Generalizations of Sherman’s inequality via Fink’s identity and Green’s function
New generalizations of Sherman’s inequality for $n$-convex functions are obtained by using Fink’s identity and Green’s function. By using inequalities for the Chebyshev functional, we establish some new Ostrowski- and Gruss-type inequalities related to these generalizations.
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| author | Ivelic, Bradanovic S. Latif, M. A. Pečarić, J. E. Івелич, Браданович С. Латіф, М. А. Печарик, Й. Е. |
| author_facet | Ivelic, Bradanovic S. Latif, M. A. Pečarić, J. E. Івелич, Браданович С. Латіф, М. А. Печарик, Й. Е. |
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| description | New generalizations of Sherman’s inequality for $n$-convex functions are obtained by using Fink’s identity and Green’s function. By using inequalities for the Chebyshev functional, we establish some new Ostrowski- and Gruss-type inequalities
related to these generalizations. |
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UDC 517.5
S. Ivelić Bradanović (Univ. Split, Croatia),
N. Latif (Govt. College Univ., Pakistan),
J. Pečarić (Univ. Zagreb, Croatia)
GENERALIZATIONS OF SHERMAN’S INEQUALITY
VIA FINK’S IDENTITY AND GREEN’S FUNCTION
УЗАГАЛЬНЕННЯ НЕРIВНОСТI ШЕРМАНА
ЗА ДОПОМОГОЮ ТОТОЖНОСТI ФIНКА ТА ФУНКЦIЇ ГРIНА
New generalizations of Sherman’s inequality for n-convex functions are obtained by using Fink’s identity and Green’s
function. By using inequalities for the Chebyshev functional, we establish some new Ostrowski- and Grüss-type inequalities
related to these generalizations.
Отримано новi узагальнення нерiвностi Шермана для n-опуклих функцiй за допомогою тотожностi Фiнка та функцiї
Грiна. За допомогою нерiвностей для функцiонала Чебишова встановлено деякi новi нерiвностi типу Островського
та Грюсса, пов’язанi з цими узагальненнями.
1. Introduction. S. Sherman [9] obtained generalization of the well known majorization theorem,
proved by G. H. Hardy et al. [4], which can be stated as follows: For every convex function \phi :
[\alpha , \beta ] \rightarrow \BbbR , the inequality
m\sum
i=1
bi\phi (yi) \leq
l\sum
j=1
aj\phi (xj) (1.1)
holds, where \bfx \in [\alpha , \beta ]l, \bfy \in [\alpha , \beta ]m, \bfa \in [0,\infty )l, \bfb \in [0,\infty )m and
\bfy = \bfx \bfA T and \bfa = \bfb \bfA (1.2)
is satisfied for some row stochastic matrix \bfA \in \scrM ml(\BbbR ), i.e., matrix with
aij \geq 0 for all i = 1, . . . ,m, j = 1, . . . , l,
l\sum
j=1
aij = 1 for all i = 1, . . . ,m,
while \bfA T denotes the transpose of \bfA . If \phi is concave, then the reverse inequality in (1.1) holds.
Some related results can be found in [1, 6, 7].
Sherman’s result holds for convex functions under assumption of non negativity of entries of
vectors \bfa , \bfb and matrix \bfA . The main purpose of this paper is to present generalizations of Sher-
man’s theorem for convex function of higher order (n-convex functions) which are in a special case
convex in the usual sense. Moreover, obtained generalizations hold for real choice, not necessary
nonnegative, of vectors \bfa , \bfb and matrix \bfA . For more details about n-convexity see [8].
The techniques that we use are based on the classical real analysis and an application of Fink’s
identity and Green’s function which we introduce in the sequel.
c\bigcirc S. IVELIĆ BRADANOVIĆ, N. LATIF, J. PEČARIĆ, 2018
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 8 1033
1034 S. IVELIĆ BRADANOVIĆ, N. LATIF, J. PEČARIĆ
Theorem 1.1 [3]. Let n \geq 1 and \phi : [\alpha , \beta ] \rightarrow \BbbR be such that \phi (n - 1) is absolutely continuous
on [\alpha , \beta ]. Then
\phi (x) =
n
\beta - \alpha
\beta \int
\alpha
\phi (t) dt -
n - 1\sum
w=1
n - w
w!
\phi (w - 1)(\alpha )(x - \alpha )w - \phi (w - 1)(\beta )(x - \beta )w
\beta - \alpha
+
+
1
(n - 1)!(\beta - \alpha )
\beta \int
\alpha
(x - t)n - 1k(t, x)\phi (n)(t) dt, (1.3)
where
k(t, x) =
\left\{ t - \alpha , \alpha \leq t \leq x \leq \beta ,
t - \beta , \alpha \leq x < t \leq \beta .
