Refinements of Jessen’s functional
We obtain new refinements of Jessen’s functional defined by means of positive linear functionals. The accumulated results are then applied to weighted generalized and power means. We also obtain new refinements of numerous classical inequalities such as the arithmetic-geometric mean inequality, Youn...
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| author | Barbir, A. Himmelreich, Kruli´c K. Pečarić, J. E. Барбір, А. Хіммельрайх, Крилік К. Печарик, Й. Е. |
| author_facet | Barbir, A. Himmelreich, Kruli´c K. Pečarić, J. E. Барбір, А. Хіммельрайх, Крилік К. Печарик, Й. Е. |
| author_sort | Barbir, A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:30:55Z |
| description | We obtain new refinements of Jessen’s functional defined by means of positive linear functionals. The accumulated
results are then applied to weighted generalized and power means. We also obtain new refinements of numerous classical
inequalities such as the arithmetic-geometric mean inequality, Young’s inequality, and H¨older’s inequality. |
| first_indexed | 2026-03-24T02:14:38Z |
| format | Article |
| fulltext |
UDC 517.5
A. Barbir, K. Krulić Himmelreich, J. Pečarić (Univ. Zagreb, Croatia)
REFINEMENTS OF JESSEN’S FUNCTIONAL*
УТОЧНЕННЯ ФУНКЦIОНАЛА ЙЄССЕНА
We obtain new refinements of Jessen’s functional defined by means of positive linear functionals. The accumulated
results are then applied to weighted generalized and power means. We also obtain new refinements of numerous classical
inequalities such as the arithmetic-geometric mean inequality, Young’s inequality, and Hölder’s inequality.
Отримано новi уточнення функцiонала Йєссена, визначенi у термiнах додатних лiнiйних функцiоналiв. Отриманi
результати застосовано до зважених узагальнених та степеневих середнiх. Також отримано новi уточнення чис-
ленних класичних нерiвностей, таких як нерiвнiсть для арифметично-геометричних середнiх, нерiвностi Янга та
Гельдера.
1. Introduction. Let us denote with \scrP n the set of all real n-tuples \bfp = (p1, . . . , pn) such that
Pk :=
\sum k
i=1
pi, k = 1, . . . , n, with 0 \leq Pk \leq Pn, k = 1, . . . , n - 1, and Pn > 0. Let I be an
interval in \BbbR and \Phi : I \rightarrow \BbbR a convex function. If \bfx = (x1, . . . , xn) is a monotonic (increasing or
decreasing) n-tuple in In and \bfp is in \scrP n, then Jensen – Steffensen’s inequality (for more details see
[17, p. 57])
\Phi
\Biggl(
1
Pn
n\sum
i=1
pixi
\Biggr)
\leq 1
Pn
n\sum
i=1
pi\Phi (xi) (1.1)
holds. Now, we define a functional as the difference between the right-hand side and the left-hand
side of (1.1) multiplied by Pn
J(\Phi ,\bfx ,\bfp ) =
n\sum
i=1
pi\Phi (xi) - Pn\Phi
\left( \sum n
i=1
pixi
Pn
\right) . (1.2)
We call it discrete Jensen – Steffensen’s functional. For a fixed function \Phi and n-tuple \bfx , J(\Phi ,\bfx , \cdot )
can be considered as a function on the set \scrP n. Because of (1.1) we have that J(\Phi ,\bfx ,\bfp ) \geq 0 for all
\bfp in \scrP n.
Inequality (1.1) can be observed under stricter conditions on \bfp to obtain the well known Jensen’s
inequality. Let \Phi : I \subset \BbbR \rightarrow \BbbR be a convex function, \bfx = (x1, x2, . . . , xn) \in In, n \geq 2, and
\bfp = (p1, p2, . . . , pn) is positive n-tuple of real numbers with Pn =
\sum n
i=1
pi.
In this case, observing the difference between the right-hand side and the left-hand side of Jensen’s
inequality, Dragomir et al. (see [9]) introduced and investigated discrete Jensen’s functional
Jn(\Phi ,\bfx ,\bfp ) =
n\sum
i=1
pi\Phi (xi) - Pn\Phi
\left( \sum n
i=1
pixi
Pn
\right) . (1.3)
Let \scrP 0
n denote the set of all nonnegative n-tuples of real numbers with Pn =
\sum n
i=1
pi > 0.
Obviously, \scrP 0
n \subset \scrP n. For a fixed function \Phi and n-tuple \bfx , Jn(\Phi ,\bfx , \cdot ) can be considered as a
function on the set \scrP 0
n.
* This work has been supported by Croatian Science Foundation under the project 5435.
c\bigcirc A. BARBIR, K. KRULIĆ HIMMELREICH, J. PEČARIĆ, 2016
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7 879
880 A. BARBIR, K. KRULIĆ HIMMELREICH, J. PEČARIĆ
Dragomir et al. (see [9]) obtained that such functional is superadditive on the set of positive real
n-tuples, that is
Jn(\Phi ,\bfx ,\bfp + \bfq ) \geq Jn(\Phi ,\bfx ,\bfp ) + Jn(\Phi ,\bfx ,\bfq ). (1.4)
Further, above functional is also increasing in the same setting, that is,
Jn(\Phi ,\bfx ,\bfp ) \geq Jn(\Phi ,\bfx ,\bfq ) \geq 0, (1.5)
where \bfp \geq \bfq (i.e., pi \geq qi, i = 1, 2, . . . , n). Monotonicity property of discrete Jensen’s functional
was proved few years before (see [13, p. 717]). Above mentioned properties provided refinements of
numerous classical inequalities. For more details about such extensions see [9]. Krnić et al. proved the
superadditivity property and monotonicity property of the functional (1.1), for more details see [12].
It is well known that Jensen’s inequality can be regarded in a more general manner, including
positive linear functionals acting on linear class of real valued functions.
More precisely, let E be nonempty set and let \scrL (E,\BbbR ) be any linear class of real-valued functions
f : E \rightarrow \BbbR satisfying following properties:
(L1) f, g \in \scrL (E,\BbbR ) \Rightarrow \alpha f + \beta g \in \scrL (E,\BbbR ) for all \alpha , \beta \in \BbbR ;
(L2) 1 \in \scrL (E,\BbbR ), that is, if f(t) = 1 for all t \in E, then f \in \scrL (E,\BbbR ).
We also consider positive linear functionals A : \scrL (E,\BbbR ) \rightarrow \BbbR . That is, we assume that
(A1) A(\alpha f + \beta g) = \alpha A(f) + \beta A(g) for f, g \in \scrL (E,\BbbR ), \alpha , \beta \in \BbbR ;
(A2) f \in \scrL (E,\BbbR ), f(t) \geq 0 for all t \in E \Rightarrow A(f) \geq 0.
Further, if
(A3) A(1) = 1
also holds, we say that A is normalized positive linear functional or A(f) is linear mean defined on
\scrL (E,\BbbR ).
Jessen’s generalization of Jensen’s inequality (see [17, p. 47, 48]), in view of positive functionals,
claims that
\Phi (A(f)) \leq A(\Phi (f)), (1.6)
where \Phi is continuous convex function on interval I \subseteq \BbbR , f attains its values on the interval I,
A is normalized positive linear functional, and f \in \scrL (E,\BbbR ) such that \Phi (f) \in \scrL (E,\BbbR ). Jessen’s
inequality was extensively studied during the eighties and early nineties of the last century (see
[7, 8, 10, 14 – 16, 18]).
In this paper we define Jessen’s functional including positive functional. Before we define such
functional, we have to establish some basic notation see [11].
Let \scrF (I,\BbbR ) be the linear space of all real functions on interval I \subseteq \BbbR , let \scrL (E,\BbbR ) be the
linear class of real functions, defined on nonempty set E, satisfying properties (L1) and (L2), and
let \scrL +
0 (E,\BbbR ) \subset \scrL (E,\BbbR ) be subset of nonnegative functions in \scrL (E,\BbbR ). Further, let \scrI (\scrL (E,\BbbR ),\BbbR )
denotes the space of positive linear functionals on \scrL (E,\BbbR ), that is, we assume that such functionals
satisfy properties (A1) and (A2).
As a generalization of Jensen’s functional, with respect to positive functional, we define J :
\scrF (I,\BbbR )\times \scrL (E,\BbbR )\times \scrL +
0 (E,\BbbR )\times \scrI (\scrL (E,\BbbR ),\BbbR ) \rightarrow \BbbR as
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
REFINEMENTS OF JESSEN’S FUNCTIONAL 881
J (\Phi , f, p;A) = A (p\Phi (f)) - A(p)\Phi
\biggl(
A(pf)
A(p)
\biggr)
. (1.7)
Clearly, definition (1.7) is deduced from relation (1.6) and it also contains definition (1.3) of discrete
Jensen’s functional. We call (1.7) Jessen’s functional.
Remark 1.1. In above definition (1.7) we suppose pf, p\Phi (f) \in \scrL (E,\BbbR ). Then, it is easy to see
that \Phi
\biggl(
A(pf)
A(p)
\biggr)
is well defined provided that A(p) \not = 0. Namely, A1(f) =
A(pf)
A(p)
\in \scrI (\scrL (E,\BbbR ),\BbbR )
is normalized positive functional, that is, A1(1) = 1. Suppose I = [a, b]. Clearly, a \leq f(t) \leq b
for all t \in E. Since f(t) - a \geq 0, by using properties (A1), (A2), and (A3) we have A1(f) - a =
= A1(f) - A1(a) = A1(f - a) \geq 0, hence A1(f) \geq a. Similarly, A1(f) \leq b wherefrom we conclude
that
A(pf)
A(p)
belongs to interval I.
Conditions similar to those in Remark 1.1 will usually be omitted, so Jessen’s functional (1.7)
will initially assumed to be well defined.
