#### Levinson’s type generalization of the Jensen inequality and its converse for real Stieltjes measure

Mikic´ et al. Journal of Inequalities and Applications
Levinson's type generalization of the Jensen inequality and its converse for real Stieltjes measure
Rozarija Mikic´ 0
Josip Pecˇaric´ 0
Mirna Rodic´ 0
0 Faculty of Textile Technology, University of Zagreb , Prilaz baruna Filipovic ́a 28a, Zagreb, 10000 , Croatia
We derive the Levinson type generalization of the Jensen and the converse Jensen inequality for real Stieltjes measure, not necessarily positive. As a consequence, also the Levinson type generalization of the Hermite-Hadamard inequality is obtained. Similarly, we derive the Levinson type generalization of Giaccardi's inequality. The obtained results are then applied for establishing new mean-value theorems. The results from this paper represent a generalization of several recent results.
Jensen's inequality; converse Jensen's inequality; Hermite-Hadamard's inequality; Giaccardi's inequality; Levinson's inequality; Green function; mean-value theorems
1 Introduction and preliminary results
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The following theorem states the well-known Levinson inequality.
≥ and let pi, xi, yi, i = , . . . , n be such
Theorem . ([]) Let f : , c → R satisfy f
that pi > , in= pi = , ≤ xi ≤ c and
x + y = x + y = · · · = xn + yn.
Then the following inequality is valid:
pif (xi) – f (x¯) ≤
pif (yi) – f (y¯),
where x¯ =
in= pixi and y¯ =
in= piyi denote the weighted arithmetic means.
Numerous papers have been devoted to extensions and generalizations of this result, as
well as to weakening the assumptions under which inequality (.) is valid (see for instance
[–], and []).
A function f : I → R is called k-convex if [x, . . . , xk]f ≥ for all choices of k + distinct
points x, x, . . . , xk ∈ I. If the kth derivative of a convex function exists, then f (k) ≥ , but
f (k) may not exist (for properties of divided differences and k-convex functions see []).
max{x, . . . , xn} ≤ max{y, . . . , yn},
then (.) holds.
(ii) Pečarić [] proved that inequality (.) is valid when one weakens the previous
assumption (.) to
xi + xn–i+ ≤ c and
(iii) Mercer [] made a significant improvement by replacing condition (.) with a
weaker one, i.e. he proved that inequality (.) holds under the following conditions:
≥ ,
pi = ,
a ≤ xi, yi ≤ b,
max{x, . . . , xn} ≤ max{y, . . . , yn},
pi(xi – x¯) =
(iv) Witkowski [] showed that it is enough to assume that f is -convex in Mercer’s
assumptions. Furthermore, Witkowski weakened the assumption (.) and showed
that equality can be replaced by inequality in a certain direction.
Furthermore, Baloch, Pečarić, and Praljak in their paper [] introduced a new class of
functions Kc(a, b) that extends -convex functions and can be interpreted as functions
that are ‘-convex at point c ∈ a, b ’. They showed that Kc(a, b) is the largest class of
functions for which Levinson’s inequality (.) holds under Mercer’s assumptions, i.e. that
f ∈ Kc(a, b) if and only if inequality (.) holds for arbitrary weights pi > , in= pi = and
sequences xi and yi that satisfy xi ≤ c ≤ yi for i = , , . . . , n.
We give the definition of the class Kc(a, b) extended to an arbitrary interval I.
Definition . Let f : I → R and c ∈ I◦, where I◦ is the interior of I. We say that f ∈ Kc(I)
(f ∈ Kc(I)) if there exists a constant D such that the function F(x) = f (x) – D x is concave
(convex) on –∞, c] ∩ I and convex (concave) on [c, +∞ ∩ I.
Remark . For the class Kc(a, b) the following useful results hold (see []):
Jakšetić, Pečarić, and Praljak in [] gave the following Levinson type generalization of
the Jensen-Boas inequality.
Theorem . ([]) Let c ∈ I◦ and let f : [a, b] → R and g : [a, b] → R be
continuous monotonic functions (either increasing or decreasing) with ranges I ∩ –∞, c] and
I ∩ [c, +∞ , respectively. Let λ : [a, b] → R and μ : [a, b] → R be continuous or of
bounded variation satisfying
On the other hand, in [] Pečarić, Perić, and Rodić Lipanović generalized the Jensen
inequality (.) for a real Stieltjes measure. They considered the Green function G defined
on [α, β] × [α, β] by
which is convex and continuous with respect to both s and t. The function G is continuous
under s and continuous under t, and it can easily be shown by integrating by parts that any
function ϕ : [α, β] → R, ϕ ∈ C([α, β]), can be represented by
ϕ(x) = ββ –– αx ϕ(α) + βx –– αα ϕ(β) +
G(x, s)ϕ (s) ds.