(1.4)
The sum in (1.3) is zero when n = 1.
Green’s function G : [\alpha , \beta ]\times [\alpha , \beta ] \rightarrow \BbbR is defined by
G(t, s) =
\left\{
(t - \beta )(s - \alpha )
\beta - \alpha
, \alpha \leq s \leq t,
(s - \beta )(t - \alpha )
\beta - \alpha
, t \leq s \leq \beta .
(1.5)
This function is convex and continuous with respect to both variables s and t. Furthermore, for any
function \phi \in C2([\alpha , \beta ]), it can be easily shown integration by parts that the next identity is valid
\phi (x) =
\beta - x
\beta - \alpha
\phi (\alpha ) +
x - \alpha
\beta - \alpha
\phi (\beta ) +
\beta \int
\alpha
G(x, s)\phi \prime \prime (s) ds. (1.6)
For more details see [10].
To establish some new Ostrowski- and Grüss-type inequalities related to obtained generalizations,
we use recent results for the Chebyshev functional, which for two Lebesgue integrable functions f, g :
[\alpha , \beta ] \rightarrow \BbbR is defined by
T (f, g) :=
1
\beta - \alpha
\beta \int
\alpha
f(t)g(t) dt - 1
\beta - \alpha
\beta \int
\alpha
f(t) dt
1
\beta - \alpha
\beta \int
\alpha
g(t) dt.
Theorem 1.2 ([2], Theorem 1). Let f : [\alpha , \beta ] \rightarrow \BbbR be Lebesgue integrable and g : [\alpha , \beta ] \rightarrow \BbbR
be absolutely continuous with (\cdot - \alpha )(\beta - \cdot )(g\prime )2 \in L1[\alpha , \beta ]. Then
| T (f, g)| \leq 1\surd
2
[T (f, f)]1/2
1\surd
\beta - \alpha
\left( \beta \int
\alpha
(x - \alpha )(\beta - x)[g\prime (x)]2dx
\right) 1/2
. (1.7)
The constant
1\surd
2
in (1.7) is the best possible.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 8
GENERALIZATIONS OF SHERMAN’S INEQUALITY VIA FINK’S IDENTITY AND GREEN’S FUNCTION 1035
Theorem 1.3 ([2], Theorem 2). Let g : [\alpha , \beta ] \rightarrow \BbbR be monotonic nondecreasing and f : [\alpha , \beta ] \rightarrow
\rightarrow \BbbR be absolutely continuous with f \prime \in L\infty [\alpha , \beta ]. Then
| T (f, g)| \leq 1
2(\beta - \alpha )
\| f \prime \| \infty
\beta \int
\alpha
(x - \alpha )(\beta - x) dg(x). (1.8)
The constant
1
2
in (1.8) is the best possible.
With \| \cdot \| p , 1 \leq p \leq \infty , we denote the usual Lebesgue norms on space Lp[\alpha , \beta ].
Through the paper, we consider simultaneously two aspect, i.e., represent two types of results, in
first case results obtained by using only Fink’s identity and in another case results obtained by using
Fink’s identity with Green’s function.
2. Main results. We start with two identities which are very useful for us to obtain generaliza-
tions.
Theorem 2.1. Let \bfx \in [\alpha , \beta ]l, \bfy \in [\alpha , \beta ]m, \bfa \in \BbbR l and \bfb \in \BbbR m be such that (1.2) holds for
some matrix \bfA \in \scrM ml(\BbbR ) whose entries satisfy the condition
\sum l
j=1
aij = 1 for i = 1, . . . ,m.
Let k(\cdot , \cdot ) and G(\cdot , \cdot ) be defined as in (1.4) and (1.5), respectively. Let \phi : [\alpha , \beta ] \rightarrow \BbbR be such that
\phi (n - 1) is absolutely continuous on [\alpha , \beta ].
(i) For n \geq 1, the identity
l\sum
j=1
aj\phi (xj) -
m\sum
i=1
bi\phi (yi) =
=
1
\beta - \alpha
n - 1\sum
w=2
n - w
w!
\phi (w - 1)(\beta )
\left( l\sum
j=1
aj(xj - \beta )w -
m\sum
i=1
bi(yi - \beta )w
\right) -
- 1
\beta - \alpha
n - 1\sum
w=2
n - w
w!
\phi (w - 1)(\alpha )
\left( l\sum
j=1
aj(xj - \alpha )w -
m\sum
i=1
bi(yi - \alpha )w
\right) +
+
1
(n - 1)!(\beta - \alpha )
\beta \int
\alpha
\left[ l\sum
j=1
aj(xj - t)n - 1k(t, xj) -
m\sum
i=1
bi(yi - t)n - 1k(t, yi)
\right] \phi (n)(t) dt (2.1)
holds.
