On the Jensen functional and superquadraticity

arXiv:1502.07859v1 [math.CA] 27 Feb 2015

ON THE JENSEN FUNCTIONAL AND SUPERQUADRATICITY FLAVIA-CORINA MITROI-SYMEONIDIS AND NICUS ¸ OR MINCULETE Abstract. In this note we give a recipe which describes upper and lower bounds for the Jensen functional under superquadraticity conditions. Some results involve the Chebychev functional. We give a more general definition of these functionals and establish the analogue results.

1. Introduction The object of this paper is to derive some results related to the Jensen functional in the framework of superquadratic functions. We are interested in finding bounds both for the discrete and continuous case. For the reader’s convenience, let us briefly state known facts regarding the principal tools, the superquadraticity and the Jensen functional. See S. Abramovich and S. S. Dragomir [1] for details and proofs. Definition 1 ([2, Definition 2.1]). A function f defined on an interval I = [0, a] or [0, ∞) is superquadratic if for each x in I there exists a real number C(x) such that (1.1)

f (y) − f (x) ≥ f (|y − x|) + C(x)(y − x)

for all y ∈ I. We say that f is a subquadratic function if −f is superquadratic. The set of superquadratic functions is closed under addition and positive scalar multiplication. Example 1 ([3]). The function f (x) = xp , p ≥ 2 is superquadratic with C(x) = p f ′ (x) = pxp−1 . Similarly, g (x) = − 1 + x1/p , p > 0 is superquadratic with C(x) = 0. Also h(x) = x2 log x with C(x) = h′ (x) = x(2 log x + 1) is a superquadratic function (but not monotonic and not convex). Example 2 ([11]). Some elementary functions are not superquadratic, such as f (x) = x and f (x) = exp x. Lemma 1 ([2, Lemma 2.2]). Let f be a superquadratic function with C(x) defined as above. (i) Then f (0) ≤ 0. (ii) If f (0) = f ′ (0) = 0, then C(x) = f ′ (x), whenever f is differentiable at x > 0. (iii) If f ≥ 0, then f is convex and f (0) = f ′ (0) = 0. Theorem 1 ([2, Theorem 2.3]). The inequality Z Z Z (1.2) f ϕdµ ≤ f (ϕ (s)) − f ϕ (s) − ϕdµ dµ (s)

2000 Mathematics Subject Classification. Primary 26B25; Secondary 26D15. Key words and phrases. Jensen functional, Chebychev functional, superquadratic function. 1

2

FLAVIA-CORINA MITROI-SYMEONIDIS AND NICUS ¸ OR MINCULETE

holds for all probability measures µ and all nonnegative, µ-integrable functions ϕ if and only if f is superquadratic. Definition 2 ([1]). Let f be a real valued P function defined on an interval I, n x1 , ..., xn ∈ I and p1 , ..., pn ∈ (0, 1) such that i=1 pi = 1. The Jensen functional is defined by ! n n X X pi xi pi f (xi ) − f (1.3) J (f, p, x) = i=1

i=1

and the Chebychev functional is defined by   n n X X pj xj  f (xi ) . pi xi − (1.4) T (f, p, x) = j=1

i=1

See [6] and [10]. The discrete form of Theorem 1 is as follows.

Proposition 1 ([1, Lemma 2]). Let xi ≥ 0, i = 1, ..., n, and pi > 0, i = 1, ..., n, Pn with i=1 pi = 1. If f is superquadratic, then   n n X X  pj xj  . pi f xi − J (f, p, x) ≥ j=1 i=1

Theorem Pn 2 ([1, Theorem 4]). Assume that Pnxi ∈ I, i = 1, ..., n, pi > 0 are such that i=1 pi = 1 and ri > 0 are such that i=1 ri = 1. We denote pi pi m = min , M = max . i=1,...,n ri i=1,...,n ri Then J (f, p, x) − mJ (f, r, x)   ! n n n X X X ≥ mf (pi − mri ) f  xi − pj xj  (ri − pi ) xi + i=1

i=1

and

j=1

M J (f, r, x) − J (f, p, x)   ! n n n X X X  (M ri − pi ) f xi − rj xj  (ri − pi ) xi + ≥ f i=1

i=1

j=1

for all superquadratic functions f .

