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Levinson-type inequalities via new Green functions and Montgomery identity

  • Muhammad Adeel EMAIL logo , Khuram Ali Khan , Ðilda Pečarić and Josip Pečarić
Published/Copyright: July 1, 2020
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Open Mathematics
From the journal Open Mathematics Volume 18 Issue 1

Abstract

In this study, Levinson-type inequalities are generalized by using new Green functions and Montgomery identity for the class of k-convex functions (k ≥ 3). Čebyšev-, Grüss- and Ostrowski-type new bounds are found for the functionals involving data points of two types. Moreover, a new functional is introduced based on f divergence and then some estimates for new functional are obtained. Some inequalities for Shannon entropies are obtained too.

Keywords: Levinson's inequality; Montgomery identity; information theory
MSC 2010: 26D20; 26D15; 94A20

1 Introduction and preliminaries

The theory of convex functions has encountered a fast advancement. This can be attributed to a few causes: first, applications of convex functions are directly involved in modern analysis; second, many important inequalities are results of applications of convex functions, and convex functions are closely related to inequalities (see [1]). Divided differences are found to be very helpful when we are dealing with functions having different degrees of smoothness. The following definition of divided difference is given in [1, p. 14].

1.1 kth-order divided difference

The kth-order divided difference of a function f : [ ζ 1 , ζ 2 ] → ℝ at mutually distinct points x 0,…,x k ∈ [ζ 1,ζ 2] is defined recursively by

(1) [ x ρ ; f ] = f ( x ρ ) , ρ = 0 , … , k , [ x 0 , … , x k ; f ] = [ x 1 , … , x k ; f ] − [ x 0 , … , x k − 1 ; f ] x k − x 0 .

It is easy to see that (1) is equivalent to

[ x 0 , … , x k ; f ] = ∑ ρ =0 k f ( x ρ ) q ′ ( x ρ ) , where q ( x ) = ∏ j =0 k ( x − x j ) .

The following definition of a real valued convex function is characterized by the kth-order divided difference (see [1, p. 15]).

Definition 1.1

A function f : [ ζ 1 , ζ 2 ] → ℝ is said to be k-convex (k ≥ 0) if and only if for all choices of (k + 1) distinct points x 0,…,x k ∈ [ζ1, ζ2], [ x 0 , … , x k ; f ] ≥ 0 holds.

If this inequality is reversed, then f is said to be k-concave. If the inequality is strict, then f is said to be a strictly k-convex (k-concave) function.

Note that 0-convex functions, 1-convex functions and 2-convex functions are non-negative functions, increasing functions and simply convex functions, respectively.

1.2 Criteria for k convex functions

In [1, p. 16], criteria to examine the n-convexity of a function f are given as:

“If f ( k ) exists, then f is k-convex if and only if f ( k ) ≥ 0 ”.

In [2], (see also [3, p. 32, Theorem 1]) Ky Fan’s inequality is generalized by Levinson for 3-convex functions as follows:

Theorem A

Let f : ℑ = ( 0 , 2 α ) → ℝ be such that f ( 3 ) ( t ) ≥ 0 . Let x ρ ∈ (0, α) and p ρ > 0. Then,

(2) 1 P n ∑ ρ =1 n p ρ f ( x ρ ) − f ( 1 P n ∑ ρ =1 n p ρ x ρ ) ≤ 1 P n ∑ ρ =1 n p ρ f (2 α − x ρ ) − f ( 1 P n ∑ ρ =1 n p ρ (2 α − x ρ ) ) .

Functional form of (2) is defined as follows:

(3) J 1 ( f ( ⋅ ) ) = 1 P n ∑ i =1 n p ρ f (2 a − x ρ ) − f ( 1 P n ∑ ρ =1 n p ρ (2 a − x ρ ) ) − 1 P n ∑ ρ =1 n p ρ f ( x ρ ) + f ( 1 P n ∑ ρ =1 n p ρ x ρ ) ≥ 0 .

In [4], Popoviciu noted that (2) is valid on (0, 2a) for 3-convex functions, while in [5] (see also [3, p. 32, Theorem 2]) Bullen provided a different proof of Popoviciu’s result and also the converse of (2).

Theorem B

(a) Let f : ℑ = [ ζ 1 , ζ 2 ] → ℝ be such that f (3) ( t ) ≥ 0 and x ρ , y ρ , ∈ [ζ 1, ζ 2] for ρ = 1, 2,…,n such that

(4) max { x 1 … x n } ≤ min { y 1 … y n } , x 1 + y 1 = ⋯ = x n + y n

and p ρ > 0 then

(5) 1 P n ∑ ρ = 1 n p ρ f ( x ρ ) − f ( 1 P n ∑ ρ = 1 n p ρ x ρ ) ≤ 1 P n ∑ ρ = 1 n p ρ f ( y ρ ) − f ( 1 P n ∑ ρ = 1 n p ρ y ρ ) .

(b) f is 3-convex if f is continuous, p ρ > 0 and ( 5 ) holds for all x ρ , y ρ satisfying ( 4 ).

Functional form of (5) is defined as follows:

(6) J 2 ( f ( ⋅ ) ) = 1 P n ∑ ρ = 1 n p ρ f ( y ρ ) − f ( 1 P n ∑ ρ = 1 n p ρ y ρ ) − 1 P n ∑ ρ = 1 n p ρ f ( x ρ ) + f ( 1 P n ∑ ρ = 1 n p ρ x ρ ) ≥ 0 .

Remark 1.1

It is essential to take note of that under the suppositions of Theorems A and B, if the function f is 3-convex, then J i ( f ( ⋅ ) ) ≥ 0 for i = 1, 2.

In [6], (see also [3, p. 32, Theorem 4]) the following result holds.

Theorem C

Let f : ℑ = [ ζ 1 , ζ 2 ] → ℝ be a 3-convex function, p ρ > 0, and let x ρ , y ρ ∈ [ζ 1, ζ 2] be such that x ρ + y ρ = 2 c ˘ , for ρ = 1,…,n, x ρ + x n − ρ + 1 ≤ 2 c ˘ and p ρ x ρ + p n − ρ + 1 x n − ρ + 1 p ρ + p n − ρ + 1 ≤ c ˘ . Then, ( 5 ) holds.

In [7], Mercer replaced the condition of symmetric distribution of points x ρ and y ρ with symmetric variances of points x ρ and y ρ .

Theorem D

Let f : ℑ = [ ζ 1 , ζ 2 ] → ℝ be a 3-convex function, p ρ are positive such that ∑ ρ =1 n p ρ = 1 . Also, let x ρ , y ρ satisfy max {x 1,…,x n } ≤ min{y 1,…,y n } and

(7) ∑ ρ =1 n p ρ ( x ρ − ∑ ρ =1 n p ρ x ρ ) 2 = ∑ ρ =1 n p ρ ( y ρ − ∑ ρ =1 n p ρ y ρ ) 2 ,

then ( 5 ) holds.

In [8], Adeel et al. generalized Levinson’s inequality for 3-convex functions; they also obtained some results for information theory. Similar results can be found in [9,10]. In [11], Witkowski showed the Levinson inequality with random variables. Furthermore, he showed that it is enough to assume that f is 3-convex and that assumption (7) can be weakened to inequality in a certain direction.

In [12], Pečarić et al. provided the probabilistic version of Levinson’s inequality (2) under Mercer’s assumption of equal variances for the family of 3-convex functions at a point. They showed that this is the largest family of continuous functions for which inequality (5) holds. The operator version of probabilistic Levinson’s inequality was discussed in [13]. In [14], Pavic provided geometrical interpretation of different inequalities involving convex functions. Many other researchers generalized different inequalities for different classes of convex functions [15,16,17,18].

For our main results, we use the generalized Montgomery identity via Taylor’s formula given in [19].

Theorem E

Let k ∈ ℕ , f : ℑ → ℝ be such that f ( k − 1) is absolutely continuous, ℑ is an open interval contained in ℝ , ζ 1 , ζ 2 ∈ ℑ such that ζ 1 < ζ 2 . Then, the following identity holds:

(8) f ( s ) = 1 ζ 2 − ζ 1 ∫ ζ 1 ζ 2 f ( v ) d v + ∑ l =0 k − 2 f ( l + 1) ( ζ 1 ) l ! ( l + 2) ( s − ζ 1 ) l + 2 ζ 2 − ζ 1 − ∑ l =0 k − 2 f ( l + 1) ( ζ 2 ) l ! ( l + 2) ( s − ζ 2 ) l + 2 ζ 2 − ζ 1 + 1 ( k − 1)! ∫ ζ 1 ζ 2 R k ( s , v ) f ( k ) ( v ) d v ,

where

(9) R k ( s , v ) = { − ( s − v ) k k ( ζ 2 − ζ 1 ) + s − ζ 1 ζ 2 − ζ 1 ( ( s − v ) k − 1 ) , ζ 1 ≤ v ≤ s ; − ( s − v ) k k ( ζ 2 − ζ 1 ) + s − ζ 2 ζ 2 − ζ 1 ( ( s − v ) k − 1 ) , s < v ≤ ζ 2 .

When k = 1, identity (8) reduces to the well-known Montgomery identity (see [20]).

f ( s ) = 1 ζ 2 − ζ 1 ∫ ζ 1 ζ 2 f ( v ) d v + ∫ ζ 1 ζ 2 P ( s , v ) f ′ ( v ) d v ,

where P(s, v) is the Peano kernel defined as

(10) P ( s , v ) = { v − ζ 1 ζ 2 − ζ 1 , ζ 1 ≤ v ≤ s ; v − ζ 2 ζ 2 − ζ 1 , s < v ≤ ζ 2 .