Using that fact, the authors in [] gave the conditions under which inequality (.) holds
for a real Stieltjes measure, which is not necessarily positive nor increasing. This result is
stated in the following theorem.
Theorem . ([]) Let g : [a, b] → R be continuous function and [α, β] interval such that
the image of g is a subset of [α, β]. Let λ : [a, b] → R be continuous function or the function
of bounded variation, such that λ(a) = λ(b) and
Then the following two statements are equivalent:
, s ≤
Furthermore, the statements () and () are also equivalent if we change the sign of
inequality in both (.) and (.).
Corollary . ([]) Let the conditions from the previous theorem hold. Then the following
two statements are equivalent:
Moreover, the statements ( ) and ( ) are also equivalent if we change the sign of inequality
in both statements ( ) and ( ).
The main aim of our paper is to give a Levinson type generalization of the result from
Theorem .. In that way, a generalization of Theorem . for real Stieltjes measure, not
necessarily positive nor increasing, will also be obtained.
2 Main results
In order to simplify the notation, throughout this paper we use the following notation:
The following theorem states our main result.
Theorem . Let f : [a, b] → R and g : [a, b] → R be continuous functions, [α, β] ⊆
R an interval and c ∈ α, β such that f ([a, b]) ⊆ [α, c] and g([a, b]) ⊆ [c, β]. Let
λ : [a, b] → R and μ : [a, b] → R be continuous functions or functions of bounded
variation such that λ(a) = λ(b) and μ(a) = μ(b) and such that
C := ab f (x) dλ(x) – f¯ = ab g(x) dμ(x) – g¯
holds.
If for all s ∈ [α, c] and for all s ∈ [c, β] we have
G(f¯, s) ≤
G(g¯, s) ≤
where the function G is defined in (.), then for every continuous function ϕ ∈ Kc([α, β])
we have
The statement also holds if we reverse all signs of inequalities in (.) and (.).
When we rearrange the previous inequality, we get
and after rearranging we get
– g¯ ≤
Since the function φ is continuous and convex on [c, β] and for all s ∈ [c, β] (.) holds,
from Theorem . it follows that
Inequality (.) follows directly by combining inequalities (.) and (.), and taking into
account the condition (.).
Remark . It is obvious from the proof of the previous theorem that if we replace the
equality (.) by a weaker condition
then (.) becomes
– ϕ(f¯) ≤ C ≤ C ≤
Since the function ϕ belongs to class Kc([α, β]), we have ϕ–(c) ≤ D ≤ ϕ+(c) (see []), so if
additionally ϕ is convex (resp. concave), the condition (.) can be further weakened to
– f¯ ≤
Remark . It is easy to see that Theorem . further generalizes the Levinson type
generalization of the Jensen-Boas inequality given in Theorem .. Namely, if in Theorem .
we set the functions f and g to be monotonic, and the functions λ and μ to satisfy
for all uk ∈ vk–, vk , v = a, vn = b, and μ(a) < μ(b), then since the function G is
continuous and convex in both variables, we can apply the Jensen inequality and see that for
all s ∈ [α, c] and s ∈ [c, β] inequalities (.) hold, so we get exactly Theorem ..
3 Discrete case
In this section we give the results for the discrete case. The proofs are similar to those in
the integral case given in the previous section, so we will state these results without the
proofs.
In Levinson’s inequality (.) and its generalizations (see []) we see that pi (i = , . . . , n)
are positive real numbers. Here, we will give a generalization of that result, allowing pi to
also be negative, with the sum not equal to zero, but with a supplementary demand on pi
and xi given by using the Green function G defined in (.).
Here we use the common notation: for real n-tuples (x, . . . , xn) and (p, . . . , pn) we set
Pk = ik= pi, P¯k = Pn – Pk– (k = , . . . , n) and x¯ = Pn in= pixi. Analogously, for real
mtuples (y, . . . , ym) and (q, . . . , qm) we define Qk , Q¯k (k = , . . . , m) and y¯.
We already know from the first section that we can represent any function ϕ : [α, β] →
R, ϕ ∈ C([α, β]), in the form (.), where the function G is defined in (.), and by some
calculation it is easy to show that the following holds:
Pn i=
piϕ(xi) =
Pn i=
piG(xi, s) ϕ (s) ds.
