Generalizations of Steffensen’s Inequality by Lidstone’s Polynomials
We obtain generalizations of Steffensen’s inequality by using Lidstone’s polynomials. Furthermore, the functionals associated with the obtained generalizations are used to generate n-exponentially and exponentially convex functions, as well as the new Stolarsky-type means.
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| author | Pečarić, J. E. Perušić, A. Smoljak, K. Печарик, Й. Е. Перучик, А. Столяк, К. |
| author_facet | Pečarić, J. E. Perušić, A. Smoljak, K. Печарик, Й. Е. Перучик, А. Столяк, К. |
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| description | We obtain generalizations of Steffensen’s inequality by using Lidstone’s polynomials. Furthermore, the functionals associated with the obtained generalizations are used to generate n-exponentially and exponentially convex functions, as well as the new Stolarsky-type means. |
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UDC 517.5
J. Pečarić (Univ. Zagreb, Croatia),
A. Perušić (Univ. Rijeka, Croatia),
K. Smoljak (Univ. Zagreb, Croatia)
GENERALIZATIONS OF STEFFENSEN’S INEQUALITY
BY LIDSTONE’S POLYNOMIAL*
УЗАГАЛЬНЕННЯ НЕРIВНОСТI СТЕФФЕНСЕНА
ДЛЯ ПОТЕНЦIАЛIВ ЛIДСТОУНА
We obtain generalizations of Steffensen’s inequality by using Lidstone’s polynomial. Furthermore, the functionals associated
with the obtained generalizations are used to generate n-exponentially and exponentially convex functions, as well as the
new Stolarsky-type means.
Отримано узагальнення нерiвностi Стеффенсена за допомогою потенцiалiв Лiдстоуна. Крiм того, функцiонали, що
вiдповiдають отриманим узагальненням, також застосовуються для одержання як n-експоненцiально та експонен-
цiально опуклих функцiй, так i нових середнiх Столярського.
1. Introduction. Since its appearance in 1918 Steffensen’s inequality is still the subject of the
investigation and generalization by many mathematicians. The well-known Steffensen inequality
reads [10]:
Theorem 1.1. Suppose that f is decreasing and g is integrable on [a, b] with 0 ≤ g ≤ 1 and
λ =
∫ b
a
g(t)dt. Then we have
b∫
b−λ
f(t)dt ≤
b∫
a
f(t)g(t)dt ≤
a+λ∫
a
f(t)dt.
The inequalities are reversed for f increasing.
In 1929 G. J. Lidstone [5] introduced a generalization of Taylor’s series, today known as Lidstone
series. It approximates a given function in the neighborhood of two points instead of one. Such
series have been studied by H. Poritsky [8], J. M. Wittaker [13], I. J. Schoenberg [9], R. P. Boas [3]
and others.
Definition 1.1. Let f ∈ C∞([0, 1]), then Lidstone series has the form
∞∑
k=0
(
f (2k)(0)Λk(1− x) + f (2k)(1)Λk(x)
)
,
where Λn is Lidstone polynomial of degree 2n+ 1 defined by the relations
Λ0(t) = t,
Λ′′n(t) = Λn−1(t),
Λn(0) = Λn(1) = 0, n ≥ 1.
* The research of authors was supported by Ministry of Science, Education and Sports, under the Research Grants
117-1170889 (first and third authors) and 114-0000000-3145 (second author).
c© J. PEČARIĆ, A. PERUŠIĆ, K. SMOLJAK, 2015
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 11 1525
1526 J. PEČARIĆ, A. PERUŠIĆ, K. SMOLJAK
Another explicit representations of Lidstone polynomial are given in [1] and [13]. Some of those
representations are given by
Λn(t) = (−1)n
2
π2n+1
∞∑
k=1
(−1)k+1
k2n+1
sin kπt, n ≥ 1,
Λn(t) =
1
6
[
6t2n+1
(2n+ 1)!
− t2n−1
(2n− 1)!
]
−
−
n−2∑
k=0
2(22k+3 − 1)
(2k + 4)!
B2k+4
t2n−2k−3
(2n− 2k − 3)!
, n = 1, 2, . . . ,
Λn(t) =
22n+1
(2n+ 1)!
B2n+1
(
1 + t
2
)
, n = 1, 2, . . . ,
where B2k+4 is the (2k + 4)th Bernoulli number and B2n+1
(
1 + t
2
)
is a Bernoulli polynomial.
In [12] Widder proved the following fundamental lemma:
Lemma 1.1. If f ∈ C2n([0, 1]), then
f(t) =
n−1∑
k=0
[
f (2k)(0)Λk(1− t) + f (2k)(1)Λk(t)
]
+
1∫
0
Gn(t, s)f (2n)(s)ds,
where
G1(t, s) = G(t, s) =
(t− 1)s, if s < t,
(s− 1)t, if t ≤ s,
is the homogeneous Green’s function of the differential operator
d2
ds2
on [0, 1], and with the successive
iterates of G(t, s)
Gn(t, s) =
1∫
0
G1(t, p)Gn−1(p, s)dp, n ≥ 2.
The aim of this paper is to generalize Steffensen’s inequality using Lidstone’s polynomial. In
Section 2 we obtain difference of integrals on two intervals from which we obtain some general
inequality. This general inequality is used in Section 3 to obtain new generalizations of Steffensen’s
inequality for (2n)-convex functions. In Section 4 we give estimation of the difference of the left-
hand and right-hand sides of obtained generalizations. In Section 5 we consider three functionals
associated with new generalizations and use them to generate n-exponentially and exponentially
convex functions. In Section 6 we apply results from Section 5 to some families of functions to
obtain new Stolarsky-type means related to these functionals.