The error function e ℱ ( t ) can be represented in terms of the Green functions G ℱ , k ( t , s ) of the boundary value problem

z ( k ) ( t ) = 0 , z ( i ) ( a 1 ) = 0 , 0 ≤ i ≤ p , z ( i ) ( a 2 ) = 0 , p + 1 ≤ i ≤ k − 1 ,

e F ( t ) = ∫ ζ 1 ζ 2 G F , k ( t , s ) f ( k ) ( s ) d s , t ∈ [ ζ 1 , ζ 2 ] ,

where

(11) G F , k ( t , s ) = 1 ( k − 1 ) ! { ∑ i = 0 p ( k − 1 i ) ( t − a 1 ) i ( a 1 − s ) k − i − 1 , ζ 1 ≤ s ≤ t ; − ∑ i = p + 1 k − 1 ( k − 1 i ) ( t − a 1 ) i ( a 1 − s ) k − i − 1 , t ≤ s ≤ ζ 2 .

In [21], the following result holds.

Theorem F

Let f ∈ C k [ ζ 1 , ζ 2 ] , and let P F be its “two-point right focal” interpolating polynomial. Then, for ζ 1 ≤ a 1 < a 2 ≤ ζ 2 and 0 ≤ p ≤ k − 2,

(12) f ( t ) = P F ( t ) + e F ( t ) = ∑ i = 0 p ( t − a 1 ) i i ! f ( i ) ( a 1 ) + ∑ j = 0 k − p − 2 ( ∑ i = 0 j ( t − a 1 ) p + 1 + i ( a 1 − a 2 ) j − i ( p + 1 + i ) ! ( j − i ) ! ) f ( p + 1 + j ) ( a 2 ) + ∫ ζ 1 ζ 2 G F , k ( t , s ) f ( k ) ( s ) d s ,

where G F,k (t,s) is the Green function, defined by ( 11 ).

Let f ∈ C k [ ζ 1 , ζ 2 ] , and let P F be its “two-point right focal” interpolating polynomial for ζ 1 ≤ a 1 < a 2 ≤ ζ 2. Then, for k = 3 and p = 0, (12) becomes

(13) f ( t ) = f ( a 1 ) + ( t − a 1 ) f ( 1 ) ( a 1 ) + ( t − a 1 ) ( a 1 − a 2 ) f ( 2 ) ( a 2 ) + ( t − a 1 ) 2 2 f ( 2 ) ( a 2 ) + ∫ ζ 1 ζ 2 G 1 ( t , s ) f ( 3 ) ( s ) d s ,

where

(14) G 1 ( t , s ) = { 1 2 ( a 1 − s ) 2 , ζ 1 ≤ s ≤ t ; − ( t − a 1 ) ( a 1 − s ) − 1 2 ( t − a 1 ) 2 , t ≤ s ≤ ζ 2 .

For k = 3 and p = 1, (12) becomes

(15) f ( t ) = f ( a 1 ) + ( t − a 1 ) f ( 1 ) ( a 1 ) + ( t − a 1 ) 2 2 f ( 2 ) ( a 2 ) + ∫ ζ 1 ζ 2 G 2 ( t , s ) f ( 3 ) ( s ) d s ,

where

(16) G 2 ( t , s ) = { 1 2 ( a 1 − s ) 2 + ( t − a 1 ) ( a 1 − s ) , ζ 1 ≤ s ≤ t ; − 1 2 ( t − a 1 ) 2 , t ≤ s ≤ ζ 2 .

In [22], Mehmood et al. generalized Popoviciu-type inequalities via new Green functions and Montgomery identity. All generalizations that exist in the literature are only for one type of data points. But in this study, motivated by the above discussion Levinson-type inequalities are generalized via new Green functions and Montgomery identity involving two types of data points for higher order convex functions. Čebyšev-, Grüss- and Ostrowski-type new bounds are also found for the functionals involving data points of two types.

2 Main results

Motivated by identity (6), we construct the following identities.

2.1 Generalization of Bullen-type inequalities for higher order convex functions

First, we define the following functional:

ℱ : let f : ℑ 1 = [ ζ 1 , ζ 2 ] → ℝ be a function, x 1,…,x n and y 1 , … , y m ∈ ℑ 1 such that max {x 1,…,x n } ≤ min{y 1,…,y m } and

(17) ∑ ρ =1 n p ρ ( x ρ − ∑ ρ =1 n p ρ x ρ ) 2 = ∑ ϱ =1 m q ϱ ( y ϱ − ∑ ϱ =1 m q ϱ y ϱ ) 2 .

Also, let ( p 1 , … , p n ) ∈ ℝ n and ( q 1 , … , q m ) ∈ ℝ m be such that ∑ ρ = 1 n p ρ = 1 , ∑ ϱ =1 m q ϱ =1 and x ρ , y ϱ , ∑ ρ =1 n p ρ x ρ , ∑ ϱ =1 m p ϱ y ϱ ∈ ℑ 1 . Then,

(18) J ˘ ( f ( ⋅ ) ) = ∑ ϱ =1 m q ϱ f ( y ϱ ) − f ( ∑ ϱ =1 m q ϱ y ϱ ) − ∑ ρ =1 n p ρ f ( x ρ ) + f ( ∑ ρ =1 n p ρ x ρ ) .

Theorem 2.1

Assume ℱ . Let f ∈ C k [ ζ 1 , ζ 2 ] be such that f ( k − 1) is absolutely continuous. Then, we have the following new identities for t = 1, 2:

(i)

(19) J ˘ ( f ( ⋅ ) ) = 1 2 [ ∑ ϱ =1 m q ϱ y ϱ 2 − ( ∑ ϱ =1 m q ϱ y ϱ ) 2 − ∑ ρ =1 n p ρ x ρ 2 + ( ∑ ρ =1 n p ρ x ρ ) 2 ] f (2) ( ζ 2 ) + 2 ( f ( ζ 1 ) − f ( ζ 2 ) ζ 2 − ζ 1 ) ∫ ζ 1 ζ 2 J ˘ ( G t ( ⋅ , s ) ) d s + ∫ ζ 1 ζ 2 J ˘ ( G t ( ⋅ , s ) ) ∑ l =3 k − 1 l ( l − 1) ( l − 1)! [ f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ] d s + 1 ( k − 4)! ∫ ζ 1 ζ 2 f ( k ) ( v ) ( ∫ ζ 1 ζ 2 J ˘ ( G t ( ⋅ , s ) ) R ˜ k − 3 ( s , v ) d s ) d v ,

where

(20) R ˜ k − 3 ( s , v ) = { 2 ( s − v ) k − 3 ( k − 3 ) ( ζ 2 − ζ 1 ) + s − ζ 1 ζ 2 − ζ 1 ( ( s − v ) k − 4 ) , ζ 1 ≤ v ≤ s ; 2 ( s − v ) k − 3 ( k − 3 ) ( ζ 2 − ζ 1 ) + s − ζ 2 ζ 2 − ζ 1 ( ( s − v ) k − 4 ) , s < v ≤ ζ 2 ,

and

(21) J ˘ ( G t ( ⋅ , s ) ) = ∑ ϱ =1 m q ϱ G t ( y ϱ , s ) − G t ( ∑ ϱ =1 m q ϱ y ϱ , s ) − ∑ ρ =1 n p ρ G t ( x ρ , s ) + G t ( ∑ ρ =1 n p ρ x ρ , s ) ,

for t = 1, 2, G t (·,s) is defined in ( 14 ) and ( 16 ), respectively.

(ii)

(22) J ˘ ( f ( ⋅ ) ) = 1 2 [ ∑ ϱ =1 m q ϱ y ϱ 2 − ( ∑ ϱ =1 m q ϱ y ϱ ) 2 − ∑ ρ =1 n p ρ x ρ 2 + ( ∑ ρ =1 n p ρ x ρ ) 2 ] f (2) ( ζ 2 ) + 2 ( f ′ ( ζ 2 ) − f ′ ( ζ 1 ) ζ 2 − ζ 1 ) ∫ ζ 1 ζ 2 J ˘ ( G t ( ⋅ , s ) ) d s + ∫ ζ 1 ζ 2 J ˘ ( G t ( ⋅ , s ) ) ∑ l =4 k − 1 1 ( l − 2)( l − 4)! [ f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ] d s + 1 ( k − 4)! ∫ ζ 1 ζ 2 f ( k ) ( v ) ( ∫ ζ 1 ζ 2 J ˘ ( G t ( ⋅ , s ) ) R k − 3 ( s , v ) d s ) d v ,

where

(23) R k − 3 ( s , v ) = { − ( s − v ) k − 3 ( k − 3 ) ( ζ 2 − ζ 1 ) + s − ζ 1 ζ 2 − ζ 1 ( ( s − v ) k − 4 ) , ζ 1 ≤ v ≤ s ; − ( s − v ) k − 3 ( k − 3 ) ( ζ 2 − ζ 1 ) + s − ζ 2 ζ 2 − ζ 1 ( ( s − v ) k − 4 ) , s < v ≤ ζ 2 .

Proof

  1. Applying (18) to identities (13) and (15) along with their respective new Green functions, by means of simple calculations and following the properties of J ˘ ( f ( ⋅ ) ) , we get

    (24) J ˘ ( f ( ⋅ ) ) = 1 2 [ ∑ ϱ =1 m q ϱ y ϱ 2 − ( ∑ ϱ =1 m q ϱ y ϱ ) 2 − ∑ ρ =1 n p ρ x ρ 2 + ( ∑ ρ =1 n p ρ x ρ ) 2 ] f (2) ( ζ 2 ) + ∫ ζ 1 ζ 2 J ˘ ( G t ( ⋅ , s ) ) f (3) ( s ) d s .

    Differentiating (8) with respect to “s”, we have

    (25) f ( 3 ) ( s ) = 2 ( f ″ ( ζ 1 ) − f ″ ( ζ 2 ) ζ 2 − ζ 1 ) + ∑ l = 3 k − 1 l ( l − 1 ) ( l − 1 ) ! ( f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ) + 1 ( k − 4 ) ! ∫ ζ 1 ζ 2 R ˜ k − 3 ( s , v ) f ( k ) ( v ) d v .