Using that fact the authors in [] derived the analogous results of Theorem . and
Corollary . for discrete case, and here, similarly as in the previous section, we get the
following results.
Theorem . Let [α, β] ⊆ R be an interval and c ∈ α, β . Let xi ∈ [a, b] ⊆ [α, c], pi ∈ R
(i = , . . . , n) be such that Pn = and x¯ ∈ [α, c], and let yj ∈ [a, b] ⊆ [c, β], qj ∈ R (j =
, . . . , m) be such that Qm = and y¯ ∈ [c, β] and let
C :=
Pn i=
pixi – x¯ =
Qm j=
where the function G is defined in (.), then for every continuous function ϕ ∈ Kc([α, β])
we have
Pn i=
qjϕ(yj) – ϕ(y¯),
where D is the constant from Definition ..
Inequality (.) is reversed if we change the signs of inequalities in (.).
Remark . Theorem . is the generalization of Levinson’s type inequality given in [].
Namely, since the function G is convex in both variables, in the case when all pi > and
qj > we can apply the Jensen inequality and we see that for all s ∈ [α, c] and s ∈ [c, β]
inequalities (.) hold. Now from Theorem . and Corollary . we get the result from [].
Remark . We can replace the equality from the condition (.) by a weaker condition
in the analogous way as in Remark . from the previous section.
4 Converses of the Jensen inequality
The Jensen inequality for convex functions implies a whole series of other classical
inequalities. One of the most famous ones amongst them is the so-called
Edmundson-LahRibarič inequality which states that, for a positive measure μ on [, ] and a convex
function φ on [m, M] (–∞ < m < M < +∞), if f is any μ-measurable function on [, ] such
that m ≤ f (x) ≤ M for x ∈ [, ], one has
≤ MM–– mf¯ φ(m) + Mf¯ ––mm φ(M),
where f¯ = f dμ/ dμ.
It was obtained in . by Lah and Ribarič in their paper []. Since then, there have
been many papers written on the subject of its generalizations and converses (for instance,
see [] and []).
In [] the authors gave a Levinson type generalization of inequality (.) for positive
measures. In this section we will obtain a similar result involving signed measures, with a
supplementary demand by using the Green function G defined in (.). In order to do so,
we first need to state a result from [], which gives us a version of the
Edmundson-LahRibarič inequality for signed measures.
Theorem . ([]) Let g : [a, b] → R be continuous function and [α, β] be an interval such
that the image of g is a subset of [α, β]. Let m, M ∈ [α, β] (m = M) be such that m ≤ g(t) ≤
M for all t ∈ [a, b]. Let λ : [a, b] → R be continuous function or the function of bounded
variation, and λ(a) = λ(b). Then the following two statements are equivalent:
M – g¯ ϕ(m) + g¯ – m ϕ(M)
≤ M – m M – m
≤ MM–– mg¯ G(m, s) + Mg¯ ––mm G(M, s)
Furthermore, the statements () and () are also equivalent if we change the sign of
inequality in both (.) and (.).
Corollary . ([]) Let the conditions from the previous theorem hold. Then the following
two statements are equivalent:
Moreover, the statements ( ) and ( ) are also equivalent if we change the sign of inequality
in both statements ( ) and ( ).
In the following theorem we give the Levinson type generalization of the upper result,
and we use a similar method to Section of this paper.
Theorem . Let f : [a, b] → R and g : [a, b] → R be continuous functions, [α, β] ⊆
R an interval and c ∈ α, β such that f ([a, b]) = [m, M] ⊆ [α, c] and g([a, b]) =
[m, M] ⊆ [c, β], where m = M and m = M. Let λ : [a, b] → R and μ : [a, b] → R
be continuous functions or functions of bounded variation such that λ(a) = λ(b) and
μ(a) = μ(b) and
C :=
≤ MM––mf¯ G(m, s) + Mf¯ ––mm G(M, s)
M – g¯ G(m, s) + g¯ – m G(M, s),
≤ M – m M – m
where the function G is defined in (.), then for every continuous function ϕ ∈ Kc([α, β])
we have
M – f¯ ϕ(m) + f¯ – m ϕ(M) – ab ϕ(f (x)) dλ(x)
M – m M – m ab dλ(x)
g – m ϕ(M) – ab ϕ(g(x)) dμ(x) .
M – m ab dμ(x)
The statement also holds if we reverse all signs of inequalities in (.), (.) and (.).
