Inequalities of the Edmundson-Lah-Ribarč type for n-convex functions with applications
UDC 517.5 We derive some Edmundson – Lah – Ribarič type inequalities for positive linear functionals and $n$-convex functions. Main results are applied to the generalized $f$ -divergence functional. Examples with Zipf – Mandelbrot law are used to illustrate the results.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507091335118848 |
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| author | Mikić , R. Pečarić, D. Pečarić, J. Mikić , R. Pečarić, D. Pečarić, J. |
| author_facet | Mikić , R. Pečarić, D. Pečarić, J. Mikić , R. Pečarić, D. Pečarić, J. |
| author_sort | Mikić , R. |
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UDC 517.5
We derive some Edmundson – Lah – Ribarič type inequalities for positive linear functionals and $n$-convex functions. Main results are applied to the generalized $f$ -divergence functional. Examples with Zipf – Mandelbrot law are used to illustrate the results. |
| doi_str_mv | 10.37863/umzh.v73i1.721 |
| first_indexed | 2026-03-24T02:03:48Z |
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DOI: 10.37863/umzh.v73i1.721
UDC 517.5
R. Mikić (Univ. Zagreb, Croatia),
D. Pečarić (Catholic Univ. Croatia, Zagreb, Croatia),
J. Pečarić (RUDN Univ., Moscow, Russia)
INEQUALITIES OF THE EDMUNDSON – LAH – RIBARIČ TYPE
FOR \bfitn -CONVEX FUNCTIONS WITH APPLICATIONS*
НЕРIВНОСТI ТИПУ ЕДМУНДСОНА – ЛАХА – РИБАРИЧА
ДЛЯ \bfitn -ОПУКЛИХ ФУНКЦIЙ ТА ЇХ ЗАСТОСУВАННЯ
We derive some Edmundson – Lah – Ribarič type inequalities for positive linear functionals and n-convex functions. Main
results are applied to the generalized f -divergence functional. Examples with Zipf – Mandelbrot law are used to illustrate
the results.
Отримано нерiвностi типу Едмундсона – Лаха – Рибарича для додатних лiнiйних функцiоналiв та n-опуклих функ-
цiй. Основнi результати застосовуються до узагальнених f -дивергентних функцiоналiв. Наведено приклади, в яких
використовується закон Зiпфа – Мандельброта.
1. Introduction. Let E be a nonempty set and let L be a vector space of real-valued functions f :
E \rightarrow \BbbR having the properties:
(\mathrm{L}1) f, g \in L\Rightarrow (af + bg) \in L for all a, b \in \BbbR ;
(\mathrm{L}2) \bfone \in L, i.e., if f(t) = 1 for every t \in E, then f \in L.
We also consider positive linear functionals A : L\rightarrow \BbbR . That is, we assume that:
(\mathrm{A}1) A(af + bg) = aA(f) + bA(g) for f, g \in L and a, b \in \BbbR ;
(\mathrm{A}2) f \in L, f(t) \geq 0 for every t \in E \Rightarrow A(f) \geq 0 (A is positive).
Since it was proved, the famous Jensen inequality and its converses have been extensively studied
by many authors and have been generalized in numerous directions. Jessen [17] gave the following
generalization of Jensen’s inequality for convex functions (see also [30, p.47]).
Theorem 1.1 [17]. Let L satisfy properties (\mathrm{L}1) and (\mathrm{L}2) on a nonempty set E, and assume
that f is a continuous convex function on an interval I \subset \BbbR . If A is a positive linear functional with
A(1) = 1, then for all g \in L such that f(g) \in L we have A(g) \in I and
f(A(g)) \leq A(f(g)). (1.1)
The following result is one of the most famous converses of the Jensen inequality known as
the Edmundson – Lah – Ribarič inequality, and it was proved in [3] by Beesack and Pečarić (see also
[30, p.98]).
Theorem 1.2 [3]. Let f be convex on the interval I = [a, b] such that - \infty < a < b < \infty .
Let L satisfy conditions (\mathrm{L}1) and (\mathrm{L}2) on E and let A be any positive linear functional on L with
A(1) = 1. Then for every g \in L such that f(g) \in L (so that a \leq g(t) \leq b for all t \in E ) we have
A(f(g)) \leq b - A(g)
b - a
f(a) +
A(g) - a
b - a
f(b). (1.2)
* This paper was supported by the Ministry of Education and Science of the Russian Federation (the Agreement number
02.a03.21.0008).
c\bigcirc R. MIKIĆ, D. PEČARIĆ, J. PEČARIĆ, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1 89
90 R. MIKIĆ, D. PEČARIĆ, J. PEČARIĆ
For some recent results on the converses of the Jensen inequality, the reader is referred to [7, 19,
20, 27, 29, 31].
Unlike the results from the above mentioned papers, which require convexity of the involved
functions, the main objective of this paper is to obtain inequalities of the Edmundson – Lah – Ribarič
type that hold for n-convex functions, which will also be a generalization of the results from [24, 25].
Definition of n-convex functions is characterized by nth order divided differences. The nth
order divided difference of a function f : [a, b]\rightarrow \BbbR at mutually distinct points t0, t1, . . . , tn \in [a, b]
is defined recursively by
[ti]f = f(ti), i = 0, . . . , n,
[t0, . . . , tn]f =
[t1, . . . , tn]f - [t0, . . . , tn - 1]f
tn - t0
.
The value [t0, . . . , tn]f is independent of the order of the points t0, . . . , tn.
Definition of divided differences can be extended to include the cases in which some or all the
points coincide (see, e.g., [2, 30]):
f [a, . . . , a\underbrace{} \underbrace{}
n times
] =
1
(n - 1)!
f (n - 1)(a), n \in \BbbN .
A function f : [a, b]\rightarrow \BbbR is said to be n-convex (n \geq 0) if and only if for all choices of (n+1)
distinct points t0, t1, . . . , tn \in [a, b], we have [t0, . . . , tn]f \geq 0.
The results in this paper are obtained by utilizing Hermite’s interpolating polynomial, so first we
need to give a definition and some properties (see [2]).
Let - \infty < a < b <\infty and let a \leq a1 < a2 < . . . < ar \leq b, where r \geq 2, be given points. For
f \in \scrC n([a, b]) there exists a unique polynomial PH(t), called Hermite’s interpolating polynomial, of
degree (n - 1) fulfilling Hermite’s conditions
P
(i)
H (aj) = f (i)(aj) : 0 \leq i \leq kj , 1 \leq j \leq r,
r\sum
j=1
kj + r = n.
Among other special cases, these conditions include type (m,n - m) conditions, which will be of
special interest to us:
(r = 2, 1 \leq m \leq n - 1, k1 = m - 1, k2 = n - m - 1)
P (i)
mn(a) = f (i)(a), 0 \leq i \leq m - 1,
P (i)
mn(b) = f (i)(b), 0 \leq i \leq n - m - 1.
To give a development of the interpolating polynomial in terms of divided differences, first let us
assume that the function f is also defined at a point t \not = aj , 1 \leq j \leq r. In [2] it is shown that
f(t) = P (t) +R(t), (1.3)
where
P (t) = f(a1) + (t - a1)f [a1, a2] + (t - a1)(t - a2)f [a1, a2, a3] + . . .
. . .+ (t - a1) . . . (t - ar - 1)f [a1, . . . , ar] (1.4)
and
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INEQUALITIES OF THE EDMUNDSON – LAH – RIBARIČ TYPE FOR n-CONVEX FUNCTIONS . . . 91
R(t) = (t - a1) . . . (t - ar)f [t, a1, . . . , ar]. (1.5)
In case of (m,n - m) conditions, (1.4) and (1.5) become
Pmn(t) = f(a) + (t - a)f [a, a] + . . .+ (t - a)m - 1f [a, . . . , a\underbrace{} \underbrace{}
m times
]+
+(t - a)mf [a, . . . , a\underbrace{} \underbrace{}
m times
; b] + (t - a)m(t - b)f [a, . . . , a\underbrace{} \underbrace{}
m times
; b, b] + . . .