    Using (25) in (24) and following the properties of J ˘ ( f ( ⋅ ) ) , we get

    J ˘ ( f ( ⋅ ) ) = 1 2 [ ∑ ϱ =1 m q ϱ y ϱ 2 − ( ∑ ϱ =1 m q ϱ y ϱ ) 2 − ∑ ρ =1 n p ρ x ρ 2 + ( ∑ ρ =1 n p ρ x ρ ) 2 ] f (2) ( ζ 2 ) + ∫ ζ 1 ζ 2 J ˘ ( G t ( ⋅ , s ) ) [ 2 ( f ″ ( ζ 1 ) − f ″ ( ζ 2 ) ζ 2 − ζ 1 ) + ∑ l =3 k − 1 l ( l − 1) ( l − 1)! ( f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ) + 1 ( k − 4)! ∫ ζ 1 ζ 2 R ˜ k − 3 ( s , v ) f ( k ) ( v ) d v ] d s .

    J ˘ ( f ( ⋅ ) ) = 1 2 [ ∑ ϱ =1 m q ϱ y ϱ 2 − ( ∑ ϱ =1 m q ϱ y ϱ ) 2 − ∑ ρ =1 n p ρ x ρ 2 + ( ∑ ρ =1 n p ρ x ρ ) 2 ] f (2) ( ζ 2 ) + 2 ( f ″ ( ζ 1 ) − f ″ ( ζ 2 ) ζ 2 − ζ 1 ) ∫ ζ 1 ζ 2 J ˘ ( G t ( ⋅ , s ) ) d s + ∫ ζ 1 ζ 2 J ˘ ( G t ( ⋅ , s ) ) [ ∑ l =3 k − 1 l ( l − 1) ( l − 1)! ( f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ) ] d s + 1 ( k − 4)! ∫ ζ 1 ζ 2 J ˘ ( G t ( ⋅ , s ) ) ( ∫ ζ 1 ζ 2 R ˜ k − 3 ( s , v ) f ( k ) ( v ) d v ) d s .

    Execute Fubini’s theorem in the last term to get (19) for t = 1, 2.

  2. Using formula (8) on the function f (3) , replacing k by k − 3 and rearranging the indices, we have

(26) f (3) ( s ) = 2 ( f ″ ( ζ 2 ) − f ″ ( ζ 1 ) ζ 2 − ζ 1 ) + ∑ l =4 k − 1 l ( l − 2)( l − 4)! ( f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ) + 1 ( k − 4)! ∫ ζ 1 ζ 2 R k − 3 ( s , v ) f ( k ) ( v ) d v .

Using (26) in (24) and Fubini’s theorem we conclude (22) for t = 1, 2.□

As an application of new identities (19) and (22), the next result gives generalization of Bullen-type inequalities involving new Green functions for k-convex (k ≥ 3) functions.

Theorem 2.2

Assuming the conditions of Theorem 2.1 with f as the k-convex function, then for t = 1, 2 we have the following two results:

If

(27) ∫ ζ 1 ζ 2 J ˘ ( G t ( ⋅ , s ) ) R ˜ k − 3 ( s , v ) d s ≥ 0 , v ∈ ℑ 1 ,

then

(28) J ˘ ( f ( ⋅ ) ) ≥ 1 2 [ ∑ ϱ =1 m q ϱ y ϱ 2 − ( ∑ ϱ =1 m q ϱ y ϱ ) 2 − ∑ ρ =1 n p ρ x ρ 2 + ( ∑ ρ =1 n p ρ x ρ ) 2 ] f (2) ( ζ 2 ) + 2 ( f ′ ′ ( ζ 1 ) − f ′ ′ ( ζ 2 ) ζ 2 − ζ 1 ) ∫ ζ 1 ζ 2 J ˘ ( G t ( ⋅ , s ) ) d s + ∫ ζ 1 ζ 2 J ˘ ( G t ( ⋅ , s ) ) ∑ l =3 k − 1 l ( l − 1) ( l − 1)! [ f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ] d s ,

and if

(29) ∫ ζ 1 ζ 2 J ˘ ( G t ( ⋅ , s ) ) R k − 3 ( s , v ) d s ≥ 0 , v ∈ ℑ 1 ,

then

(30) J ˘ ( f ( ⋅ ) ) ≥ 1 2 [ ∑ ϱ =1 m q ϱ y ϱ 2 − ( ∑ ϱ =1 m q ϱ y ϱ ) 2 − ∑ ρ =1 n p ρ x ρ 2 + ( ∑ ρ =1 n p ρ x ρ ) 2 ] f (2) ( ζ 2 ) + 2 ( f ″ ( ζ 2 ) − f ″ ( ζ 1 ) ζ 2 − ζ 1 ) ∫ ζ 1 ζ 2 J ˘ ( G t ( ⋅ , s ) ) d s + ∫ ζ 1 ζ 2 J ˘ ( G t ( ⋅ , s ) ) ∑ l =4 k − 1 1 ( l − 2)( l − 4)! [ f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ] d s .

Proof

As function f is k-convex (k ≥ 3) and is k-times differentiable, so

f ( k ) ( s ) ≥ 0 ∀   s ∈ ℑ 1 ,

then by using (27) in (19) and (29) in (22), we get (28) and (30), respectively.□

Remark 2.1

In Theorem 2.2, inequalities (28) and (30) hold in the reverse direction if inequalities (27) and (29) are reversed.

If we put m = n, p ρ = q ϱ and by using positive weights in (18), then J ˘ ( ⋅ ) converted to the functional J 2 (·) defined in (6) also in this case, (19), (21), (22), (27), (28), (29) and (30) become

(31) J 2 ( f ( ⋅ ) ) = 2 ( f ″ ( ζ 1 ) − f ″ ( ζ 2 ) ζ 2 − ζ 1 ) ∫ ζ 1 ζ 2 J 2 ( G t ( ⋅ , s ) ) d s + ∫ ζ 1 ζ 2 J 2 ( G t ( ⋅ , s ) ) ∑ l =3 k − 1 l ( l − 1) ( l − 1)! [ f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ] d s + 1 ( k − 4)! ∫ ζ 1 ζ 2 f ( k ) ( v ) ( ∫ ζ 1 ζ 2 J 2 ( G t ( ⋅ , s ) ) R ˜ k − 3 ( s , v ) d s ) d v ,

(32) J 2 ( G t ( ⋅ , s ) ) = ∑ ρ =1 n p ρ G t ( y ρ , s ) − G t ( ∑ ρ =1 n p ρ y ρ , s ) − ∑ ρ =1 n p ρ G t ( x ρ , s ) + G t ( ∑ ρ =1 n p ρ x ρ , s ) ,

(33) J 2 ( f ( ⋅ ) ) = 2 ( f ″ ( ζ 2 ) − f ″ ( ζ 1 ) ζ 2 − ζ 1 ) ∫ ζ 1 ζ 2 J 2 ( G t ( ⋅ , s ) ) d s + ∫ ζ 1 ζ 2 J 2 ( G t ( ⋅ , s ) ) ∑ l =4 k − 1 1 ( l − 2)( l − 4)! [ f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ] d s + 1 ( k − 4)! ∫ ζ 1 ζ 2 f ( k ) ( v ) ( ∫ ζ 1 ζ 2 J 2 ( G t ( ⋅ , s ) ) R k − 3 ( s , v ) d s ) d v ,

(34) ∫ ζ 1 ζ 2 J 2 ( G t ( ⋅ , s ) ) R ˜ k − 3 ( s , v ) d s ≥ 0, v ∈ ℑ 1 ,

(35) J 2 ( f ( ⋅ ) ) ≥ 2 ( f ″ ( ζ 1 ) − f ″ ( ζ 2 ) ζ 2 − ζ 1 ) ∫ ζ 1 ζ 2 J 2 ( G t ( ⋅ , s ) ) d s + ∫ ζ 1 ζ 2 J 2 ( G t ( ⋅ , s ) ) ∑ l =3 k − 1 l ( l − 1) ( l − 1)! [ f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ] d s ,

(36) ∫ ζ 1 ζ 2 J 2 ( G t ( ⋅ , s ) ) R k − 3 ( s , v ) d s ≥ 0, v ∈ ℑ 1

and

(37) J 2 ( f ( ⋅ ) ) ≥ 2 ( f ″ ( ζ 2 ) − f ″ ( ζ 1 ) ζ 2 − ζ 1 ) ∫ ζ 1 ζ 2 J 2 ( G t ( ⋅ , s ) ) d s + ∫ ζ 1 ζ 2 J 2 ( G t ( ⋅ , s ) ) ∑ l =4 k − 1 1 ( l − 2)( l − 4)! [ f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ] d s .

Theorem 2.3

Let f : ℑ 1 = [ ζ 1 , ζ 2 ] → ℝ be a k-convex (k ≥ 3) function. Also, let f ( k − 1) be an absolutely continuous function and (p 1,…,p n ) are positive real numbers such that ∑ ρ = 1 n p ρ = P n . Then, for the functional J 2(·) defined in ( 6 ), we have the following:

  1. For t = 1, 2, inequalities ( 35 ) and ( 37 ) hold provided that k is odd.

  2. For fixed t = 1, 2, let inequality ( 35 ) be satisfied and

(38) ( f ″ ( ζ 1 ) − f ″ ( ζ 2 ) ) + ∑ l =3 k − 1 l ( l − 1) ( l − 1)! [ f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ] ≥ 0 ∀ s ∈ ℑ 1 ,

or ( 37 ) be satisfied and

(39) ∑ l = 4 k − 1 1 ( l − 2 ) ( l − 4 ) ! [ f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ] ≥ 0 .

