Let
be a selfadjoint linear operator on a complex Hilbert space
. The *Gelfand map* establishes a
-isometrically isomorphism
between the set
of all *continuous functions* defined on the *spectrum* of
denoted
and the
-algebra
generated by
and the identity operator
on
as follows (see e.g., [1, page 3]):

For any
and any
we have

(i)

(ii)
and

(iii)

(iv)
and
where
and
for

With this notation we define

and we call it the *continuous functional calculus* for a selfadjoint operator

If

is a selfadjoint operator and

is a real valued continuous function on

, then

for any

implies that

that is,

is a positive operator on

Moreover, if both

and

are real valued functions on

then the following important property holds:

in the operator order of

For a recent monograph devoted to various inequalities for functions of selfadjoint operators, see [1] and the references therein. For other results, see [2–4].

The following result that provides an operator version for the Jensen inequality is due to [5] (see also [1, page 5]).

Theorem 1.1 (Mond and Pečarić, 1993, [5]).

Let

be a selfadjoint operator on the Hilbert space

and assume that

for some scalars

with

If

is a convex function on

then

for each
with

As a special case of Theorem 1.1 we have the following Hölder-McCarthy inequality.

Theorem 1.2 (Hölder-McCarthy, 1967, [6]).

Let
be a selfadjoint positive operator on a Hilbert space
. Then

(i)
for all
and
with

(ii)
for all
and
with

(iii)if
is invertible, then
for all
and
with

The following theorem is a multiple operator version of Theorem 1.1 (see e.g., [1, page 5]).

Theorem 1.3.

Let

be selfadjoint operators with

,

for some scalars

and

with

. If

is a convex function on

, then

The following particular case is of interest. Apparently it has not been stated before either in the monograph [1] or in the research papers cited therein.

Corollary 1.4.

Let

be selfadjoint operators with

,

for some scalars

If

with

then

for any
with

Proof.

It follows from Theorem 1.3 by choosing
where
with

Remark 1.5.

The above inequality can be used to produce some norm inequalities for the sum of positive operators in the case when the convex function

is nonnegative and monotonic nondecreasing on

Namely, we have

The inequality (1.4) reverses if the function is concave on
.

As particular cases we can state the following inequalities:

for

and

for

If
are positive definite for each
, then (1.5) also holds for

If one uses the inequality (1.4) for the exponential function, then one obtains the inequality

where
are positive operators for each

In Section
of the monograph [1] there are numerous and interesting converses of the Jensen type inequality from which we would like to mention one of the simplest (see [4] and [1, page 61]).

Theorem 1.6.

Let

be selfadjoint operators with

,

, for some scalars

and

with

. If

is a strictly convex function twice differentiable on

, then for any positive real number

one has

The case of equality was also analyzed but will be not stated in here.

The main aim of the present paper is to provide different reverses of the Jensen inequality where some upper bounds for the nonnegative difference

will be provided. Applications for some particular convex functions of interest are also given.