. . .+ (t - a)m(t - b)n - m - 1f [a, . . . , a\underbrace{} \underbrace{}
m times
; b, b, . . . , b\underbrace{} \underbrace{}
(n - m) times
] (1.6)
and
Rm(t) = (t - a)m(t - b)n - mf [t; a, . . . , a\underbrace{} \underbrace{}
m times
; b, b, . . . , b\underbrace{} \underbrace{}
(n - m) times
]. (1.7)
This paper is organized as follows. Main results, that are inequalities of the Edmundson – Lah –
Ribarič type for n-convex functions, are given in Section 2. Application of the main results to the
generalized f -divergence functional is given in Section 3. Finally, in Section 4 the results for the
generalized f -divergence are applied to Zipf – Mandelbrot law.
2. Results. Throughout this paper, whenever mentioning the interval [a, b], we assume that
- \infty < a < b <\infty holds.
Let L satisfy conditions (\mathrm{L}1) and (\mathrm{L}2) on a nonempty set E, let A be any positive linear
functional on L with A(\bfone ) = 1, and let g \in L be any function such that g(E) \subseteq [a, b]. For a given
function f : [a, b]\rightarrow \BbbR denote
LR(f, g, a, b, A) = A(f(g)) - b - A(g)
b - a
f(a) - A(g) - a
b - a
f(b). (2.1)
Following representations of the left-hand side in the Edmundson – Lah – Ribarič inequality are
obtained by using Hermite’s interpolating polynomials in terms of divided differences (1.6).
Lemma 2.1. Let L satisfy conditions (\mathrm{L}1) and (\mathrm{L}2) on a nonempty set E and let A be any
positive linear functional on L with A(\bfone ) = 1. Let f \in \scrC n([a, b]) and let g \in L be any function
such that f \circ g \in L. Then the following identities hold:
LR(f, g, a, b, A) =
n - 1\sum
k=2
f
\bigl[
a; b, . . . , b\underbrace{} \underbrace{}
k times
\bigr]
A
\bigl[
(g - a\bfone )(g - b\bfone )k - 1
\bigr]
+A(R1(g)), (2.2)
LR(f, g, a, b, A) = f [a, a; b]A[(g - a\bfone )(g - b\bfone )]+
+
n - 2\sum
k=2
f
\bigl[
a, a; b, . . . , b\underbrace{} \underbrace{}
k times
\bigr]
A
\bigl[
(g - a\bfone )2(g - b\bfone )k - 1
\bigr]
+A(R2(g)), (2.3)
LR(f, g, a, b, A) = (A(g) - a) (f [a, a] - f [a, b]) +
m - 1\sum
k=2
f (k)(a)
k!
A
\bigl[
(g - a\bfone )k
\bigr]
+
+
n - m\sum
k=1
f
\bigl[
a, . . . , a\underbrace{} \underbrace{}
m times
; b, . . . , b\underbrace{} \underbrace{}
k times
\bigr]
A
\bigl[
(g - a\bfone )m(g - b\bfone )k - 1
\bigr]
+A(Rm(g)), (2.4)
where m \geq 3 and Rm(\cdot ) is defined in (1.7).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
92 R. MIKIĆ, D. PEČARIĆ, J. PEČARIĆ
Proof. From representation (1.3) of every function f \in \scrC n([a, b]) and its Hermite interpolating
polynomial of type (m,n - m) conditions in terms of divided differences (1.6) we have
f(t) = f(a) + (t - a)f [a, a] + . . .+ (t - a)m - 1f [a, . . . , a\underbrace{} \underbrace{}
m times
]+
+(t - a)mf [a, . . . , a\underbrace{} \underbrace{}
m times
; b] + (t - a)m(t - b)f [a, . . . , a\underbrace{} \underbrace{}
m times
; b, b] + . . .
. . .+ (t - a)m(t - b)n - m - 1f [a, . . . , a\underbrace{} \underbrace{}
m times
; b, b, . . . , b\underbrace{} \underbrace{}
(n - m) times
] +Rm(t), (2.5)
where Rm(\cdot ) is defined in (1.7). After some straightforward calculations, for different choices of
1 \leq m \leq n - 1, from (2.5) we get the following:
for m = 1 it holds
LR(f,\bfone , a, b, \mathrm{i}\mathrm{d}) = (t - a)(t - b)f [a; b, b] + (t - a)(t - b)2f [a; b, b, b] + . . .
. . .+ (t - a)(t - b)n - 2f
\bigl[
a; b, b, . . . , b\underbrace{} \underbrace{}
(n - 1) times
\bigr]
+R1(t), (2.6)
for m = 2 it holds
LR(f,\bfone , a, b, \mathrm{i}\mathrm{d}) = (t - a)(t - b)f [a, a; b] + (t - a)2(t - b)f [a, a; b, b] + . . .
. . .+ (t - a)2(t - b)n - 3f
\bigl[
a, a; b, b, . . . , b\underbrace{} \underbrace{}
(n - 2) times
\bigr]
+R2(t), (2.7)
for 3 \leq m \leq n - 1 it holds
LR(f,\bfone , a, b, \mathrm{i}\mathrm{d}) = (t - a) (f [a, a] - f [a, b]) + . . .+ (t - a)m - 1f [a, . . . , a\underbrace{} \underbrace{}
m times
]+
+(t - a)mf [a, . . . , a\underbrace{} \underbrace{}
m times
; b] + (t - a)m(t - b)f [a, . . . , a\underbrace{} \underbrace{}
m times
; b, b] + . . .
. . .+ (t - a)m(t - b)n - m - 1f [a, . . . , a\underbrace{} \underbrace{}
m times
; b, b, . . . , b\underbrace{} \underbrace{}
(n - m) times
] +Rm(t). (2.8)
Since f \circ g \in L it holds g(E) \subseteq [a, b], so we can replace t with g(t) in (2.6), (2.7) and (2.8), and
obtain
LR(f, g, a, b, \mathrm{i}\mathrm{d}) =
n - 1\sum
k=2
(g(t) - a)(g(t) - b)k - 1f
\bigl[
a; b, . . . , b\underbrace{} \underbrace{}
k times
\bigr]
+R1(g(t)),
LR(f, g, a, b, \mathrm{i}\mathrm{d}) = (g(t) - a)(g(t) - b)f [a, a; b]+
+
n - 2\sum
k=2
(g(t) - a)2(g(t) - b)k - 1f
\bigl[
a, a; b, . . . , b\underbrace{} \underbrace{}
k times
\bigr]
+R2(g(t))
and
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INEQUALITIES OF THE EDMUNDSON – LAH – RIBARIČ TYPE FOR n-CONVEX FUNCTIONS . . . 93
LR(f, g, a, b, \mathrm{i}\mathrm{d}) = (g(t) - a) (f [a, a] - f [a, b]) +
m\sum
k=3
(g(t) - a)k - 1f [a, . . . , a\underbrace{} \underbrace{}
k times
]+
+
n - m\sum
k=1
(g(t) - a)m(g(t) - b)k - 1f
\bigl[
a, . . . , a\underbrace{} \underbrace{}
m times
; b, . . . , b\underbrace{} \underbrace{}
k times
\bigr]
+Rm(g(t)).
Identities (2.2), (2.3) and (2.4) follow by applying positive normalized linear functional A to the
previous equalities, respectively.
Lemma 2.1 is proved.
Lemma 2.2. Let L satisfy conditions (\mathrm{L}1) and (\mathrm{L}2) on a nonempty set E and let A be any
positive linear functional on L with A(\bfone ) = 1. Let f \in \scrC n([a, b]) and let g \in L be any function
such that f \circ g \in L. Then the following identities hold:
LR(f, g, a, b, A) =
n - 1\sum
k=2
f
\bigl[
b; a, . . . , a\underbrace{} \underbrace{}
k times
\bigr]
A
\bigl[
(g - b\bfone )(g - a\bfone )k - 1
\bigr]
+A(R\ast
1(g)), (2.9)
LR(f, g, a, b, A) = f [b, b; a]A
\bigl[
(g - b\bfone )(g - a\bfone )
\bigr]
+
+
n - 2\sum
k=2
f
\bigl[
b, b; a, . . . , a\underbrace{} \underbrace{}
k times
\bigr]
A
\bigl[
(g - b\bfone )2(g - a\bfone )k - 1
\bigr]
+A(R\ast
2(g)), (2.10)
LR(f, g, a, b, A) = (b - A(g))
\bigl(
f [a, b] - f [b, b]
\bigr)
+
m - 1\sum
k=2
f (k)(b)
k!