2. Difference of integrals on two intervals. If [a, b] ∩ [c, d] 6= ∅ we have four possible cases
for two intervals [a, b] and [c, d]. First case is [c, d] ⊂ [a, b], second case is [a, b] ∩ [c, d] = [c, b] and
other two cases are obtained by changing a ↔ c, b ↔ d. Hence, in the following theorem we will
only observe first two cases.
In this paper by T [a,b]
w,n we will denote
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 11
GENERALIZATIONS OF STEFFENSEN’S INEQUALITY BY LIDSTONE’S POLYNOMIAL 1527
T [a,b]
w,n =
n−1∑
k=0
(b− a)2k
b∫
a
w(x)
[
f (2k)(a)Λk
(
b− x
b− a
)
+ f (2k)(b)Λk
(
x− a
b− a
)]
dx.
Theorem 2.1. Let f : [a, b]∪ [c, d]→ R be of class C2n on [a, b]∪ [c, d] for some n ≥ 1. Let w :
[a, b]→ [0,∞〉 and u : [c, d]→ [0,∞〉. Then if [a, b] ∩ [c, d] 6= ∅ we have
b∫
a
w(t)f(t)dt−
d∫
c
u(t)f(t)dt− T [a,b]
w,n + T [c,d]
u,n =
max{b,d}∫
a
Kn(s)f (2n)(s)ds, (2.1)
where, in case [c, d] ⊆ [a, b],
Kn(s) =
(b− a)2n−1
b∫
a
w(x)Gn
(
x− a
b− a
,
s− a
b− a
)
dx, s ∈ [a, c],
(b− a)2n−1
b∫
a
w(x)Gn
(
x− a
b− a
,
s− a
b− a
)
dx−
−(d− c)2n−1
d∫
c
u(x)Gn
(
x− c
d− c
,
s− c
d− c
)
dx, s ∈ 〈c, d],
(b− a)2n−1
b∫
a
w(x)Gn
(
x− a
b− a
,
s− a
b− a
)
dx, s ∈ 〈d, b],
(2.2)
and, in case [a, b] ∩ [c, d] = [c, b],
Kn(s) =
(b− a)2n−1
b∫
a
w(x)Gn
(
x− a
b− a
,
s− a
b− a
)
dx, s ∈ [a, c],
(b− a)2n−1
b∫
a
w(x)Gn
(
x− a
b− a
,
s− a
b− a
)
dx −
−(d− c)2n−1
d∫
c
u(x)Gn
(
x− c
d− c
,
s− c
d− c
)
dx, s ∈ 〈c, b] ,
−(d− c)2n−1
d∫
c
u(x)Gn
(
x− c
d− c
,
s− c
d− c
)
dx, s ∈ 〈b, d].
(2.3)
Proof. From Widder’s lemma for f ∈ C2n([a, b]) we have the following identity:
f(x) =
n−1∑
k=0
(b− a)2k
[
f (2k)(a)Λk
(
b− x
b− a
)
+ f (2k)(b)Λk
(
x− a
b− a
)]
+
+(b− a)2n−1
b∫
a
Gn
(
x− a
b− a
,
s− a
b− a
)
f (2n)(s)ds. (2.4)
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 11
1528 J. PEČARIĆ, A. PERUŠIĆ, K. SMOLJAK
Multiplying identity (2.4) by w(x), then integrating from a to b and using Fubini’s theorem we obtain
b∫
a
w(x)f(x)dx =
n−1∑
k=0
(b− a)2k
b∫
a
w(x)
[
f (2k)(a)Λk
(
b− x
b− a
)
+ f (2k)(b)Λk
(
x− a
b− a
)]
dx+
+(b− a)2n−1
b∫
a
f (2n)(s)
b∫
a
w(x)Gn
(
x− a
b− a
,
s− a
b− a
)
dx
ds. (2.5)
Now subtracting identities (2.5) for interval [a, b] and [c, d] we get (2.1).
Theorem 2.2. Let f : [a, b]∪ [c, d]→ R be (2n)-convex on [a, b]∪ [c, d], w : [a, b]→ [0,∞〉, u :
[c, d]→ [0,∞〉. Then if [a, b] ∩ [c, d] 6= ∅ and
Kn(s) ≥ 0, (2.6)
we have
b∫
a
w(t)f(t)dt− T [a,b]
w,n ≥
d∫
c
u(t)f(t)dt− T [c,d]
u,n , (2.7)
where, in case [c, d] ⊆ [a, b], Kn(s) is defined by (2.2) and, in case [a, b] ∩ [c, d] = [c, b], Kn(s) is
defined by (2.3).
Proof. Since f is (2n)-convex, withouth loss of generality we can assume that f is (2n)-times
differentiable and f (2n) ≥ 0 see [7, p. 16 and 293]. Now we can apply Theorem 2.1 to obtain (2.7).
3. Generalization of Steffensen’s inequality by Lidstone’s polynomial. For a special choice
of weights and intervals in previous section we obtain a generalization of Steffensen’s inequality.
Theorem 3.1. Let f : [a, b] ∪ [a, a+ λ] → R be (2n)-convex on [a, b] ∪ [a, a + λ] and w :
[a, b]→ [0,∞〉. Then if
Kn(s) ≥ 0, (3.1)
we have
b∫
a
w(t)f(t)dt− T [a,b]
w,n ≥
a+λ∫
a
f(t)dt− T [a,a+λ]
1,n , (3.2)
where, in case a ≤ a+ λ ≤ b,
Kn(s) =
(b− a)2n−1
b∫
a
w(x)Gn
(
x− a
b− a
,
s− a
b− a
)
dx−
−λ2n−1
a+λ∫
a
Gn
(
x− a
λ
,
s− a
λ
)
dx, s ∈ [a, a+ λ],
(b− a)2n−1
b∫
a
w(x)Gn
(
x− a
b− a
,
s− a
b− a
)
dx, s ∈ 〈a+ λ, b] ,
(3.3)
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 11
GENERALIZATIONS OF STEFFENSEN’S INEQUALITY BY LIDSTONE’S POLYNOMIAL 1529
and, in case a < b ≤ a+ λ,
Kn(s) =
(b− a)2n−1
b∫
a
w(x)Gn
(
x− a
b− a
,
s− a
b− a
)
dx−
−λ2n−1
a+λ∫
a
Gn
(
x− a
λ
,
s− a
λ
)
dx, s ∈ [a, b],
−λ2n−1
a+λ∫
a
Gn
(
x− a
λ
,
s− a
λ
)
dx, s ∈ 〈b, a+ λ] .
