MathDB

Let

n\geqslant2

be a natural number. Show that there is no real number

c{}

for which

\exp\left(\frac{T+S}{2}\right)\leqslant c\cdot \frac{\exp(T)+\exp(S)}{2}

is satisfied for any self-adjoint

n\times n

complex matrices

T{}

and

S{}

. (If

A{}

and

B{}

are self-adjoint

n\times n

matrices,

A\leqslant B

means that

B-A

is positive semi-definite.)

Problem Statement