Remark 1.2. If \Phi is continuous convex function on interval I, then Jessen’s functional is non-
negative, i.e.,
J (\Phi , f, p;A) \geq 0. (1.8)
It follows directly from Jessen’s relation (1.6) applied on normalized positive functional
A1(f) =
A(pf)
A(p)
\in \scrI (\scrL (E,\BbbR ),\BbbR ).
On the other hand, if \Phi is continuous concave function, then the sign of inequality in (1.8) is reversed.
Recently, Krnić et al. (see [11]) gave basic properties of Jessen’s functional. That proporties
are superadditivity and monotonicity. Monotonocity property applies to functions p, q \in \scrL +
0 (E,\BbbR )
where p \geq q means pi \geq qi, i = 1, 2, . . . , n.
Theorem 1.1. Suppose \Phi : I \subset \BbbR \rightarrow \BbbR is continuous convex function. Let f \in \scrL (E,\BbbR ),
p, q \in \scrL +
0 (E,\BbbR ), A \in \scrI (\scrL (E,\BbbR ),\BbbR ), such that Jessen’s functional (1.7) is well defined. Then
functional (1.7) possess the following properties:
(i) J (\Phi , f, \cdot ;A) is superadditive on \scrL +
0 (E,\BbbR ), i.e.,
J (\Phi , f, p+ q;A) \geq J (\Phi , f, p;A) + J (\Phi , f, q;A) . (1.9)
(ii) If p, q \in \scrL +
0 (E,\BbbR ) with p \geq q, then
J (\Phi , f, p;A) \geq J (\Phi , f, q;A) \geq 0, (1.10)
i.e., J (\Phi , f, \cdot ;A) is increasing on \scrL +
0 (E,\BbbR ).
(iii) If \Phi is continuous concave function, then the signs of inequality in (1.9) and (1.10) are
reversed, i.e., J(\Phi , f, \cdot ;A) is subadditive and decreasing on \scrL +
0 (E,\BbbR ).
As the first consequence of Theorem 1.1, they obtain monotonicity property of Jessen’s functional
which includes the function that attains minimum and maximum value on its domain.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
882 A. BARBIR, K. KRULIĆ HIMMELREICH, J. PEČARIĆ
Corollary 1.1. Let \Phi be continuous convex function on real interval, f \in \scrL (E,\BbbR ), and A \in
\in \scrI (\scrL (E,\BbbR ),\BbbR ). Suppose p \in \scrL +
0 (E,\BbbR ) attains minimum and maximum value on the set E. If the
functional (1.7) is well defined, then the following series of inequalities hold:\biggl[
\mathrm{m}\mathrm{a}\mathrm{x}
x\in E
p(x)
\biggr]
j (\Phi , f, 1;A) \geq J (\Phi , f, p;A) \geq
\biggl[
\mathrm{m}\mathrm{i}\mathrm{n}
x\in E
p(x)
\biggr]
j (\Phi , f, 1;A) , (1.11)
where
j (\Phi , f, 1;A) = A (\Phi (f)) - A(1)\Phi
\biggl(
A(f)
A(1)
\biggr)
. (1.12)
Further, if \Phi is continuous concave function, then the signs of inequality in (1.11) are reversed.
Now we consider the discrete case of Corollary 1.1. We suppose E = \{ 1, 2, . . . , n\} and
\scrL (E,\BbbR ) is the class of real n-tuples. If we consider discrete functional A \in \scrI (\scrL (E,\BbbR ),\BbbR ) defined
by A(\bfx ) =
\sum n
i=1
xi, where \bfx = (x1, x2, . . . , xn), then the functional (1.7) becomes discrete
functional (1.3) from paper [9] and relation (1.11) takes form
\mathrm{m}\mathrm{a}\mathrm{x}
1\leq i\leq n
\{ pi\} S\Phi (\bfx ) \geq Jn(\Phi ,\bfx ,\bfp ) \geq \mathrm{m}\mathrm{i}\mathrm{n}
1\leq i\leq n
\{ pi\} S\Phi (\bfx ), (1.13)
where the functional Jn(\Phi ,\bfx ,\bfp ) is defined by (1.3) and
S\Phi (\bfx ) =
n\sum
i=1
\Phi (xi) - n\Phi
\left( \sum n
i=1
xi
n
\right) .
In this paper we give refinements of Theorem 1.1 and Corollary 1.1.
2. Main results.
Theorem 2.1. Suppose \Phi : I \subset \BbbR \rightarrow \BbbR is continuous convex function. Let
f, p, q \in \scrL (E,\BbbR ), A \in \scrI (\scrL (E,\BbbR ),\BbbR ), A(p), A(q) \geq 0,
A(pf)
A(p)
,
A(qf)
A(q)
\in I
such that Jessen’s functional (1.7) is well defined. Then the following holds:
(i) \mathrm{m}\mathrm{i}\mathrm{n}\{ A(p), A(q)\}
\biggl[
\Phi
\biggl(
A(pf)
A(p)
\biggr)
+\Phi
\biggl(
A(qf)
A(q)
\biggr)
- 2\Phi
\biggl(
A(pf)
2A(p)
+
A(qf)
2A(q)
\biggr) \biggr]
\leq
\leq J (\Phi , f, p+ q;A) - J (\Phi , f, p;A) - J (\Phi , f, q;A) \leq
\leq \mathrm{m}\mathrm{a}\mathrm{x}\{ A(p), A(q)\}
\biggl[
\Phi
\biggl(
A(pf)
A(p)
\biggr)
+\Phi
\biggl(
A(qf)
A(q)
\biggr)
- 2\Phi
\biggl(
A(pf)
2A(p)
+
A(qf)
2A(q)
\biggr) \biggr]
. (2.1)
(ii) If A(p) \geq A(q) \geq 0 and
A(pf) - A(qf)
A(p) - A(q)
\in I, then
J (\Phi , f, p;A) - J (\Phi , f, q;A) \geq
\geq \mathrm{m}\mathrm{i}\mathrm{n}\{ A(p) - A(q), A(q)\}
\biggl[
\Phi
\biggl(
A(pf) - A(qf)
A(p) - A(q)
\biggr)
+\Phi
\biggl(
A(qf)
A(q)
\biggr)
-
- 2\Phi
\biggl\{
1
2
\biggl[
A(pf) - A(qf)
A(p) - A(q)
+
A(qf)
A(q)
\biggr] \biggr\} \biggr]
. (2.2)
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
REFINEMENTS OF JESSEN’S FUNCTIONAL 883
Proof. From relation (1.13) in case n = 2 we have
\mathrm{m}\mathrm{i}\mathrm{n}\{ \=p, \=q\}
\biggl[
\Phi (x) + \Phi (y) - 2\Phi
\biggl(
x+ y
2
\biggr) \biggr]
\leq
\leq \=p\Phi (x) + \=q\Phi (y) - (\=p+ \=q)\Phi
\biggl(
\=px+ \=qy
\=p+ \=q
\biggr)
\leq
\leq \mathrm{m}\mathrm{a}\mathrm{x}\{ \=p, \=q\}
\biggl[
\Phi (x) + \Phi (y) - 2\Phi
\biggl(
x+ y
2
\biggr) \biggr]
(2.3)
holds. If we supstitute \=p with A(p), \=q with A(q), x with
A(pf)
A(p)
and y with
A(qf)
A(q)
in (2.3) we
obtain
\mathrm{m}\mathrm{i}\mathrm{n}\{ A(p), A(q)\}
\biggl[
\Phi
\biggl(
A(pf)
A(p)
\biggr)
+\Phi
\biggl(
A(qf)
A(q)
\biggr)
- 2\Phi
\biggl(
A(pf)
2A(p)
+
A(qf)
2A(q)
\biggr) \biggr]
\leq
\leq A(p)\Phi
\biggl(
A(pf)
A(p)
\biggr)
+A(q)\Phi
\biggl(
A(qf)
A(q)
\biggr)
- (A(p) +A(q))\Phi
\biggl(
A(pf) +A(qf)
A(p) +A(q)
\biggr)
\leq
\leq \mathrm{m}\mathrm{a}\mathrm{x}\{ A(p), A(q)\}
\biggl[
\Phi
\biggl(
A(pf)
A(p)
\biggr)
+\Phi
\biggl(
A(qf)
A(q)
\biggr)
- 2\Phi
\biggl(
A(pf)
2A(p)
+
A(qf)
2A(q)
\biggr) \biggr]
. (2.4)
From the definition of Jessen’s functional (1.7) we get
J (\Phi , f, p+ q;A) - J (\Phi , f, p;A) - J (\Phi , f, q;A) =
= A(p)\Phi
\biggl(
A(pf)
A(p)
\biggr)
+A(q)\Phi
\biggl(
A(qf)
A(q)
\biggr)
- (A(p) +A(q))\Phi
\biggl(
A(pf) +A(qf)
A(p) +A(q)
\biggr)
. (2.5)
So, by combining relations (2.4) and (2.5) we have (2.1).
(ii) Functional J(\Phi , f, \cdot , A) is superadditive and increasing on \scrL (E,\BbbR ) and satisfied rela-
tion (2.1). So for A(p) \geq A(q) \geq 0 and
A(pf) - A(gf)
A(p) - A(q)
\in I the following holds:
J (\Phi , f, p;A) - J (\Phi , f, p - q;A) - J (\Phi , f, q;A) \geq
\geq \mathrm{m}\mathrm{i}\mathrm{n}\{ A(p - q), A(q)\}
\biggl[
\Phi
\biggl(
A((p - q)f)
A(p - q)
\biggr)
+\Phi
\biggl(
A(qf)
A(q)
\biggr)
-
- 2\Phi
\biggl(
A((p - q)f)
2A(p - q)
+
A(qf)
2A(q)
\biggr) \biggr]
\geq
\geq \mathrm{m}\mathrm{i}\mathrm{n}\{ A(p) - A(q), A(q)\}
\biggl[
\Phi
\biggl(
A(pf) - A(qf)
A(p) - A(q)
\biggr)
+\Phi
\biggl(
A(qf)
A(q)
\biggr)
-
- 2\Phi
\biggl\{
1
2
\biggl[
A(pf) - A(qf)
A(p) - A(q)
+
A(qf)
A(q)
\biggr] \biggr\} \biggr]
. (2.6)
Since J (\Phi , f, p - q;A) \geq 0 we obtain (2.2).
