1. Introduction
Recall that the Jensen functional
is defined on an interval
by
where
,
and
is a positive weight sequence.
If
h is a convex function on
I then the inequality
holds for each
and any positive weight sequence
p.
If
h is a concave function on
I then the above inequality is reversed. Those inequalities play a fundamental role in many parts of mathematical analysis and applications. For example, the well-known
inequality, Holder’s inequality, Ky Fan inequality, etc., are proven by the help of Jensen’s inequality (cf. [
1,
2,
3,
4,
5,
6]).
Our aim in this paper is to find the simplest constant
C such that
for any choice of
and thus make this inequality symmetrical.
This will be done by assuming that
, and we shall find some
for the generalized Jensen functional
that is, the bounds not depending on
p or
x but only on
and functions
g and
h.
In this sense, a typical result is given by the part of Theorem 1 (below).
For
, let
be convex and
be an increasing function. Then
Our global bounds will be entirely presented in terms of elementary means.
Recall that the
mean is a map
, with a property
for each
.
In the sequel we shall use the class of so-called Stolarsky (or extended) two-parametric mean values, defined for positive values of
by the following
In this form it was introduced by Keneth Stolarsky in [
7].
Most of the classical two variable means are special cases of the class
E. For example,
is the arithmetic mean;
is the geometric mean;
is the logarithmic mean;
is the identric mean, etc.
More generally, the
r-th power mean
is equal to
.
Using the class of Stolarsky means enables our results to be presented in a condensed and applicable way. For example, we give some results regarding
inequalities, where
are the generalized arithmetic, geometric and harmonic means, respectively.
Let
. Then
where
stands for the arithmetic, geometric, harmonic, logarithmic and identric means of positive numbers
a and
b, respectively.
All bounds above are the best possible.
2. Results and Proofs
Our results concerning global bounds for the generalized Jensen functional are given in the following two assertions.
Theorem 1. 1. For continuous functions let be convex and be an increasing function or be concave and be a decreasing function. Then 2. If is convex and is a decreasing function or is concave and is an increasing function. Then Proof. We shall prove only the part 1. The proof of part 2 of this theorem is analogous.
Therefore, if
h is a convex function on
J we have
. Since
g is an increasing function, it follows that
Similarly, if
h is a concave function on
J we have
. Since
g is a decreasing function, it follows again that
On the other hand, since , there exist non-negative numbers , such that .
Hence,
where we denoted
.
The second case with concave h and decreasing g leads to the same result. □
Note that the function is continuous in p and non-negative with . Therefore, exists. Another and sometimes difficult problem is to evaluate its exact value (see Open Problem below).
For this cause, we give an estimation of with a unique maximum, which could be easily calculated. This method can be applied to the second part of Theorem 1, as well.
Theorem 2. 1. Under the conditions of the first part of Theorem 1, assume firstly that g is a convex function on J. Thenwhere . 2. Assuming that is a concave function, we have Now, both maximums can be easily determined by the standard technique.
Proof. By the first part of Theorem 1, we found that there exists
such that
If additionally
g is convex on
J, then
Consequently, if
is a concave function on
J, we have
and
□
3. Applications
The results above are the source of a number of interesting inequalities. For instance, taking
in Theorem 1, we are enabled to determine converses of the quotient
Or, taking
, we can estimate the difference
where
is the quasi-arithmetic mean and
is the generalized arithmetic mean.
We shall specialize this argument for the class of generalized power means
of order
, where
Some important particular cases are
that is, the generalized harmonic, geometric and arithmetic means, respectively.
It is well-known that
is monotone increasing in
(cf. [
4]).
Therefore,
represents the famous
inequality.
As an application of Theorem 1, we shall estimate the difference .
Theorem 3. Let .
Proof. Let .
If
, then
h is a convex function and
g is monotone increasing on
. Hence, by the first part of Theorem 1, we obtain
This maximum is easy to calculate and we obtain
In cases and one should apply the second part of Theorem 1, since then h is concave and g is increasing in the first case and h is convex and g is decreasing in the second case. Proceeding as above, the result follows. □
As a consequence, we obtain some converses of the inequality.
Corollary 1. Let .
Proof. Putting
, we obtain
□
Corollary 2. Let .
Proof. After somewhat laborous calculation using Taylor series, the result follows. □
Remark 1. Estimating the Jensen functionalfor and then changing variables , we obtain the same result. Open problem Find the exact upper global bound for The next proposition gives global bounds for the quotient of two power means.
Theorem 4. For and , we have Both bounds are the best possible.
Proof. Applying the method from the proof of Theorem 1, we obtain
In the cases or , we have that the function is convex.
Therefore, it follows that
By standard means we obtain that this maximum satisfies the equation
that is,
Hence,
and we finally obtain
In the third case, for
, we have
since
It is obvious that 1 is the best possible lower global bound. To prove that
is also the best possible global bound, denote by
an arbitrary upper bound. Then the relation
holds for any
p and
x.
Putting
, we obtain
and the proof is complete. □
Some important consequences of this theorem are given in the following
In the last two corollaries we used the identity
Finally, putting in Corollary 4 and in Corollary 5, since , we obtain global converses of the inequality.
Therefore, a sort of tight symmetry is established for these inequalities.
4. Conclusions
We give a method for two-sided estimations of the generalized Jensen functional , with applications to the general means. In particular, sharp converses of the famous
inequality are obtained. Further investigations can be undertaken on more general settings, i.e., or even , with properly chosen functions and .