Proof Let ϕ ∈ Kc([α, β]) be continuous function on [α, β] and let φ(x) = ϕ(x) – D x, where
D is the constant from Definition ..
M – f¯ φ(m) + f¯ – m φ(M).
≥ M – m M – m
When we rearrange the previous inequality, we get
M – f¯ ϕ(m) + f¯ – m ϕ(M) – ab ϕ(f (x)) dλ(x)
M – m M – m ab dλ(x)
and after rearranging we get
M – g¯ φ(m) + g¯ – m φ(M),
≤ M – m M – m
Since the function φ is continuous and convex on [c, β] and for all s ∈ [c, β] (.) holds,
from Theorem . it follows that
Inequality (.) follows directly by combining inequalities (.) and (.), and taking into
account the condition (.).
Remark . It is obvious from the proof of the previous theorem that if we replace the
equality (.) by a weaker condition
then (.) becomes
M – f¯ ϕ(m) + f¯ – m ϕ(M) – ab ϕ(f (x)) dλ(x)
M – m M – m ab dλ(x)
Since ϕ–(c) ≤ D ≤ ϕ+(c) (see []), if additionally ϕ is convex (resp. concave), the condition
(.) can be further weakened to
5 Discrete form of the converses of the Jensen inequality
In this section we give the Levinson type generalization for converses of Jensen’s inequality
in discrete case. The proofs are similar to those in the integral case given in the previous
section, so we give these results with the proofs omitted.
As we can represent any function ϕ : [α, β] → R, ϕ ∈ C([α, β]), in the form (.), where
the function G is defined in (.), by some calculation it is easy to show that the following
holds:
Pn i=
piϕ(xi) – bb –– ax¯ ϕ(a) – bx¯ –– aa ϕ(b)
Pn i=
piG(xi, s) – bb –– ax¯ G(a, s) – bx¯ –– aa G(b, s) ϕ (s) ds.
Theorem . ([]) Let –∞ < a ≤ A ≤ c ≤ b ≤ B < +∞. If xi ∈ [a, A], yj ∈ [b, B], pi > ,
qj > for i = , . . . , n and j = , . . . , m are such that in= pi = jm= qj = and
pixi = BB –– by¯ b + By¯ –– bb B –
where x¯ =
in= pixi and y¯ =
jm= qjyj, then for every f ∈ Kc(a, B) we have
pif (xi) ≤ BB –– by¯ f (b) + By¯ –– bb f (B) –
Our first result is a generalization of the result from [] stated above, in which it is
allowed for pi, qj to also be negative, with the sum not equal to zero, but with supplementary
demands on pi, qj and xi, yj given by using the Green function G defined in (.).
C := bb––ax¯ a + bx¯––aa b –
= bb––ay¯ a + by¯––aa b –
Qm j=
Pn i=
Qm j=
b – y¯ G(a, s) + y¯ – a G(b, s),
qjG(yj, s) ≤ b – a b – a
Pn i=
D b – y¯ ϕ(a) + y¯ – a ϕ(b) –
≤ C ≤ b – a b – a
Qm j=
The statement also holds if we reverse all signs of the inequalities in (.), (.), and (.).
Remark . If we set all pi, qj to be positive, then Theorem . becomes the result from
[] which is stated above in Theorem ..
Remark . We can replace the equality from the condition (.) by a weaker condition
in the analogous way as in Remark . from the previous chapter.
6 The Hermite-Hadamard inequality
The classical Hermite-Hadamard inequality states that for a convex function ϕ : [a, b] → R
the following estimation holds:
Its weighted form is proved by Fejér in []: If ϕ : [a, b] → R is a convex function and
p : [a, b] → R nonnegative integrable function, symmetric with respect to the middle point
(a + b)/, then the following estimation holds:
p(x) dx ≤
ϕ(x)p(x) dx ≤ ϕ(a) + ϕ(b)
Fink in [] discussed the generalization of (.) by separately looking the left and right
side of the inequality and considering certain signed measures. In their paper [], the
authors gave a complete characterization of the right side of the Hermite-Hadamard
inequality.
Rodić Lipanović, Pečarić, and Perić in [] obtained the complete characterization for
the left and the right side of the generalized Hermite-Hadamard inequality for the real
Stieltjes measure.
In this section a Levinson type generalization of the Hermite-Hadamard inequality for
signed measures will be given as a consequence of the results given in Sections and .
Here we use the following notation:
The statement also holds if we reverse all signs of the inequalities in (.) and (.).