Definition 3. Assume that x = (x1 , ..., xn ) ∈ I n , p = (p1 , ..., pn ) are such that Pn Pk pi > 0, i=1 pi = 1, q = (q1 , ..., qk ) are such that qi > 0, i=1 qi = 1 (1 ≤ k ≤ n). We define   ! n k n X X X Jk (f, p, q, x) := pi1 ...pik f  qj xij  − f pi xi . i1 ,...,ik =1

j=1

i=1

Obviously J1 (f, p, q, x) = J (f, p, x) . We quote now some results that we refine in the following section.

ON THE JENSEN FUNCTIONAL AND SUPERQUADRATICITY

3

Proposition 2 ([1, Theorem 6]). Let xi ≥ 0, i = 1, ..., n, and pi > 0, i = 1, ..., n, P P such that ni=1 pi = 1 and qi > 0, i = 1, ..., k, ki=1 qi = 1 (1 ≤ k ≤ n). If f is superquadratic, then   k n X X  Jk (f, p, q, x) ≥ pi1 ...pik f pj xj  . qj xij − j=1 i1 ,...,ik =1 j=1 n X

Notice that the Proposition 2 is a slightly more general assertion than Proposition 1 above. Theorem 3 ([1, Theorem 7]). Assume that x = (x1 , x2 , ..., xn ) ∈ I n , p = (p1 , p2 , ..., pn ) are P P such that pi > 0, ni=1 pi = 1, q = (q1 , q2 , ..., qk ) are suchPthat qi > 0, ki=1 qi = 1 (1 ≤ k ≤ n) and r = (r1 , r2 , ..., rn ) are such that ri > 0, ni=1 ri = 1. We denote m=

min

1≤i1 ,...,ik ≤n

pi1 ...pik ri1 ...rik

and M =

max

1≤i1 ,...,ik ≤n

pi1 ...pik ri1 ...rik

.

If f is superquadratic then

≥

Jk (f, p, q, x) − mJk (f, r, q, x) ! n X (ri − pi ) xi mf i=1

  X n X k  pj xj  qj xij − (pi1 ...pik − mri1 ...rik ) f + j=1 j=1 i1 ,...,ik =1 n X

and

M Jk (f, r, q, x) − Jk (f, p, q, x) ! n X (ri − pi ) xi ≥ f i=1

  k n X X rj xj  . qj xij − (M ri1 ...rik − pi1 ...pik ) f  + j=1 j=1 i1 ,...,ik =1 n X

Results involving Jensen’s and Chebychev’s inequalities are sometimes stated in terms of probability measures rather than summation or Lebesgue integration. Then some of our results can be derived from the ones above applied to a product measure, as Gord Sinnamon pointed out during some useful discussions. Section 3 contains a definition of such functionals and analogue results. Their study is done for the discrete and integral case, not in probabilistic terms. The distinction between summation and integration is not artificial, but useful for different areas of study as information theory and transport theory. For convex, strong convex and superquadratic functions the interested reader can also find relevant results in [8] and [9].

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2. Main results 2.1. More on the Jensen functional. Theorem 4. Let f be a superquadratic function defined on P an interval I = [0, a] or n [0, ∞), x1 , x2 , ..., xn ∈ I and p1 , p2 , ..., pn ∈ (0, 1) such that i=1 pi = 1, λ ∈ [0, 1] . Then (2.1)       n n n n n X X X X X pj xj  . pi f λ xi − pj xj + λxi  − f  pj xj  ≥ pi f (1 − λ) j=1 i=1 j=1 j=1 i=1

Proof. Let f be a superquadratic function with C(x) defined as above. Then replacing y by (1 − λ) x + λy in (1.1) we deduce the inequality f ((1 − λ) x + λy) − f (x) ≥ f (λ|y − x|) + λC(x)(y − x). Pn In this inequality we put x = j=1 pj xj and y = xi . Multiplying by pi and summing over i we get the conclusion. For λ = 1 we recover the result of Proposition 1. As an immediat consequence of this result, due to the convexity of positive superquadratic functions, we get the following lower bound of interest: Corollary 1. Let f ≥ 0 be a superquadratic function defined on an P interval I = n [0, a] or [0, ∞), x1 , x2 , ..., xn ∈ I and p1 , p2 , ..., pn ∈ (0, 1) such that i=1 pi = 1, λ ∈ [0, 1] . Then   n n X X 1 pj xj  . pi f  xi − J (f, p, x) ≥ 2 2 j=1 i=1 Proof. In (2.1) we consider λ = (2.2)

n X i=1

pi f

Pn

j=1

pj xj + xi 2

1 2

!

:

  n X 1    pi f −f pj xj ≥ pj xj  . xi − 2 i=1 j=1 j=1 

n X

Since by Jensen’s inequality one has     n 1  X f pj xj  + f (xi ) ≥ f 2 j=1



n X

Pn

j=1

pj xj + xi 2

!