Theorem 2.1 is proved.
Observe that we can obtain that (2.1) and (2.2) hold also for p, q \in \scrL +
0 (E,\BbbR ). That result is
given in the following corollary.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
884 A. BARBIR, K. KRULIĆ HIMMELREICH, J. PEČARIĆ
Corollary 2.1. Suppose \Phi : I \subset \BbbR \rightarrow \BbbR is continuous convex function. Let f \in \scrL (E,\BbbR ),
p, q \in \scrL +
0 (E,\BbbR ), A \in \scrI (\scrL (E,\BbbR ),\BbbR ), A(p), A(q) \geq 0 such that Jessen’s functional (1.7) is well
defined. Then the inequality (2.1) holds. If p \geq q and A(p) \geq A(q) \geq 0, then (2.2) holds.
Now, we obtain consequence of Corollary 2.1.
Corollary 2.2. Let \Phi be continuous convex function on real interval, f \in \scrL (E,\BbbR ), and A \in
\in \scrI (\scrL (E,\BbbR ),\BbbR ). Suppose p \in \scrL +
0 (E,\BbbR ) attains minimum and maximum value on the set E. If the
functional (1.7) is well defined, p(x)A(1) \leq A(p) \leq p(x)A(1), A(p), A(1) \geq 0, then the following
series of inequalities hold: \Bigl[
\mathrm{m}\mathrm{a}\mathrm{x}
x\in E
p(x)
\Bigr]
j (\Phi , f, 1;A) - J (\Phi , f, p;A) \geq
\geq \mathrm{m}\mathrm{i}\mathrm{n}\{ p(x)A(1) - A(p), A(p)\}
\biggl[
\Phi
\biggl(
p(x)A(f) - A(pf)
p(x)A(1) - A(p)
\biggr)
+\Phi
\biggl(
A(pf)
A(p)
\biggr)
-
- 2\Phi
\biggl\{
1
2
\biggl[
p(x)A(f) - A(pf)
p(x)A(1) - A(p)
+
A(pf)
A(p)
\biggr] \biggr\} \biggr]
, (2.7)
J (\Phi , f, p;A) -
\Bigl[
\mathrm{m}\mathrm{i}\mathrm{n}
x\in E
p(x)
\Bigr]
j (\Phi , f, 1;A) \geq
\geq \mathrm{m}\mathrm{i}\mathrm{n}\{ A(p) - p(x)A(1), p(x)A(1)\}
\biggl[
\Phi
\biggl(
A(pf) - p(x)A(f)
A(p) - p(x)A(1)
\biggr)
+
+ \Phi
\biggl(
A(f)
A(1)
\biggr)
- 2\Phi
\biggl\{
1
2
\biggl[
A(pf) - p(x)A(f)
A(p) - p(x)A(1)
+
A(f)
A(1)
\biggr] \biggr\} \biggr]
, (2.8)
where p(x) = \mathrm{m}\mathrm{a}\mathrm{x}x\in E p(x), p(x) = \mathrm{m}\mathrm{i}\mathrm{n}x\in E p(x) and
j (\Phi , f, 1;A) = A (\Phi (f)) - A(1)\Phi
\biggl(
A(f)
A(1)
\biggr)
.
Proof. Since p \in \scrL +
0 (E,\BbbR ) attains minimum and maximum value on its domain E, then
\mathrm{m}\mathrm{a}\mathrm{x}
x\in E
p(x) \geq p(x) \geq \mathrm{m}\mathrm{i}\mathrm{n}
x\in E
p(x),
so we can consider two constant functions
p(x) = \mathrm{m}\mathrm{a}\mathrm{x}
x\in E
p(x) and p(x) = \mathrm{m}\mathrm{i}\mathrm{n}
x\in E
p(x).
Now, double application of property (2.2) yields required result since
J (\Phi , f, p;A) = p(x)j (\Phi , f, 1;A) and J
\bigl(
\Phi , f, p;A
\bigr)
= p(x)j (\Phi , f, 1;A) .
Corollary 2.2 is proved.
Remark 2.1. Let’s rewrite relations (2.7) and (2.8) from Corollary 2.2 in a discrete form. We
suppose E = \{ 1, 2, . . . , n\} and \scrL (E,\BbbR ) is the class of real n-tuples. If we consider discrete
functional A \in \scrI (\scrL (E,\BbbR ),\BbbR ) defined by A(\bfx ) =
\sum n
i=1
xi, where \bfx = (x1, x2, . . . , xn), then the
functional (1.7) becomes discrete functional (1.3) from paper [9] and relation (2.7) takes form
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
REFINEMENTS OF JESSEN’S FUNCTIONAL 885
\mathrm{m}\mathrm{a}\mathrm{x}
1\leq i\leq n
\{ pi\} S\Phi (\bfx ) - Jn(\Phi ,\bfx ,\bfp ) \geq
\geq \mathrm{m}\mathrm{i}\mathrm{n}
\biggl\{
n \mathrm{m}\mathrm{a}\mathrm{x}
1\leq i\leq n
\{ pi\} - Pn, Pn
\biggr\} \left[ \Phi
\left( \mathrm{m}\mathrm{a}\mathrm{x}1\leq i\leq n\{ pi\}
\sum n
i=1
xi -
\sum n
i=1
pixi
n\mathrm{m}\mathrm{a}\mathrm{x}1\leq i\leq n\{ pi\} - Pn
\right) +
+ \Phi
\left( \sum n
i=1
pixi
Pn
\right) - 2\Phi
\left\{ 1
2
\left[ \mathrm{m}\mathrm{a}\mathrm{x}1\leq i\leq n\{ pi\}
\sum n
i=1
xi -
\sum n
i=1
pixi
n\mathrm{m}\mathrm{a}\mathrm{x}1\leq i\leq n\{ pi\} - Pn
+
\sum n
i=1
pixi
Pn
\right] \right\}
\right]
and relation (2.8)
Jn(\Phi ,\bfx ,\bfp ) - \mathrm{m}\mathrm{i}\mathrm{n}
1\leq i\leq n
\{ pi\} S\Phi (\bfx ) \geq
\geq \mathrm{m}\mathrm{i}\mathrm{n}
\biggl\{
Pn - n \mathrm{m}\mathrm{i}\mathrm{n}
1\leq i\leq n
\{ pi\} , n \mathrm{m}\mathrm{i}\mathrm{n}
1\leq i\leq n
\{ pi\}
\biggr\} \left[ \Phi
\left( \sum n
i=1
pixi - \mathrm{m}\mathrm{i}\mathrm{n}1\leq i\leq n\{ pi\}
\sum n
i=1
xi
Pn - n\mathrm{m}\mathrm{i}\mathrm{n}1\leq i\leq n\{ pi\}
\right) +
+ \Phi
\left( \sum n
i=1
xi
n
\right) - 2\Phi
\left\{ 1
2
\left[ \sum n
i=1
pixi - \mathrm{m}\mathrm{i}\mathrm{n}1\leq i\leq n\{ pi\}
\sum n
i=1
xi
Pn - n\mathrm{m}\mathrm{i}\mathrm{n}1\leq i\leq n\{ pi\}
+
\sum n
i=1
xi
n
\right] \right\}
\right] ,
where the functional Jn(\Phi ,\bfx ,\bfp ) is defined by (1.3) and
S\Phi (\bfx ) =
n\sum
i=1
\Phi (xi) - n\Phi
\Biggl( \sum n
i=1
xi
n
\Biggr)
.
3. Applications to weighted generalized and power means. In this section we apply our
basic results from previous section to weighted generalized and power means with respect to positive
functional A \in \scrI (\scrL (E,\BbbR ),\BbbR ).
We recall weighted generalized mean with respect to positive linear functional A \in \scrI (\scrL (E,\BbbR ),\BbbR )
and continuous and strictly monotone function \chi \in \scrF (I,\BbbR ), which is defined as
M\chi (f, p;A) = \chi - 1
\biggl(
A (p\chi (f))
A(p)
\biggr)
, f \in \scrL (E,\BbbR ), p \in \scrL +
0 (E,\BbbR ). (3.1)
We assume that (3.1) is well defined, that is, A(p) \not = 0 and p\chi (f) \in \scrL (E,\BbbR ). Similarly as in the
previous section, such conditions will usually be omitted, so weighted generalized mean (3.1) will
initially assumed to be well defined.
Now we define functional
J\tau (\chi \circ \psi - 1, \psi (f), p;A) = A(p)
\Bigl[
\chi (M\chi (f, p;A)) - \chi (M\psi (f, p;A))
\Bigr]
(3.2)
where \psi : I \rightarrow \BbbR is continuous and strictly monotone function such that \psi (f), p\psi (f) \in \scrL (E,\BbbR ). It
is Jessen’s functional (1.7) where the convex function \Phi is replaced with \chi \circ \psi - 1 and f \in \scrL (E,\BbbR )
with \psi (f) \in \scrL (E,\BbbR ).