Then, we have

(40) J 2 ( f ( ⋅ ) ) ≥ 0 .

Proof

It is clear that Green functions G t (·,s) defined in (14) and (16) are 3-convex functions. Since weights p ρ are positive, applying Theorem B and by using Remark 1.1, we have J 2 (G t (·, s)) ≥ 0 for fixed t = 1, 2.

  1. R ˜ k ( s , v ) ≥ 0 and R k (s, v) ≥ 0 for k = 5, 7,…, so (33) and (35) hold. As f is k-convex, hence by following Theorem 2.2 we get (34) and (36).

  2. Using (38) in (35) and (39) in (37) we get (40) for fixed t = 1, 2.

Next, we have a generalized form of Levinson-type inequality for 2n points given in [6] (see also [3]).□

Theorem 2.4

Let f ∈ C k [ ζ 1 , ζ 2 ] be such that f ( k − 1 ) is absolutely continuous and (p 1,…,p n ) be the positive real numbers such that ∑ ρ = 1 n p ρ = 1 . Also, let x ρ + y ϱ =2 c ˘ and x ρ + x n − ρ + 1 ≤ 2 c ˘ , p ρ x ρ + p n − ρ + 1 x n − ρ + 1 p ρ + p n − ρ + 1 ≤ c ˘ . Then, for the functional J 2(·) defined in ( 6 ), we have the following:

  1. For t = 1, 2, inequalities ( 35 ) and ( 37 ) hold provided that k is odd.

  2. For fixed t = 1, 2, let inequality ( 35 ) be satisfied and inequality ( 38 ) is valid or ( 37 ) be satisfied and inequality ( 38 ) is valid then we have inequality ( 40 ).

Proof

Proof is similar to that of Theorem 2.3.□

In [7], Mercer made a significant improvement by replacing the condition x 1 + y 1 = … = x n + y n of symmetric distribution with the weaker one that the variances of the two sequences are equal.

In the next result, Levinson-type inequality is given (for positive weights) under Mercer’s condition given in (7).

Corollary 2.1

Let f : ℑ 1 = [ ζ 1 , ζ 2 ] → ℝ be such that f ( k − 1) is absolutely continuous, x ρ , y ρ satisfy max{x 1…x n } ≤ min{y 1,…,y n } and ( 7 ). Also, let ( p 1 , … , p n ) ∈ ℝ n such that ∑ ρ = 1 n p ρ = 1 . Then, ( 31 ) and ( 33 ) are valid.

Proof

Proof is similar to that of Theorem 2.1 and by using the conditions given in the statement we get (31) and (33).□

Theorem 2.5

Let f : ℑ 1 = [ ζ 1 , ζ 2 ] → ℝ be a k-convex (k > 3) function. Also, let f ( k − 1) be an absolutely continuous function and (p 1,…,p n ) are positive real numbers such that ∑ ρ = 1 n p ρ = 1 , max{x 1 ,…,x n } ≤ min{y 1,…,y n } and

(41) ∑ ρ = 1 n p ρ ( x ρ − ∑ ρ = 1 n p ρ x ρ ) 2 = ∑ ρ = 1 n p ρ ( y ρ − ∑ ρ = 1 n p ρ y ρ ) 2 .

Then,

  1. For t = 1, 2, inequalities ( 35 ) and ( 37 ) hold provided that k is odd.

  2. For fixed t = 1, 2, let inequality ( 35 ) be satisfied and

(42) ( f ′ ′ ( ζ 1 ) − f ′ ′ ( ζ 2 ) ) + ∑ l =3 k − 1 l ( l − 1) ( l − 1)! [ f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ] ≥ 0 ∀ s ∈ ℑ 1 ,

or ( 37 ) be satisfied and

(43) ∑ l =4 k − 1 1 ( l − 2)( l − 4)! [ f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ] ≥ 0 .

Then, we have

(44) J 2 ( f ( ⋅ ) ) ≥ 0 .

Proof

It is clear that Green functions G t (·,s) defined in (14) and (16) are 3-convex functions. Since weights are positive, applying Theorem D, we have J 2 (G t (·,s)) ≥ 0 for fixed t = 1, 2.

  1. R ˜ k ( s , v ) ≥ 0 and R k (s,v) ≥ 0 for k = 5, 7,…, so (34) and (36) hold. As f is k-convex, hence by following Theorem 2.2 we get (35) and (37).

  2. Using (42) in (35) and (43) in (37) we get (44) for fixed t = 1, 2.□

2.2 Levinson-type inequality for k-convex (k ≥ 3) functions

Motivated by identity (3), we construct the following identities.

First, we defined the following functional:

ℋ : let f : ℑ 2 = [0,2 a ] → ℝ be a function, x 1,…,x n ∈ (0,a), ( p 1 , … , p n ) ∈ ℝ n , ( q 1 , … , q m ) ∈ ℝ m are real numbers such that ∑ ρ =1 n p ρ = 1 and ∑ ϱ =1 m q ϱ = 1 . Also, let x ρ , ∑ ϱ =1 m q ϱ (2 a − x ϱ ) and ∑ ρ =1 n p ρ ∈ ℑ 2 . Then,

(45) J ˜ ( f ( ⋅ ) ) = ∑ ϱ =1 m q ϱ f (2 a − x ϱ ) − f ( ∑ ϱ =1 m q ϱ (2 a − x ϱ ) ) − ∑ ρ =1 n p ρ f ( x ρ ) + f ( ∑ ρ =1 n p ρ x ρ ) .

Theorem 2.6

Assume ℋ . Let f ∈ C k [0,2 a ] ( k ≥ 3) be such that f ( k − 1) is absolutely continuous. Then, we have the following new identities for t = 1, 2 and 0 ≤ ζ 1 < ζ 2 ≤ 2a:

(i)

(46) J ˜ ( f ( ⋅ ) ) = 1 2 [ ∑ ϱ =1 m q ϱ (2 a − x ϱ ) 2 − ( ∑ ϱ =1 m q ϱ (2 a − x ϱ ) ) 2 − ∑ ρ =1 n p ρ x ρ 2 + ( ∑ ρ =1 n p ρ x ρ ) 2 ] f (2) ( ζ 2 ) + 2 ( f ′ ′ ( ζ 1 ) − f ′ ′ ( ζ 2 ) ζ 2 − ζ 1 ) × ∫ ζ 1 ζ 2 J ˜ ( G t ( ⋅ , s ) ) d s + ∫ ζ 1 ζ 2 J ˜ ( G t ( ⋅ , s ) ) × ∑ l =3 k − 1 l ( l − 1) ( l − 1)! [ f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ] d s + 1 ( k − 4)! ∫ ζ 1 ζ 2 f ( k ) ( v ) ( ∫ ζ 1 ζ 2 J ˜ ( G t ( ⋅ , s ) ) R ˜ k − 3 ( s , v ) d s ) d v ,

where

R ˜ k − 3 ( s , v ) = { 2 ( s − v ) k − 3 ( k − 3 ) ( ζ 2 − ζ 1 ) + s − ζ 1 ζ 2 − ζ 1 ( ( s − v ) k − 4 ) , ζ 1 ≤ v ≤ s ; 2 ( s − v ) k − 3 ( k − 3 ) ( ζ 2 − ζ 1 ) + s − ζ 2 ζ 2 − ζ 1 ( ( s − v ) k − 4 ) , s < v ≤ ζ 2 ,

and

(47) J ˜ ( G t ( ⋅ , s ) ) = ∑ ϱ =1 m q ϱ G t (2 a − x ϱ , s ) − G t ( ∑ ϱ =1 m q ϱ (2 a − x ϱ , s ) ) − ∑ ρ =1 n p ρ G t ( x ρ , s ) + G t ( ∑ ρ =1 n p ρ x ρ , s ) ,

for t = 1, 2, G t (·,s) is defined in ( 14 ) and ( 16 ), respectively.

(ii)

(48) J ˜ ( f ( ⋅ ) ) = 1 2 [ ∑ ϱ =1 m q ϱ (2 a − x ϱ ) 2 − ( ∑ ϱ =1 m q ϱ (2 a − x ϱ ) ) 2 − ∑ ρ =1 n p ρ x ρ 2 + ( ∑ ρ =1 n p ρ x ρ ) 2 ] f (2) ( ζ 2 ) + 2 ( f ″ ( ζ 2 ) − f ″ ( ζ 1 ) ζ 2 − ζ 1 ) ∫ ζ 1 ζ 2 J ˜ ( G t ( ⋅ , s ) ) d s + ∫ ζ 1 ζ 2 J ˜ ( G t ( ⋅ , s ) ) ∑ l =4 k − 1 1 ( l − 2)( l − 4)! [ f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ] d s + 1 ( k − 4)! ∫ ζ 1 ζ 2 f ( k ) ( v ) ( ∫ ζ 1 ζ 2 J ˜ ( G t ( ⋅ , s ) ) R k − 3 ( s , v ) d s ) d v .

Proof

Replace ℑ 1 with ℑ 2 and y ϱ with (2 a − x ϱ ) in Theorem 2.1, we get the required result.□

As an application of the above obtained identities (45) and (47), the next Theorem gives generalization of Levinson’s inequality (for real weights) for k-convex (k ≥ 3) functions involving new Green functions.