,

we get (2.3)         n n n n n X X X 1 X  X 1      pi f pi f pj xj + f (xi ) − f pj xj  . pj xj ≥ xi − 2 i=1 2 i=1 j=1 j=1 j=1 This completes the proof.

The interested reader can refine our last result by using refinements of Jensen’s inequality instead of the classic result.


5

2.2. The discrete case. Motivated by the above results, introduce in a natural other functionals. Definition 4. Assume that we have a real valued function f defined on interval Pan ni I, the real numbers pij , i = 1, ..., k and j = 1, ..., ni such that pij > 0, j=1 pij = 1 for all i = 1, ..., k (we denote pi = (pi1 , pi2 , ..., pini )), xi = (xi1 , xi2 , ..., xini ) ∈ I ni Pk for all i = 1, ..., k and q = (q1 , q2 , ..., qk ) , qi > 0 such that i=1 qi = 1. We define the generalized Jensen functional by ! n1X ,...,nk k X qi xiji p1j1 ...pkjk f Jk (f, p1 , ..., pk , q, x1 , ..., xk ) : = i=1

j1 ,...,jk =1

  ni k X X pij xij  . −f  qi i=1

j=1

and the generalized Chebychev functional by:

=

Tk (f, p1 , ..., pk , q, x1 , ..., xk )   n1X ,...,nk ni k X X p1j1 ...pkjk qi xiji − pij xij  f

j1 ,...,jk =1

i=1

j=1

k X

qi xiji

i=1

!

.

We easily notice also that for k = 1 this definition reduces to Definition 2. In [8] the following estimation is obtained: Remark 1. If f is a convex function then we have p1j1 ...pkjk Jk (f, r1 , ..., rk , q, x1 , ..., xk ) min 1≤j1 ≤n1 r1j1 ...rkjk ... 1≤jk ≤nk

≤ ≤

Jk (f, p1 , ..., pk , q, x1 , ..., xk ) p1j1 ...pkjk max Jk (f, r1 , ..., rk , q, x1 , ..., xk ) . 1≤j1 ≤n1 r1j1 ...rkjk

... 1≤jk ≤nk

In this paper, we investigate upper and lower bounds that we have if the function f is superquadratic. Now we extend the earlier results. The following lemma describes the behaviour of the functional under the superquadraticity condition: Lemma 2. Let pi , xi , q be as in Definition 4. If f is superquadratic then we have ! k n1X ,...,nk X ¯ , qi xiji − x p1j1 ...pkjk f Jk (f, p1 , ..., pk , q, x1 , ..., xk ) ≥ j1 ,...,jk =1

where x ¯= tion).

Pk

i=1 qi

Pni

j=1

i=1

pij xij (we will keep this notation throughout this subsec-

Proof. Straightforward from the definition of superquadratic functions. Using the same recipe as in the proof of Corollary 1, we get:

6


Corollary 2. Let pi , xi , q be as in Definition 4. Let f ≥ 0 be a superquadratic function defined on an interval I = [0, a] or [0, ∞), λ ∈ [0, 1] . Then ! k n1X ,...,nk 1 X ¯ . qi xiji − x p1j1 ...pkjk f Jk (f, p1 , ..., pk , q, x1 , ..., xk ) ≥ 2 2 j1 ,...,jk =1

i=1

The next result of the paper can be expressed as:

Theorem 5. Let f, pi , xi and q be as inP Definition 4 and the positive real numbers i rij , i = 1, ..., k and j = 1, ..., ni such that nj=1 rij = 1 for all i = 1, ..., k. We denote ri

=

m

=

(ri1 , ri2 , ..., rini ) for all i = 1, ..., k, p1j1 ...pkjk , min 1≤j1 ≤n1 r1j1 ...rkjk

... 1≤jk ≤nk

M

=

max

1≤j1 ≤n1 ... 1≤jk ≤nk

p1j1 ...pkjk r1j1 ...rkjk

.