Recently Krnić et al. in [11] proved that this functional J\tau (\chi \circ \psi - 1, \psi (f), \cdot ;A) is superadditive
and increasing on \scrL +
0 (E,\BbbR ) if \chi \circ \psi - 1 is a convex function. Now we can generalize their result.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
886 A. BARBIR, K. KRULIĆ HIMMELREICH, J. PEČARIĆ
Theorem 3.1. Let \chi , \psi \in \scrF (I,\BbbR ) be continuous and strictly monotone functions such that
the function \chi \circ \psi - 1 is convex. Suppose f \in \scrL (E,\BbbR ), p, q \in \scrL +
0 (E,\BbbR ), A \in \scrI (\scrL (E,\BbbR ),\BbbR ),
A(p), A(q) \geq 0 are such that the functional J\tau (\chi \circ \psi - 1, \psi (f), \cdot ;A) is well defined. Then, func-
tional (3.2) satisfies the following properties:
(i) \mathrm{m}\mathrm{i}\mathrm{n}\{ A(p), A(q)\}
\biggl[
\chi \circ \psi - 1
\biggl(
A(p\psi (f))
A(p)
\biggr)
+ \chi \circ \psi - 1
\biggl(
A(q\psi (f))
A(q)
\biggr)
-
- 2\chi \circ \psi - 1
\biggl(
A(p\psi (f))
2A(p)
+
A(q\psi (f))
2A(q)
\biggr) \biggr]
\leq
\leq J\tau (\chi \circ \psi - 1, \psi (f), p+ q;A) - J\tau (\chi \circ \psi - 1, \psi (f), p;A) - J\tau (\chi \circ \psi - 1, \psi (f), q;A) \leq
\leq \mathrm{m}\mathrm{a}\mathrm{x}\{ A(p), A(q)\}
\biggl[
\chi \circ \psi - 1
\biggl(
A(p\psi (f))
A(p)
\biggr)
+ \chi \circ \psi - 1
\biggl(
A(q\psi (f))
A(q)
\biggr)
-
- 2\chi \circ \psi - 1
\biggl(
A(p\psi (f))
2A(p)
+
A(q\psi (f))
2A(q)
\biggr) \biggr]
.
(ii) If p, q \in \scrL +
0 (E,\BbbR ) with p \geq q and A(p) \geq A(q) \geq 0, then
J\tau (\chi \circ \psi - 1, \psi (f), p;A) - J\tau (\chi \circ \psi - 1, \psi (f), q;A) \geq
\geq \mathrm{m}\mathrm{i}\mathrm{n}\{ A(p) - A(q), A(q)\}
\biggl[
\chi \circ \psi - 1
\biggl(
A(p\psi (f)) - A(q\psi (f))
A(p) - A(q)
\biggr)
+ \chi \circ \psi - 1
\biggl(
A(q\psi (f))
A(q)
\biggr)
-
- 2\chi \circ \psi - 1
\biggl\{
1
2
\biggl[
A(p\psi (f)) - A(q\psi (f))
A(p) - A(q)
+
A(q\psi (f))
A(q)
\biggr] \biggr\} \biggr]
.
Proof. We consider Jessen’s functional (1.7) where the convex function \Phi is replaced with
\chi \circ \psi - 1 and f \in \scrL (E,\BbbR ) with \psi (f) \in \scrL (E,\BbbR ). Also functional (3.2) can be rewritten in the form
J\tau
\bigl(
\chi \circ \psi - 1, \psi (f), p;A
\bigr)
= A
\bigl(
p \cdot
\bigl(
\chi \circ \psi - 1 (\psi (f))
\bigr) \bigr)
- A(p)\chi
\biggl(
\psi - 1
\biggl(
A (p\psi (f))
A(p)
\biggr) \biggr)
=
= A (p\chi (f)) - A(p)\chi (M\psi (f, p;A)) =
= A(p)\chi (M\chi (f, p;A)) - A(p)\chi (M\psi (f, p;A)) =
= A(p)
\bigl[
\chi (M\chi (f, p;A)) - \chi (M\psi (f, p;A))
\bigr]
.
Now, the properties (i) and (ii) follow from Theorem 1.1.
Theorem 3.1 is proved.
If in Corollary 2.2 we substitute convex function \Phi with \chi \circ \psi - 1 and f \in \scrL (E,\BbbR ) with
\psi (f) \in \scrL (E,\BbbR ) we obtain the following result.
Corollary 3.1. Suppose \chi , \psi , f,A are defined as in Theorem 3.1 and p \in \scrL +
0 (E,\BbbR ) attains
minimum and maximum value on the set E. If the function \chi \circ \psi - 1 is convex, p(x)A(1) \leq A(p) \leq
\leq p(x)A(1), A(p), A(1) \geq 0, then for the functional J\tau (\chi \circ \psi - 1, \psi (f), \cdot ;A) defined by (3.2) the
following series of inequalities hold:\Bigl[
\mathrm{m}\mathrm{a}\mathrm{x}
x\in E
p(x)
\Bigr]
J\tau
\bigl(
\chi \circ \psi - 1, \psi (f), 1;A
\bigr)
- J\tau
\bigl(
\chi \circ \psi - 1, \psi (f), p;A
\bigr)
\geq
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
REFINEMENTS OF JESSEN’S FUNCTIONAL 887
\geq \mathrm{m}\mathrm{i}\mathrm{n}\{ p(x)A(1) - A(p), A(p)\}
\biggl[
\chi \circ \psi - 1
\biggl(
p(x)A(\psi (f)) - A(p\psi (f))
p(x)A(1) - A(p)
\biggr)
+
+ \chi \circ \psi - 1
\biggl(
A(p\psi (f))
A(p)
\biggr)
- 2\chi \circ \psi - 1
\biggl\{
1
2
\biggl[
p(x)A(\psi (f)) - A(p\psi (f))
p(x)A(1) - A(p)
+
A(p\psi (f))
A(p)
\biggr] \biggr\} \biggr]
,
J\tau
\bigl(
\chi \circ \psi - 1, \psi (f), p;A
\bigr)
-
\Bigl[
\mathrm{m}\mathrm{i}\mathrm{n}
x\in E
p(x)
\Bigr]
J\tau
\bigl(
\chi \circ \psi - 1, \psi (f), 1;A
\bigr)
\geq
\geq \mathrm{m}\mathrm{i}\mathrm{n}\{ A(p) - p(x)A(1), p(x)A(1)\}
\biggl[
\chi \circ \psi - 1
\biggl(
A(p\psi (f)) - p(x)A(\psi (f))
A(p) - p(x)A(1)
\biggr)
+
+ \chi \circ \psi - 1
\biggl(
A(\psi (f))
A(1)
\biggr)
- 2\chi \circ \psi - 1
\biggl\{
1
2
\biggl[
A(p\psi (f)) - p(x)A(\psi (f))
A(p) - p(x)A(1)
+
A(\psi (f))
A(1)
\biggr] \biggr\} \biggr]
,
where p(x) = \mathrm{m}\mathrm{a}\mathrm{x}x\in E p(x), p(x) = \mathrm{m}\mathrm{i}\mathrm{n}x\in E p(x),
J\tau
\bigl(
\chi \circ \psi - 1, \psi (f), 1;A
\bigr)
= A(1) [\chi (M\chi (f ;A)) - \chi (M\psi (f ;A))]
and
M\eta (f ;A) = \eta - 1
\biggl(
A (\eta (f))
A(1)
\biggr)
, \eta = \chi , \psi .
Let r \in \BbbR and f \in \scrL +
0 (E,\BbbR ) such that f(x) > 0 for all x \in E. Generalized power mean
M [r] (f, p;A) equipped with positive functional A \in \scrI (\scrL (E,\BbbR ),\BbbR ) is defined by
M [r] (f, p;A) =
\left\{
\biggl(
A (pf r)
A(p)
\biggr) 1/r
, r \not = 0,
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
A (p \mathrm{l}\mathrm{n}(f))
A(p)
\biggr)
, r = 0,
(3.3)
where p \in \scrL +
0 (E,\BbbR ). We assume that the above expression is well defined, that is, pf r \in \scrL +
0 (E,\BbbR ),
p \mathrm{l}\mathrm{n}(f) \in \scrL (E,\BbbR ), and A(p) \not = 0.
Let now r, s \in \BbbR , s \not = 0. We define a functional
JP (\chi \circ \psi - 1, \psi (f), p;A) = A(p)
\Bigl\{ \Bigl[
M [s](f, p;A)
\Bigr] s
-
\Bigl[
M [r](f, p;A)
\Bigr] s\Bigr\}
, (3.4)
where \chi , \psi : I \rightarrow \BbbR are functions defined by \chi (x) = xs, s \not = 0, \psi (x) = xr, r \not = 0 and \psi (x) =
= \mathrm{l}\mathrm{n}x, r = 0. The first consequence of Theorem 3.1 refers to generalized power means M [r] (f, p;A) ,
r \in \BbbR . Results are given in the following corollary.
Corollary 3.2. Let s \not = 0 and r be real numbers, f, p, q \in \scrL +
0 (E,\BbbR ), f(x) > 0 for all x \in E,
and A \in \scrI (\scrL (E,\BbbR ),\BbbR ), A(p), A(q) \geq 0. The functional (3.4) has the following properties:
(i) If r \not = 0 and s > 0, s > r or s < 0, s < r, then
\mathrm{m}\mathrm{i}\mathrm{n}\{ A(p), A(q)\}
\Biggl[ \biggl(
A (pf r)
A(p)
\biggr) s/r
+
\biggl(
A (qf r)
A(q)
\biggr) s/r
- 2
\biggl(
A(pf r)
2A(p)
+
A(qf r)
2A(q)
\biggr) s/r\Biggr]
\leq
\leq JP (\chi \circ \psi - 1, \psi (f), p+ q;A) - JP (\chi \circ \psi - 1, \psi (f), p;A) - JP (\chi \circ \psi - 1, \psi (f), q;A) \leq
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
888 A. BARBIR, K. KRULIĆ HIMMELREICH, J. PEČARIĆ
\leq \mathrm{m}\mathrm{a}\mathrm{x}\{ A(p), A(q)\}
\Biggl[ \biggl(
A (pf r)
A(p)
\biggr) s/r
+
\biggl(
A (qf r)
A(q)
\biggr) s/r
- 2
\biggl(
A(pf r)
2A(p)
+
A(qf r)
2A(q)
\biggr) s/r\Biggr]
.