(3.4)
Proof. We take c = a, d = a+ λ and u(t) = 1 in Theorem 2.2.
Theorem 3.2. Let f : [a, b]∪ [b− λ, b]→ R be (2n)-convex on [a, b]∪ [b−λ, b] and w : [a, b]→
→ [0,∞〉. Then if
Kn(s) ≥ 0, (3.5)
we have
b∫
b−λ
f(t)dt− T [b−λ,b]
1,n ≥
b∫
a
w(t)f(t)dt− T [a,b]
w,n , (3.6)
where, in case a ≤ b− λ ≤ b,
Kn(s) =
−(b− a)2n−1
b∫
a
w(x)Gn
(
x− a
b− a
,
s− a
b− a
)
dx, s ∈ [a, b− λ],
λ2n−1
b∫
b−λ
Gn
(
x− b+ λ
λ
,
s− b+ λ
λ
)
dx−
−(b− a)2n−1
b∫
a
w(x)Gn
(
x− a
b− a
,
s− a
b− a
)
dx, s ∈ 〈b− λ, b] ,
(3.7)
and, in case b− λ ≤ a ≤ b,
Kn(s) =
λ2n−1
b∫
b−λ
Gn
(
x− b+ λ
λ
,
s− b+ λ
λ
)
dx, s ∈ [b− λ, a],
λ2n−1
b∫
b−λ
Gn
(
x− b+ λ
λ
,
s− b+ λ
λ
)
dx −
−(b− a)2n−1
b∫
a
w(x)Gn
(
x− a
b− a
,
s− a
b− a
)
dx, s ∈ 〈a, b] .
(3.8)
Proof. First we change a↔ c, b↔ d and w ↔ u in Theorem 2.2 and then we take c = b− λ,
d = b and u(t) = 1.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 11
1530 J. PEČARIĆ, A. PERUŠIĆ, K. SMOLJAK
4. Estimation of the difference.
Theorem 4.1. Suppose that all assumptions of Theorem 2.1 hold. Assume (p, q) is a pair of
conjugate exponents, that is 1 ≤ p, q ≤ ∞, 1/p + 1/q = 1. Let
∣∣f (2n)∣∣p : [a, b] ∪ [c, d] → R be an
R-integrable function for some n ≥ 1. Then we have∣∣∣∣∣∣
b∫
a
w(t)f(t)dt−
d∫
c
u(t)f(t)dt− T [a,b]
w,n + T [c,d]
u,n
∣∣∣∣∣∣ ≤
≤
∥∥∥f (2n)∥∥∥
p
max{b,d}∫
a
|Kn(s)|q ds
1
q
. (4.1)
The constant
(∫ max{b,d}
a
|Kn(s)|q ds
)1/q
in the inequality (4.1) is sharp for 1 < p ≤ ∞ and the
best possible for p = 1.
Proof. Using inequality (2.1) and applying Hölder’s inequality we obtain∣∣∣∣∣∣
b∫
a
w(t)f(t)dt−
d∫
c
u(t)f(t)dt− T [a,b]
w,n + T [c,d]
u,n
∣∣∣∣∣∣ =
=
∣∣∣∣∣∣∣
max{b,d}∫
a
Kn(s)f (2n)(s)ds
∣∣∣∣∣∣∣ ≤
∥∥∥f (2n)∥∥∥
p
max{b,d}∫
a
|Kn(s)|q ds
1
q
.
For the proof of the sharpness of the constant
(∫ max{b,d}
a
|Kn(s)|q ds
)1/q
we will find a function f
for which the equality in (4.1) is obtained.
For 1 < p <∞ take f to be such that
f (2n)(s) = sgnKn(s) |Kn(s)|
1
p−1 .
For p =∞ take f (2n)(s) = sgnKn(s).
For p = 1 we will prove that∣∣∣∣∣∣∣
max{b,d}∫
a
Kn(s)f (2n)(s)ds
∣∣∣∣∣∣∣ ≤ max
s∈[a,max{b,d}]
|Kn(s)|
max{b,d}∫
a
∣∣∣f (2n)(s)∣∣∣ ds
(4.2)
is the best possible inequality. Suppose that |Kn(s)| attains its maximum at s0 ∈ [a,max{b, d}].
First we assume that Kn(s0) > 0. For ε small enough we define fε(s) by
fε(s) =
0, a ≤ s ≤ s0,
1
ε (2n)!
(s− s0)2n, s0 ≤ s ≤ s0 + ε,
1
(2n)!
(s− s0)2n−1, s0 + ε ≤ s ≤ max{b, d}.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 11
GENERALIZATIONS OF STEFFENSEN’S INEQUALITY BY LIDSTONE’S POLYNOMIAL 1531
Then for ε small enough∣∣∣∣∣∣∣
max{b,d}∫
a
Kn(s)f (2n)(s)ds
∣∣∣∣∣∣∣ =
∣∣∣∣∣∣
s0+ε∫
s0
Kn(s)
1
ε
ds
∣∣∣∣∣∣ =
1
ε
s0+ε∫
s0
Kn(s) ds.