A[(g - b\bfone )k]+
+
n - m\sum
k=1
f
\bigl[
b, . . . , b\underbrace{} \underbrace{}
m times
; a, . . . , a\underbrace{} \underbrace{}
k times
\bigr]
A[(g - b\bfone )m(g - a\bfone )k - 1] +A(R\ast
m(g)), (2.11)
where m \geq 3 and
A(R\ast
m(g)) = A
\Bigl[
f
\bigl[
g; b\bfone , . . . , b\bfone \underbrace{} \underbrace{}
m times
; a\bfone , . . . , a\bfone \underbrace{} \underbrace{}
(n - m) times
\bigr]
(g - b\bfone )m(g - a\bfone )n - m
\Bigr]
. (2.12)
Proof. Let us define an auxiliary function F : [a, b]\rightarrow \BbbR with
F (t) = f(a+ b - t).
Since f \in \scrC n([a, b]) we immediately have F \in \scrC n([a, b]), so we can apply (2.6), (2.7) and (2.8) to
F and obtain respectively
LR(F,\bfone , a, b, \mathrm{i}\mathrm{d}) =
n - 1\sum
k=2
F
\bigl[
a; b, . . . , b\underbrace{} \underbrace{}
k times
\bigr]
(t - a)(t - b)k - 1 +R1(t), (2.13)
LR(F,\bfone , a, b, \mathrm{i}\mathrm{d}) = F [a, a; b](t - a)(t - b)+
+
n - 2\sum
k=2
F
\bigl[
a, a; b, . . . , b\underbrace{} \underbrace{}
k times
\bigr]
(t - a)2(t - b)k - 1 +R2(t), (2.14)
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
94 R. MIKIĆ, D. PEČARIĆ, J. PEČARIĆ
LR(F,\bfone , a, b, \mathrm{i}\mathrm{d}) = (t - a) (F [a, a] - F [a, b]) +
m - 1\sum
k=2
F (k)(a)
k!
(t - a)k+
+
n - m\sum
k=1
F
\bigl[
a, . . . , a\underbrace{} \underbrace{}
m times
; b, . . . , b\underbrace{} \underbrace{}
k times
\bigr]
(t - a)m(t - b)k - 1 +Rm(t). (2.15)
We can calculate divided differences of the function F in terms of divided differences of the func-
tion f :
F
\bigl[
a, . . . , a\underbrace{} \underbrace{}
k times
; b, . . . , b\underbrace{} \underbrace{}
i times
\bigr]
= ( - 1)k+i - 1f
\bigl[
b, . . . , b\underbrace{} \underbrace{}
k times
; a, . . . , a\underbrace{} \underbrace{}
i times
\bigr]
.
Now (2.13), (2.14) and (2.15) become
LR(F,\bfone , a, b, \mathrm{i}\mathrm{d}) =
n - 1\sum
k=2
( - 1)kf
\bigl[
b; a, . . . , a\underbrace{} \underbrace{}
k times
\bigr]
(t - a)(t - b)k - 1 + \=R1(t), (2.16)
LR(F,\bfone , a, b, \mathrm{i}\mathrm{d}) = ( - 1)2f [b, b; a](t - a)(t - b)+
+
n - 2\sum
k=2
( - 1)k+1f
\bigl[
b, b; a, . . . , a\underbrace{} \underbrace{}
k times
\bigr]
(t - a)2(t - b)k - 1 + \=R2(t), (2.17)
LR(F,\bfone , a, b, \mathrm{i}\mathrm{d}) = (t - a) ( - f [b, b] + f [a, b]) +
m - 1\sum
k=2
( - 1)kf (k)(b)
k!
(t - a)k+
+
n - m\sum
k=1
( - 1)m+k - 1f
\bigl[
b, . . . , b\underbrace{} \underbrace{}
m times
; a, . . . , a\underbrace{} \underbrace{}
k times
\bigr]
(t - a)m(t - b)k - 1 + \=Rm(t), (2.18)
where
\=Rm(t) = (t - a)m(t - b)n - m( - 1)nf
\bigl[
a+ b - t; b, . . . , b\underbrace{} \underbrace{}
m times
; a, a, . . . , a\underbrace{} \underbrace{}
(n - m) times
\bigr]
.
Let g \in L be any function such that f \circ g \in L, that is, a \leq g(t) \leq b for every t \in E. Let us define
a function \=g(t) = a+ b - g(t). Trivially, we have a \leq \=g(t) \leq b and \=g \in L. Since
LR(F, \=g, a, b, \mathrm{i}\mathrm{d}) = f(a+ b - (a+ b - g(t))) - b - (a+ b - g(t))
b - a
f(a+ b - a) -
- a+ b - g(t) - a
b - a
f(a+ b - b) = LR(f, g, a, b, \mathrm{i}\mathrm{d}),
after putting \=g(t) in (2.16), (2.17) and (2.18) instead of t, we get
LR(f, g, a, b, \mathrm{i}\mathrm{d}) =
n - 1\sum
k=2
( - 1)kf
\bigl[
b; a, . . . , a\underbrace{} \underbrace{}
k times
\bigr]
(b - g(t))(a - g(t))k - 1 + \=R1(a+ b - g(t)),
LR(f, g, a, b, \mathrm{i}\mathrm{d}) = ( - 1)2f [b, b; a](b - g(t))(a - g(t))+
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INEQUALITIES OF THE EDMUNDSON – LAH – RIBARIČ TYPE FOR n-CONVEX FUNCTIONS . . . 95
+
n - 2\sum
k=2
( - 1)k+1f
\bigl[
b, b; a, . . . , a\underbrace{} \underbrace{}
k times
\bigr]
(b - g(t))2(a - g(t))k - 1 + \=R2(a+ b - g(t)),
LR(f, g, a, b, \mathrm{i}\mathrm{d}) = (b - g(t)) ( - f [b, b] + f [a, b]) +
m - 1\sum
k=2
( - 1)kf (k)(b)
k!
(b - g(t))k+
+
n - m\sum
k=1
( - 1)m+k - 1f
\bigl[
b, . . . , b\underbrace{} \underbrace{}
m times
; a, . . . , a\underbrace{} \underbrace{}
k times
\bigr]
(b - g(t))m(a - g(t))k - 1 + \=Rm(a+ b - g(t)).
Identities (2.9), (2.10) and (2.11) follow after applying a normalized positive linear functional A to
previous equalities, respectively.
Lemma 2.2 is proved.
Our first result is an upper bound for the difference in the Edmundson – Lah – Ribarič inequality,
expressed by Hermite’s interpolating polynomials in terms of divided differences.
Theorem 2.1. Let L satisfy conditions (\mathrm{L}1) and (\mathrm{L}2) on a nonempty set E and let A be any
positive linear functional on L with A(\bfone ) = 1. Let f \in \scrC n([a, b]) and let g \in L be any function
such that f \circ g \in L. If the function f is n-convex and if n and m \geq 3 are of different parity, then
LR(f, g, a, b, A) \leq (A(g) - a) (f [a, a] - f [a, b]) +
m - 1\sum
k=2
f (k)(a)
k!
A
\bigl[
(g - a\bfone )k
\bigr]
+
+
n - m\sum
k=1
f
\bigl[
a, . . . , a\underbrace{} \underbrace{}
m times
; b, . . . , b\underbrace{} \underbrace{}
k times
\bigr]
A
\bigl[
(g - a\bfone )m(g - b\bfone )k - 1
\bigr]
. (2.19)
Inequality (2.19) also holds when the function f is n-concave and n and m are of equal parity. In
case when the function f is n-convex and n and m are of equal parity, or when the function f is
n-concave and n and m are of different parity, the inequality sign in (2.19) is reversed.