(ii) For n \geq 3, the identity
l\sum
j=1
aj\phi (xj) -
m\sum
i=1
bi\phi (yi) =
n - 3\sum
w=0
n - w - 2
(\beta - \alpha )w!
\times
\times
\beta \int
\alpha
\left( l\sum
j=1
ajG(xj , s) -
m\sum
i=1
biG(yi, s)
\right) \Bigl( \phi (w+1)(\beta )(s - \beta )w - \phi (w+1)(\alpha )(s - \alpha )w
\Bigr)
ds+
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 8
1036 S. IVELIĆ BRADANOVIĆ, N. LATIF, J. PEČARIĆ
+
1
(n - 3)!(\beta - \alpha )
\beta \int
\alpha
\phi (n)(t)
\left( \beta \int
\alpha
\left( l\sum
j=1
ajG(xj , s) -
m\sum
i=1
biG(yi, s)
\right) (s - t)n - 3k(t, s)ds
\right) dt
(2.2)
holds.
Proof. (i) By using (1.3) in the difference
\sum l
j=1
aj\phi (xj) -
\sum m
i=1
bi\phi (yi), we get
l\sum
j=1
aj\phi (xj) -
m\sum
i=1
bi\phi (yi) =
=
1
\beta - \alpha
n - 1\sum
w=1
n - w
w!
\phi (w - 1)(\beta )
\left( l\sum
j=1
aj(xj - \beta )w -
m\sum
i=1
bi(yi - \beta )w
\right) -
- 1
\beta - \alpha
n - 1\sum
w=1
n - w
w!
\phi (w - 1)(\alpha )
\left( l\sum
j=1
aj(xj - \alpha )w -
m\sum
i=1
bi(yi - \alpha )w
\right) +
+
1
(n - 1)!(\beta - \alpha )
\beta \int
\alpha
\left[ l\sum
j=1
aj(xj - t)n - 1k(t, xj) -
m\sum
i=1
bi(yi - t)n - 1k(t, yi)
\right] \phi (n)(t) dt.
Since under assumption (1.2) we have
l\sum
j=1
aj(xj - \alpha ) -
m\sum
i=1
bi(yi - \alpha ) =
l\sum
j=1
aj(xj - \beta ) -
m\sum
i=1
bi(yi - \beta ) = 0,
the identity (2.1) immediately follows.
(ii) By using (1.2), and (1.6) in the difference
\sum l
j=1
aj\phi (xj) -
\sum m
i=1
bi\phi (yi), we obtain
l\sum
j=1
aj\phi (xj) -
m\sum
i=1
bi\phi (yi) =
\beta \int
\alpha
\left( l\sum
j=1
ajG(xj , s) -
m\sum
i=1
biG(yi, s)
\right) \phi \prime \prime (s) ds. (2.3)
Applying Fink’s identity (1.3) to \phi \prime \prime we get
\phi \prime \prime (s) =
n - 3\sum
w=0
n - w - 2
w!
\phi (w+1)(\beta )(s - \beta )w - \phi (w+1)(\alpha )(s - \alpha )w
\beta - \alpha
+
+
1
(n - 3)!(\beta - \alpha )
\beta \int
\alpha
(s - t)n - 3k(t, s)\phi (n)(t) dt. (2.4)
By an easy calculation, using (2.3) and (2.4) we have
l\sum
j=1
aj\phi (xj) -
m\sum
i=1
bi\phi (yi) =
\beta \int
\alpha
\left[ \left( l\sum
j=1
ajG(xj , s) -
m\sum
i=1
biG(yi, s)
\right) \times
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 8
GENERALIZATIONS OF SHERMAN’S INEQUALITY VIA FINK’S IDENTITY AND GREEN’S FUNCTION 1037
\times
\left( n - 3\sum
w=0
n - w - 2
w!
\cdot \phi
(w+1)(\beta )(s - \beta )w - \phi (w+1)(\alpha )(s - \alpha )w
\beta - \alpha
+
+
1
(n - 3)!(\beta - \alpha )
\beta \int
\alpha
(s - t)n - 3k(t, s)\phi (n)(t) dt
\right) \right] ds.
After interchanging the order of summation and integration and applying Fubini’s theorem we
get (2.2).
The following generalizations of Sherman’s theorem for n-convex functions hold.
Theorem 2.2. Let all the assumptions of Theorem 2.1 be satisfied. Additionally, let \phi be n-
convex on [\alpha , \beta ].
