LAUSR

Let $${\mathscr {H}}$$H be a Hilbert space, J be an open interval and $$B_J({\mathscr {H}})$$BJ(H) be the set of all self-adjoint operators on $${\mathscr {H}}$$H with spectra in J. Suppose that $$\phi :B_J({\mathscr {H}})\rightarrow B({\mathscr {H}})$$ϕ:BJ(H)→B(H) is an operator map satisfying $$\begin{aligned} \phi (C^*AC+aD^*D)\le C^*\phi (A)C+\phi (a.1_{\mathscr {H}})D^*D, \end{aligned}$$ϕ(C∗AC+aD∗D)≤C∗ϕ(A)C+ϕ(a.1H)D∗D,for every $$A\in B_J({\mathscr {H}})$$A∈BJ(H) with finite spectrum, every $$a\in J$$a∈J and all operators C, D on $${\mathscr {H}}$$H with $$C^*C+D^*D=1_{\mathscr {H}}$$C∗C+D∗D=1H. We prove that there exists a real convex function f on J such that $$\phi (A)=f(A)$$ϕ(A)=f(A) for every $$A\in B_J({\mathscr {H}})$$A∈BJ(H) with finite spectrum and $$\phi (A)\le f(A)$$ϕ(A)≤f(A) for every $$A\in B_J({\mathscr {H}})$$A∈BJ(H). If, moreover, $$\phi $$ϕ is monotone, then the equality is obtained for all $$A\in B_J({\mathscr {H}})$$A∈BJ(H). We apply these results to investigate the relation between the above inequality and an inequality of Jensen-type by Jensen–Mercer operator inequality.