Proof. We start with the representation of the left-hand side in the Edmundson – Lah – Ribarič
inequality (2.4) with a special focus on the last term:
A(R(g)) = A
\left( (g - a\bfone )m (g - b\bfone )n - m f
\bigl[
g; a\bfone , . . . , a\bfone \underbrace{} \underbrace{}
m times
; b\bfone , . . . , b\bfone \underbrace{} \underbrace{}
(n - m) times
\bigr] \right) .
Since A is positive, it preserves the sign, so we need to study the sign of the expression
(g(t) - a)m (g(t) - b)n - m f
\bigl[
g(t); a, . . . , a\underbrace{} \underbrace{}
m times
; b, b, . . . , b\underbrace{} \underbrace{}
(n - m) times
\bigr]
.
Since a \leq g(t) \leq b for every t \in E, we have (g(t) - a)m \geq 0 for every t \in E and any choice
of m. For the same reason we have (g(t) - b) \leq 0. Trivially it follows that (g(t) - b)n - m \leq 0 when
n and m are of different parity, and (g(t) - b)n - m \geq 0 when n and m are of equal parity.
If the function f is n-convex, then f
\bigl[
g(t); a, . . . , a\underbrace{} \underbrace{}
m times
; b, b, . . . , b\underbrace{} \underbrace{}
(n - m) times
\bigr]
\geq 0, and if the function f is
n-concave, then f
\bigl[
g(t); a, . . . , a\underbrace{} \underbrace{}
m times
; b, b, . . . , b\underbrace{} \underbrace{}
(n - m) times
\bigr]
\leq 0.
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96 R. MIKIĆ, D. PEČARIĆ, J. PEČARIĆ
Now (2.19) easily follows from (2.1).
Theorem 2.1 is proved.
Following result provides us with a similar upper bound for the difference in the Edmundson –
Lah – Ribarič inequality, and it is obtained from Lemma 2.2.
Theorem 2.2. Let L satisfy conditions (\mathrm{L}1) and (\mathrm{L}2) on a nonempty set E and let A be any
positive linear functional on L with A(\bfone ) = 1. Let f \in \scrC n([a, b]) and let g \in L be any function
such that f \circ g \in L. If the function f is n-convex and if m \geq 3 is odd, then
LR(f, g, a, b, A) \leq (b - A(g))
\bigl(
f [a, b] - f [b, b]
\bigr)
+
m - 1\sum
k=2
f (k)(b)
k!
A[(g - b\bfone )k]+
+
n - m\sum
k=1
f
\bigl[
b, . . . , b\underbrace{} \underbrace{}
m times
; a, . . . , a\underbrace{} \underbrace{}
k times
\bigr]
A
\bigl[
(g - b\bfone )m(g - a\bfone )k - 1
\bigr]
. (2.20)
Inequality (2.20) also holds when the function f is n-concave and m is even. In case when the
function f is n-convex and m is even, or when the function f is n-concave and m is odd, the
inequality sign in (2.20) is reversed.
Proof. Similarly as in the proof of the previous theorem, we start with the representation of
the left-hand side in the Edmundson – Lah – Ribarič inequality (2.11) with a special focus on the last
term:
A(R\ast
m(g)) = A
\left( f
\bigl[
g; b\bfone , . . . , b\bfone \underbrace{} \underbrace{}
m times
; a\bfone , . . . , a\bfone \underbrace{} \underbrace{}
(n - m) times
\bigr]
(g - b\bfone )m(g - a\bfone )n - m
\right) .
As before, because of the positivity of the linear functional A, we only need to study the sign of the
expression:
(g(t) - b)m(g(t) - a)n - mf
\bigl[
g(t); b, . . . , b\underbrace{} \underbrace{}
m times
; a, a, . . . , a\underbrace{} \underbrace{}
(n - m) times
\bigr]
.
Since a \leq g(t) \leq b for every t \in E, we have (g(t) - a)n - m \geq 0 for every t \in E and any
choice of m. For the same reason we have (g(t) - b) \leq 0. Trivially it follows that (g(t) - b)m \leq 0
when m is odd, and (g(t) - b)m \geq 0 when m is even.
If the function f is n-convex, then its nth order divided differences are greater of equal to zero,
and if the function f is n-concave, then its nth order divided differences are less or equal to zero.
Now (2.20) easily follows from Lemma 2.2.
Theorem 2.2 is proved.
Corollary 2.1. Let L satisfy conditions (\mathrm{L}1) and (\mathrm{L}2) on a nonempty set E and let A be any
positive linear functional on L with A(\bfone ) = 1. Let n be an odd number, let f \in \scrC n([a, b]), and let
g \in L be any function such that f \circ g \in L. If the function f is n-convex and if m \geq 3 is odd, then
(A(g) - a) (f [a, a] - f [a, b]) +
m - 1\sum
k=2
f (k)(a)
k!
A
\bigl[
(g - a\bfone )k
\bigr]
+
+
n - m\sum
k=1
f
\bigl[
a, . . . , a\underbrace{} \underbrace{}
m times
; b, . . . , b\underbrace{} \underbrace{}
k times
\bigr]
A
\bigl[
(g - a\bfone )m(g - b\bfone )k - 1
\bigr]
\leq
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INEQUALITIES OF THE EDMUNDSON – LAH – RIBARIČ TYPE FOR n-CONVEX FUNCTIONS . . . 97
\leq LR(f, g, a, b, A) \leq (b - A(g))
\bigl(
f [a, b] - f [b, b]
\bigr)
+
m - 1\sum
k=2
f (k)(b)
k!
A[(g - b\bfone )k]+
+
n - m\sum
k=1
f
\bigl[
b, . . . , b\underbrace{} \underbrace{}
m times
; a, . . . , a\underbrace{} \underbrace{}
k times
\bigr]
A[(g - b\bfone )m(g - a\bfone )k - 1]. (2.21)
Inequality (2.21) also holds when the function f is n-concave and m is even. In case when the
function f is n-convex and m is even, or when the function f is n-concave and m is odd, the
inequality signs in (2.21) are reversed.
Remark 2.1. In [25] (Theorem 2.3) is proved that for a 3-convex functions we have
(A(g) - a)
\biggl[
f \prime (a) - f(b) - f(a)
b - a
\biggr]
+
f \prime \prime (a)
2
A[(g - a\bfone )2] \leq
\leq LR(f, g, a, b, A) \leq (b - A(g))
\biggl[
f(b) - f(a)
b - a
- f \prime (b)
\biggr]
+
f \prime \prime (b)
2
A[(b\bfone - g)2]
and if the function f is 3-concave, then the inequality signs are reversed. It is obvious that inequali-
ties (2.21) from Corollary 2.1 provide us with a generalization of the result stated above.
Next result gives us an upper and a lower bound for the difference in the Edmundson – Lah –
Ribarič inequality expressed by Hermite’s interpolating polynomials in terms of divided differences,
and it is obtained from Lemma 2.1.
Theorem 2.3. Let L satisfy conditions (\mathrm{L}1) and (\mathrm{L}2) on a nonempty set E and let A be any
positive linear functional on L with A(\bfone ) = 1. Let f \in \scrC n([a, b]) and let g \in L be any function
such that f \circ g \in L. If the function f is n-convex and if n is odd, then
n - 1\sum
k=2
f
\bigl[
a; b, . . . , b\underbrace{} \underbrace{}
k times
\bigr]
A
\bigl[
(g - a\bfone )(g - b\bfone )k - 1
\bigr]
\leq LR(f, g, a, b, A) \leq
\leq f [a, a; b]A[(g - a\bfone )(g - b\bfone )] +
n - 2\sum
k=2
f
\bigl[
a, a; b, . . . , b\underbrace{} \underbrace{}
k times
\bigr]
A
\bigl[
(g - a\bfone )2(g - b\bfone )k - 1
\bigr]
. (2.22)
Inequalities (2.22) also hold when the function f is n-concave and n is even. In case when the
function f is n-convex and n is even, or when the function f is n-concave and n is odd, the
inequality signs in (2.22) are reversed.
Proof. From the discussion about positivity and negativity of the term A(Rm(g)) in the proof
of Theorem 2.1, for m = 1 it follows that
A(R1(g)) \geq 0 when the function f is n-convex and n is odd, or when f is n-concave and n
even;
A(R1(g)) \leq 0 when the function f is n-concave and n is odd, or when f is n-convex and n
even.