Corollary . Let [α, β] ⊆ R be an interval and c ∈ α, β and let [a, b] ⊆ [α, c] and
[a, b] ⊆ [c, β]. Let λ : [a, b] → R and μ : [a, b] → R be continuous functions or
functions of bounded variation such that λ(a) = λ(b), μ(a) = μ(b) and x˜ ∈ [α, c], y˜ ∈ [c, β],
and such that
C := ab x dλ(x) – x˜ = ab y dμ(y) – y˜.
ab dλ(x) ab dμ(y)
G(x˜, s) ≤
G(y˜, s) ≤
Remark . Let the conditions from the previous corollary hold.
(i) If for all s ∈ [α, c] and s ∈ [c, β] inequalities (.) hold, then for every continuous
function ϕ ∈ Kc([α, β]) the reverse inequalities in (.) hold.
(ii) If for all s ∈ [α, c] and s ∈ [c, β] the reversed inequalities in (.) hold, then for
every continuous function ϕ ∈ Kc([α, β]) (.) holds.
Note that for the Levinson type generalization of the left-side inequality of the
generalized Hermite-Hadamard inequality it is necessary to demand that x˜ ∈ [α, c] and y˜ ∈ [c, β].
Remark . If in Remark . we put f (x) = x and g(x) = x, we can obtain weaker conditions
instead of equality (.) under which inequality (.) holds.
Similarly, from the results given in the fourth section we get the Levinson type
generalization of the right-side inequality of the generalized Hermite-Hadamard inequality.
Here we allow that the mean value x˜ goes outside of the interval [α, c] and y˜ outside of the
interval [c, β].
Corollary . Let [α, β] ⊆ R be an interval and c ∈ α, β , and let [a, b] ⊆ [α, c] and
[a, b] ⊆ [c, β]. Let λ : [a, b] → R and μ : [a, b] → R be continuous functions or
functions of bounded variation such that λ(a) = λ(b) and μ(a) = μ(b) and such that
C := b – x˜ a + x˜ – a b – ab x dλ(x)
b – a b – a ab dλ(x)
≤ bb––ax˜ G(a, s) + bx˜––aa G(b, s),
b – y˜ G(a, s) + y˜ – a G(b, s),
≤ b – a b – a
where the function G is defined in (.), then for every continuous function ϕ ∈ Kc([α, β])
we have
b – x˜ ϕ(a) + x˜ – a ϕ(b) – ab ϕ(x) dλ(x)
b – a b – a ab dλ(x)
≤ D C ≤ bb––ay˜ ϕ(a) + by˜––aa ϕ(b) – abaϕb(dy)μd(μy)(y) .
The statement also holds if we reverse all signs of the inequalities in (.), (.) and (.).
Remark . If in Remark . we put f (x) = x and g(x) = x, we can obtain analogous weaker
conditions instead of equality (.) under which inequality (.) holds.
It is easy to see that for λ(x) = x and μ(x) = x the conditions (.), (.) and (.) are
always fulfilled. In that way we can obtain a Levinson type generalization of both sides in
the classical weighted Hermite-Hadamard inequality.
Corollary . Let [α, β] ⊆ R be an interval and c ∈ α, β , and let [a, b] ⊆ [α, c] and
[a, b] ⊆ [c, β].
(i) If C := (b – a) = (b – a) holds, then for every continuous function
ϕ ∈ Kc([α, β])
ϕ(x) dx – ϕ
ϕ(x) dx – ϕ
(ii) If C := (b – a) = (b – a) holds, then for every continuous function
ϕ ∈ Kc([α, β])
7 The inequalities of Giaccardi and Petrovic´
The well-known Petrović inequality [] for convex function f : [, a] → R is given by
f (xi) ≤ f
xi + (n – )f (),
where xi (i = , . . . , n) are nonnegative numbers such that x, . . . , xn, in= xi ∈ [, a].
The following generalization of (.) is given by Giaccardi (see [] and []).
Theorem . (Giaccardi, []) Let p = (p, . . . , pn) be a nonnegative n-tuple and x =
(x, . . . , xn) be a real n-tuple such that
pjxj – xi ≥ , i = , . . . , n,
pixi ∈ [a, b] and
pixi = x.
If f : [a, b] → R is a convex function, then
A =
pif (xi) ≤ Af
pi – f (x),
B =
In this section we will use an analogous technique as in the previous sections to obtain
a Levinson type generalization of the Giaccardi inequality for n-tuples p of real numbers
which are not necessarily nonnegative. As a simple consequence, we will obtain a Levinson
type generalization of the original Giaccardi inequality (.). In order to do so, we first need
to state two results from [].