Now from inequality (4.2) we have
1
ε
s0+ε∫
s0
Kn(s)ds ≤ Kn(s0)
s0+ε∫
s0
1
ε
ds = Kn(s0).
Since
lim
ε→0
1
ε
s0+ε∫
s0
Kn(s)ds = Kn(s0)
the statement follows. In case Kn(s0) < 0 we define
fε(s) =
1
(2n)!
(s− s0 − ε)2n−1, a ≤ s ≤ s0,
− 1
ε (2n)!
(s− s0 − ε)2n, s0 ≤ s ≤ s0 + ε,
0, s0 + ε ≤ s ≤ max{b, d},
and the rest of the proof is the same as above.
Theorem 4.2. Suppose that all assumptions of Theorem 3.1 hold. Assume (p, q) is a pair of
conjugate exponents, that is 1 ≤ p, q ≤ ∞, 1/p+ 1/q = 1. Let
∣∣f (2n)∣∣p : [a, b] ∪ [a, a+ λ]→ R be
an R-integrable function for some n ≥ 1. Let Kn(s) be defined by (3.3) in case a ≤ a+ λ ≤ b and
by (3.4) in case a < b ≤ a+ λ. Then we have∣∣∣∣∣∣
b∫
a
w(t)f(t)dt−
a+λ∫
a
f(t)dt− T [a,b]
w,n + T
[a,a+λ]
1,n
∣∣∣∣∣∣ ≤
≤
∥∥∥f (2n)∥∥∥
p
max{b,a+λ}∫
a
|Kn(s)|q ds
1
q
. (4.3)
The constant
(∫ max{b,a+λ}
a
|Kn(s)|q ds
)1/q
in the inequality (4.3) is sharp for 1 < p ≤ ∞ and the
best possible for p = 1.
Proof. We take c = a, d = a+ λ and u(t) = 1 in Theorem 4.1.
Theorem 4.3. Suppose that all assumptions of Theorem 3.2 hold. Assume (p, q) is a pair of
conjugate exponents, that is 1 ≤ p, q ≤ ∞, 1/p + 1/q = 1. Let
∣∣f (2n)∣∣p : [a, b] ∪ [b− λ, b] → R be
an R-integrable function for some n ≥ 1. Let Kn(s) be defined by (3.7) in case a ≤ b − λ ≤ b and
by (3.8) in case b− λ ≤ a ≤ b. Then we have
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 11
1532 J. PEČARIĆ, A. PERUŠIĆ, K. SMOLJAK∣∣∣∣∣∣
b∫
b−λ
f(t)dt−
b∫
a
w(t)f(t)dt− T [b−λ,b]
1,n + T [a,b]
w,n
∣∣∣∣∣∣ ≤
≤
∥∥∥f (2n)∥∥∥
p
b∫
min{a,b−λ}
|Kn(s)|q ds
1
q
. (4.4)
The constant
(∫ b
min{a,b−λ}
|Kn(s)|q ds
)1/q
in the inequality (4.4) is sharp for 1 < p ≤ ∞ and the
best possible for p = 1.
Proof. First we change a↔ c, b↔ d and w ↔ u in Theorem 2.1 and then we take c = b− λ,
d = b and u(t) = 1. The rest of the proof is similar to the proof of Theorem 4.1.
5. Mean value theorems and exponential convexity. Motivated by inequalities (2.7), (3.2)
and (3.6) under assumptions of Theorems 2.2, 3.1 and 3.2, respectively, we define following linear
functionals:
L1(f) =
b∫
a
w(t)f(t)dt−
d∫
c
u(t)f(t)dt−
−
n−1∑
k=0
(b− a)2k
b∫
a
w(x)
[
f (2k)(a)Λk
(
b− x
b− a
)
+ f (2k)(b)Λk
(
x− a
b− a
)]
dx+
+
n−1∑
k=0
(d− c)2k
d∫
c
u(x)
[
f (2k)(c)Λk
(
d− x
d− c
)
+ f (2k)(d)Λk
(
x− c
d− c
)]
dx, (5.1)
L2(f) =
b∫
a
w(t)f(t)dt−
a+λ∫
a
f(t)dt−
−
n−1∑
k=0
(b− a)2k
b∫
a
w(x)
[
f (2k)(a)Λk
(
b− x
b− a
)
+ f (2k)(b)Λk
(
x− a
b− a
)]
dx+
+
n−1∑
k=0
λ2k
a+λ∫
a
[
f (2k)(a)Λk
(
a+ λ− x
λ
)
+ f (2k)(a+ λ)Λk
(
x− a
λ
)]
dx, (5.2)
L3(f) =
b∫
b−λ
f(t)dt−
b∫
a
w(t)f(t)dt−
−
n−1∑
k=0
λ2k
b∫
b−λ
[
f (2k)(b− λ)Λk
(
b− x
λ
)
+ f (2k)(b)Λk
(
x− b+ λ
λ
)]
dx+
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GENERALIZATIONS OF STEFFENSEN’S INEQUALITY BY LIDSTONE’S POLYNOMIAL 1533
+
n−1∑
k=0
(b− a)2k
b∫
a
w(x)
[
f (2k)(a)Λk
(
b− x
b− a
)
+ f (2k)(b)Λk
(
x− a
b− a
)]
dx. (5.3)
Also, we define I1 = [a, b] ∪ [c, d], I2 = [a, b] ∪ [a, a+ λ], and I3 = [a, b] ∪ [b− λ, b].
Remark 5.1. Under assumptions of Theorems 2.2, 3.1 and 3.2 respectively, it holds Li(f) ≥ 0,
i = 1, 2, 3, for all (2n)-convex functions f.