Now the identity (2.2) gives us
LR(f, g, a, b, A) \geq f [a; b, b]A[(g - a\bfone )(g - b\bfone )] + f [a; b, b, b]A[(g - a\bfone )(g - b\bfone )2] + . . .
. . .+ f
\bigl[
a; b, b, . . . , b\underbrace{} \underbrace{}
(n - 1) times
\bigr]
A
\bigl[
(g - a\bfone )(g - b\bfone )n - 2
\bigr]
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98 R. MIKIĆ, D. PEČARIĆ, J. PEČARIĆ
for A(R1(g)) \geq 0, and in case A(R1(g)) \leq 0 the inequality sign is reversed.
In the same manner, for m = 2 it follows that
A(R2(g)) \leq 0 when the function f is n-convex and n is odd, or when f is n-concave and n
even;
A(R2(g)) \geq 0 when the function f is n-concave and n is odd, or when f is n-convex and n
even.
In this case the identity (2.3) for A(R2(g)) \leq 0 gives us
LR(f, g, a, b, A) \leq f [a, a; b]A[(g - a\bfone )(g - b\bfone )] + f [a, a; b, b]A[(g - a\bfone )2(g - b\bfone )] + . . .
. . .+ f
\bigl[
a, a; b, b, . . . , b\underbrace{} \underbrace{}
(n - 2) times
\bigr]
A
\bigl[
(g - a\bfone )2(g - b\bfone )n - 3
\bigr]
and in case A(R2(g)) \geq 0 the inequality sign is reversed.
When we combine the two results from above, we get exactly (2.22).
Theorem 2.3 is proved.
By utilizing Lemma 2.2 we can get similar bounds for the difference in the Edmundson – Lah –
Ribarič inequality that hold for all n \in \BbbN , not only the odd ones.
Theorem 2.4. Let L satisfy conditions (\mathrm{L}1) and (\mathrm{L}2) on a nonempty set E and let A be any
positive linear functional on L with A(\bfone ) = 1. Let f \in \scrC n([a, b]) and let g \in L be any function
such that f \circ g \in L. If the function f is n-convex, then
f [b, b; a]A
\bigl[
(g - b\bfone )(g - a\bfone )
\bigr]
+
n - 2\sum
k=2
f
\bigl[
b, b; a, . . . , a\underbrace{} \underbrace{}
k times
\bigr]
A
\bigl[
(g - b\bfone )2(g - a\bfone )k - 1
\bigr]
\leq
\leq LR(f, g, a, b, A) \leq
n - 1\sum
k=1
f
\bigl[
b; a, . . . , a\underbrace{} \underbrace{}
k times
\bigr]
A
\bigl[
(g - b\bfone )(g - a\bfone )k - 1
\bigr]
. (2.23)
If the function f is n-concave, the inequality signs in (2.23) are reversed.
Proof. We return to the discussion about positivity and negativity of the term A(R\ast
m(g)) in the
proof of Theorem 2.2. For m = 1 we have
(g(t) - b)1(g(t) - a)n - 1 \leq 0 \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{y} t \in E,
so A(R\ast
1(g)) \geq 0 when the function f is n-concave, and A(R\ast
1(g)) \leq 0 when the function f is
n-convex. Now the identity (2.9) for a n-convex function f gives us
LR(f, g, a, b, A) \geq f [b, b; a]A
\bigl[
(g - b\bfone )(g - a\bfone )
\bigr]
+ f [b, b; a, a]A[(g - b\bfone )2(g - a\bfone )] + . . .
. . .+ f
\bigl[
b, b; a, a, . . . , a\underbrace{} \underbrace{}
(n - 2) times
\bigr]
A[(g - b\bfone )2(g - a\bfone )n - 3]
and if the function f is n-concave, the inequality sign is reversed.
Similarly, for m = 2 we have
(g(t) - b)2(g(t) - a)n - 2 \geq 0 \mathrm{f}\mathrm{o}\mathrm{r} \mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{y} t \in E,
so A(R\ast
2(g)) \geq 0 when the function f is n-convex, and A(R\ast
2(g)) \leq 0 when the function f is
n-concave. In this case the identity (2.10) for a n-convex function f gives us
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INEQUALITIES OF THE EDMUNDSON – LAH – RIBARIČ TYPE FOR n-CONVEX FUNCTIONS . . . 99
LR(f, g, a, b, A) \leq f [b; a, a]A
\bigl[
(g - b\bfone )(g - a\bfone )
\bigr]
+ f [b; a, a, a]A
\bigl[
(g - b\bfone )(g - a\bfone )2
\bigr]
+ . . .
. . .+ f
\bigl[
b; a, a, . . . , a\underbrace{} \underbrace{}
(n - 1) times
\bigr]
A
\bigl[
(g - b\bfone )(g - a\bfone )n - 2
\bigr]
and if the function f is n-concave, the inequality sign is reversed.
When we combine the two results from above, we get exactly (2.23).
Theorem 2.4 is proved.
Remark 2.2. Since
f [a; b, b] =
1
b - a
\biggl(
f \prime (b) - f(b) - f(a)
b - a
\biggr)
,
f [a, a; b] =
1
b - a
\biggl(
f \prime (b) - f(b) - f(a)
b - a
\biggr)
,
when we take n = 3 in (2.22) or (2.23), we get that
A[(g - a\bfone )(g - b\bfone )]
b - a
\biggl(
f \prime (b) - f(b) - f(a)
b - a
\biggr)
\leq
\leq LR(f, g, a, b, A) \leq A[(g - a\bfone )(g - b\bfone )]
b - a
\biggl(
f \prime (b) - f(b) - f(a)
b - a
\biggr)
(2.24)
holds for a 3-convex function, and for a 3-concave function the inequality signs are reversed. In-
equalities (2.24) are proved in [25] (Theorem 2.1), so it follows that Theorem 2.3 and Theorem 2.4
give a generalization of a result from [25].
3. Applications to Csiszár divergence. Let us denote the set of all finite discrete probability dis-
tributions by \BbbP , that is we say \bfitp = (p1, . . . , pr) \in \BbbP if pi \in [0, 1] for i = 1, . . . , r and
\sum r
i=1
pi = 1.
Numerous theoretic divergence measures between two probability distributions have been intro-
duced and comprehensively studied. Their applications can be found in the analysis of contingency
tables [13], in approximation of probability distributions [8, 22], in signal processing [18], and in
pattern recognition [4, 6].
Csiszár [9 – 10] introduced the f -divergence functional as
Df (\bfitp , \bfitq ) =
r\sum
i=1
qif
\biggl(
pi
qi
\biggr)
, (3.1)
where f : [0,+\infty \rangle is a convex function, and it represent a “distance function” on the set of probability
distributions \BbbP .
A great number of theoretic divergences are special cases of Csiszár f -divergence for different
choices of the function f.
As in Csiszár [10], we interpret undefined expressions by
f(0) = \mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow 0+
f(t), 0 \cdot f
\biggl(
0
0
\biggr)
= 0,
0 \cdot f
\Bigl( a
0
\Bigr)
= \mathrm{l}\mathrm{i}\mathrm{m}
\epsilon \rightarrow 0+
\epsilon \cdot f
\Bigl( a
\epsilon
\Bigr)
= a \cdot \mathrm{l}\mathrm{i}\mathrm{m}
t\rightarrow \infty
f(t)
t
.
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100 R. MIKIĆ, D. PEČARIĆ, J. PEČARIĆ
In this section our intention is to derive mutual bounds for the generalized f -divergence func-
tional in described setting. In such a way, we will obtain some new reverse relations for the genera-
lized f -divergence functional that correspond to the class of n-convex functions. It is a generalization
of the results obtained in [25]. Throughout this section, when mentioning the interval [a, b], we as-
sume that [a, b] \subseteq \BbbR +. For a n-convex function f : [a, b] \rightarrow \BbbR we give the following definition of
generalized f -divergence functional:
\~Df (\bfitp , \bfitq ) =
r\sum
i=1
qif
\biggl(
pi
qi
\biggr)
. (3.2)
The first result in this section is carried out by virtue of our Theorem 2.1.