(i) If n \geq 1 and
l\sum
j=1
aj(xj - t)n - 1k(t, xj) -
m\sum
i=1
bi(yi - t)n - 1k(t, yi) \geq 0, \alpha \leq t \leq \beta , (2.5)
then
l\sum
j=1
aj\phi (xj) -
m\sum
i=1
bi\phi (yi) \geq
\geq 1
\beta - \alpha
n - 1\sum
w=2
n - w
w!
\phi (w - 1)(\beta )
\left( l\sum
j=1
aj(xj - \beta )w -
m\sum
i=1
bi(yi - \beta )w
\right) -
- 1
\beta - \alpha
n - 1\sum
w=2
n - w
w!
\phi (w - 1)(\alpha )
\left( l\sum
j=1
aj(xj - \alpha )w -
m\sum
i=1
bi(yi - \alpha )w
\right) . (2.6)
If the reverse inequality in (2.5) holds, then the reverse inequality in (2.6) holds.
(ii) If n \geq 3 and
\beta \int
\alpha
\left( l\sum
j=1
ajG(xj , s) -
m\sum
i=1
biG(yi, s)
\right) (s - t)n - 3k(t, s) ds \geq 0, \alpha \leq s, t \leq \beta , (2.7)
then
l\sum
j=1
aj\phi (xj) -
m\sum
i=1
bi\phi (yi) \geq
\geq
n - 3\sum
w=0
n - w - 2
(\beta - \alpha )w!
\beta \int
\alpha
\left( l\sum
j=1
ajG(xj , s) -
m\sum
i=1
biG(yi, s)
\right) \times
\times
\Bigl(
\phi (w+1)(\beta )(s - \beta )w - \phi (w+1)(\alpha )(s - \alpha )w
\Bigr)
ds. (2.8)
If the reverse inequality in (2.7) holds, then the reverse inequality in (2.8) holds.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 8
1038 S. IVELIĆ BRADANOVIĆ, N. LATIF, J. PEČARIĆ
Proof. (i) Since \phi is n-convex on [\alpha , \beta ], we may assume without loss of generality that \phi is
n-times differentiable and \phi (n)(t) \geq 0, t \in [\alpha , \beta ] (see [8, p. 16]).
By using this fact and the assumption (2.5), applying Theorem 2.1, we obtain (2.6).
(ii) Analogous to the part (i).
The following generalizations under Sherman’s assumption of non negativity are also valid.
Theorem 2.3. Let all the assumptions of Theorem 2.1 be satisfied. Additionally, let \bfa , \bfb and \bfA
be nonnegative and \phi be n-convex on [\alpha , \beta ].
(i) If n is even and n \geq 2, then (2.6) holds. Moreover, if the function
\=F (\cdot ) = 1
\beta - \alpha
n - 1\sum
w=2
n - w
w!
\Bigl[
\phi (w - 1)(\beta )(\cdot - \beta )w - \phi (w - 1)(\alpha )(\cdot - \alpha )w
\Bigr]
(2.9)
is convex on [\alpha , \beta ], then (1.1) holds.
(ii) If n is even and n \geq 4, then (2.8) holds. Moreover, if the function
\~F (\cdot ) =
n - 3\sum
w=0
n - w - 2
(\beta - \alpha )w!
\beta \int
\alpha
G(\cdot , s)
\Bigl[
\phi (w+1)(\beta )(s - \beta )w - \phi (w+1)(\alpha )(s - \alpha )w
\Bigr]
ds, (2.10)
where s \in [\alpha , \beta ], is convex on [\alpha , \beta ], then (1.1) holds.
Proof. (i) Consider the function s : [\alpha , \beta ] \rightarrow \BbbR defined by
s(x) = (x - t)n - 1k(t, x) =
\left\{ (x - t)n - 1(t - \alpha ), \alpha \leq t \leq x \leq \beta ,
(x - t)n - 1(t - \beta ), \alpha \leq x < t \leq \beta .
Since
s\prime \prime (x) =
\left\{ (n - 1)(n - 2)(x - t)n - 3(t - \alpha ), \alpha \leq t \leq x \leq \beta ,
(n - 1)(n - 2)(x - t)n - 3(t - \beta ), \alpha \leq x < t \leq \beta ,
it follows that for even n \geq 2, s is convex on [\alpha , \beta ]. Then by Sherman’s theorem, the inequality (2.5)
holds. Therefore, by Theorem 2.2, the inequality (2.6) holds. Changing the order of summation, the
right-hand side of (2.6) can be written in the form
l\sum
j=1
aj \=F (xj) -
m\sum
i=1
bi \=F (yi),
where \=F is defined as in (2.9). If \=F is convex, then by Sherman’s theorem we have
l\sum
j=1
aj \=F (xj) -
m\sum
i=1
bi \=F (yi) \geq 0,
i.e., the right-hand side of (2.6) is nonnegative, so the inequality (1.1) immediately follows.