Theorem . ([]) Let xi ∈ [a, b] ⊆ [α, β], a = b, pi ∈ R (i = , . . . , n) be such that Pn = .
Then the following two statements are equivalent:
Pn i=
Pn i=
pif (xi) ≤ bb –– ax¯ f (a) + bx¯ –– aa f (b)
piG(xi, s) ≤ bb –– ax¯ G(a, s) + bx¯ –– aa G(b, s)
Moreover, the statements () and () are also equivalent if we change the sign of the
inequality in both inequalities, in (.) and in (.).
Corollary . ([]) Under the conditions from the previous theorem, the following two
statements are also equivalent:
( ) For every continuous concave function ϕ : [α, β] → R the reverse inequality in (.)
holds.
Moreover, the statements ( ) and ( ) are also equivalent if we change the sign of inequality
in both statements ( ) and ( ).
Our first result is a Levinson type generalization of the Giaccardi inequality for n-tuples
p and m-tuples q of arbitrary real numbers instead of nonnegative real numbers.
Theorem . Let [α, β] ⊆ R be an interval and c ∈ α, β such that [a, b] ⊆ [α, c] and
[a, b] ⊆ [c, β]. Let p and x be n-tuples of real numbers, and let q and y be m-tuples of
real numbers such that Pn = in= pi = , Qm = im= qi = , and
pixi – xj ≥ (j = , . . . , n);
pixi ∈ [a, b];
pixi = x;
qiyi – yj ≥ (j = , . . . , m);
qiyi ∈ [a, b];
qiyi = y.
C := A
= A
A =
A =
i=
m
i=
m
B =
B =
piG(xi, s) ≤ AG
pixi, s + B
pi – G(x, s),
qjG(yj, s) ≤ AG
qj – G(y, s),
pi – ϕ(x) –
qj – ϕ(y) –
The statement also holds if we reverse all signs of the inequalities in (.), (.), and (.).
Remark . One needs to notice that if we set pi (i = , . . . , n) and qj (j = , . . . , m) to be
positive, Theorem . becomes the Levinson type generalization of the original Giaccardi
inequality (.).
Remark . As in the previous sections, we can replace the equality (.) by a weaker
condition
and then (.) becomes
pi – ϕ(x) –
piϕ(xi) ≤ C
≤ C ≤ Aϕ
qj – ϕ(y) –
Since ϕ–(c) ≤ D ≤ ϕ+(c) (see []), if additionally ϕ is convex (resp. concave), the condition
(.) can be further weakened to
≤ A
8 Mean-value theorems
Let f : [a, b] → R and g : [a, b] → R be continuous functions, [α, β] ⊆ R an interval
and c ∈ α, β such that f ([a, b]) = [m, M] ⊆ [α, c] and g([a, b]) = [m, M] ⊆ [c, β].
Let λ : [a, b] → R and μ : [a, b] → R be continuous functions or functions of bounded
variation such that λ(a) = λ(b) and μ(a) = μ(b), and let ϕ ∈ Kc([α, β]) be a continuous
function.
Motivated by the results obtained in previous sections, we define the following linear
functionals which, respectively, represent the difference between the right and the left
side of inequalities (.) and (.):
J (ϕ) = ab ϕ(g(x)) dμ(x) – ϕ(g¯) – ab ϕ(f (x)) dλ(x) + ϕ(f¯),
ELR(ϕ) =
M – g¯ ϕ(m) + g – m ϕ(M) – ab ϕ(g(x)) dμ(x)
M – m M – m
– M – f¯ ϕ(m) – f¯ – m ϕ(M) + ab ϕ(f (x)) dλ(x) ,
M – m M – m
where m = M and m = M.
We have:
(i) J (ϕ) ≥ , when (.) holds and for all s ∈ [α, c], s ∈ [c, β] (.) holds;
(ii) ELR(ϕ) ≥ , when (.) holds, and for all s ∈ [α, c] (.) holds and for all s ∈ [c, β]
(.) holds.
In the following two theorems we give the mean-value theorems of the Lagrange and
Cauchy type, respectively.