Theorem 3.1. Let [a, b] \subset \BbbR be an interval such that a \leq 1 \leq b. Let f \in \scrC n([a, b]) and let
\bfitp = (p1, . . . , pr) and \bfitq = (q1, . . . , qr) be probability distributions such that pi/qi \in [a, b] for every
i = 1, . . . , r. If the function f is n-convex and if n and 3 \leq m \leq n - 1 are of different parity, then
b - 1
b - a
f(a) +
1 - a
b - a
f(b) - \~Df (\bfitp , \bfitq ) \leq
\leq (1 - a) (f [a, a] - f [a, b]) +
m - 1\sum
k=2
f (k)(a)
k!
r\sum
i=1
(pi - aqi)
k
qk - 1
i
+
+
n - m\sum
k=1
f
\bigl[
a, . . . , a\underbrace{} \underbrace{}
m times
; b, . . . , b\underbrace{} \underbrace{}
k times
\bigr] r\sum
i=1
(pi - aqi)
m(pi - aqi)
k - 1
qm+k - 2
i
. (3.3)
Inequality (3.3) also holds when the function f is n-concave and n and m are of equal parity. In
case when the function f is n-convex and n and m are of equal parity, or when the function f is
n-concave and n and m are of different parity, the inequality sign in (3.3) is reversed.
Proof. Let \bfitx = (x1, . . . , xr) be such that xi \in [a, b] for i = 1, . . . , r. In the relation (2.19) we
can replace
g \leftarrow \rightarrow \bfitx and A(\bfitx ) =
r\sum
i=1
pixi.
In that way we get
b - \=x
b - a
f(a) +
\=x - a
b - a
f(b) -
r\sum
i=1
pif(xi) \leq
\leq (\=x - a) (f [a, a] - f [a, b]) +
m - 1\sum
k=2
f (k)(a)
k!
r\sum
i=1
pi(xi - a)k+
+
n - m\sum
k=1
f
\bigl[
a, . . . , a\underbrace{} \underbrace{}
m times
; b, . . . , b\underbrace{} \underbrace{}
k times
\bigr] r\sum
i=1
pi(xi - a)m(xi - b)k - 1,
where \=x =
\sum n
i=1
pixi. In the previous relation we can set
pi = qi and xi =
pi
qi
,
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INEQUALITIES OF THE EDMUNDSON – LAH – RIBARIČ TYPE FOR n-CONVEX FUNCTIONS . . . 101
and after calculating
\=x =
n\sum
i=1
qi
pi
qi
=
n\sum
i=1
pi = 1
we get (3.3).
Theorem 3.1 is proved.
By utilizing Theorem 2.2 in the analogous way as above, we get an Edmundson – Lah – Ribarič
type inequality for the generalized f -divergence functional (3.2) which does not depend on parity of
n, and it is given in the following theorem.
Theorem 3.2. Let [a, b] \subset \BbbR be an interval such that a \leq 1 \leq b. Let f \in \scrC n([a, b]) and let
\bfitp = (p1, . . . , pr) and \bfitp = (q1, . . . , qr) be probability distributions such that pi/qi \in [a, b] for every
i = 1, . . . , r. If the function f is n-convex and if 3 \leq m \leq n - 1 is odd, then
b - 1
b - a
f(a) +
1 - a
b - a
f(b) - \~Df (\bfitp , \bfitq ) \leq
\leq (b - 1)
\bigl(
f [a, b] - f [b, b]
\bigr)
+
m - 1\sum
k=2
f (k)(b)
k!
r\sum
i=1
(pi - bqi)
k
qk - 1
i
+
+
n - m\sum
k=1
f
\bigl[
b, . . . , b\underbrace{} \underbrace{}
m times
; a, . . . , a\underbrace{} \underbrace{}
k times
\bigr] r\sum
i=1
(pi - bqi)
m(pi - aqi)
k - 1
qm+k - 2
i
. (3.4)
Inequality (3.4) also holds when the function f is n-concave and m is even. In case when the
function f is n-convex and m is even, or when the function f is n-concave and m is odd, the
inequality sign in (3.4) is reversed.
Another generalization of the Edmundson – Lah – Ribarič inequality, which provides us with a
lower and an upper bound for the generalized f -divergence functional, is given in the following
theorem.
Theorem 3.3. Let [a, b] \subset \BbbR be an interval such that a \leq 1 \leq b. Let f \in \scrC n([a, b]) and let
\bfitp = (p1, . . . , pr) and \bfitp = (q1, . . . , qr) be probability distributions such that pi/qi \in [a, b] for every
i = 1, . . . , r. If the function f is n-convex and if n is odd, then we have
n - 1\sum
k=2
f [a; b, b, . . . , b\underbrace{} \underbrace{}
k times
]
r\sum
i=1
(pi - aqi)(pi - bqi)
k - 1
qk - 1
i
\leq b - 1
b - a
f(a) +
1 - a
b - a
f(b) - \~Df (\bfitp , \bfitq ) \leq
\leq f [a, a; b]
r\sum
i=1
(pi - aqi)(pi - bqi)
qi
+
n - 2\sum
k=2
f
\bigl[
a, a; b, . . . , b\underbrace{} \underbrace{}
k times
\bigr] r\sum
i=1
(pi - aqi)
2(pi - bqi)
k - 1
qki
. (3.5)
Inequalities (3.5) also hold when the function f is n-concave and n is even. In case when the
function f is n-convex and n is even, or when the function f is n-concave and n is odd, the
inequality signs in (3.5) are reversed.
Proof. We start with inequalities (2.22), and follow the steps from the proof of Theorem 3.1.
By utilizing Theorem 2.4 in an analogue way, we can get similar bounds for the generalized
f -divergence functional that hold for all n \in \BbbN , not only the odd ones.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
102 R. MIKIĆ, D. PEČARIĆ, J. PEČARIĆ
Theorem 3.4. Let [a, b] \subset \BbbR be an interval such that a \leq 1 \leq b. Let f \in \scrC n([a, b]) and let
\bfitp = (p1, . . . , pr) and \bfitp = (q1, . . . , qr) be probability distributions such that pi/qi \in [a, b] for every
i = 1, . . . , r. If the function f is n-convex, then we have
f [b, b; a]
r\sum
i=1
(pi - aqi)(pi - bqi)
qi
+
n - 2\sum
k=2
f [b, b; a, a, . . . , a\underbrace{} \underbrace{}
k times
]
r\sum
i=1
(pi - aqi)
k - 1(pi - bqi)
2
qki
\leq
\leq b - 1
b - a
f(a) +
1 - a
b - a
f(b) - \~Df (\bfitp , \bfitq ) \leq
n - 1\sum
k=2
f
\bigl[
b; a, . . . , a\underbrace{} \underbrace{}
k times
\bigr] r\sum
i=1
(pi - aqi)
k - 1(pi - bqi)
qk - 1
i
. (3.6)
If the function f is n-concave, the inequality signs in (3.6) are reversed.
Example 3.1. Let \bfitp = (p1, . . . , pr) and \bfitp = (q1, . . . , qr) be probability distributions.
Kullback – Leibler divergence of the probability distributions \bfitp and \bfitq is defined as
DKL(\bfitp , \bfitq ) =
r\sum
i=1
qi \mathrm{l}\mathrm{o}\mathrm{g}
qi
pi
,
and the corresponding generating function is f(t) = t \mathrm{l}\mathrm{o}\mathrm{g} t, t > 0. We can calculate
f (n)(t) = ( - 1)n(n - 2)!t - (n - 1).
It is clear that this function is (2n - 1)-concave and (2n)-convex for any n \in \BbbN .
Hellinger divergence of the probability distributions \bfitp and \bfitq is defined as
DH(\bfitp , \bfitq ) =
1
2
n\sum
i=1
(
\surd
qi -
\surd
pi)
2,
and the corresponding generating function is f(t) =
1
2
(1 -
\surd
t)2, t > 0. We see that
f (n)(t) = ( - 1)n (2n - 3)!!
2n
t -
2n - 1
2 ,
so function f is (2n - 1)-concave and (2n)-convex for any n \in \BbbN .