(ii) Further, the function G(\cdot , s), s \in [\alpha , \beta ], is convex on [\alpha , \beta ] and by Sherman’s theorem we
obtain
l\sum
j=1
ajG(xj , s) -
m\sum
i=1
biG(yi, s) \geq 0.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 8
GENERALIZATIONS OF SHERMAN’S INEQUALITY VIA FINK’S IDENTITY AND GREEN’S FUNCTION 1039
It is easy to see that for even n > 3, the inequality
\beta \int
\alpha
\left( l\sum
j=1
ajG(xj , s) -
m\sum
i=1
biG(yi, s)
\right) (s - t)n - 3k(t, s) ds \geq 0, \alpha \leq s \leq t, (2.11)
holds, while for every n \geq 3, we get
\beta \int
\alpha
\left( l\sum
j=1
ajG(xj , s) -
m\sum
i=1
biG(yi, s)
\right) (s - t)n - 3k(t, s) ds \geq 0, t \leq s \leq \beta .
Now applying Theorem 2.3, we have the inequality (2.8).
The rest of proof is analog to the part (ii), whereby instead of \=F we consider the function \~F
defined by (2.10).
Remark 2.1. Let all the assumptions of the previous theorem be satisfied.
(i) For even n \geq 2, the inequality (2.6) holds. Further, the function s \mapsto \rightarrow (s - \alpha )w is convex on
[\alpha , \beta ] for every w, while s \mapsto \rightarrow (s - \beta )w is convex on [\alpha , \beta ] for even w and concave for odd w.
If for even w, \phi (w - 1)(\alpha ) \leq 0 and \phi (w - 1)(\beta ) \geq 0 and for odd w, \phi (w - 1)(\alpha ) \leq 0 and
\phi (w - 1)(\beta ) \leq 0, then the right-hand side of (2.6) is nonnegative. Therefore, (1.1) immediately
follows.
(ii) For even n \geq 4, the inequality (2.8) holds. Further, when \alpha \leq s \leq \beta , we have (s - \alpha )w \geq 0
for every w while (s - \beta )w \geq 0 for even w and (s - \beta )w \leq 0 for odd w.
If for even w, \phi (w+1)(\alpha ) \leq 0 and \phi (w+1)(\beta ) \geq 0 and for odd w, \phi (w+1)(\alpha ) \leq 0 and
\phi (w+1)(\beta ) \leq 0, then the right-hand side of (2.8) is nonnegative and the inequality (1.1) immedi-
ately follows.
3. The Ostrowski- and Grüss-type inequalities. To avoid many notations, we define the
functions \scrB , \~\scrB : [\alpha , \beta ] \rightarrow \BbbR by
\scrB (t) =
l\sum
j=1
aj(xj - t)n - 1k(t, xj) -
m\sum
i=1
bi(yi - t)n - 1k(t, yi), n \geq 1, (3.1)
\~\scrB (t) =
\beta \int
\alpha
\left( l\sum
j=1
ajG(xj , s) -
m\sum
i=1
biG(yi, s)
\right) (s - t)n - 3k(t, s)ds, \alpha \leq s \leq \beta , n \geq 3,
where \bfx \in [\alpha , \beta ]l, \bfy \in [\alpha , \beta ]m, \bfa \in \BbbR l and \bfb \in \BbbR m are such that (1.2) holds for some matrix
\bfA \in \scrM ml(\BbbR ) whose entries satisfy the condition
\sum l
j=1
aij = 1 for i = 1, . . . ,m, and G(\cdot , \cdot ) and
k(\cdot , \cdot ) are defined by (1.5) and (1.4), respectively.
We also consider the Chebyshev functionals defined by
T (\scrB ,\scrB ) = 1
\beta - \alpha
\beta \int
\alpha
\scrB 2(t) dt -
\left( 1
\beta - \alpha
\beta \int
\alpha
\scrB (t) dt
\right) 2
,
T ( \~\scrB , \~\scrB ) = 1
\beta - \alpha
\beta \int
\alpha
\~\scrB 2(t) dt -
\left( 1
\beta - \alpha
\beta \int
\alpha
\~\scrB (t) dt
\right) 2
.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 8
1040 S. IVELIĆ BRADANOVIĆ, N. LATIF, J. PEČARIĆ
Theorem 3.1. Let \bfx \in [\alpha , \beta ]l, \bfy \in [\alpha , \beta ]m, \bfa \in \BbbR l and \bfb \in \BbbR m be such that (1.2) holds for
some matrix \bfA \in \scrM ml(\BbbR ) whose entries satisfy the condition
\sum l
j=1
aij = 1 for i = 1, . . . ,m.