First we will state and prove mean value theorems for defined functionals.
Theorem 5.1. Let f : Ii → R, i = 1, 2, 3, be such that f ∈ C2n(Ii). If inequalities in (2.6),
i = 1, (3.1), i = 2, and (3.5), i = 3, hold, then there exist ξi ∈ Ii such that
Li(f) = f (2n)(ξi)Li(ϕ), i = 1, 2, 3, (5.4)
where ϕ(x) =
x2n
(2n)!
.
Proof. Let us denote m = min f (2n) and M = max f (2n). For a given function f ∈ C2n(Ii) we
define functions F1, F2 : Ii → R with
F1(x) =
Mx2n
(2n)!
− f(x) and F2(x) = f(x)− mx2n
(2n)!
.
Now F
(2n)
1 (x) = M−f (2n)(x) ≥ 0, x ∈ Ii, so we conclude Li(F1) ≥ 0 and then Li(f) ≤M ·Li(ϕ).
Similarly, from F
(2n)
2 (x) = f (2n)(x)−m ≥ 0 we conclude m · Li(ϕ) ≤ Li(f).
If Li(ϕ) = 0, (5.4) holds for all ξi ∈ Ii. Otherwise, m ≤ Li(f)
Li(ϕ)
≤ M. Since f (2n)(x) is
continuous on Ii there exist ξi ∈ Ii such that (5.4) holds.
Theorem 5.2. Let f, g : Ii → R, i = 1, 2, 3, be such that f, g ∈ C2n(Ii) and g(2n)(x) 6= 0 for
every x ∈ Ii. If inequalities in (2.6), i = 1, (3.1), i = 2, and (3.5), i = 3, hold, then there exist
ξi ∈ Ii such that
Li(f)
Li(g)
=
f (2n)(ξi)
g(2n)(ξi)
, i = 1, 2, 3. (5.5)
Proof. We define functions φi(x) = f(x)Li(g) − g(x)Li(f), i = 1, 2, 3. Applying Theo-
rem 5.1 on φi, there exist ξi ∈ Ii such that Li(φi) = φ
(2n)
i (ξi)Li(ϕ). Since Li(φi) = 0 it follows
f (2n)(ξi)Li(g)− g(2n)(ξi)Li(f) = 0 and (5.5) is proved.
Now we will use previously defined functionals to construct exponentially convex functions. We
will start this part of the section with some definitions and properties which are used in our results
(see [6]).
Definition 5.1. A function ψ : I → R is n-exponentially convex in the Jensen sense on I if
n∑
i,j=1
ξiξj ψ
(
xi + xj
2
)
≥ 0,
holds for all choices ξ1, . . . , ξn ∈ R and all choices x1, . . . , xn ∈ I. A function ψ : I → R is
n-exponentially convex if it is n-exponentially convex in the Jensen sense and continuous on I.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 11
1534 J. PEČARIĆ, A. PERUŠIĆ, K. SMOLJAK
Remark 5.2. It is clear from the definition that 1-exponentially convex functions in the Jensen
sense are in fact nonnegative functions. Also, n-exponentially convex functions in the Jensen sense
are k-exponentially convex in the Jensen sense for every k ∈ N, k ≤ n.
Definition 5.2. A function ψ : I → R is exponentially convex in the Jensen sense on I if it is
n-exponentially convex in the Jensen sense for all n ∈ N.
A function ψ : I → R is exponentially convex if it is exponentially convex in the Jensen sense and
continuous.
Remark 5.3. It is known that ψ : I → R is log-convex in the Jensen sense if and only if
α2ψ(x) + 2αβψ
(
x+ y
2
)
+ β2ψ(y) ≥ 0,
holds for every α, β ∈ R and x, y ∈ I. It follows that a positive function is log-convex in the Jensen
sense if and only if it is 2-exponentially convex in the Jensen sense.
A positive function is log-convex if and only if it is 2-exponentially convex.
Proposition 5.1. If f is a convex function on I and if x1 ≤ y1, x2 ≤ y2, x1 6= x2, y1 6= y2, then
the following inequality is valid:
f(x2)− f(x1)
x2 − x1
≤ f(y2)− f(y1)
y2 − y1
.
If the function f is concave, the inequality is reversed.
Lemma 5.1. A function Φ is log-convex on an interval I if and only if, for all a, b, c ∈ I,
a < b < c, it holds
[Φ(b)]c−a ≤ [Φ(a)]c−b[Φ(c)]b−a.
Definition 5.3. Let f be a real-valued function defined on the segment [a, b]. The divided dif-
ference of order n of the function f at distinct points x0, . . . , xn ∈ [a, b], is defined recursively (see
[2, 7]) by
f [xi] = f(xi), i = 0, . . . , n,
and
f [x0, . . . , xn] =
f [x1, . . . , xn]− f [x0, . . . , xn−1]
xn − x0
.
The value f [x0, . . . , xn] is independent of the order of the points x0, . . . , xn.
The definition may be extended to include the case in which some (or all) of the points coincide.
Assuming that f (j−1)(x) exists, we define
f [x, . . . , x︸ ︷︷ ︸
j times
] =
f (j−1)(x)
(j − 1)!
. (5.6)
An elegant method of producing n-exponentially convex and exponentially convex functions is
given in [4]. We use this method to prove the n-exponential convexity for above defined functionals.
In the sequel the notion log denotes the natural logarithm function.
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GENERALIZATIONS OF STEFFENSEN’S INEQUALITY BY LIDSTONE’S POLYNOMIAL 1535
Theorem 5.3. Let Ω = {fp : p ∈ J}, where J is an interval in R, be a family of functions defined
on an interval Ii, i = 1, 2, 3, in R such that the function p 7→ fp[x0, . . . , x2m] is n-exponentially
convex in the Jensen sense on J for every (2m + 1) mutually different points x0, . . . , x2m ∈ Ii,
i = 1, 2, 3. Let Li, i = 1, 2, 3, be linear functionals defined by (5.1) – (5.3). Then p 7→ Li(fp) is
n-exponentially convex function in the Jensen sense on J.