Harmonic divergence of the probability distributions \bfitp and \bfitq is defined as
DHa(\bfitp , \bfitq ) =
n\sum
i=1
2piqi
pi + qi
,
and the corresponding generating function is f(t) =
2t
1 + t
. We can calculate
f (n)(t) = 2( - 1)n+1n!(1 + t) - (n+1).
Two cases need to be considered:
if t < - 1, then the function f is n-convex for every n \in \BbbN ;
if t > - 1, then the function f is (2n)-concave and (2n - 1)-convex for any n \in \BbbN .
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INEQUALITIES OF THE EDMUNDSON – LAH – RIBARIČ TYPE FOR n-CONVEX FUNCTIONS . . . 103
Jeffreys divergence of the probability distributions \bfitp and \bfitq is defined as
DJ(\bfitp , \bfitq ) =
1
2
n\sum
i=1
(qi - pi) \mathrm{l}\mathrm{o}\mathrm{g}
qi
pi
,
and the corresponding generating function is f(t) = (1 - t) \mathrm{l}\mathrm{o}\mathrm{g}
1
t
, t > 0. After calculating, we see
that
f (n)(t) = ( - 1)n+1t - n(n - 1)! (1 + nt).
Obviously, this function is (2n - 1)-convex and (2n)-concave for any n \in \BbbN .
It is clear that all of the results from this section can be applied to the special types of divergences
mentioned in this example.
4. Examples with Zipf and Zipf – Mandelbrot law. Zipf’s law [33, 34] has a significant
application in a wide variety of scientific disciplines — from astronomy to demographics to software
structure to economics to zoology, and even to warfare [12]. It is one of the basic laws in information
science and bibliometrics, but it is also often used in linguistics. Typically one is dealing with integer-
valued observables (numbers of objects, people, cities, words, animals, corpses) and the frequency
of their occurrence.
Probability mass function of Zipf’s law with parameters N \in \BbbN and s > 0 is
f(k;N, s) =
1/ks
HN,s
, where HN,s =
N\sum
i=1
1
is
.
Benoit Mandelbrot in 1966 gave an improvement of Zipf law for the count of the low-rank words.
Various scientific fields use this law for different purposes, for example information sciences use it
for indexing [11, 32], ecological field studies in predictability of ecosystem [26], in music it is used
to determine aesthetically pleasing music [23].
Zipf – Mandelbrot law is a discrete probability distribution with parameters N \in \BbbN , q, s \in \BbbR
such that q \geq 0 and s > 0, possible values \{ 1, 2, . . . , N\} and probability mass function
f(i;N, q, s) =
1/(i+ q)s
HN,q,s
, where HN,q,s =
N\sum
i=1
1
(i+ q)s
. (4.1)
Let \bfitp and \bfitq be Zipf – Mandelbrot laws with parameters N \in \BbbN , q1, q2 \geq 0 and s1, s2 > 0,
respectively, and let us denote
HN,q1,s1 = H1, HN,q2,s2 = H2,
a\bfitp ,\bfitq := \mathrm{m}\mathrm{i}\mathrm{n}
\biggl\{
pi
qi
\biggr\}
=
H2
H1
\mathrm{m}\mathrm{i}\mathrm{n}
\biggl\{
(i+ q2)
s2
(i+ q1)s1
\biggr\}
, (4.2)
b\bfitp ,\bfitq := \mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
pi
qi
\biggr\}
=
H2
H1
\mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
(i+ q2)
s2
(i+ q1)s1
\biggr\}
.
In this section we utilize the results regarding Csiszár divergence from the previous section in
order to obtain different inequalities for the Zipf – Mandelbrot law. The following results are special
cases of Theorems 3.1, 3.2, 3.3 and 3.4, respectively, and they gives us Edmundson – Lah – Ribarič
type inequality for the generalized f -divergence of the Zipf – Mandelbrot law.
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104 R. MIKIĆ, D. PEČARIĆ, J. PEČARIĆ
Corollary 4.1. Let \bfitp and \bfitq be Zipf – Mandelbrot laws with parameters N \in \BbbN , q1, q2 \geq 0 and
s1, s2 > 0, respectively, and let H1, H2, a\bfitp ,\bfitq and a\bfitp ,\bfitq be defined in (4.2). Let f \in \scrC n([a\bfitp ,\bfitq , b\bfitp ,\bfitq ])
be a n-convex function. If n and 3 \leq m \leq n - 1 are of different parity, then
b\bfitp ,\bfitq - 1
b\bfitp ,\bfitq - a\bfitp ,\bfitq
f(a\bfitp ,\bfitq ) +
1 - a\bfitp ,\bfitq
b\bfitp ,\bfitq - a\bfitp ,\bfitq
f(b\bfitp ,\bfitq ) - \~Df (\bfitp , \bfitq ) \leq
\leq (1 - a\bfitp ,\bfitq )
\bigl(
f \prime (a\bfitp ,\bfitq ) - f [a\bfitp ,\bfitq , b\bfitp ,\bfitq ]
\bigr)
+
m - 1\sum
k=2
f (k)(a\bfitp ,\bfitq )
H2k!
r\sum
i=1
\biggl(
H2(i+ q2)
s2
H1(i+ q1)s1
- a\bfitp ,\bfitq
\biggr) k
(i+ q2)s2
+
+
n - m\sum
k=1
f [a\bfitp ,\bfitq , . . . , a\bfitp ,\bfitq \underbrace{} \underbrace{}
m times
; b\bfitp ,\bfitq , . . . , b\bfitp ,\bfitq \underbrace{} \underbrace{}
k times
]
r\sum
i=1
\biggl(
H2(i+ q2)
s2
H1(i+ q1)s1
- a\bfitp ,\bfitq
\biggr) m\biggl(
H2(i+ q2)
s2
H1(i+ q1)s1
- b\bfitp ,\bfitq
\biggr) k - 1
H2(i+ q2)s2
.
This inequality also holds when the function f is n-concave and n and m are of equal parity. In
case when the function f is n-convex and n and m are of equal parity, or when the function f is
n-concave and n and m are of different parity, the inequality sign is reversed.
Corollary 4.2. Let \bfitp and \bfitq be Zipf – Mandelbrot laws with parameters N \in \BbbN , q1, q2 \geq 0 and
s1, s2 > 0, respectively, and let H1, H2, a\bfitp ,\bfitq and a\bfitp ,\bfitq be defined in (4.2). Let f \in \scrC n([a\bfitp ,\bfitq , b\bfitp ,\bfitq ])
be a n-convex function and let 3 \leq m \leq n - 1 be of different parity. Then
b\bfitp ,\bfitq - 1
b\bfitp ,\bfitq - a\bfitp ,\bfitq
f(a\bfitp ,\bfitq ) +
1 - a\bfitp ,\bfitq
b\bfitp ,\bfitq - a\bfitp ,\bfitq
f(b\bfitp ,\bfitq ) - \~Df (\bfitp , \bfitq ) \leq
\leq (b\bfitp ,\bfitq - 1)
\bigl(
f [a\bfitp ,\bfitq , b\bfitp ,\bfitq ] - f \prime (b\bfitp ,\bfitq )
\bigr)
+
m - 1\sum
k=2
f (k)(b\bfitp ,\bfitq )
H2k!
r\sum
i=1
\biggl(
H2(i+ q2)
s2
H1(i+ q1)s1
- b\bfitp ,\bfitq
\biggr) k
(i+ q2)s2
+
+
n - m\sum
k=1
f
\Bigl[
b\bfitp ,\bfitq , . . . , b\bfitp ,\bfitq \underbrace{} \underbrace{}
m times
; a\bfitp ,\bfitq , . . . , a\bfitp ,\bfitq \underbrace{} \underbrace{}
k times
\Bigr] r\sum
i=1
\biggl(
H2(i+ q2)
s2
H1(i+ q1)s1
- b\bfitp ,\bfitq
\biggr) m\biggl(
H2(i+ q2)
s2
H1(i+ q1)s1
- a\bfitp ,\bfitq
\biggr) k - 1
H2(i+ q2)s2
.
The inequality above also holds when the function f is n-concave and m is even. In case when
the function f is n-convex and m is even, or when the function f is n-concave and m is odd, the
inequality sign is reversed.
