Around Choi inequalities for positive linear maps

We show an inequality for unital positive linear maps ΦΦ interpolating Choi’s inequality (p=0)(p=0) with a recent result of Bourin and Ricard (r=1)(r=1):LetA∈A+,0⩽p⩽qandqq+p⩽r⩽2qq+p.Then|Φ(Ap)rΦ(Aq)r|⩽Φ(A(p+q)r).