Let \scrB , \~\scrB be defined as in (3.1). Let \phi : [\alpha , \beta ] \rightarrow \BbbR be such that \phi (n) is absolutely continuous with
(\cdot - \alpha )(\beta - \cdot )(\phi (n+1))2 \in L1[\alpha , \beta ].
(i) For n \geq 1, we have
l\sum
j=1
aj\phi (xj) -
m\sum
i=1
bi\phi (yi) =
=
1
\beta - \alpha
n - 1\sum
w=2
n - w
w!
\phi (w - 1)(\beta )
\left( l\sum
j=1
aj(xj - \beta )w -
m\sum
i=1
bi(yi - \beta )w
\right) -
- 1
\beta - \alpha
n - 1\sum
w=2
n - w
w!
\phi (w - 1)(\alpha )
\left( l\sum
j=1
aj(xj - \alpha )w -
m\sum
i=1
bi(yi - \alpha )w
\right) +
+
\phi (n - 1)(\beta ) - \phi (n - 1)(\alpha )
(\beta - \alpha )2(n - 1)!
\beta \int
\alpha
\scrB (t) dt+Rn(\phi ;\alpha , \beta ), (3.2)
where the remainder satisfies
\bigm| \bigm| Rn(\phi ;\alpha , \beta )
\bigm| \bigm| \leq 1\sqrt{}
2(\beta - \alpha )(n - 1)!
\bigl[
T (\scrB ,\scrB )
\bigr] 1/2\left( \beta \int
\alpha
(t - \alpha )(\beta - t)[\phi (n+1)(t)]2 dt
\right) 1/2
. (3.3)
(ii) For n \geq 3, we get
l\sum
j=1
aj\phi (xj) -
m\sum
i=1
bi\phi (yi) =
=
n - 3\sum
w=0
n - w - 2
(\beta - \alpha )w!
\beta \int
\alpha
\scrG (s)
\Bigl(
\phi (w+1)(\beta )(s - \beta )w - \phi (w+1)(\alpha )(s - \alpha )w
\Bigr)
ds+
+
\phi (n - 1)(\beta ) - \phi (n - 1)(\alpha )
(n - 3)!(\beta - \alpha )2
\beta \int
\alpha
\~\scrB (t) dt+ \~Rn(\phi ;\alpha , \beta ), (3.4)
where \scrG (s) =
\sum l
j=1
ajG(xj , s) -
\sum m
i=1
biG(yi, s) for G(\cdot , \cdot ) defined by (1.5), and the remainder
satisfies
\bigm| \bigm| \~Rn(\phi ;\alpha , \beta )
\bigm| \bigm| \leq 1\sqrt{}
2(\beta - \alpha )(n - 3)!
\bigl[
T ( \~\scrB , \~\scrB )
\bigr] 1/2\left( \beta \int
\alpha
(t - \alpha )(\beta - t)[\phi (n+1)(t)]2dt
\right) 1/2
. (3.5)
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 8
GENERALIZATIONS OF SHERMAN’S INEQUALITY VIA FINK’S IDENTITY AND GREEN’S FUNCTION 1041
Proof. Our proof proceeds similarly to the proof of Theorem 9 in [1].
By using Theorem 1.3, we obtain the Grüss-type inequality.
Theorem 3.2. Let \bfx \in [\alpha , \beta ]l, \bfy \in [\alpha , \beta ]m, \bfa \in \BbbR l and \bfb \in \BbbR m be such that (1.2) holds for
some matrix \bfA \in \scrM ml(\BbbR ) whose entries satisfy the condition
\sum l
j=1
aij = 1 for i = 1, . . . ,m.
Let \scrB , \~\scrB be defined as in (3.1). Let \phi : [\alpha , \beta ] \rightarrow \BbbR be such that \phi (n) is absolutely continuous and
\phi (n+1) \geq 0 on [\alpha , \beta ].
(i) For n \geq 1, the representation (3.2) holds and the remainder Rn(\phi ;\alpha , \beta ) satisfies
| Rn(\phi ;\alpha , \beta )| \leq
1
(n - 1)!
\bigm\| \bigm\| \scrB \prime \bigm\| \bigm\|
\infty
\Biggl[
\phi (n - 1)(\beta ) + \phi (n - 1)(\alpha )
2
- \phi (n - 2)(\beta ) - \phi (n - 2)(\alpha )
\beta - \alpha
\Biggr]
. (3.6)
(ii) For n \geq 3, the representation (3.4) holds and the remainder \~Rn(\phi ;\alpha , \beta ) satisfies
\bigm| \bigm| \bigm| \~Rn(\phi ;\alpha , \beta )
\bigm| \bigm| \bigm| \leq 1
(n - 3)!