Theorem . Let f : [a, b] → R and g : [a, b] → R be continuous functions, [α, β] ⊆
R an interval and c ∈ α, β such that f ([a, b]) = [m, M] ⊆ [α, c] and g([a, b]) =
Proof Since ϕ (x) is continuous on [α, β], it attains its minimum and maximum value on
[α, β], i.e. there exist m = minx∈[α,β] ϕ (x) and M = maxx∈[α,β] ϕ (x). The functions ϕ, ϕ :
[α, β] → R defined by
J (ϕ) =
ELR(ϕ) =
ϕ(x) = M x – ϕ(x)
are -convex because ϕ (x) ≥ and ϕ (x) ≥ , so from Remark . it follows that they
belong to the class Kc([α, β]). From Theorem . it follows that J (ϕ) ≥ and J (ϕ) ≥ ,
and from Theorem . it follows that ELR(ϕ) ≥ and ELR(ϕ) ≥ and so we get
where ϕ˜(x) = x. Since the function ϕ˜ is -convex, we have ϕ˜ ∈ Kc([α, β]), so by
applying Theorem . (resp. Theorem .) we get J (ϕ˜) ≥ (resp. ELR(ϕ˜) ≥ ). If J (ϕ˜) =
(resp. ELR(ϕ˜) = ), then (.) implies J (ϕ) = (resp. (.) implies ELR(ϕ) = ), so (.)
(resp. (.)) holds for every ξ ∈ [α, β]. Otherwise, dividing (.) by J (ϕ˜) > (resp. (.)
by ELR(ϕ˜) > ) we get
resp. m ≤
and continuity of ϕ ensures the existence of ξ ∈ [α, β] satisfying (.) (resp. ξ ∈ [α, β]
satisfying (.)).
Theorem . Let the conditions from Theorem . hold. Let ϕ, ψ ∈ C([α, β]). If J (ψ ) =
and ELR(ψ ) = , then there exist ξ, ξ ∈ [α, β] such that
where ϕ˜(x) = x. Now we have J (ϕ˜) = , because otherwise we would have J (ψ ) = ,
which is a contradiction with the assumption J (ψ ) = . So we have
χ (ξ) = J (ψ )ϕ (ξ) – J (ϕ)ψ (ξ) = ,
and this gives us the first claim of the theorem. The second claim is proved in an analogous
manner, by observing the linear functional ELR instead of J .
Remark . Note that if we set the functions f , g, λ, and μ from our theorems to fulfill
the conditions from Jensen’s integral inequality or Jensen-Steffensen’s, or Jensen-Brunk’s,
or Jensen-Boas’ inequality, then - applying that inequality on the function G which is
continuous and convex in both variables - we see that in these cases for all s ∈ [α, c], s ∈ [c, β]
inequalities in (.) hold, and so from our results we directly get the results from the paper
[].
Remark . If in the definition of the functional J (resp. ELR) we set f (x) = x and g(x) =
x, then we get a functional that represents the difference between the right and the left
side of the left-hand part (resp. right-hand part) of the generalized Hermite-Hadamard
inequality. In the same manner, adequate results of Lagrange and Cauchy type for those
functionals can be derived directly from Theorem . and Theorem ..
8.1 Discrete case
Qm j=
Pn i=
JD (ϕ) =
piϕ(xi) + ϕ(x¯) – ϕ(y¯),
– bb––ax¯ ϕ(a) – bx¯––aa ϕ(b) +
where a = b and a = b;
G(ϕ) = Aϕ
qj – ϕ(y) –
pi – ϕ(x) +
D(ϕ) =
Qm j=
Pn i=
ELRD (ϕ) =
Pn i=
Qm j=
G(ϕ) =
j=
Theorem . Let the conditions of Theorem . hold and let ϕ, ψ ∈ C([α, β]). If JD (ψ ) =
, ELRD (ψ ) = , and G(ψ ) = , then there exist ξ, ξ, ξ ∈ [α, β] such that all of the
following statements hold:
As a consequence of the previous two theorems, we now give some further results in
which we give explicit conditions on pi, xi (i = , . . . , n) and qj, yj (j = , . . . , m) for (.) and
(.) to hold, where using the properties of the function G we can skip the supplementary
conditions on that function.
Proof Note that pi, qj > implies that x¯ ∈ [α, c] and y¯ ∈ [c, β], so we can set the interval
[a, b] to be [α, c] and [a, b] to be [c, β]. The function G is convex, so by Jensen’s
inequality we see that the inequalities in (.) hold for all s ∈ [α, c], s ∈ [c, β]. Now we can
apply Theorem . and Theorem . to get the statements of this corollary.