If the function p 7→ Li(fp) is continuous on J, then it is n-exponentially convex on J.
Proof. For ξj ∈ R, j = 1, . . . , n, and pj ∈ J, j = 1, . . . , n, we define the function
g(x) =
n∑
j,k=1
ξjξkf pj+pk
2
(x).
Using the assumption that the function p 7→ fp[x0, . . . , x2m] is n-exponentially convex in the Jensen
sense, we have
g[x0, . . . , x2m] =
n∑
j,k=1
ξjξkf pj+pk
2
[x0, . . . , x2m] ≥ 0,
which in turn implies that g is a (2m)-convex function on J, so Li(g) ≥ 0, i = 1, 2, 3. Hence
n∑
j,k=1
ξjξkLi
(
f pj+pk
2
)
≥ 0.
We conclude that the function p 7→ Li(fp) is n-exponentially convex on J in the Jensen sense.
If the function p 7→ Li(fp) is also continuous on J, then p 7→ Li(fp) is n-exponentially convex
by definition.
The following corollaries are an immediate consequences of the above theorem:
Corollary 5.1. Let Ω = {fp : p ∈ J}, where J is an interval in R, be a family of functions defined
on an interval Ii, i = 1, 2, 3, in R, such that the function p 7→ fp[x0, . . . , x2m] is exponentially convex
in the Jensen sense on J for every (2m+ 1) mutually different points x0, . . . , x2m ∈ Ii, i = 1, 2, 3.
Let Li, i = 1, 2, 3, be linear functionals defined by (5.1) – (5.3). Then p 7→ Li(fp) is exponentially
convex function in the Jensen sense on J. If the function p 7→ Li(fp) is continuous on J, then it is
exponentially convex on J.
Corollary 5.2. Let Ω = {fp : p ∈ J}, where J is an interval in R, be a family of functions defined
on an interval Ii, i = 1, 2, 3, in R, such that the function p 7→ fp[x0, . . . , x2m] is 2-exponentially
convex in the Jensen sense on J for every (2m + 1) mutually different points x0, . . . , x2m ∈ Ii,
i = 1, 2, 3. Let Li, i = 1, 2, 3, be linear functionals defined by (5.1) – (5.3). Then the following
statements hold:
(i) If the function p 7→ Li(fp) is continuous on J, then it is 2-exponentially convex function on
J. If p 7→ Li(fp) is additionally strictly positive, then it is also log-convex on J. Furthermore, the
following inequality holds true:
[Li(fs)]
t−r ≤ [Li(fr)]
t−s [Li(ft)]
s−r (5.7)
for every choice r, s, t ∈ J, such that r < s < t.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 11
1536 J. PEČARIĆ, A. PERUŠIĆ, K. SMOLJAK
(ii) If the function p 7→ Li(fp) is strictly positive and differentiable on J, then for every p, q, u, v ∈
∈ J, such that p ≤ u and q ≤ v, we have
µp,q(Li,Ω) ≤ µu,v(Li,Ω), (5.8)
where
µp,q(Li,Ω) =
(
Li(fp)
Li(fq)
) 1
p−q
, p 6= q,
exp
d
dp
Li(fp)
Li(fp)
, p = q,
(5.9)
for fp, fq ∈ Ω.
Proof. (i) This is an immediate consequence of Theorem 5.3, Remark 5.3 and Lemma 5.1.
(ii) Since p 7→ Li(fp) is continuous and strictly positive, by (i) we have that p 7→ Li(fp) is
log-convex on J, that is, the function p 7→ logLi(fp) is convex on J. Applying Proposition 5.1 we
get
logLi(fp)− logLi(fq)
p− q
≤ logLi(fu)− logLi(fv)
u− v
(5.10)
for p ≤ u, q ≤ v, p 6= q, u 6= v. Hence, we conclude that
µp,q(Li,Ω) ≤ µu,v(Li,Ω).
Cases p = q and u = v follow from (5.10) as limit cases.
Remark 5.4. Note that the results from above theorem and corollaries still hold when two of
the points x0, . . . , x2m ∈ Ii, i = 1, 2, 3, coincide, say x1 = x0, for a family of differentiable
functions fp such that the function p 7→ fp[x0, . . . , x2m] is n-exponentially convex in the Jensen
sense (exponentially convex in the Jensen sense, log-convex in the Jensen sense), and furthermore,
they still hold when all 2m + 1 points coincide for a family of 2m differentiable functions with the
same property. The proofs use (5.6) and suitable characterization of convexity.
6. Applications to Stolarsky type means. In this section we will apply general results from
previous section to several families of functions which fulfil conditions of obtained general results to
get other exponentially convex functions and Stolarsky means.
Example 6.1. Consider a family of functions
Ω1 = {fp : R→ [0,∞) : p ∈ R}
defined by
fp(x) =
epx
p2n
, p 6= 0,
x2n
(2n)!