Theorem 2.7

Assuming the conditions of Theorem 2.6 with f as the k-convex function, then for t = 1, 2 we have the following two results:

If

(49) ∫ ζ 1 ζ 2 J ˜ ( G t ( ⋅ , s ) ) R ˜ k − 3 ( s , v ) d s ≥ 0, v ∈ ℑ 2 ,

then

(50) J ˜ ( f ( ⋅ ) ) ≥ 1 2 [ ∑ ϱ =1 m q ϱ (2 a − x ϱ ) 2 − ( ∑ ϱ =1 m q ϱ (2 a − x ϱ ) ) 2 − ∑ ρ =1 n p ρ x ρ 2 + ( ∑ ρ =1 n p ρ x ρ ) 2 ] f (2) ( ζ 2 ) + 2 ( f ″ ( ζ 1 ) − f ″ ( ζ 2 ) ζ 2 − ζ 1 ) ∫ ζ 1 ζ 2 J ˜ ( G t ( ⋅ , s ) ) d s + ∫ ζ 1 ζ 2 J ˜ ( G t ( ⋅ , s ) ) ∑ l =3 k − 1 l ( l − 1) ( l − 1)! [ f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ] d s ,

and if

(51) ∫ ζ 1 ζ 2 J ˜ ( G t ( ⋅ , s ) ) R k − 3 ( s , v ) d s ≥ 0, v ∈ ℑ 1 ,

then

(52) J ˜ ( f ( ⋅ ) ) ≥ 1 2 [ ∑ ϱ =1 m q ϱ (2 a − x ϱ ) 2 − ( ∑ ϱ =1 m q ϱ (2 a − x ϱ ) ) 2 − ∑ ρ =1 n p ρ x ρ 2 + ( ∑ ρ =1 n p ρ x ρ ) 2 ] f (2) ( ζ 2 ) + 2 ( f ″ ( ζ 2 ) − f ″ ( ζ 1 ) ζ 2 − ζ 1 ) ∫ ζ 1 ζ 2 J ˜ ( G t ( ⋅ , s ) ) d s + ∫ ζ 1 ζ 2 J ˜ ( G t ( ⋅ , s ) ) ∑ l =4 k − 1 1 ( l − 2)( l − 4)! [ f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ] d s .

Proof

Proof is similar to that of Theorem 2.2.

If we put m = n, p ρ = q ϱ and use positive weights in (44), then J ˜ ( ⋅ ) converted to the functional J 1(·) defined in (2), also in this case (46), (47), (48), (49), (50), (51) and (52) become

(53) J 1 ( f ( ⋅ ) ) = 2 ( f ″ ( ζ 1 ) − f ″ ( ζ 2 ) ζ 2 − ζ 1 ) ∫ ζ 1 ζ 2 J 1 ( G t ( ⋅ , s ) ) d s + ∫ ζ 1 ζ 2 J 1 ( G t ( ⋅ , s ) ) ∑ l = 3 k − 1 l ( l − 1 ) ( l − 1 ) ! [ f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ] d s + 1 ( k − 4 ) ! ∫ ζ 1 ζ 2 f ( k ) ( v ) ( ∫ ζ 1 ζ 2 J 1 ( G t ( ⋅ , s ) ) R ˜ k − 3 ( s , v ) d s ) d v ,

(54) J 1 ( G t ( ⋅ , s ) ) = ∑ ρ = 1 n p ρ G t ( 2 a − x ρ , s ) − G t ( ∑ ρ = 1 n p ρ 2 a − x ρ , s ) − ∑ ρ = 1 n p ρ G t ( x ρ , s ) + G t ( ∑ ρ = 1 n p ρ x ρ , s ) ,

(55) J 1 ( f ( ⋅ ) ) = 2 ( f ″ ( ζ 2 ) − f ′ ′ ( ζ 1 ) ζ 2 − ζ 1 ) ∫ ζ 1 ζ 2 J 1 ( G t ( ⋅ , s ) ) d s + ∫ ζ 1 ζ 2 J 1 ( G t ( ⋅ , s ) ) ∑ l =4 k − 1 1 ( l − 2)( l − 4)! [ f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ] d s + 1 ( k − 4)! ∫ ζ 1 ζ 2 f ( k ) ( v ) ( ∫ ζ 1 ζ 2 J 1 ( G t ( ⋅ , s ) ) R k − 3 ( s , v ) d s ) d v ,

(56) ∫ ζ 1 ζ 2 J 1 ( G t ( ⋅ , s ) ) R ˜ k − 3 ( s , v ) d s ≥ 0, v ∈ ℑ 1

(57) J 1 ( f ( ⋅ ) ) ≥ 2 ( f ″ ( ζ 1 ) − f ″ ( ζ 2 ) ζ 2 − ζ 1 ) ∫ ζ 1 ζ 2 J 1 ( G t ( ⋅ , s ) ) d s + ∫ ζ 1 ζ 2 J 1 ( G t ( ⋅ , s ) ) ∑ l =3 k − 1 l ( l − 1) ( l − 1)! [ f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ] d s ,

(58) ∫ ζ 1 ζ 2 J 1 ( G t ( ⋅ , s ) ) R k − 3 ( s , v ) d s ≥ 0, v ∈ ℑ 1

and

(59) J 1 ( f ( ⋅ ) ) ≥ 2 ( f ″ ( ζ 2 ) − f ″ ( ζ 1 ) ζ 2 − ζ 1 ) ∫ ζ 1 ζ 2 J 1 ( G t ( ⋅ , s ) ) d s + ∫ ζ 1 ζ 2 J 1 ( G t ( ⋅ , s ) ) ∑ l =4 k − 1 1 ( l − 2)( l − 4)! [ f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ] d s .

□

Theorem 2.8

Let f ∈ C k [ ζ 1 , ζ 2 ] be such that f ( k − 1) is absolutely continuous. Also, let (p 1,…,p n ) be positive real numbers such that ∑ ρ = 1 n p ρ = 1 . Then, for the functional J 1(·) defined in ( 2 ), we have the following:

  1. For t = 1, 2, inequalities ( 57 ) and ( 59 ) hold, provided that n is odd.

  2. For fixed k = 1, 2, let inequality ( 57 ) be satisfied and

(60) ( f ″ ( ζ 1 ) − f ″ ( ζ 2 ) ) + ∑ l =3 k − 1 l ( l − 1) ( l − 1)! [ f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ] ≥ 0 ∀ s ∈ ℑ 2 ,

or ( 59 ) be satisfied and

(61) ∑ l =4 k − 1 1 ( l − 2)( l − 4)! [ f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ] .

Then,

J 1 ( f ( ⋅ ) ) ≥ 0 .

Proof

Proof is similar to that of Theorem 2.3.□

3 Bounds for identities related to generalization of Levinson’s inequality

For two Lebesgue integrable functions f 1 , f 2 : [ ζ 1 , ζ 1 ] → ℝ , we consider the Čebyšev functional

(62) Θ ( f 1 , f 2 ) = 1 ζ 2 − ζ 1 ∫ ζ 1 ζ 2 f 1 ( t ) f 2 ( t ) d t − 1 ζ 2 − ζ 1 ∫ ζ 1 ζ 2 f 1 ( t ) d t ⋅ 1 ζ 2 − ζ 1 ∫ ζ 1 ζ 2 f 2 ( t ) d t ,

where the integrals are assumed to exist.

Theorem 3.1

[23] Let f 1 : [ ζ 1 , ζ 2 ] → ℝ be a Lebesgue integrable function and f 2 : [ ζ 1 , ζ 2 ] → ℝ be an absolutely continuous function with ( ⋅ , − ζ 1 ) ( ⋅ , − ζ 2 ) [ f 2 ′ ] 2 ∈ L [ ζ 1 , ζ 2 ] . Then, we have the inequality

(63) | Θ ( f 1 , f 2 ) | ≤ 1 2 [ Θ ( f 1 , f 1 ) ] 1 2 1 ζ 2 − ζ 1 ( ∫ ζ 1 ζ 2 ( x − ζ 1 ) ( ζ 2 − x ) [ f 2 ′ ( x ) ] 2 d x ) 1 2 .

The constant 1 2 is the best possible value.

Theorem 3.2

[23] Let f 1 : [ ζ 1 , ζ 2 ] → ℝ be an absolutely continuous with f 1 ′ ∈ L ∞ [ ζ 1 , ζ 2 ] and let f 2 : [ ζ 1 , ζ 2 ] → ℝ be monotonic non-decreasing on [ ζ 1 , ζ 2 ] . Then, we have the inequality

(64) | Θ ( f 1 , f 2 ) | ≤ 1 2 ( ζ 2 − ζ 1 ) ∥ f 1 ′ ∥ ∞ ∫ ζ 1 ζ 2 ( x − ζ 1 ) ( ζ 2 − x ) d f 2 ( x ) .

The constant 1 2 is the best possible value.

Now we consider Theorems 3.1 and 3.2 to generalize results given in the previous section.

In order to avoid many notions let us denote

(65) ℛ t ˜ ( v ) = ∫ ζ 1 ζ 2 J ˘ ( G t ( ⋅ , s ) ) R ˜ k − 3 ( s , v ) d s , v ∈ [ ζ 1 , ζ 2 ] ,

and

(66) ℛ t ( v ) = ∫ ζ 1 ζ 2 J ˘ ( G t ( ⋅ , s ) ) R k − 3 ( s , v ) d s , v ∈ [ ζ 1 , ζ 2 ] .

First we intend to give the Ostrowski-type inequalities related to generalizations of the Bullen inequality.