\bigm\| \bigm\| \~\scrB \prime \bigm\| \bigm\|
\infty
\Biggl[
\phi (n - 1)(\beta ) + \phi (n - 1)(\alpha )
2
- \phi (n - 2)(\beta ) - \phi (n - 2)(\alpha )
\beta - \alpha
\Biggr]
. (3.7)
Proof. Our proof proceeds similarly to the proof of Theorem 10 in [1].
We present the Ostrowski-type inequality related to the identity (2.1).
Theorem 3.3. Suppose that all assumptions of Theorem 2.1 hold. Furthermore, let \scrB , \~\scrB be
defined as in (3.1). Let (p, q) be a pair of conjugate exponents, that is 1 \leq p, q \leq \infty , 1/p+1/q = 1
and \phi (n) \in Lp[\alpha , \beta ].
(i) For n \geq 1, we have \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
l\sum
j=1
aj\phi (xj) -
m\sum
i=1
bi\phi (yi) -
- 1
\beta - \alpha
n - 1\sum
w=2
n - w
w!
\phi (w - 1)(\beta )
\left( l\sum
j=1
aj(xj - \beta )w -
m\sum
i=1
bi(yi - \beta )w
\right) +
+
1
\beta - \alpha
n - 1\sum
w=2
n - w
w!
\phi (w - 1)(\alpha )
\left( l\sum
j=1
aj(xj - \alpha )w -
m\sum
i=1
bi(yi - \alpha )w
\right) \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq 1
(n - 1)!(\beta - \alpha )
\left( \beta \int
\alpha
| \scrB (t)| q dt
\right) 1/q \bigm\| \bigm\| \phi (n)
\bigm\| \bigm\|
p
. (3.8)
The constant on the right-hand side of (3.8) is sharp for 1 < p \leq \infty and the best possible for p = 1.
(ii) For n \geq 3, we get
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 8
1042 S. IVELIĆ BRADANOVIĆ, N. LATIF, J. PEČARIĆ\bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
l\sum
j=1
aj\phi (xj) -
m\sum
i=1
bi\phi (yi) -
-
n - 3\sum
w=0
n - w - 2
(\beta - \alpha )w!
\beta \int
\alpha
\scrG (s)
\Bigl(
\phi (w+1)(\beta )(s - \beta )w - \phi (w+1)(\alpha )(s - \alpha )w
\Bigr)
ds
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \leq
\leq 1
(n - 3)!(\beta - \alpha )
\left( \beta \int
\alpha
\bigm| \bigm| \bigm| \~\scrB (t)\bigm| \bigm| \bigm| q dt
\right) 1/q \bigm\| \bigm\| \phi (n)
\bigm\| \bigm\|
p
, (3.9)
where \scrG (s) =
\sum l
j=1
ajG(xj , s) -
\sum m
i=1
biG(yi, s) for G(\cdot , \cdot ) defined by (1.5). The constant on
the right-hand side of (3.9) is sharp for 1 < p \leq \infty and the best possible for p = 1.
Proof. Our proof proceeds similarly to the proof of Theorem 11 in [1].
4. Some applications. Under the assumptions of Theorem 2.2, using the inequality (2.6) and
(2.8), we can define two linear functionals
A1(\phi ) =
l\sum
j=1
aj\phi (xj) -
m\sum
i=1
bi\phi (yi) -
- 1
\beta - \alpha
n - 1\sum
w=2
n - w
w!
\phi (w - 1)(\beta )
\left( l\sum
j=1
aj(xj - \beta )w -
m\sum
i=1
bi(yi - \beta )w
\right) +
+
1
\beta - \alpha
n - 1\sum
w=2
n - w
w!
\phi (w - 1)(\alpha )
\left( l\sum
j=1
aj(xj - \alpha )w -
m\sum
i=1
bi(yi - \alpha )w
\right)
and
A2(\phi ) =
l\sum
j=1
aj\phi (xj) -
m\sum
i=1
bi\phi (yi) -
-
n - 3\sum
w=0
n - w - 2
(\beta - \alpha )w!
\beta \int
\alpha
\scrG (s)
\Bigl(
\phi (w+1)(\beta )(s - \beta )w - \phi (w+1)(\alpha )(s - \alpha )w
\Bigr)
ds,
where \scrG (s) =
\sum l
j=1
ajG(xj , s) -
\sum m
i=1
biG(yi, s) for G(\cdot , \cdot ) defined by (1.5).