If f is a superquadratic function, then: Jk (f, p1 , ..., pk , q, x1 , ..., xk ) − mJk (f, r1 , ..., rk , q, x1 , ..., xk )   k ni X X (rij − pij ) xij  qi ≥ mf  i=1 j=1 ! k n1X ,...,nk X ¯ qi xiji − x (p1j1 ...pkjk − mr1j1 ...rkjk ) f + i=1

j1 ,...,jk =1

and

≥

M Jk (f, r1 , ..., rk , q, x1 , ..., xk ) − Jk (f, p1 , ..., pk , q, x1 , ..., xk )   k ni X X (rij − pij ) xij  qi f  i=1 j=1 ! k n1X ,...,nk X ¯ . qi xiji − x (M r1j1 ...rkjk − p1j1 ...pkjk ) f + i=1

j1 ,...,jk =1

Proof. We prove only the first inequality. Obviously

Jk (f, p1 , ..., pk , q, x1 , ..., xk ) − mJk (f, r1 , ..., rk , q, x1 , ..., xk ) ! n1X ,...,nk k X qi xiji (p1j1 ...pkjk − mr1j1 ...rkjk ) f = i=1

j1 ,...,jk =1



+mf 

Since x ¯=

n1X ,...,nk

j1 ,...,jk =1

k X i=1

qi

ni X j=1



x) . rij xij  − f (¯

(p1j1 ...pkjk − mr1j1 ...rkjk )

k X i=1

qi xiji + m

k X i=1

qi

ni X j=1

rij xij


7

we conclude by Lemma 2 that Jk (f, p1 , ..., pk , q, x1 , ..., xk ) − mJk (f, r1 , ..., rk , q, x1 , ..., xk ) ! n1X ,...,nk k X ¯ qi xiji − x (p1j1 ...pkjk − mr1j1 ...rkjk ) f ≥ i=1 j1 ,...,jk =1   k ni X X rij xij − x ¯  qi +mf  i=1 j=1 =

n1X ,...,nk

(p1j1 ...pkjk − mr1j1 ...rkjk ) f

j1 ,...,jk =1

! k X qi xiji − x¯ i=1

  k ni X X (rij − pij ) xij  . qi +mf  i=1 j=1

The proof of the other inequality goes likewise and we omit the details.

The following particular case is of interest. Remark 2. Let p1 = ... = pk = p and x1 = ... = xk = x. In this case we see that Lemma 2 and Theorem 5 are recovering Proposition 2, respectively Theorem 3 . Also for k = 1 Lemma 2 and Theorem 5 recapture Proposition 1, respectively Theorem 2. According to [2, Lemma 2.2], if a superquadratic function is also nonnegative then it is convex. We may conclude that in this particular case Theorem 5 is a refinement of the result stated in Remark 1. Different results are obtained by using the Chebychev functional: ˜ such that Lemma ˜ M P3. Let f : [0, ∞) → R. If there exist real numbers m, k ˜ m ˜ ≤f i=1 qi xiji ≤ M , for all ji ∈ 1, ..., ni , i = 1, ..., k, then (2.4) k n1X ,...,nk X ˜ −m M ˜ |Tk (f, p1 , ..., pk , q, x1 , ..., xk )| ≤ qi xiji − x¯ . p1j1 ...pkjk 2 i=1 j ,...,j =1 1

k

This lemma is a discrete version of a result due to P. Cerone and S. S. Dragomir [4, Theorem 2]. See also [5, Lemma 5.58]. Proof. Notice that f

k X i=1

qi xiji

!

for all ji ∈ 1, ..., ni , i = 1, ..., k. Since n1X ,...,nk

j1 ,...,jk =1

˜ +m ˜ −m M ˜ M ˜ − , ≤ 2 2

p1j1 ...pkjk

k X i=1

qi xiji − x¯

!

= 0,

8


we have Tk (f, p1 , ..., pk , q, x1 , ..., xk ) =

n1X ,...,nk

p1j1 ...pkjk

k X

qi xiji − x¯

i=1

j1 ,...,jk =1

whence it follows that

!

f

k X

qi xiji

i=1

!

˜ +m M ˜ − 2

!

,

|Tk (f, p1 , ..., pk , q, x1 , ..., xk )| k ! n1X ,...,nk k X X ˜ +m M ˜ qi xiji − ¯ f qi xiji − x p1j1 ...pkjk ≤ 2 i=1 i=1 j1 ,...,jk =1 k n1X ,...,nk X ˜ −m M ˜ ¯ . qi xiji − x p1j1 ...pkjk ≤ 2 i=1

j1 ,...,jk =1

We close this subsection with a proposition that gives us an upper bound for the Jensen functional under the superquadraticity condition, via the above result on the Chebyshev functional. Proposition 3. Let f : [0, ∞) → R be a superquadratic P function. If for C(x) there k ˜ ˜ exist real numbers m, ˜ M such that m ˜ ≤C i=1 qi xiji ≤ M , for all ji ∈ 1, ..., ni , i = 1, ..., k, then we have: Jk (f, p1 , ..., pk , q, x1 , ..., xk ) ≤