Theorem 3.3

Assuming the conditions of Theorem 2.1, moreover, assume (r, s) is a pair of conjugate exponents that is 1 ≤ r, s ≤ ∞ such that 1 r + 1 s = 1 . Let | f ( k ) | r : [ ζ 1 , ζ 2 ] → ℝ be the Riemann integrable function. Then,

(67) | J ˘ ( f ( ⋅ ) ) − 1 2 [ ∑ ϱ =1 m q ϱ y ϱ 2 − ( ∑ ϱ =1 m q ϱ y ϱ ) 2 − ∑ ρ =1 n p ρ x ρ 2 + ( ∑ ρ =1 n p ρ x ρ ) 2 ] f (2) ( ζ 2 ) − 2 ( f ″ ( ζ 1 ) − f ″ ( ζ 2 ) ζ 2 − ζ 1 ) ∫ ζ 1 ζ 2 J ˘ ( G t ( ⋅ , s ) ) d s − ∫ ζ 1 ζ 2 J ˘ ( G t ( ⋅ , s ) ) ∑ l =3 k − 1 l ( l − 1) ( l − 1)! [ f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ] d s | ≤ 1 ( k − 4)! | | f ( k ) | | r ( ∫ ζ 1 ζ 2 | ∫ ζ 1 ζ 2 J ˘ ( G t ( ⋅ , s ) ) R ˜ k − 3 ( s , v ) d s | s d v ) 1 s ,

and

(68) | J ˘ ( f ( ⋅ ) ) − 1 2 [ ∑ ϱ =1 m q ϱ y ϱ 2 − ( ∑ ϱ =1 m q ϱ y ϱ ) 2 − ∑ ρ =1 n p ρ x ρ 2 + ( ∑ ρ =1 n p ρ x ρ ) 2 ] f (2) ( ζ 2 ) − 2 ( f ″ ( ζ 2 ) − f ″ ( ζ 1 ) ζ 2 − ζ 1 ) ∫ ζ 1 ζ 2 J ˘ ( G t ( ⋅ , s ) ) d s − ∫ ζ 1 ζ 2 J ˘ ( G t ( ⋅ , s ) ) ∑ l =3 k − 1 l ( l − 1) ( l − 1)! [ f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ] d s | ≤ 1 ( k − 4)! ∥ f ( k ) ∥ r ( ∫ ζ 1 ζ 2 | ∫ ζ 1 ζ 2 J ˘ ( G t ( ⋅ , s ) ) R k − 3 ( s , v ) d s | s d v ) 1 s .

The constants on the right-hand side (R.H.S.) of ( 67 ) and ( 68 ) are sharp for 1 < r ≤ ∞ and best possible r = 1.

Proof

Fix t = 1, 2. Rearrange identity (19) in such a way

(69) | J ˘ ( f ( ⋅ ) ) − 1 2 [ ∑ ϱ =1 m q ϱ y ϱ 2 − ( ∑ ϱ =1 m q ϱ y ϱ ) 2 − ∑ ρ =1 n p ρ x ρ 2 + ( ∑ ρ =1 n p ρ x ρ ) 2 ] f (2) ( ζ 2 ) − 2 ( f ″ ( ζ 1 ) − f ″ ( ζ 2 ) ζ 2 − ζ 1 ) ∫ ζ 1 ζ 2 J ˘ ( G t ( ⋅ , s ) ) d s − ∫ ζ 1 ζ 2 J ˘ ( G t ( ⋅ , s ) ) ∑ l =3 k − 1 l ( l − 1) ( l − 1)! [ f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ] d s | ≤ 1 ( k − 4)! | ∫ ζ 1 ζ 2 ℛ t ˜ ( v ) f k ( v ) d v | .

Employing the classical Holder’s inequality to R.H.S. of (68) yields (66). The proof for sharpness is similar to Theorem 3.5 given in [13].

The proof of (67) is similar to that of (68).□

The Grüss-type inequalities can be obtained by using Theorem 3.2.

Theorem 3.4

Assume ℱ . Let f ∈ C k [ ζ 1 , ζ 2 ] and f ( k ) be an absolutely continuous with ( ⋅ − ζ 1 ) ( ζ 2 − ⋅ ) [ f ( k + 1) ] 2 ∈ L [ ζ 1 , ζ 2 ] . Also, ℛ t ˜ ( v ) and ℛ t ( v ) are defined in ( 65 ) and ( 66 ), respectively. Then, we have

(70) J ˘ ( f ( ⋅ ) ) − 1 2 [ ∑ ϱ =1 m q ϱ y ϱ 2 − ( ∑ ϱ =1 m q ϱ y ϱ ) 2 − ∑ ρ =1 n p ρ x ρ 2 + ( ∑ ρ =1 n p ρ x ρ ) 2 ] f (2) ( ζ 2 ) − 2 ( f ″ ( ζ 1 ) − f ″ ( ζ 2 ) ζ 2 − ζ 1 ) ∫ ζ 1 ζ 2 J ˘ ( G t ( ⋅ , s ) ) d s − ∫ ζ 1 ζ 2 J ˘ ( G t ( ⋅ , s ) ) ∑ l =3 k − 1 l ( l − 1) ( l − 1)! [ f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ] d s − f ( k − 1) ( ζ 2 ) − f ( k − 1) ( ζ 1 ) ( ζ 2 − ζ 1 ) ( k − 4)! ∫ ζ 1 ζ 2 ℛ t ˜ ( v ) d v = ℜ p ( ζ 1 , ζ 2 , ℛ t ˜ , f ( k ) ) ,

and

(71) J ˘ ( f ( ⋅ ) ) − 1 2 [ ∑ ϱ =1 m q ϱ y ϱ 2 − ( ∑ ϱ =1 m q ϱ y ϱ ) 2 − ∑ ρ =1 n p ρ x ρ 2 + ( ∑ ρ =1 n p ρ x ρ ) 2 ] f (2) ( ζ 2 ) − 2 ( f ″ ( ζ 2 ) − f ″ ( ζ 1 ) ζ 2 − ζ 1 ) ∫ ζ 1 ζ 2 J ˘ ( G t ( ⋅ , s ) ) d s − ∫ ζ 1 ζ 2 J ˘ ( G t ( ⋅ , s ) ) × ∑ l =3 k − 1 l ( l − 1) ( l − 1)! [ f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ] d s − f ( k − 1) ( ζ 2 ) − f ( k − 1) ( ζ 1 ) ( ζ 2 − ζ 1 ) ( k − 4)! ∫ ζ 1 ζ 2 ℛ t ( v ) d v = ℜ p ( ζ 1 , ζ 2 , ℛ t , f ( k ) ) ,

where the remainder ℜ p ( ζ 1 , ζ 2 , ℛ t ˜ , f ( k ) ) satisfies the bound

(72) | ℜ p ( ζ 1 , ζ 2 , ℛ t ˜ , f ( k ) ) | ≤ ( ζ 2 − ζ 1 ) 1 2 2 ( k − 4)! [ Θ ( ℛ t ˜ ( v ) , ℛ t ˜ ( v ) ) ] 1 2 | ∫ ζ 1 ζ 2 ( v − ζ 1 ) ( ζ 2 − v ) [ f ( k + 1) ( v ) ] 2 d v | 1 2 ,

and

(73) | ℜ p ( ζ 1 , ζ 2 , ℛ t , f ( k ) ) | ≤ ( ζ 2 − ζ 1 ) 1 2 2 ( k − 4)! [ Θ ( ℛ t ( v ) , ℛ t ( v ) ) ] 1 2 | ∫ ζ 1 ζ 2 ( v − ζ 1 ) ( ζ 2 − v ) [ f ( k + 1) ( v ) ] 2 d v | 1 2 ,

respectively.

Proof

Fix t = 1, 2. Using the Čebyšev functional for f 1 = ℛ t ˜ and f 2 = f ( k ) , by comparing (70) with (19), we have

ℜ p ( ζ 1 , ζ 2 , ℛ t ˜ , f ( k ) ) = ζ 2 − ζ 1 ( k − 4)! Θ ( ℛ t ˜ , f ( k ) ) .

Now applying Theorem 3.1 for the corresponding functions, we get the required bound.

Similarly, by comparing (71) with identity (22), we get the respective bound.□

Theorem 3.5

Assume ℱ . Let f ∈ C k [ ζ 1 , ζ 2 ] and f ( k ) be an absolutely continuous and f ( k + 1) ≥ 0 on [ζ1,ζ 2] with ℛ t ˜ ( v ) and ℛ t ( v ) defined in ( 65 ) and ( 66 ), respectively. Then, in representation ( 70 ) the remainder ℜ p ( ζ 1 , ζ 2 , ℛ t ˜ , f ( k ) ) satisfies the estimation

(74) | ℜ ( ζ 1 , ζ 2 , ℛ t ˜ , f ( k ) ) | ≤ ( ζ 2 − ζ 1 ) ∥ ℛ ˜ t ′ ∥ ∞ ( k − 4)! [ f ( k − 1) ( ζ 2 ) + f ( k − 1) ( ζ 1 ) 2 − f ( k − 2) ( ζ 2 ) − f ( k − 2) ( ζ 1 ) ζ 2 − ζ 1 ] ,

whereas in representation ( 71 ) the remainder ℜ p ( ζ 1 , ζ 2 , ℜ t , f ( k ) ) satisfies the estimation

(75) | ℜ ( ζ 1 , ζ 2 , ℛ t , f ( k ) ) | ≤ ( ζ 2 − ζ 1 ) ∥ ℛ t ′ ˜ ∥ ∞ ( k − 4)! [ f ( k − 1) ( ζ 2 ) + f ( k − 1) ( ζ 1 ) 2 − f ( k − 2) ( ζ 2 ) − f ( k − 2) ( ζ 1 ) ζ 2 − ζ 1 ] .

Proof

Fix t = 1, 2. We have established

ℜ p ( ζ 1 , ζ 2 , ℛ t ˜ , f ( k ) ) = ζ 2 − ζ 1 ( k − 4 ) ! Θ ( ℛ t ˜ , f ( k ) ) .

Now setting f 1 ↦ ℜ ˜ t and f 2 ↦ f ( k ) in Theorem 3.2, we get

(76) | ℜ ( ζ 1 , ζ 2 , ℛ t , f ( k ) ) | = 1 ( k − 4)! | Θ ( ℛ t ˜ , f ( k ) ) | ≤ ∥ ℛ t ′ ∥ ∞ 2( ζ 2 − ζ 1 ) ( k − 4)! ∫ ζ 1 ζ 2 ( v − ζ 1 ) ( ζ 2 − v ) f ( k + 1) v d v .