(ii) If r \not = 0 and s > 0, s > r or s < 0, s < r, then for p, q \in \scrL +
0 (E,\BbbR ) with p \geq q and
A(p) \geq A(q) \geq 0, holds inequality
JP (\chi \circ \psi - 1, \psi (f), p;A) - JP (\chi \circ \psi - 1, \psi (f), q;A) \geq
\geq \mathrm{m}\mathrm{i}\mathrm{n}\{ A(p - q), A(q)\}
\Biggl[ \biggl(
A(pf r) - A(qf r)
A(p) - A(q)
\biggr) s/r
+
\biggl(
A(qf r)
A(q)
\biggr) s/r
-
- 21 -
s
r
\biggl(
A(pf r) - A(qf r)
A(p) - A(q)
+
A(qf r)
A(q)
\biggr) s/r\Biggr]
.
(iii) If r = 0, then
\mathrm{m}\mathrm{i}\mathrm{n}\{ A(p), A(q)\}
\biggl[
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
sA(p \mathrm{l}\mathrm{n} f)
A(p)
\biggr)
+ \mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
sA(q \mathrm{l}\mathrm{n} f)
A(q)
\biggr)
-
- 2 \mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
sA(p \mathrm{l}\mathrm{n} f)
2A(p)
+
sA(q \mathrm{l}\mathrm{n} f)
2A(q)
\biggr) \biggr]
\leq
\leq JP (\chi \circ \psi - 1, \psi (f), p+ q;A) - JP (\chi \circ \psi - 1, \psi (f), p;A) - JP (\chi \circ \psi - 1, \psi (f), q;A) \leq
\leq \mathrm{m}\mathrm{a}\mathrm{x}\{ A(p), A(q)\}
\biggl[
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
sA(p \mathrm{l}\mathrm{n} f)
A(p)
\biggr)
+ \mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
sA(q \mathrm{l}\mathrm{n} f)
A(q)
\biggr)
-
- 2 \mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
sA(p \mathrm{l}\mathrm{n} f)
2A(p)
+
sA(q \mathrm{l}\mathrm{n} f)
2A(q)
\biggr) \biggr]
.
(iv) If r = 0, then for p, q \in \scrL +
0 (E,\BbbR ) with p \geq q and A(p) \geq A(q) \geq 0, holds inequality
JP (\chi \circ \psi - 1, \psi (f), p;A) - JP (\chi \circ \psi - 1, \psi (f), q;A) \geq
\geq \mathrm{m}\mathrm{i}\mathrm{n}\{ A(p - q), A(q)\}
\biggl[
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
sA(p \mathrm{l}\mathrm{n} f) - sA(q \mathrm{l}\mathrm{n} f)
A(p) - A(q)
\biggr)
+ \mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
sA(q \mathrm{l}\mathrm{n} f)
A(q)
\biggr)
-
- 2 \mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
s
2
\biggl[
A(p \mathrm{l}\mathrm{n} f) - A(q \mathrm{l}\mathrm{n} f)
A(p) - A(q)
+
A(q \mathrm{l}\mathrm{n} f)
A(q)
\biggr] \biggr\} \biggr]
.
Proof. The proof is direct use of Theorem 3.1. We have to consider two cases depending on
whether r \not = 0 or r = 0.
If r \not = 0, we define \chi (x) = xs and \psi (x) = xr. Then \chi \circ \psi - 1(x) = xs/r and
\bigl(
\chi \circ \psi - 1
\bigr) \prime \prime
(x) =
=
s(s - r)
r2
xs/r - 2. Thus, \chi \circ \psi - 1 is convex if s > 0, s > r or s < 0, s < r. On the other hand,
\chi \circ \psi - 1 is concave if s > 0, s < r or s < 0, s > r.
If r = 0, we put \chi (x) = xs and \psi (x) = \mathrm{l}\mathrm{n}x. Then, \chi \circ \psi - 1(x) = esx is convex under
assumption s \not = 0. Results follow immediately from Theorem 3.1.
Corollary 3.2 is proved.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
REFINEMENTS OF JESSEN’S FUNCTIONAL 889
Corollary 3.3. Suppose s \not = 0 and r be real numbers such that r \not = 0, s > 0, s > r or s < 0,
s < r and p \in \scrL +
0 (E,\BbbR ) attains minimum and maximum value on the set E. If p(x)A(1) \leq
\leq A(p) \leq p(x)A(1), A(p), A(1) \geq 0, then for the functional JP (\chi \circ \psi - 1, \psi (f), \cdot ;A) defined by
(3.4) the following series of inequalities hold:\biggl[
\mathrm{m}\mathrm{a}\mathrm{x}
x\in E
p(x)
\biggr]
JP
\bigl(
\chi \circ \psi - 1, \psi (f), 1;A
\bigr)
- JP
\bigl(
\chi \circ \psi - 1, \psi (f), p;A
\bigr)
\geq
\geq \mathrm{m}\mathrm{i}\mathrm{n}\{ p(x)A(1) - A(p), A(p)\}
\Biggl[ \biggl(
p(x)A(f r) - A(pf r)
p(x)A(1) - A(p)
\biggr) s/r
+
+
\biggl(
A(pf r)
A(p)
\biggr) s/r
- 21 - s/r
\biggl(
p(x)A(f r) - A(pf r)
p(x)A(1) - A(p)
+
A(pf r)
A(p)
\biggr) s/r\Biggr]
, (3.5)
JP
\bigl(
\chi \circ \psi - 1, \psi (f), p;A
\bigr)
-
\biggl[
\mathrm{m}\mathrm{i}\mathrm{n}
x\in E
p(x)
\biggr]
JP
\bigl(
\chi \circ \psi - 1, \psi (f), 1;A
\bigr)
\geq
\geq \mathrm{m}\mathrm{i}\mathrm{n}\{ A(p) - p(x)A(1), p(x)A(1)\}
\Biggl[ \biggl(
A(pf r) - p(x)A(f r)
A(p) - p(x)A(1)
\biggr) s/r
+
+
\biggl(
A(f r)
A(1)
\biggr) s/r
- 21 - s/r
\biggl(
A(pf r) - p(x)A(f r)
A(p) - p(x)A(1)
+
A(f r)
A(1)
\biggr) s/r\Biggr]
. (3.6)
If r = 0, then \biggl[
\mathrm{m}\mathrm{a}\mathrm{x}
x\in E
p(x)
\biggr]
JP
\bigl(
\chi \circ \psi - 1, \psi (f), 1;A
\bigr)
- JP
\bigl(
\chi \circ \psi - 1, \psi (f), p;A
\bigr)
\geq
\geq \mathrm{m}\mathrm{i}\mathrm{n}\{ p(x)A(1) - A(p), A(p)\}
\biggl[
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
s
p(x)A(\mathrm{l}\mathrm{n} f) - A(p \mathrm{l}\mathrm{n} f)
p(x)A(1) - A(p)
\biggr)
+
+\mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
sA(p \mathrm{l}\mathrm{n} f)
A(p)
\biggr)
- 2 \mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
s
2
\biggl[
p(x)A(\mathrm{l}\mathrm{n} f) - A(p \mathrm{l}\mathrm{n} f)
p(x)A(1) - A(p)
+
A(p \mathrm{l}\mathrm{n} f)
A(p)
\biggr] \biggr\} \biggr]
, (3.7)
JP
\bigl(
\chi \circ \psi - 1, \psi (f), p;A
\bigr)
-
\biggl[
\mathrm{m}\mathrm{i}\mathrm{n}
x\in E
p(x)
\biggr]
JP
\bigl(
\chi \circ \psi - 1, \psi (f), 1;A
\bigr)
\geq
\geq \mathrm{m}\mathrm{i}\mathrm{n}\{ A(p) - p(x)A(1), p(x)A(1)\}
\biggl[
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
s
A(p \mathrm{l}\mathrm{n} f) - p(x)A(\mathrm{l}\mathrm{n} f)
A(p) - p(x)A(1)
\biggr)
+
+\mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
sA(\mathrm{l}\mathrm{n} f)
A(1)
\biggr)
- 2 \mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
s
2
\biggl[
A(p \mathrm{l}\mathrm{n} f) - p(x)A(\mathrm{l}\mathrm{n} f)
A(p) - p(x)A(1)
+
A(\mathrm{l}\mathrm{n} f)
A(1)
\biggr] \biggr\} \biggr]
, (3.8)
where p(x) = \mathrm{m}\mathrm{a}\mathrm{x}x\in E p(x), p(x) = \mathrm{m}\mathrm{i}\mathrm{n}x\in E p(x),
JP
\bigl(
\chi \circ \psi - 1, \psi (f), 1;A
\bigr)
= A(1)
\Bigl\{ \Bigl[
M [s](f ;A)
\Bigr] s
-
\Bigl[
M [r](f ;A)
\Bigr] s\Bigr\}
and
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
890 A. BARBIR, K. KRULIĆ HIMMELREICH, J. PEČARIĆ
M [t] (f ;A) =
\left\{
\Biggl(
A
\bigl(
f t
\bigr)
A(1)
\Biggr) 1/t
, t \not = 0,
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
A (\mathrm{l}\mathrm{n}(f))
A(1)
\biggr)
, t = 0,
t = r, s. (3.9)
Now, we consider a discrete variant of relations (3.5) – (3.8). As in Remark 2.1, we suppose
E = \{ 1, 2, . . . , n\} , n \in \BbbN , and \scrL (E,\BbbR ) is a class of real n-tuples. We consider discrete functional
A \in \scrI (\scrL (E,\BbbR ),\BbbR ) defined by A(\bfx ) =
\sum n
i=1
xi, where \bfx = (x1, x2, . . . , xn). Clearly, A(\bfone ) =
=
\sum n
i=1
1 = n.