, p = 0.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 11
GENERALIZATIONS OF STEFFENSEN’S INEQUALITY BY LIDSTONE’S POLYNOMIAL 1537
Here,
d2nfp
dx2n
(x) = epx > 0 which shows that fp is (2n)-convex on R for every p ∈ R and
p 7→ d2nfp
dx2n
(x) is exponentially convex by definition. Using analogous arguing as in the proof of
Theorem 5.3 we also have that p 7→ fp[x0, . . . , x2m] is exponentially convex (and so exponentially
convex in the Jensen sense). Using Corollary 5.1 we conclude that p 7→ Li(fp), i = 1, 2, 3, are
exponentially convex in the Jensen sense. It is easy to verify that this mapping is continuous
(although mapping p 7→ fp is not continuous for p = 0), so it is exponentially convex. For this family
of functions, µp,q(Li,Ω1), i = 1, 2, 3, from (5.9), becomes
µp,q(Li,Ω1) =
(
Li(fp)
Li(fq)
) 1
p−q
, p 6= q,
exp
(
Li(id · fp)
Li(fp)
− 2n
p
)
, p = q 6= 0,
exp
(
1
2n+ 1
Li(id · f0)
Li(f0)
)
, p = q = 0,
where id is the identity function. Also, by Corollary 5.2 it is monotonic function in parameters p
and q.
Theorem 5.2 applied on functions fp, fq ∈ Ω1 and functionals Li, i = 1, 2, 3, implies that there
exist ξi ∈ Ii such that
e(p−q)ξi =
Li(fp)
Li(fq)
so it follows that:
Mp,q(Li,Ω1) = log µp,q(Li,Ω1), i = 1, 2, 3,
satisfies
min{a, b− λ, c} ≤Mp,q(Li,Ω1) ≤ max{a+ λ, b, d}, i = 1, 2, 3.
So, Mp,q(Li,Ω1) is monotonic mean.
Example 6.2. Consider a family of functions
Ω2 = {gp : (0,∞)→ R : p ∈ R}
defined by
gp(x) =
xp
p(p− 1) . . . (p− 2n+ 1)
, p /∈ {0, 1, . . . , 2n− 1},
xj log x
(−1)2n−1−jj!(2n− 1− j)!
, p = j ∈ {0, 1, . . . , 2n− 1}.
Here,
d2ngp
dx2n
(x) = xp−2n > 0 which shows that gp is (2n)-convex for x > 0 and p 7→ d2ngp
dx2n
(x) is
exponentially convex by definition. Arguing as in Example 6.1 we get that the mappings p 7→ Li(gp),
i = 1, 2, 3, are exponentially convex. For this family of functions µp,j(Li,Ω2), i = 1, 2, 3, from
(5.9), is now equal to
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 11
1538 J. PEČARIĆ, A. PERUŠIĆ, K. SMOLJAK
µp,q(Li,Ω2) =
(
Li(gp)
Li(gq)
) 1
p−q
, p 6= q,
exp
(
(−1)2n−1(2n− 1)!
Li(g0gp)
Li(gp)
+
2n−1∑
i=0
1
i− p
)
, p = q /∈ {0, 1, . . . , 2n− 1},
exp
(−1)2n−1(2n− 1)!
Li(g0gp)
2Li(gp)
+
2n−1∑
i=0
i 6=p
1
i− p
, p = q ∈ {0, 1, . . . , 2n− 1}.
Again, using Theorem 5.2 we conclude that
min{a, b− λ, c} ≤
(
Li(gp)
Li(gq)
) 1
p−q
≤ max{a+ λ, b, d}, i = 1, 2, 3.
So, µp,q(Li,Ω2), i = 1, 2, 3 is mean.
Example 6.3. Consider a family of functions
Ω3 = {φp : (0,∞)→ (0,∞) : p ∈ (0,∞)}
defined by
φp(x) =
p−x
(log p)2n
, p 6= 1,
x2n
(2n)!
, p = 1.
Since
d2nφp
dx2n
(x) = p−x is the Laplace transform of a nonnegative function (see [11]) it is exponentially
convex. Obviously φp are (2n)-convex functions for every p > 0. For this family of functions,
µp,q(Li,Ω3), i = 1, 2, 3, from (5.9) is equal to
µp,q(Li,Ω3) =
(
Li(φp)
Li(φq)
) 1
p−q
, p 6= q,
exp
(
−Li(id · φp)
p Li(φp)
− 2n
p log p
)
, p = q 6= 1,
exp
(
− 1
2n+ 1
Li(id · φ1)
Li(φ1)
)
, p = q = 1,
where id is the identity function. This is monotone function in parameters p and q by (5.8). Using
Theorem 5.2 it follows that
Mp,q(Li,Ω3) = −L(p, q) logµp,q(Li,Ω3), i = 1, 2, 3,
satisfies
min{a, b− λ, c} ≤Mp,q(Li,Ω3) ≤ max{a+ λ, b, d}.
So Mp,q(Li,Ω3) is monotonic mean. L(p, q) is logarithmic mean defined by
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 11
GENERALIZATIONS OF STEFFENSEN’S INEQUALITY BY LIDSTONE’S POLYNOMIAL 1539
L(p, q) =
p− q
log p− log q
, p 6= q,
p, p = q.
Example 6.4. Consider a family of functions
Ω4 = {ψp : (0,∞)→ (0,∞) : p ∈ (0,∞)}
defined by
ψp(x) =
e−x
√
p
pn
.
Since
d2nψp
dx2n
(x) = e−x
√
p is the Laplace transform of a nonnegative function (see [11]) it is exponen-
tially convex. Obviously ψp are (2n)-convex functions for every p > 0. For this family of functions,
µp,q(Li,Ω4), i = 1, 2, 3, from (5.9) is equal to
µp,q(Li,Ω4) =
(
Li(ψp)
Li(ψq)
) 1
p−q
, p 6= q,
exp
(
− Li(id · ψp)
2
√
pLi(ψp)
− n
p
)
, p = q,
where id is the identity function. This is monotone function in parameters p and q by (5.8). Using
Theorem 5.2 it follows that
Mp,q(Li,Ω4) = −
(√
p+
√
q
)
logµp,q(Li,Ω4), i = 1, 2, 3,
satisfies min{a, b− λ, c} ≤Mp,q(Li,Ω4) ≤ max{a+ λ, b, d}, so Mp,q(Li,Ω4) is monotonic mean.