Simplifying the integral on R.H.S. of (75) we get the required results.□

Remark 3.1

Similar work can be done for Levinson’s inequality (2) (one type of data points) for k-convex functions.

Remark 3.2

We can give related mean value theorems by using non-negative functionals (19), (22), (46) and (47), and we can construct the new families of k-exponentially convex functions and Cauchy means related to these functionals as given in Section 4 of [24].

4 Application to information theory

The idea of Shannon entropy is the central job of information speculation now and again implied as a measure of uncertainty. The entropy of random variable is described with respect to probability distribution and can be shown to be a decent measure of random. The assignment of Shannon entropy is to assess the typical least number of bits expected to encode a progression of pictures subject to the letters, including the size and the repetition of the symbols.

Divergences between probability distributions have been familiar with the measure of the difference between them. An assortment of sorts of divergences exists, for example, the f -divergences (especially, Kullback–Leibler divergences, Hellinger distance and total variation distance), Rényi divergences, Jensen–Shannon divergences, etc. (see [28,29]). There are a lot of papers overseeing inequalities and entropies, see, e.g., [8,25,26,27] and references therein. The Jensen inequality is an essential job in a bit of these inequalities. Regardless, Jensen’s inequality manages one kind of data points and Levinson’s inequality deals two types of data points.

4.1 Csiszár divergence

In [30,31], Csiszár provided the following definition.

Definition 4.1

Let f be a convex function from ℝ + to ℝ + . Let r ˜ , k ˜ ∈ ℝ + n be such that ∑ v =1 n r v = 1 and ∑ v =1 n k v =1 . Then, the f -divergence functional is defined by

I f ( r ˜ , k ˜ ) ≔ ∑ v =1 n k v f ( r v k v ) .

By defining the following:

f (0) ≔ lim x → 0 + f ( x ) ; 0 f ( 0 0 ) ≔ 0 ; 0 f ( a x ) ≔ lim x → 0 + x f ( a x ) , a > 0,

he stated that nonnegative probability distributions can also be used.

Using the definition of the f -divergence functional, Horváth et al. [32] provided the following functional.

Definition 4.2

Let I be an interval contained in ℝ and f : I → ℝ be a function. Also, let r ˜ = ( r 1 , … , r n ) ∈ ℝ n and k ˜ = ( k 1 , … , k n ) ∈ (0, ∞ ) n be such that

r v k v ∈ I , v = 1 , … , n .

Then,

(77) I ˆ f ( r ˜ , k ˜ ) ≔ ∑ v =1 n k v f ( r v k v ) .

We apply Theorem 2.2 for k-convex functions to I ˆ f ( r ˜ , k ˜ ) .

Theorem 4.1

Let r ˜ = ( r 1 , … , r n ) ∈ ℝ n , w ˜ = ( w 1 , … , w m ) ∈ ℝ m , k ˜ = ( k 1 , … , k n ) ∈ (0,  ∞ ) n and t ˜ = ( t 1 , … , t m ) ∈ (0,  ∞ ) m such that

r v k v ∈ I , v = 1 , … , n ,

and

w u t u ∈ I , u = 1 , … , m .

Also, let f ∈ C k [ ζ 1 , ζ 2 ] be such that f is the k-convex function. Then,

(i) For k = 5, 7,…

(78) J c i s ( f ( ⋅ ) ) ≥ 1 2 [ 1 ∑ u =1 m t u ∑ u =1 m ( w u ) 2 t u − ( ∑ u =1 m w u ∑ u =1 m t u ) 2 − 1 ∑ v =1 n k v ∑ v =1 n ( r v ) 2 k v + ( ∑ v =1 n r v ∑ v =1 n k v ) 2 ] f (2) ( ζ 2 ) + 2 ( f ′ ( ζ 1 ) − f ′ ( ζ 2 ) ζ 2 − ζ 1 ) ∫ ζ 1 ζ 2 J ( G t ( ⋅ , s ) ) d s + ∫ ζ 1 ζ 2 J ( G t ( ⋅ , s ) ) ∑ l =3 k − 1 l ( l − 1) ( l − 1)! [ f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ] d s

and

(79) J c i s ( f ( ⋅ ) ) ≥ 1 2 [ 1 ∑ u =1 m t u ∑ u =1 m ( w u ) 2 t u − ( ∑ u =1 m w u ∑ u =1 m t u ) 2 − 1 ∑ v =1 n k v ∑ v =1 n ( r v ) 2 k v + ( ∑ v =1 n r v ∑ v =1 n k v ) 2 ] f (2) ( ζ 2 ) + 2 ( f ″ ( ζ 2 ) − f ″ ( ζ 1 ) ζ 2 − ζ 1 ) ∫ ζ 1 ζ 2 J ( G t ( ⋅ , s ) ) d s + ∫ ζ 1 ζ 2 J ( G t ( ⋅ , s ) ) ∑ l =4 k − 1 1 ( l − 2)( l − 4)! [ f ( l ) ( ζ 1 ) ( s − ζ 1 ) l − 2 − f ( l ) ( ζ 2 ) ( s − ζ 2 ) l − 2 ζ 2 − ζ 1 ] d s ,

where

(80) J c i s ( f ( ⋅ ) ) = 1 ∑ u =1 m t u I ˆ f ( w ˜ , t ˜ ) − f ( ∑ u =1 m w u ∑ u =1 m t u ) − 1 ∑ v =1 n k v I ˆ f ( r ˜ , k ˜ ) + f ( ∑ v =1 n r v ∑ v =1 n k v )

and

(81) J ( G t ( ⋅ , s ) ) = ∑ ρ =1 m t u ∑ u =1 m t u G t ( w u t u , s ) − G t ( ∑ ρ =1 m w u ∑ u =1 m t u , s ) − ∑ ρ =1 n k v ∑ v =1 n k v G t ( r v k v , s ) + G t ( ∑ ρ =1 n r v ∑ v =1 n k v , s ) .

(ii) For k = 4, 6,…, ( 78 ) and ( 79 ) hold in the reverse direction.

Proof

It is clear that Green functions G t (·,s) defined in (14) and (16) are 3-convex functions, therefore, by using Remark 1.1, J(G t (·,s)) ≥ 0 for fixed t = 1, 2.

  1. From (20) and (23), R ˜ k ( s , v ) ≥ 0 and R k (s,v) ≥ 0 for k = 5,7,…, so (27) and (29) hold. Hence, using p ρ = k v ∑ v = 1 n k v , x ρ = r v k v , q ϱ = t u ∑ u =1 m t u , y ϱ = w u t u in Theorem 2.2, (28) and (30) become (78) and (79), respectively, where I ˆ f ( r ˜ , k ˜ ) is defined in (77) and

    (82) I ˆ f ( w ˜ , t ˜ ) ≔ ∑ u =1 m t u f ( w u t u ) .

  2. From (20) and (23), R ˜ k ( s , v ) ≤ 0 and R k (s,v) ≤ 0 for k = 4, 6,…, so inequalities in (27) and (29) are reversed. Therefore, by using Remark 2.1, (28) and (30) hold in the reverse direction. Using the same substitutions as in (i), we get inequalities (78) and (79) in the reverse direction.□

4.2 Shannon entropy

Definition 4.3

(see [32]). The Shannon entropy of the positive probability distribution k ˜ = ( k 1 , … , k n ) is defined by

(83) S ≔ − ∑ v =1 n k v log ( k v ) .

Corollary 4.1

Let k ˜ = ( k 1 , … , k n ) and t ˜ = ( t 1 , … , t m ) be positive probability distributions. Also, let r ˜ = ( r 1 , … , r n ) ∈ (0,  ∞ ) n and w ˜ = ( w 1 , … , w m ) ∈ (0,  ∞ ) m .

(i) If base of log is greater than 1 and k = odd(k = 3, 5,…), then

(84) J s ( ⋅ ) ≥ − 1 2( ζ 2 ) 2 [ 1 ∑ u =1 m t u ∑ u =1 m ( w u ) 2 t u − ( ∑ u =1 m w u ∑ u =1 m t u ) 2 − 1 ∑ v =1 n k v ∑ v =1 n ( r v ) 2 k v + ( ∑ v =1 n r v ∑ v =1 n k v ) 2 ] − 2( ζ 1 + ζ 2 ) ( ζ 1 ζ 2 ) 2 ∫ ζ 1 ζ 2 J ( G t ( ⋅ , s ) ) d s + 1 ζ 2 − ζ 1 ∫ ζ 1 ζ 2 J ( G t ( ⋅ , s ) ) ∑ l =3 k − 1 ( − 1) l − 1 l ( l − 1) [ ( s − ζ 1 ) l − 2 ( ζ 1 ) l − ( s − ζ 2 ) l − 2 ( ζ 2 ) l ] d s

and

(85) J s ( ⋅ ) ≥ − 1 2( ζ 2 ) 2 [ 1 ∑ u =1 m t u ∑ u =1 m ( w u ) 2 t u − ( ∑ u =1 m w u ∑ u =1 m t u ) 2 − 1 ∑ v =1 n k v ∑ v =1 n ( r v ) 2 k v + ( ∑ v =1 n r v ∑ v =1 n k v ) 2 ] − ( ζ 1 + ζ 2 ) ( ζ 1 ζ 2 ) 2 ∫ ζ 1 ζ 2 J ( G t ( ⋅ , s ) ) d s + 1 ζ 2 − ζ 1 ∫ ζ 1 ζ 2 J ( G t ( ⋅ , s ) ) ∑ l =4 k − 1 ( − 1) l − 1 ( l − 1)( l − 3) [ ( s − ζ 1 ) l − 2 ( ζ 1 ) l − ( s − ζ 2 ) l − 2 ( ζ 2 ) l ] d s ,

where

(86) J s ( ⋅ ) = ∑ u =1 m t u log ( w u ) + S ˜ − log ( ∑ u =1 m w u ) − ∑ v =1 n k v log ( r v ) + S + log ( ∑ v =1 n r v )

and J(G t (·,s)) is defined in ( 81 ).