Generalized power mean (3.3) in discrete case becomes
Mr(\bfx ,\bfp ) =
\left\{
\left( \sum n
i=1
pix
r
i
Pn
\right) 1/r
, r \not = 0,
\Bigl( \prod n
i=1
xpii
\Bigr) 1/Pn
, r = 0,
where xi, pi \geq 0, i = 1, . . . , n. For r = 1 we obtain arithmetic mean An(\bfx ,\bfp ) = M1(\bfx ,\bfp ) =
=
\biggl(
1
Pn
\sum n
i=1
pixi
\biggr)
, while for r = 0 geometric mean Gn(\bfx ,\bfp ) = M0(\bfx ,\bfp ) =
\Bigl( \prod n
i=1
xpii
\Bigr) 1/Pn
.
Now, if we take constant n-tuples
\bfp =
\biggl(
\mathrm{m}\mathrm{a}\mathrm{x}
1\leq i\leq n
\{ pi\} , . . . , \mathrm{m}\mathrm{a}\mathrm{x}
1\leq i\leq n
\{ pi\}
\biggr)
or \bfp =
\biggl(
\mathrm{m}\mathrm{i}\mathrm{n}
1\leq i\leq n
\{ pi\} , . . . , \mathrm{m}\mathrm{i}\mathrm{n}
1\leq i\leq n
\{ pi\}
\biggr)
expression for arithemetic and geometric mean reduce to
A0
n(\bfx ) =
1
n
n\sum
i=1
xi, and G0
n(\bfx ) =
\Biggl(
n\prod
i=1
xi
\Biggr) 1/n
and inequalities (3.7) and (3.8) for s = 1 and r = 0 can be rewritten as
n \mathrm{m}\mathrm{a}\mathrm{x}
1\leq i\leq n
\{ pi\}
\bigl[
A0
n(\bfx ) - G0
n(\bfx )
\bigr]
- Pn [An(\bfx ,\bfp ) - Gn(\bfx ,\bfp )] \geq
\geq \mathrm{m}\mathrm{i}\mathrm{n}\{ n \mathrm{m}\mathrm{a}\mathrm{x}
1\leq i\leq n
\{ pi\} - Pn, Pn\} \times
\times
\biggl[
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
\mathrm{m}\mathrm{a}\mathrm{x}1\leq i\leq n\{ pi\} \mathrm{l}\mathrm{n}(G0
n(\bfx ))
n - \mathrm{l}\mathrm{n}(Gn(\bfx ,\bfp ))
Pn
n\mathrm{m}\mathrm{a}\mathrm{x}1\leq i\leq n\{ pi\} - Pn
\biggr)
+Gn(\bfx ,\bfp ) -
- 2 \mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
1
2
\biggl[
\mathrm{m}\mathrm{a}\mathrm{x}1\leq i\leq n\{ pi\} \mathrm{l}\mathrm{n}(G0
n(\bfx ))
n - \mathrm{l}\mathrm{n}(Gn(\bfx ,\bfp ))
Pn
n\mathrm{m}\mathrm{a}\mathrm{x}1\leq i\leq n\{ pi\} - Pn
+ \mathrm{l}\mathrm{n}Gn(\bfx ,\bfp )
\biggr] \biggr\} \biggr]
, (3.10)
Pn [An(\bfx ,\bfp ) - Gn(\bfx ,\bfp )] - n \mathrm{m}\mathrm{i}\mathrm{n}
1\leq i\leq n
\{ pi\}
\bigl[
A0
n(\bfx ) - G0
n(\bfx )
\bigr]
\geq
\geq \mathrm{m}\mathrm{i}\mathrm{n}\{ Pn - n \mathrm{m}\mathrm{i}\mathrm{n}
1\leq i\leq n
\{ pi\} , n \mathrm{m}\mathrm{i}\mathrm{n}
1\leq i\leq n
\{ pi\} \} \times
\times
\biggl[
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
\mathrm{l}\mathrm{n}(Gn(\bfx ,\bfp ))
Pn - \mathrm{m}\mathrm{i}\mathrm{n}1\leq i\leq n\{ pi\} \mathrm{l}\mathrm{n}(G0
n(\bfx ))
n
Pn - n\mathrm{m}\mathrm{i}\mathrm{n}1\leq i\leq n\{ pi\}
\biggr)
+G0
n(\bfx ) -
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
REFINEMENTS OF JESSEN’S FUNCTIONAL 891
- 2 \mathrm{e}\mathrm{x}\mathrm{p}
\biggl\{
1
2
\biggl[
\mathrm{l}\mathrm{n}(Gn(\bfx ,\bfp ))
Pn - \mathrm{m}\mathrm{i}\mathrm{n}1\leq i\leq n\{ pi\} \mathrm{l}\mathrm{n}(G0
n(\bfx ))
n
Pn - n\mathrm{m}\mathrm{i}\mathrm{n}1\leq i\leq n\{ pi\}
+ \mathrm{l}\mathrm{n}G0(\bfx )
\biggr] \biggr\} \biggr]
. (3.11)
Some variants of inequalities (3.10 ) and (3.11) were studied in papers [1 – 6].
Remark 3.1. Young’s inequality follows directly from arithmetic-geometric mean inequality, so
relations (3.10) and (3.11) provide refinements of Young’s inequality. Let \bfx = (x1, x2, . . . , xn) and
\bfp = (p1, p2, . . . , pn) be positive n-tuples such that
\sum n
i=1
1
pi
= 1. We denote
\bfx \bfp = (xp11 , x
p2
2 , . . . , x
pn
n ) and \bfp - 1 =
\biggl(
1
p1
,
1
p2
, . . . ,
1
pn
\biggr)
.
Then series of inequalities (3.10) and (3.11) can be rewritten in the form
n \mathrm{m}\mathrm{a}\mathrm{x}
1\leq i\leq n
\biggl\{
1
pi
\biggr\} \bigl[
A0
n(\bfx
\bfp ) - G0
n(\bfx
\bfp )
\bigr]
- [An(\bfx
\bfp ,\bfp - 1) - Gn(\bfx
\bfp ,\bfp - 1)] \geq
\geq \mathrm{m}\mathrm{i}\mathrm{n}
\biggl\{
n \mathrm{m}\mathrm{a}\mathrm{x}
1\leq i\leq n
\biggl\{
1
pi
\biggr\}
- 1, 1
\biggr\}
\times
\times
\left[ \mathrm{e}\mathrm{x}\mathrm{p}
\left( \mathrm{m}\mathrm{a}\mathrm{x}1\leq i\leq n
\biggl\{
1
pi
\biggr\}
\mathrm{l}\mathrm{n}(G0
n(\bfx
\bfp ))n - \mathrm{l}\mathrm{n}Gn(\bfx
\bfp ,\bfp - 1)
n\mathrm{m}\mathrm{a}\mathrm{x}1\leq i\leq n
\biggl\{
1
pi
\biggr\}
- 1
\right) + Gn(\bfx
\bfp ,\bfp - 1) -
- 2 \mathrm{e}\mathrm{x}\mathrm{p}
\left\{
1
2
\left[ \mathrm{m}\mathrm{a}\mathrm{x}1\leq i\leq n
\biggl\{
1
pi
\biggr\}
\mathrm{l}\mathrm{n}(G0
n(\bfx
\bfp ))n - \mathrm{l}\mathrm{n}Gn(\bfx
\bfp ,\bfp - 1)
n\mathrm{m}\mathrm{a}\mathrm{x}1\leq i\leq n
\biggl\{
1
pi
\biggr\}
- 1
+ \mathrm{l}\mathrm{n}Gn(\bfx
\bfp ,\bfp - 1)
\right]
\right\}
\right] , (3.12)
An(\bfx
\bfp ,\bfp - 1) - Gn(\bfx
\bfp ,\bfp - 1) - n \mathrm{m}\mathrm{i}\mathrm{n}
1\leq i\leq n
\biggl\{
1
pi
\biggr\} \bigl[
A0
n(\bfx
\bfp ) - G0
n(\bfx
\bfp )
\bigr]
\geq
\geq \mathrm{m}\mathrm{i}\mathrm{n}
\biggl\{
1 - n \mathrm{m}\mathrm{i}\mathrm{n}
1\leq i\leq n
\biggl\{
1
pi
\biggr\}
, n \mathrm{m}\mathrm{i}\mathrm{n}
1\leq i\leq n
\biggl\{
1
pi
\biggr\} \biggr\}
\times
\times
\left[ \mathrm{e}\mathrm{x}\mathrm{p}
\left( \mathrm{l}\mathrm{n}Gn(\bfx
\bfp ,\bfp - 1) - \mathrm{m}\mathrm{i}\mathrm{n}1\leq i\leq n
\biggl\{
1
pi
\biggr\}
\mathrm{l}\mathrm{n}(G0
n(\bfx
\bfp ))n
1 - n\mathrm{m}\mathrm{i}\mathrm{n}1\leq i\leq n
\biggl\{
1
pi
\biggr\}
\right) + G0
n(\bfx
\bfp ) -
- 2 \mathrm{e}\mathrm{x}\mathrm{p}
\left\{
1
2
\left[ \mathrm{l}\mathrm{n}Gn(\bfx
\bfp ,\bfp - 1) - \mathrm{m}\mathrm{i}\mathrm{n}1\leq i\leq n
\biggl\{
1
pi
\biggr\}
\mathrm{l}\mathrm{n}(G0
n(\bfx
\bfp ))n
1 - n\mathrm{m}\mathrm{i}\mathrm{n}1\leq i\leq n
\biggl\{
1
pi
\biggr\} + \mathrm{l}\mathrm{n}(G0(\bfx
\bfp ))
\right]
\right\}
\right] . (3.13)
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
892 A. BARBIR, K. KRULIĆ HIMMELREICH, J. PEČARIĆ
Note that Corollaries 3.2 and 3.3 do not cover the case when s = 0 and r \not = 0. This case should
be considered separately. Let r \not = 0 be real number, f, p, q \in \scrL +
0 (E,\BbbR ), f(x) > 0 for all x \in E,
and A \in \scrI (\scrL (E,\BbbR ),\BbbR ). Then we define a functional
JP (\chi \circ \psi - 1, \psi (f), p;A) = A(p)
\biggl\{
A (p \mathrm{l}\mathrm{n} f)
A(p)
- \mathrm{l}\mathrm{n}
\Bigl[
M [r](f, p;A)
\Bigr] \biggr\}
, (3.14)
where \chi (x) = \mathrm{l}\mathrm{n}x and \psi (x) = xr.