Corollary . Let (x, . . . , xn) be monotonic n-tuple, xi ∈ [α, c] (i = , . . . , n) and (y, . . . , ym)
be monotonic m-tuple, yj ∈ [c, β] (j = , . . . , m). Let (p, . . . , pn) be a real n-tuple such that
≤ Pk ≤ Pn (k = , . . . , n), Pn > ,
and (q, . . . , qm) be a real m-tuple such that
≤ Qk ≤ Qm
(k = , . . . , m), Qm > .
Proof Suppose that x ≥ x ≥ · · · ≥ xn. We have
Pn(x – x¯) =
pi(x – xi) =
(xj– – xj)(Pn – Pj–) ≥
so it follows that x ≥ x¯ . Furthermore,
Pn(x¯ – xn) =
pi(xi – xn) =
(xj – xj+)Pj ≥ ,
and Theorem ..
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Acknowledgements
This research is supported by Croatian Science Foundation under the project 5435.
1. Steffensen , JF: On certain inequalities and methods of approximation . J. Inst. Actuar . 51 , 274 - 297 ( 1919 )
2. Boas , RP: The Jensen-Steffensen inequality . Publ. Elektroteh. Fak. Univ. Beogr., Ser. Mat. Fiz . 302 - 319 , 1 - 8 ( 1970 )
3. Pecˇaric´, JE, Proschan , F, Tong, YL : Convex Functions, Partial Orderings, and Statistical Applications. Mathematics in Science and Engineering , vol. 187 . Academic Press, San Diego ( 1992 )
4. Levinson , N: Generalisation of an inequality of Ky Fan . J. Math. Anal. Appl . 8 , 133 - 134 ( 1964 )
5. Baloch , IA, Pecˇaric´ , J, Praljak, M: Generalization of Levinson's inequality . J. Math. Inequal. 9 ( 2 ), 571 - 586 ( 2015 )
6. Bullen , PS: An inequality of N. Levinson. Publ. Elektroteh. Fak. Univ. Beogr., Ser. Mat. Fiz . 421 - 460 , 109 - 112 ( 1985 )
7. Mercer , AMcD: Short proof of Jensen's and Levinson's inequality . Math. Gaz. 94 , 492 - 495 ( 2010 )
8. Pecˇaric´ , J: On an inequality of N . Levinson. Publ. Elektroteh. Fak. Univ. Beogr., Ser. Mat. Fiz . 678 - 715 , 71 - 74 ( 1980 )
9. Witkowski , A: On Levinson's inequality . RGMIA Research Report Collection 15 , Art. 68 ( 2012 )
10. Jakšetic´ , J, Pecˇaric´, J, Praljak, M: Generalized Jensen-Steffensen inequality and exponential convexity . J. Math. Inequal . 9 ( 4 ), 1287 - 1302 ( 2015 )
11. Pecˇaric´, J, Peric´, I, Rodic´ Lipanovic´, M: Uniform treatment of Jensen type inequalities . Math. Rep. 16 ( 66 )(2), 183 - 205 ( 2014 )
12. Lah , P, Ribaricˇ, M: Converse of Jensen's inequality for convex functions . Publ. Elektroteh. Fak. Univ. Beogr., Ser. Mat. Fiz . 412 - 460 , 201 - 205 ( 1973 )
13. Edmundson , HP: Bounds on the expectation of a convex function of a random variable . The Rand Corporation, Paper No. 982 ( 1956 )
14. Jakšic´ , R, Pecˇaric´ , J: Levinson's type generalization of the Edmundson-Lah-Ribaricˇ inequality . Mediterr . J. Math. 13 ( 1 ), 483 - 496 ( 2016 )
15. Fejér , L: Über die Fourierreihen, II. Math. Naturwiss. Anz. Ungar. Akad. Wiss. 24 , 369 - 390 ( 1906 )
16. Fink , AM: A best possible Hadamard inequality . Math. Inequal. Appl . 1 , 223 - 230 ( 1998 )
17. Florea , A, Niculescu, CP: A Hermite-Hadamard inequality for convex-concave symmetric functions . Bull. Math. Soc. Sci. Math. Roum . 50 ( 98 )(2) 149 - 156 ( 2007 )
18. Petrovic´ , M: Sur une fonctionnelle . Publ. Math. Univ. Belgrade 1, 149 - 156 ( 1932 )
19. Vasic´, PM, Pecˇaric´, JE: On the Jensen inequality for monotone functions I. An. Univ. Vest. Timis¸., Ser. Mat.-Inform . 1 , 95 - 104 ( 1979 )