n1X ,...,nk

p1j1 ...pkjk

j1 ,...,jk =1

Proof. We apply

k ˜ −m M ˜ X ¯ − f qi xiji − x 2 i=1

!! k X . ¯ qi xiji − x i=1

Jk (f, p1 , ..., pk , q, x1 , ..., xk ) ≤ Tk (C, p1 , ..., pk , q, x1 , ..., xk ) ! n1X ,...,nk k X (2.5) ¯ . qi xiji − x p1j1 ...pkjk f − j1 ,...,jk =1

i=1

and the inequality (2.4) in order to get the claimed result. The proof of (2.5) can be found in [9, Theorem 3].

This proposition extends a result due to S. Abramovich and S. S. Dragomir [1, Theorem 9]. The inequality (2.5) is establishing a connection between the Jensen functional and the Chebychev functional and is interesting in itself. 2.3. The integral case. In what follows we shall concentrate on the integral analogue of some of the results from the previous section. Let pi (x) dx and ri (x) dx, i = 1, ..., k be absolutely continuous measures, where pi , ri : [a, b] ⊂ (0, ∞) → (0, ∞) Rb Rb are such that a pi (x) dx = 1, a ri (x) dx = 1. We also consider q = (q1 , q2 , ..., qk ) , Pk qi > 0 with i=1 qi = 1. We define ! k ! Z Z b k k Y X X Jk (f, p1 , ..., pk , q) := f (pi (xi ) dxi )−f qi xi qi xpi (x) dx [a,b]k

i=1

i=1

i=1

a


9

and Tk (f, p1 , ..., pk , q) =

Z

k X

[a,b]k i=1

qi

xi −

!

b

Z

xpi (x) dx f

a

k X

qi xi

i=1

!

k Y

(pi (xi ) dxi )

i=1

for all positive integers k. Before we prove the main result, we need the following lemma providing an inequality that is interesting in itself as well. For the case of superquadratic nonnegative functions (hence convex) this result is a refinement of Jensen’s inequality. Lemma 4 (the integral analogue of Lemma 2). Assume that f is superquadratic. Then ! Z k k Y X (pi (xi ) dxi ) , qi xi − x ¯ (2.6) Jk (f, p1 , ..., pk , q) ≥ f [a,b]k i=1

i=1

where

x ¯=

k X

Z

qi

b

xpi (x) dx

a

i=1

(we will keep this notation in this subsection). Proof. Straightforward from the definition of superquadratic functions.

Example 3. A particularly interesting case is pointed out by assuming, for simplicity, that pi (x) dx = dx/ (b − a) , i = 1, ..., k, when ! k Z k Y X 1 a+b dx − f q x f i i i k 2 (b − a) [a,b]k i=1 i=1 ! Z k k X 1 a + b Y ≥ dxi . q x − f i i k 2 (b − a) [a,b]k i=1

i=1

Remark 3. For the case k = 1 the lemma gives us the following inequality ! ! Z Z b Z b Z b b f (x) p (x) dx ≥ f xp (x) dx + f x − xp (x) dx p (x) dx a a a a

for every f superquadratic. This is the integral counterpart of Proposition 1, an example of [2, Theorem 2.3]. We derive the following result.

Theorem 6 (the integral analogue of Theorem 5). We denote ) (R Qk i=1 pi (xi ) dxi [t,s]k m= inf R Qk t,s∈[a,b];s6=t i=1 ri (xi ) dxi [t,s]k and

M=

sup t,s∈[a,b];s6=t

(R

Qk

i=1 pi (xi ) dxi R Qk i=1 ri (xi ) dxi [t,s]k [t,s]k

)

.

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If f is superquadratic then

≥

Jk (f, p1 , ..., pk , q) − mJk (f, r1 , ..., rk , q) ! Z b k X qi x (pi (x) − ri (x)) dx mf a i=1 ! Z k k Y X ((pi (xi ) − mri (xi )) dxi ) qi xi − x ¯ + f [a,b]k i=1

i=1

and

≥

M Jk (f, r1 , ..., rk , q) − Jk (f, p1 , ..., pk , q) ! k X Z b qi x (pi (x) − ri (x)) dx f a i=1 ! Z k k Y X ((M ri (xi ) − pi (xi )) dxi ) . qi xi − x ¯ + f [a,b]k i=1

i=1

Proof. We will prove the first inequality. Lemma 4 implies that

Jk (f, p1 , ..., pk , q) − mJk (f, r1 , ..., rk , q) ! k Z k Y X ((pi (xi ) − mri (xi )) dxi ) qi xi f = [a,b]k