Corollary 4.3. Let \bfitp and \bfitq be Zipf – Mandelbrot laws with parameters N \in \BbbN , q1, q2 \geq 0 and
s1, s2 > 0, respectively, and let H1, H2, a\bfitp ,\bfitq and a\bfitp ,\bfitq be defined in (4.2). Let f \in \scrC n([a\bfitp ,\bfitq , b\bfitp ,\bfitq ])
be a n-convex function. If n is odd, then we have
n - 1\sum
k=2
f
\Bigl[
a\bfitp ,\bfitq ; b\bfitp ,\bfitq , . . . , b\bfitp ,\bfitq \underbrace{} \underbrace{}
k times
\Bigr] r\sum
i=1
\biggl(
H2(i+ q2)
s2
H1(i+ q1)s1
- a\bfitp ,\bfitq
\biggr) \biggl(
H2(i+ q2)
s2
H1(i+ q1)s1
- b\bfitp ,\bfitq
\biggr) k - 1
H2(i+ q2)s2
\leq
\leq b\bfitp ,\bfitq - 1
b\bfitp ,\bfitq - a\bfitp ,\bfitq
f(a\bfitp ,\bfitq ) +
1 - a\bfitp ,\bfitq
b\bfitp ,\bfitq - a\bfitp ,\bfitq
f(b\bfitp ,\bfitq ) - \~Df (\bfitp , \bfitq ) \leq
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 1
INEQUALITIES OF THE EDMUNDSON – LAH – RIBARIČ TYPE FOR n-CONVEX FUNCTIONS . . . 105
\leq f [a\bfitp ,\bfitq , a\bfitp ,\bfitq ; b\bfitp ,\bfitq ]
r\sum
i=1
\biggl(
H2(i+ q2)
s2
H1(i+ q1)s1
- a\bfitp ,\bfitq
\biggr) \biggl(
H2(i+ q2)
s2
H1(i+ q1)s1
- b\bfitp ,\bfitq
\biggr)
H2(i+ q2)s2
+
+
n - 2\sum
k=2
f
\Bigl[
a\bfitp ,\bfitq , a\bfitp ,\bfitq ; b\bfitp ,\bfitq , . . . , b\bfitp ,\bfitq \underbrace{} \underbrace{}
k times
\bigr] r\sum
i=1
\biggl(
H2(i+ q2)
s2
H1(i+ q1)s1
- a\bfitp ,\bfitq
\biggr) 2\biggl( H2(i+ q2)
s2
H1(i+ q1)s1
- b\bfitp ,\bfitq
\biggr) k - 1
H2(i+ q2)s2
.
Stated inequalities also hold when the function f is n-concave and n is even. In case when the
function f is n-convex and n is even, or when the function f is n-concave and n is odd, the
inequality signs are reversed.
Corollary 4.4. Let \bfitp and \bfitq be Zipf – Mandelbrot laws with parameters N \in \BbbN , q1, q2 \geq 0 and
s1, s2 > 0, respectively, and let H1, H2, a\bfitp ,\bfitq and a\bfitp ,\bfitq be defined in (4.2). Let f \in \scrC n([a\bfitp ,\bfitq , b\bfitp ,\bfitq ])
be a n-convex function. Then we have
f [b\bfitp ,\bfitq , b\bfitp ,\bfitq ; a\bfitp ,\bfitq ]
r\sum
i=1
\biggl(
H2(i+ q2)
s2
H1(i+ q1)s1
- a\bfitp ,\bfitq
\biggr) \biggl(
H2(i+ q2)
s2
H1(i+ q1)s1
- b\bfitp ,\bfitq
\biggr)
H2(i+ q2)s2
+
+
n - 2\sum
k=2
f
\Bigl[
b\bfitp ,\bfitq , b\bfitp ,\bfitq ; a\bfitp ,\bfitq , . . . , a\bfitp ,\bfitq \underbrace{} \underbrace{}
k times
\Bigr] r\sum
i=1
\biggl(
H2(i+ q2)
s2
H1(i+ q1)s1
- a\bfitp ,\bfitq
\biggr) k - 1\biggl( H2(i+ q2)
s2
H1(i+ q1)s1
- b\bfitp ,\bfitq
\biggr) 2
H2(i+ q2)s2
\leq
\leq b - 1
b - a
f(a) +
1 - a
b - a
f(b) - \~Df (\bfitp , \bfitq ) \leq
\leq
n - 1\sum
k=2
f
\Bigl[
b\bfitp ,\bfitq ; a\bfitp ,\bfitq , . . . , a\bfitp ,\bfitq \underbrace{} \underbrace{}
k times
\Bigr] r\sum
i=1
\biggl(
H2(i+ q2)
s2
H1(i+ q1)s1
- a\bfitp ,\bfitq
\biggr) k - 1\biggl( H2(i+ q2)
s2
H1(i+ q1)s1
- b\bfitp ,\bfitq
\biggr)
H2(i+ q2)s2
.
If the function f is n-concave, the inequality signs are reversed.
Remark 4.1. By taking into consideration Example 3.1 one can see that general results from
this section can easily be applied to any of the following divergences: Kullback – Leibler divergence,
Hellinger divergence, harmonic divergence or Jeffreys divergence.
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Received 03.04.18
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| resource_txt_mv | umjimathkievua/fc/c6e4c1181ddb59df1c290c68701c36fc.pdf |
| spelling | umjimathkievua-article-7212025-03-31T08:49:21Z Inequalities of the Edmundson-Lah-Ribarč type for n-convex functions with applications Inequalities of the Edmundson-Lah-Ribarič type for n-convex functions with applications Inequalities of the Edmundson-Lah-Ribarič type for n-convex functions with applications Mikić , R. Pečarić, D. Pečarić, J. Mikić , R. Pečarić, D. Pečarić, J. Jensen inequality Edmundson-Lah-Ribarič inequality n-convex functions divided differences f-divergence Zipf-Mandelbrot law Jensen inequality Edmundson-Lah-Ribarič inequality n-convex functions divided differences f-divergence Zipf-Mandelbrot law UDC 517.5 We derive some Edmundson – Lah – Ribarič type inequalities for positive linear functionals and $n$-convex functions. Main results are applied to the generalized $f$ -divergence functional. Examples with Zipf – Mandelbrot law are used to illustrate the results. УДК 517.5 Отримано нерiвностi типу Едмундсона – Лаха – Рибарича для додатних лiнiйних функцiоналiв та $n$-опуклих функцiй. Основнi результати застосовуються до узагальнених $f $-дивергентних функцiоналiв. Наведено приклади, в якихвикористовується закон Зiпфа – Мандельброта. Institute of Mathematics, NAS of Ukraine 2021-01-22 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/721 10.37863/umzh.v73i1.721 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 1 (2021); 89 - 106 Український математичний журнал; Том 73 № 1 (2021); 89 - 106 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/721/8905 |
| spellingShingle | Mikić , R. Pečarić, D. Pečarić, J. Mikić , R. Pečarić, D. Pečarić, J. Inequalities of the Edmundson-Lah-Ribarč type for n-convex functions with applications |
| title | Inequalities of the Edmundson-Lah-Ribarč type for n-convex functions with applications |
| title_alt | Inequalities of the Edmundson-Lah-Ribarič type for n-convex functions with applications Inequalities of the Edmundson-Lah-Ribarič type for n-convex functions with applications |
| title_full | Inequalities of the Edmundson-Lah-Ribarč type for n-convex functions with applications |
| title_fullStr | Inequalities of the Edmundson-Lah-Ribarč type for n-convex functions with applications |
| title_full_unstemmed | Inequalities of the Edmundson-Lah-Ribarč type for n-convex functions with applications |
| title_short | Inequalities of the Edmundson-Lah-Ribarč type for n-convex functions with applications |
| title_sort | inequalities of the edmundson-lah-ribarč type for n-convex functions with applications |
| topic_facet | Jensen inequality Edmundson-Lah-Ribarič inequality n-convex functions divided differences f-divergence Zipf-Mandelbrot law Jensen inequality Edmundson-Lah-Ribarič inequality n-convex functions divided differences f-divergence Zipf-Mandelbrot law |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/721 |
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