Corollary 3.4. Let r < 0 be real number, let f, p, q \in \scrL +
0 (E,\BbbR ), f(x) > 0 for all x \in E and
A \in \scrI (\scrL (E,\BbbR ),\BbbR ), A(p), A(q) \geq 0. The functional (3.14) satisfies the following:
(i) \mathrm{m}\mathrm{i}\mathrm{n}\{ A(p), A(q)\} 1
r
\biggl[
\mathrm{l}\mathrm{n}
\biggl(
A(pf r)
A(p)
\biggr)
+ \mathrm{l}\mathrm{n}
\biggl(
A(qf r)
A(q)
\biggr)
- 2 \mathrm{l}\mathrm{n}
\biggl(
A(pf r)
2A(p)
+
A(qf r)
2A(q)
\biggr) \biggr]
\leq
\leq JP (\chi \circ \psi - 1, \psi (f), p+ q;A) - JP (\chi \circ \psi - 1, \psi (f), p;A) - JP (\chi \circ \psi - 1, \psi (f), q;A) \leq
\leq \mathrm{m}\mathrm{a}\mathrm{x}\{ A(p), A(q)\} 1
r
\biggl[
\mathrm{l}\mathrm{n}
\biggl(
A(pf r)
A(p)
\biggr)
+ \mathrm{l}\mathrm{n}
\biggl(
A(qf r)
A(q)
\biggr)
- 2 \mathrm{l}\mathrm{n}
\biggl(
A(pf r)
2A(p)
+
A(qf r)
2A(q)
\biggr) \biggr]
.
(ii) If p, q \in \scrL +
0 (E,\BbbR ) with p \geq q and A(p) \geq A(q) \geq 0, then
JP (\chi \circ \psi - 1, \psi (f), p;A) - JP (\chi \circ \psi - 1, \psi (f), q;A) \geq
\geq \mathrm{m}\mathrm{i}\mathrm{n}\{ A(p) - A(q), A(q)\} 1
r
\biggl[
\mathrm{l}\mathrm{n}
\biggl(
A(pf r) - A(qf r)
A(p) - A(q)
\biggr)
+ \mathrm{l}\mathrm{n}
\biggl(
A(qf r)
A(q)
\biggr)
-
- 2 \mathrm{l}\mathrm{n}
\biggl\{
1
2
\biggl[
A(pf r) - A(qf r)
A(p) - A(q)
+
A(qf r)
A(q)
\biggr] \biggr\} \biggr]
.
Proof. The proof is direct consequence of Theorem 3.1. We define \chi (x) = \mathrm{l}\mathrm{n}x and \psi (x) = xr.
Then, the function \chi \circ \psi - 1(x) =
1
r
\mathrm{l}\mathrm{n}x is convex if r < 0 and concave if r > 0.
Corollary 3.4 is proved.
The analogue of Corollary 3.3, that covers the case when s = 0 and r \not = 0, is contained in the
following result.
Corollary 3.5. Let r < 0 be real number, f \in \scrL +
0 (E,\BbbR ), f(x) > 0 for all x \in E, A \in
\in \scrI (\scrL (E,\BbbR ),\BbbR ) and p \in \scrL +
0 (E,\BbbR ) attains minimum and maximum value on its domain E. Assume
that functional (3.14) is well defined. If p(x)A(1) \leq A(p) \leq p(x)A(1), A(p), A(1) \geq 0, then for
the functional JP (\chi \circ \psi - 1, \psi (f), \cdot ;A) defined by (3.14) the following series of inequalities hold:\Bigl[
\mathrm{m}\mathrm{a}\mathrm{x}
x\in E
p(x)
\Bigr]
JP
\bigl(
\chi \circ \psi - 1, \psi (f), 1;A
\bigr)
- JP
\bigl(
\chi \circ \psi - 1, \psi (f), p;A
\bigr)
\geq
\geq \mathrm{m}\mathrm{i}\mathrm{n}\{ p(x)A(1) - A(p), A(p)\} 1
r
\biggl[
\mathrm{l}\mathrm{n}
\biggl(
p(x)A(f r) - A(pf r)
p(x)A(1) - A(p)
\biggr)
+
+ \mathrm{l}\mathrm{n}
\biggl(
A(pf r)
A(p)
\biggr)
- 2 \mathrm{l}\mathrm{n}
\biggl\{
1
2
\biggl[
p(x)A(f r) - A(pf r)
p(x)A(1) - A(p)
+
A(f r)
A(p)
\biggr] \biggr\} \biggr]
,
JP
\bigl(
\chi \circ \psi - 1, \psi (f), p;A
\bigr)
-
\Bigl[
\mathrm{m}\mathrm{i}\mathrm{n}
x\in E
p(x)
\Bigr]
JP
\bigl(
\chi \circ \psi - 1, \psi (f), 1;A
\bigr)
\geq
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
REFINEMENTS OF JESSEN’S FUNCTIONAL 893
\geq \mathrm{m}\mathrm{i}\mathrm{n}\{ A(p) - p(x)A(1), p(x)A(1)\} 1
r
\biggl[
\mathrm{l}\mathrm{n}
\biggl(
A(pf r) - p(x)A(f r)
A(p) - p(x)A(1)
\biggr)
+
+ \mathrm{l}\mathrm{n}
\biggl(
A(f r)
A(1)
\biggr)
- 2 \mathrm{l}\mathrm{n}
\biggl\{
1
2
\biggl[
A(pf r) - p(x)A(f r)
A(p) - p(x)A(1)
+
A(f r)
A(1)
\biggr] \biggr\} \biggr]
,
where p(x) = \mathrm{m}\mathrm{a}\mathrm{x}x\in E p(x), p(x) = \mathrm{m}\mathrm{i}\mathrm{n}x\in E p(x), M
[r](f ;A) is defined by (3.9) and
JP
\bigl(
\chi \circ \psi - 1, \psi (f), 1;A
\bigr)
= A(1)
\biggl(
A (\mathrm{l}\mathrm{n} f)
A(1)
- \mathrm{l}\mathrm{n}
\Bigl[
M [r](f ;A)
\Bigr] \biggr)
.
4. Applications to Hölder’s inequality. This section is devoted to Hölder’s inequality. In view
of positive functional A \in \scrI (\scrL (E,\BbbR ),\BbbR ), Hölder’s inequality claims that
A
\Biggl(
n\prod
i=1
fi
1/pi
\Biggr)
\leq
n\prod
i=1
A1/pi (fi) ,
where pi, i = 1, 2, . . . , n, are conjugate exponents, that is
\sum n
i=1
1/pi = 1, pi > 1, i = 1, 2, . . . , n,
and provided that f1, f2, . . . , fn,
\prod n
i=1
fi
1/pi \in \scrL +
0 (E,\BbbR ).
It is well known from the literature (see [13, 17]) that Hölder’s inequality can easily be obtained
from Young’s inequality. If we consider n-tuple \bfx = (x1, x2, . . . , xn), where xi = [fi/A(fi)]
1/pi ,
i = 1, 2, . . . , n, the expressions in (3.12) and (3.13), that represent the difference between arithmetic
and geometric mean, become
An(\bfx
\bfp ,\bfp - 1) - Gn(\bfx
\bfp ,\bfp - 1) =
n\sum
i=1
fi
piA(fi)
-
n\prod
i=1
fi
1/pi
A1/pi (fi)
,
A0
n(\bfx
\bfp ) - G0
n(\bfx
\bfp ) =
1
n
n\sum
i=1
fi
A (fi)
-
n\prod
i=1
fi
1/n
A1/n (fi)
.
Now, if we apply positive functional A \in \scrI (\scrL (E,\BbbR ),\BbbR ) on above expressions, and use its linearity
property, we get
A
\bigl[
An(\bfx
\bfp ,\bfp - 1) - Gn(\bfx
\bfp ,\bfp - 1)
\bigr]
=
n\sum
i=1
A (fi)
piA(fi)
-
A
\Bigl( \prod n
i=1
fi
1/pi
\Bigr)
\prod n
i=1
A1/pi (fi)
=
= 1 -
A
\Bigl( \prod n
i=1
fi
1/pi
\Bigr)
\prod n
i=1
A1/pi (fi)
,
and
A
\bigl[
A0
n(\bfx
\bfp ) - G0
n(\bfx
\bfp )
\bigr]
=
1
n
n\sum
i=1
A (fi)
A (fi)
-
A
\Bigl( \prod n
i=1
fi
1/n
\Bigr)
\prod n
i=1
A
1
n (fi)
=
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
894 A. BARBIR, K. KRULIĆ HIMMELREICH, J. PEČARIĆ
= 1 -
A
\Bigl( \prod n
i=1
fi
1/n
\Bigr)
\prod n
i=1
A1/n (fi)
.
