(f(y)\big)\qquad (x,y\in I),then Φ∘f∘φ−1\Phi\circ f\circ \varphi^{-1} and Ψ∘f∘ψ−1\Psi\circ f\circ \psi^{-1} are convex functions.
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