1. Agarwal R. P., Wong P. J. Y. Error inequalities in polynomial interpolation and their applications. – Dordrecht etc.:
Kluwer Acad. Publ., 1993.
2. Atkinson K. E. An introduction to numerical analysis. – 2nd ed. – New York etc.: Wiley, 1989.
3. Boas R. P. Representation of functions by Lidstone series // Duke Math. J. – 1943. – 10. – P. 239 – 245.
4. Jakšetić J., Pečarić J. Exponential convexity method // J. Convex Anal. – 2013. – 20, № 1. – P. 181 – 197.
5. Lidstone G. J. Notes on the extension of Aitken’s theorem (for polynomial interpolation) to the Everett types // Proc.
Edinburgh Math. Soc. – 1929. – 2, № 2. – P. 16 – 19.
6. Pečarić J., Perić J. Improvement of the Giaccardi and the Petrović inequality and related Stolarsky type means //
An. Univ. Craiova Ser. Mat. Inform. – 2012. – 39, № 1. – P. 65 – 75.
7. Pečarić J. E., Proschan F., Tong Y. L. Convex functions, partial orderings, and statistical applications // Math. Sci.
and Eng. – 1992. – 187.
8. Poritsky H. On certain polynomial and other approximations to analytic functions // Trans. Amer. Math. Soc. –
1932. – 34, № 2. – P. 274 – 331.
9. Schoenberg I. J. On certain two-point expansions of integral functions of exponential type // Bull. Amer. Math. Soc. –
1936. – 42, № 4. – P. 284 – 288.
10. Steffensen J. F. On certain inequalities between mean values and their application to actuarial problems // Skand.
aktuarietidskr. – 1918. – P. 82 – 97.
11. Widder D. V. The Laplace transform. – New Jersey: Princeton Univ. Press, 1941.
12. Widder D. V. Completly convex functions and Lidstone series // Trans. Amer. Math. Soc. – 1942. – 51. – P. 387 – 398.
13. Whittaker J. M. On Lidstone series and two-point expansions of analytic functions // Proc. London Math. Soc. –
1933-1934. – 36. – P. 451 – 469.
Received 18.04.13
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 11
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| id | umjimathkievua-article-2087 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:18:28Z |
| publishDate | 2015 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/d9/f9f2aaf8f0ecbea607b2e116ef980dd9.pdf |
| spelling | umjimathkievua-article-20872019-12-05T09:50:14Z Generalizations of Steffensen’s Inequality by Lidstone’s Polynomials Узагальнення нерівності Стеффенсена для потенціалів Лідстоуна Pečarić, J. E. Perušić, A. Smoljak, K. Печарик, Й. Е. Перучик, А. Столяк, К. We obtain generalizations of Steffensen’s inequality by using Lidstone’s polynomials. Furthermore, the functionals associated with the obtained generalizations are used to generate n-exponentially and exponentially convex functions, as well as the new Stolarsky-type means. Отримано узагальнення нєрівності Стеффенсена за допомогою потенцiалiв Лідстоуна. Kpiм того, функціонали, що відповідають отриманим узагальненням, також застосовуються для одержання як $n$-експоненціально та експоненціально-опуклих функцій, так i нових середніх Столярського. Institute of Mathematics, NAS of Ukraine 2015-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2087 Ukrains’kyi Matematychnyi Zhurnal; Vol. 67 No. 11 (2015); 1525-1539 Український математичний журнал; Том 67 № 11 (2015); 1525-1539 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2087/1189 https://umj.imath.kiev.ua/index.php/umj/article/view/2087/1190 Copyright (c) 2015 Pečarić J. E.; Perušić A.; Smoljak K. |
| spellingShingle | Pečarić, J. E. Perušić, A. Smoljak, K. Печарик, Й. Е. Перучик, А. Столяк, К. Generalizations of Steffensen’s Inequality by Lidstone’s Polynomials |
| title | Generalizations of Steffensen’s Inequality by Lidstone’s Polynomials |
| title_alt | Узагальнення нерівності Стеффенсена для потенціалів Лідстоуна |
| title_full | Generalizations of Steffensen’s Inequality by Lidstone’s Polynomials |
| title_fullStr | Generalizations of Steffensen’s Inequality by Lidstone’s Polynomials |
| title_full_unstemmed | Generalizations of Steffensen’s Inequality by Lidstone’s Polynomials |
| title_short | Generalizations of Steffensen’s Inequality by Lidstone’s Polynomials |
| title_sort | generalizations of steffensen’s inequality by lidstone’s polynomials |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2087 |
| work_keys_str_mv | AT pecaricje generalizationsofsteffensensinequalitybylidstonespolynomials AT perusica generalizationsofsteffensensinequalitybylidstonespolynomials AT smoljakk generalizationsofsteffensensinequalitybylidstonespolynomials AT pečarikje generalizationsofsteffensensinequalitybylidstonespolynomials AT peručika generalizationsofsteffensensinequalitybylidstonespolynomials AT stolâkk generalizationsofsteffensensinequalitybylidstonespolynomials AT pecaricje uzagalʹnennânerívnostísteffensenadlâpotencíalívlídstouna AT perusica uzagalʹnennânerívnostísteffensenadlâpotencíalívlídstouna AT smoljakk uzagalʹnennânerívnostísteffensenadlâpotencíalívlídstouna AT pečarikje uzagalʹnennânerívnostísteffensenadlâpotencíalívlídstouna AT peručika uzagalʹnennânerívnostísteffensenadlâpotencíalívlídstouna AT stolâkk uzagalʹnennânerívnostísteffensenadlâpotencíalívlídstouna |