For any n-convex function \phi : [\alpha , \beta ] \rightarrow \BbbR we have
Ap(\phi ) \geq 0, p = 1, 2.
By using the linearity and positivity of defined functionals, we can apply Exponentially convex
method, established in [5], in order to interpret our results in the form of exponentially or in the
special case logarithmically convex functions. As outcome we can get some new classes of two-
parameter Cauchy-type means. For such constructions we can use the same ideas as in paper [1].
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 8
GENERALIZATIONS OF SHERMAN’S INEQUALITY VIA FINK’S IDENTITY AND GREEN’S FUNCTION 1043
References
1. Agarwal R. P., Ivelić Bradanović S., Pečarić J. Generalizations of Sherman’s inequality by Lidstone’s interpolating
polynomial // J. Inequal Appl. – 2016. – 6.
2. Cerone P., Dragomir S. S. Some new Ostrowski-type bounds for the Čebyšev functional and applications // Math J.
Inequal. – 2014. – 8, № 1. – P. 159 – 170.
3. Fink A. M. Bounds of the deviation of a function from its averages // Czechoslovak Math. J. – 1992. – 42, № 117. –
P. 289 – 310.
4. Hardy G. H., Littlewood J. E., Pólya G. Inequalities. – 2nd ed. – Cambridge: Cambridge Univ. Press, 1952.
5. Jakšetić J., Pečarić J. Exponential convexity method // J. Convex Anal. – 2013. – 20, № 1. – P. 181 – 197.
6. Niezgoda M. Remarks on Sherman like inequalities for (\alpha , \beta )-convex functions // Math. Ineqal. and Appl. – 2014. –
17, № 4. – P. 1579 – 1590.
7. Niezgoda M. Vector joint majorization and generalization of Csiszár – Körner’s inequality for f -divergence // Discrete
and Appl. Math. – 2016. – 198. – P. 195 – 205.
8. Pečarić J., Proschan F., Tong Y. L. Convex functions, partial orderings and statistical applications. – New York: Acad.
Press, 1992.
9. Sherman S. On a theorem of Hardy, Littlewood, Pólya and Blackwell // Proc. Nat. Acad. Sci. USA. – 1957. – 37,
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Received 01.03.16
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 8
|
| id | umjimathkievua-article-1615 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:09:13Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/1a/eb1c33add1308f81c76c31536de3ce1a.pdf |
| spelling | umjimathkievua-article-16152019-12-05T09:21:04Z Generalizations of Sherman’s inequality via Fink’s identity and Green’s function Узагальнення нерiвностi Шермана за допомогою тотожностi фiнка та функцiї Грiна Ivelic, Bradanovic S. Latif, M. A. Pečarić, J. E. Івелич, Браданович С. Латіф, М. А. Печарик, Й. Е. New generalizations of Sherman’s inequality for $n$-convex functions are obtained by using Fink’s identity and Green’s function. By using inequalities for the Chebyshev functional, we establish some new Ostrowski- and Gruss-type inequalities related to these generalizations. Отримано новi узагальнення нерiвностi Шермана для $n$-опуклих функцiй за допомогою тотожностi Фiнка та функцiї Грiна. За допомогою нерiвностей для функцiонала Чебишова встановлено деякi новi нерiвностi типу Островського та Грюсса, пов’язанi з цими узагальненнями. Institute of Mathematics, NAS of Ukraine 2018-08-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1615 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 8 (2018); 1033-1043 Український математичний журнал; Том 70 № 8 (2018); 1033-1043 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1615/597 Copyright (c) 2018 Ivelic Bradanovic S.; Latif M. A.; Pečarić J. E. |
| spellingShingle | Ivelic, Bradanovic S. Latif, M. A. Pečarić, J. E. Івелич, Браданович С. Латіф, М. А. Печарик, Й. Е. Generalizations of Sherman’s inequality via Fink’s identity and Green’s function |
| title | Generalizations of Sherman’s inequality via Fink’s
identity and Green’s function |
| title_alt | Узагальнення нерiвностi Шермана за допомогою тотожностi фiнка та функцiї Грiна |
| title_full | Generalizations of Sherman’s inequality via Fink’s
identity and Green’s function |
| title_fullStr | Generalizations of Sherman’s inequality via Fink’s
identity and Green’s function |
| title_full_unstemmed | Generalizations of Sherman’s inequality via Fink’s
identity and Green’s function |
| title_short | Generalizations of Sherman’s inequality via Fink’s
identity and Green’s function |
| title_sort | generalizations of sherman’s inequality via fink’s
identity and green’s function |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1615 |
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