+mf

qi

i=1

≥

i=1

i=1

k X

Z

b

!

xri (x) dx a

−f

k X i=1

qi

Z

a

b

!

xpi (x) dx

! k k Y X ((pi (xi ) − mri (xi )) dxi ) f qi xi − x ¯ [a,b]k i=1 i=1 ! Z b k X +mf qi x (pi (x) − ri (x)) dx . a Z

i=1

The proof of the second inequality goes likewise and has been omitted.

Now we turn our attention to the Chebychev functional. By an essentially similar method as in the discrete case already discussed above, one can prove the following lemma. Lemma 5. We consider a superquadratic function f : [0, ∞) → R. If there exist ˜ such that m ˜ , for all x ≥ 0, then we get real numbers m, ˜ M ˜ ≤ f (x) ≤ M k k Z Y X ˜ −m M ˜ |Tk (f, p1 , ..., pk , q)| ≤ (pi (xi ) dxi ) . qi xi − x¯ 2 [a,b]k i=1 i=1

This lemma can be used to point out our last result.

Proposition 4 (the integral analogue of Proposition 3). Using the above notations, we also consider a superquadratic function f : [0, ∞) → R. If there exist real


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˜ such that m ˜ , for all x ≥ 0, then we have: numbers m, ˜ M ˜ ≤ C(x) ≤ M

≤

Jk (f, p1 , ..., pk , q) k Z ˜ −m M ˜ X qi xi − x ¯ − f k 2 [a,b] i=1

!! k k X Y (pi (xi ) dxi ) . qi xi − x¯ i=1

i=1

Acknowledgement 1. We would like to thank Professor S. S. Dragomir for suggesting the simpler proof of Lemma 3. References [1] S. Abramovich, S. S. Dragomir, Normalized Jensen Functional, Superquadracity and Related Inequalities, Internat. Ser. Numer. Math., 157(2009), 217–228. [2] S. Abramovich, G. Jameson, G. Sinnamon, Refining Jensen’s Inequality, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 47 (95) (2004), no. 1-2, 3-14. [3] S. Abramovich, S. Iveli´ c, J. Peˇ cari´ c, Refining inequalities related to convexity via superquadraticity, weaksuperquadraticity and superterzaticity, Manuscript. [4] P. Cerone, S.S. Dragomir, A Refinement of the Gr¨ uss inequality and applications, Tamkang J. Math. 38 (2007), 37–49. [5] S.S. Dragomir, A Survey on Cauchy-Bunyakovsky-Schwarz type discrete inequalities, J. Inequal. Pure and Appl. Math., 4(3) (2003), Art. 63. [6] S.S. Dragomir, Bounds for the normalised Jensen functional, Bull. Austral. Math. Soc., 74 (2006), pp. 471-478. [7] S.S. Dragomir, Inequalities for the Cebysev functional of two functions of selfadjoint operators in Hilbert spaces, Aust. J. Math. Anal. Appl., 6 (2009), Issue 1, Article 7, pp. 1-58. [8] F.-C. Mitroi, Estimating the normalized Jensen functional, J. Math. Inequal, Vol. 5, Issue 4 (2011), 507–521. [9] F.-C. Mitroi, Connection between the Jensen and the Chebychev functionals, International Series of Numerical Mathematics, 1, Volume 161, Inequalities and Applications 2010, Part 5, Pages 217-227. C. Bandle, A. Gilanyi, L. Losonczi, M. Plum (Eds.). Dedicated to the Memory of Wolfgang Walter, Birkh¨ auser, Basel, 2012. DOI:10.1007/978-3-0348-0249-9 17. [10] C. P. Niculescu, An Extension of Chebyshev’s Inequality and its Connection with Jensen’s Inequality, J. of Inequal. & Appl., 6 (2001), pp. 451-462. [11] S. G. Walker, On a Lower Bound for the Jensen Inequality, SIAM J. Math. Anal., 46(5) 2014, 3151–3157. DOI:10.1137/140954015 LUMINA - University of South-East Europe, Faculty of Engineering Sciences, S ¸ os. Colentina 64b, Sector 2, RO-021187, Bucharest, Romania E-mail address: [email protected] Transilvania University of Bras¸ov, Iuliu Maniu Street, No. Romania E-mail address: [email protected]

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