By application of functional A \in \scrI (\scrL (E,\BbbR ),\BbbR ) on the series of inequalities in (3.12) and (3.13),
the signs of inequalities do not change, since A is linear and positive. Here we will give only result
involving inequality (3.12). Analogous result can be obtained for inequality (3.13), but here we omit
the details.
Theorem 4.1. Let pi > 1, i = 1, 2, . . . , n, be conjugate exponents, fi \in \scrL +
0 (E,\BbbR ), i =
= 1, 2, . . . , n, and
\prod n
i=1
fi
1/pi ,
\prod n
i=1
fi
1/n \in \scrL +
0 (E,\BbbR ). If A \in \scrI (\scrL (E,\BbbR ),\BbbR ), then the following
series of inequalities hold:
n \mathrm{m}\mathrm{a}\mathrm{x}
1\leq i\leq n
\biggl\{
1
pi
\biggr\} \Biggl[ n\prod
i=1
A1/pi (fi) -
n\prod
i=1
A1/pi - 1/n (fi)A
\Biggl(
n\prod
i=1
fi
1/n
\Biggr)
-
-
n\prod
i=1
A1/pi (fi) - A
\Biggl(
n\prod
i=1
fi
1/pi
\Biggr) \Biggr]
\geq
\geq \mathrm{m}\mathrm{i}\mathrm{n}\{ n \mathrm{m}\mathrm{a}\mathrm{x}
1\leq i\leq n
\biggl\{
1
pi
\biggr\}
- 1, 1\}
n\prod
i=1
A1/pi (fi)\times
\times
\left[ A
\left\{ \mathrm{e}\mathrm{x}\mathrm{p}
\left(
n\mathrm{m}\mathrm{a}\mathrm{x}1\leq i\leq n
\biggl\{
1
pi
\biggr\}
\mathrm{l}\mathrm{n}
\prod n
i=1
fi
1/n
A1/n (fi)
- \mathrm{l}\mathrm{n}
\prod n
i=1
fi
1/pi
A1/pi (fi)
n\mathrm{m}\mathrm{a}\mathrm{x}1\leq i\leq n
\biggl\{
1
pi
\biggr\}
- 1
\right)
\right\} +A
\Biggl(
n\prod
i=1
fi
1/pi
\Biggr)
-
- 2A
\left\{ \mathrm{e}\mathrm{x}\mathrm{p}
\left(
n\mathrm{m}\mathrm{a}\mathrm{x}1\leq i\leq n
\biggl\{
1
pi
\biggr\}
\mathrm{l}\mathrm{n}
\prod n
i=1
fi
1/n
A1/n(fi)
- \mathrm{l}\mathrm{n}
\prod n
i=1
fi
1/pi
A1/pi(fi)
2
\biggl(
n \mathrm{m}\mathrm{a}\mathrm{x}
1\leq i\leq n
\biggl\{
1
pi
\biggr\}
- 1
\biggr) +
1
2
\mathrm{l}\mathrm{n}
n\prod
i=1
fi
1/pi
A1/pi (fi)
\right)
\right\}
\right] .
It is also well known that Hölder’s inequality can directly be deduced from Jensen’s inequality
in the case of two functions (see [13]). Let r, s \in \BbbR such that 1/r + 1/s = 1. Let f, g \in \scrL +
0 (E,\BbbR )
and A \in \scrI (\scrL (E,\BbbR ),\BbbR ). We define a functional
JH
\biggl(
\Phi ,
g
f
, f ;A
\biggr)
= rs
\Bigl[
A1/r(f)A1/s(g) - A
\Bigl(
f1/rg1/s
\Bigr) \Bigr]
,
where \Phi : I \rightarrow \BbbR is defined by \Phi (x) = - rsx1/s. It is Jessen’s functional (1.7) where the convex
function \Phi is replaced with \Phi (x) = - rsx1/s and arguments f and p respectively replaced with g/f
and f.
We obain the following result.
Theorem 4.2. Let 1/r+1/s = 1, with r > 1, let f, g \in \scrL +
0 (E,\BbbR ), and A \in \scrI (\scrL (E,\BbbR ),\BbbR ). If
the function f attains minimum and maximum value on set E, then the following series of inequalities
hold:
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
REFINEMENTS OF JESSEN’S FUNCTIONAL 895
\Bigl[
\mathrm{m}\mathrm{a}\mathrm{x}
x\in E
f(x)
\Bigr] \Biggl[
A1/r(1)A1/s
\biggl(
g
f
\biggr)
- A
\Biggl( \biggl(
g
f
\biggr) 1/s
\Biggr) \Biggr]
-
- A1/r(f)A1/s(g) - A
\Bigl(
f1/rg1/s
\Bigr)
\geq
\geq \mathrm{m}\mathrm{i}\mathrm{n}
\biggl\{ \Bigl[
\mathrm{m}\mathrm{a}\mathrm{x}
x\in E
f(x)
\Bigr]
A(1) - A(f), A(f)
\biggr\}
\times
\times
\left[ 21 - 1/s
\left(
\Bigl[
\mathrm{m}\mathrm{a}\mathrm{x}x\in E f(x)
\Bigr]
A
\biggl(
g
f
\biggr)
- A(g)\Bigl[
\mathrm{m}\mathrm{a}\mathrm{x}x\in E f(x)
\Bigr]
A(1) - A(f)
+
A(g)
A(f)
\right)
1/s
-
-
\left(
\Bigl[
\mathrm{m}\mathrm{a}\mathrm{x}x\in E f(x)
\Bigr]
A
\Bigl(
g
f
\Bigr)
- A(g)\Bigl[
\mathrm{m}\mathrm{a}\mathrm{x}x\in E f(x)
\Bigr]
A(1) - A(f)
\right) 1/s
+A1/r(f)A1/s(g)
\right] . (4.1)
Proof. We consider relation (2.7) from Corollary 2.2 with arguments f and p respectively
replaced with g/f and f, where \Phi (x) = - rsx1/s. Clearly, \Phi
\prime \prime
(x) = x1/s - 2, so \Phi is convex
function if x > 0. In this setting, Jessen’s functional (1.7) reads
JH
\biggl(
\Phi ,
g
f
, f ;A
\biggr)
= A
\biggl(
f\Phi
\biggl(
g
f
\biggr) \biggr)
- A(f)\Phi
\biggl(
A(g)
A(f)
\biggr)
=
= rs
\Bigl[
A1 - 1/s(f)A1/s(g) - A
\Bigl(
f1 - 1/sg1/s
\Bigr) \Bigr]
=
= rs
\Bigl[
A1/r(f)A1/s(g) - A
\Bigl(
f1/rg1/s
\Bigr) \Bigr]
.
Further,
JH
\biggl(
\Phi ,
g
f
, 1;A
\biggr)
= A
\biggl(
\Phi
\biggl(
g
f
\biggr) \biggr)
- A(1)\Phi
\left( A
\biggl(
g
f
\biggr)
A(1)
\right) =
= rs
\Biggl[
A1 - 1/s(1)A1/s
\biggl(
g
f
\biggr)
- A
\Biggl( \biggl(
g
f
\biggr) 1/s
\Biggr) \Biggr]
=
= rs
\Biggl[
A1/r(1)A1/s
\biggl(
g
f
\biggr)
- A
\Biggl( \biggl(
g
f
\biggr) 1/s
\Biggr) \Biggr]
.
Now, we substitute obtained expressions JH (\Phi , g/f, f ;A) and JH (\Phi , g/f, 1;A) in (2.7) and ob-
tain (4.1).
Theorem 4.2 is proved.
Remark 4.1. We can also consider relation (2.8) from Corollary 2.2 to obtain analogous result
to (4.1), but here we omit the details.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
896 A. BARBIR, K. KRULIĆ HIMMELREICH, J. PEČARIĆ
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Received 18.03.13,
after revision — 23.05.16
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 7
|
| id | umjimathkievua-article-1888 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:14:38Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/3d/63dca9557f7ae0d75fbb8b078687e53d.pdf |
| spelling | umjimathkievua-article-18882019-12-05T09:30:55Z Refinements of Jessen’s functional Уточнення функцiонала Йєссена Barbir, A. Himmelreich, Kruli´c K. Pečarić, J. E. Барбір, А. Хіммельрайх, Крилік К. Печарик, Й. Е. We obtain new refinements of Jessen’s functional defined by means of positive linear functionals. The accumulated results are then applied to weighted generalized and power means. We also obtain new refinements of numerous classical inequalities such as the arithmetic-geometric mean inequality, Young’s inequality, and H¨older’s inequality. Отримано новi уточнення функцiонала Йєссена, визначенi у термiнах додатних лiнiйних функцiоналiв. Отриманi результати застосовано до зважених узагальнених та степеневих середнiх. Також отримано новi уточнення численних класичних нерiвностей, таких як нерiвнiсть для арифметично-геометричних середнiх, нерiвностi Янга та Гельдера. Institute of Mathematics, NAS of Ukraine 2016-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1888 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 7 (2016); 879-896 Український математичний журнал; Том 68 № 7 (2016); 879-896 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1888/870 Copyright (c) 2016 Barbir A.; Himmelreich Kruli´c K.; Pečarić J. E. |
| spellingShingle | Barbir, A. Himmelreich, Kruli´c K. Pečarić, J. E. Барбір, А. Хіммельрайх, Крилік К. Печарик, Й. Е. Refinements of Jessen’s functional |
| title | Refinements of Jessen’s functional |
| title_alt | Уточнення функцiонала Йєссена |
| title_full | Refinements of Jessen’s functional |
| title_fullStr | Refinements of Jessen’s functional |
| title_full_unstemmed | Refinements of Jessen’s functional |
| title_short | Refinements of Jessen’s functional |
| title_sort | refinements of jessen’s functional |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1888 |
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