(ii) If base of log is less than 1 or k = even ( k = 4, 6, … ), then inequalities ( 84 ) and ( 85 ) are reversed.

Proof

(i) The function f → log ( x ) is k-convex for k = 3, 5, … and base of log is greater than 1. Therefore, using f = log ( x ) in Theorem 4.1 (i), we get (84) and (85), where S is defined in (83) and

S ˜ = − ∑ u = 1 m t u log ( t u ) .

(ii) The function f → log ( x ) is k-concave for k = 4, 6,…, therefore using f = log ( x ) in Theorem 4.1 (ii), we get (84) and (85) in the reverse direction.□

Corollary 4.2

Let r ˜ = ( r 1 , … , r n ) and w ˜ = ( w 1 , … , w m ) be positive probability distributions. Also, let k ˜ = ( k 1 , … , k n ) ∈ (0,  ∞ ) n and t ˜ = ( t 1 , … , t m ) ∈ (0,  ∞ ) m . If base of log is greater than 1 and k = even (k ≥ 4), then

(87) J s ( ⋅ ) ≤ − 1 2 ζ 2 [ 1 ∑ u =1 m t u ∑ u =1 m ( w u ) 2 t u − ( ∑ u =1 m w u ∑ u =1 m t u ) 2 − 1 ∑ v =1 n k v ∑ v =1 n ( r v ) 2 k v + ( ∑ v =1 n r v ∑ v =1 n k v ) 2 ] + ( − 2 ζ 1 ζ 2 ) ∫ ζ 1 ζ 2 J ( G t ( ⋅ , s ) ) d s + 1 ζ 2 − ζ 1 ∫ ζ 1 ζ 2 J ( G t ( ⋅ , s ) ) ∑ l =3 k − 1 ( − 1) l l [ ( s − ζ 1 ) l − 2 ( ζ 1 ) l − 1 − ( s − ζ 2 ) l − 2 ( ζ 2 ) l − 1 ] d s

and

(88) J s ( ⋅ ) ≤ − 1 2( ζ 2 ) 2 [ 1 ∑ u =1 m t u ∑ u =1 m ( w u ) 2 t u − ( ∑ u =1 m w u ∑ u =1 m t u ) 2 − 1 ∑ v =1 n k v ∑ v =1 n ( r v ) 2 k v + ( ∑ v =1 n r v ∑ v =1 n k v ) 2 ] + ( 2 ζ 1 ζ 2 ) ∫ ζ 1 ζ 2 J ( G t ( ⋅ , s ) ) d s + 1 ζ 2 − ζ 1 ∫ ζ 1 ζ 2 J ( G t ( ⋅ , s ) ) ∑ l =4 k − 1 ( − 1) l ( l − 3) [ ( s − ζ 1 ) l − 2 ( ζ 1 ) l − 1 − ( s − ζ 2 ) l − 2 ( ζ 2 ) l − 1 ] d s ,

where

(89) J s ( ⋅ ) = 1 ∑ u =1 m t u ( S ˜ + ∑ u =1 m w u log ( t u ) ) − 1 ∑ u =1 m t u log ( ∑ u =1 m t u ) − 1 ∑ v =1 n k v ( S + ∑ v =1 n r v log ( k v ) ) + 1 ∑ v =1 n k v log ( ∑ v =1 n k v )

and J(G t (·,s)) is defined in ( 81 ).

Proof

The function f → − x log ( x ) is k-convex (k = 4, 6,…) and base of log is greater than 1. Therefore, using f = − x log ( x ) in Theorem 4.1 (ii), we get (87) and (88), where

S ˜ = − ∑ u =1 m w u log ( w u )

and

S = − ∑ v =1 n r v log ( r v ) . □

Remark 4.1

If base is less than 1 or k = odd (k ≥ 3), then inequalities (87) and (88) are also valid.

5 Conclusion

This study is concerned with generalization of the Levinson-type inequalities (for real weights) for two types of data points implicating higher order convex functions. New Green functions and Montgomery identity are used for the class of k-convex functions, where k ≥ 3. As applications of new obtained results, Čebyšev-, Grüss- and Ostrowski-type new bounds are found. Moreover, the main results are applied to information theory via f -divergence and Shannon entropy. In future work, the main results can apply for other types of divergences and distances such as Rényi divergence, Rényi entropy and Zipf-Mandelbrot law.

Acknowledgments

The authors wish to thank the anonymous referees for their very careful reading of the manuscript and fruitful comments and suggestions. The research of Josip Pečarić is supported by the Ministry of Education and Science of the Russian Federation (Agreement number 02.a03.21.0008).

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Received: 2019-09-15
Revised: 2020-02-29
Accepted: 2020-04-15
Published Online: 2020-07-01

© 2020 Muhammad Adeel et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  57. Jordan {g,h}-derivations on triangular algebras
  58. A systolic inequality with remainder in the real projective plane
  59. A new characterization of L2(p2)
  60. Nonlinear boundary value problems for mixed-type fractional equations and Ulam-Hyers stability
  61. Asymptotic normality and mean consistency of LS estimators in the errors-in-variables model with dependent errors
  62. Some non-commuting solutions of the Yang-Baxter-like matrix equation
  63. General (p,q)-mixed projection bodies
  64. An extension of the method of brackets. Part 2
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  66. Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis
  67. Remark on subgroup intersection graph of finite abelian groups
  68. Detectable sensation of a stochastic smoking model
  69. Almost Kenmotsu 3-h-manifolds with transversely Killing-type Ricci operators
  70. Some inequalities for star duality of the radial Blaschke-Minkowski homomorphisms
  71. Results on nonlocal stochastic integro-differential equations driven by a fractional Brownian motion
  72. On surrounding quasi-contractions on non-triangular metric spaces
  73. SEMT valuation and strength of subdivided star of K 1,4
  74. Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems
  75. Gradient estimates for a weighted nonlinear parabolic equation and applications
  76. On the equivalence of three-dimensional differential systems
  77. Free nonunitary Rota-Baxter family algebras and typed leaf-spaced decorated planar rooted forests
  78. The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices
  79. Explicit determinantal formula for a class of banded matrices
  80. Dynamics of a diffusive delayed competition and cooperation system
  81. Error term of the mean value theorem for binary Egyptian fractions
  82. The integral part of a nonlinear form with a square, a cube and a biquadrate
  83. Meromorphic solutions of certain nonlinear difference equations
  84. Characterizations for the potential operators on Carleson curves in local generalized Morrey spaces
  85. Some integral curves with a new frame
  86. Meromorphic exact solutions of the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation
  87. Towards a homological generalization of the direct summand theorem
  88. A standard form in (some) free fields: How to construct minimal linear representations
  89. On the determination of the number of positive and negative polynomial zeros and their isolation
  90. Perturbation of the one-dimensional time-independent Schrödinger equation with a rectangular potential barrier
  91. Simply connected topological spaces of weighted composition operators
  92. Generalized derivatives and optimization problems for n-dimensional fuzzy-number-valued functions
  93. A study of uniformities on the space of uniformly continuous mappings
  94. The strong nil-cleanness of semigroup rings
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  96. Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow
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  98. Inequalities for the generalized trigonometric and hyperbolic functions
  99. Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions
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  101. Constructions of pseudorandom binary lattices using cyclotomic classes in finite fields
  102. Estimates of entropy numbers in probabilistic setting
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  113. Skew-symmetric and essentially unitary operators via Berezin symbols
  114. The limit Riemann solutions to nonisentropic Chaplygin Euler equations
  115. On singularities of real algebraic sets and applications to kinematics
  116. Results on analytic functions defined by Laplace-Stieltjes transforms with perfect ϕ-type
  117. New (p, q)-estimates for different types of integral inequalities via (α, m)-convex mappings
  118. Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions
  119. Boundary layer analysis for a 2-D Keller-Segel model
  120. On some extensions of Gauss’ work and applications
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  126. The second out-neighborhood for local tournaments
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  128. The H-force sets of the graphs satisfying the condition of Ore’s theorem
  129. Bipartite graphs with close domination and k-domination numbers
  130. On the sandpile model of modified wheels II
  131. Connected even factors in k-tree
  132. On triangular matroids induced by n3-configurations
  133. The domination number of round digraphs
  134. Special Issue on Variational/Hemivariational Inequalities
  135. A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion
  136. On the finite approximate controllability for Hilfer fractional evolution systems with nonlocal conditions
  137. On the well-posedness of differential quasi-variational-hemivariational inequalities
  138. An efficient approach for the numerical solution of fifth-order KdV equations
  139. Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function
  140. Karush-Kuhn-Tucker optimality conditions for a class of robust optimization problems with an interval-valued objective function
  141. An equivalent quasinorm for the Lipschitz space of noncommutative martingales
  142. Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation
  143. Special Issue on Problems, Methods and Applications of Nonlinear analysis
  144. Generalized Picone inequalities and their applications to (p,q)-Laplace equations
  145. Positive solutions for parametric (p(z),q(z))-equations
  146. Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation
  147. (p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group
  148. Quasilinear Dirichlet problems with competing operators and convection
  149. Hyers-Ulam-Rassias stability of (m, n)-Jordan derivations
  150. Special Issue on Evolution Equations, Theory and Applications
  151. Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation
  152. Three classes of decomposable distributions
https://doi.org/10.1515/math-2020-0163
Keywords for this article
Levinson's inequality; Montgomery identity; information theory
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  82. The integral part of a nonlinear form with a square, a cube and a biquadrate
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  134. Special Issue on Variational/Hemivariational Inequalities
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  150. Special Issue on Evolution Equations, Theory and Applications
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  152. Three classes of decomposable distributions
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