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Some new estimates of the ‘Jensen gap’

Abstract

Let \(( \mu,\Omega ) \) be a probability measure space. We consider the so-called ‘Jensen gap’

$$ J ( \varphi,\mu,f ) = \int_{\Omega}\varphi \bigl( f ( s ) \bigr)\,d\mu ( s ) -\varphi \biggl( \int_{\Omega }f ( s )\,d\mu ( s ) \biggr) $$

for some classes of functions φ. Several new estimates and equalities are derived and compared with other results of this type. Especially the case when φ has a Taylor expansion is treated and the corresponding discrete results are pointed out.

1 Introduction

Let \(( \Omega,\mu ) \) be a probability measure space i.e. \(\mu ( \Omega ) =1\) and let f be a μ-measurable function on Ω. If φ is convex, then Jensen’s inequality

$$ \varphi \biggl( \int_{\Omega}f ( s )\,d\mu ( s ) \biggr) \leq \int_{\Omega}\varphi \bigl( f ( s ) \bigr)\,d\mu ( s ) $$
(1.1)

holds. This inequality can be traced back to Jensen’s original papers [1, 2] and is one of the most fundamental mathematical inequalities. One reason for that is that in fact a great number of classical inequalities can be derived from (1.1), see e.g. [3] and the references given therein. The inequality (1.1) cannot in general be improved since we have equality in (1.1) when \(\varphi ( x ) \equiv x\). However, for special cases of functions (1.1) can be given in a more specific form e.g. by giving lower estimates of the so-called ‘Jensen gap’

$$ J ( \varphi,\mu,f ) = \int_{\Omega}\varphi \bigl( f ( s ) \bigr)\,d\mu ( s ) -\varphi \biggl( \int_{\Omega }f ( s )\,d\mu ( s ) \biggr) , $$

thus obtaining refined versions of (1.1).

We give a few examples of such results.

Example 1

(see [4])

Let φ be a superquadratic function i.e. \(\varphi: [ 0,\infty ) \rightarrow \mathbb{R} \) is such that there exists a constant \(C ( x ) \), \(x\geq0\), such that

$$ \varphi ( y ) \geq\varphi ( x ) +C ( x ) ( y-x ) +\varphi \bigl( \vert y-x \vert \bigr) $$

for \(y\geq0\). For such functions we have the following estimate of the Jensen gap:

$$ J ( \varphi,\mu,f ) \geq \int_{\Omega}\varphi \biggl( \biggl\vert f ( s ) - \int_{\Omega}f ( s )\,d\mu ( s ) \biggr\vert \biggr)\,d\mu ( s ) . $$

Example 2

(see [5] and [6])

We say that a function \(K ( x ) \) in γ-superconvex if \(\varphi ( x ) :=x^{-\gamma }K ( x ) \) is convex. If φ is a differentiable convex, increasing function and \(\varphi ( 0 ) = \lim_{z\rightarrow0+} z\varphi^{\prime} ( z ) =0\), then we have the following estimate of the Jensen gap:

$$ J ( K,\mu,f ) \geq\varphi ( z ) \int_{\Omega} \bigl( \bigl( f ( s ) \bigr) ^{\gamma}-z^{\gamma} \bigr)\,d\mu ( s ) +\varphi^{\prime} ( z ) \int_{\Omega} \bigl( f ( s ) \bigr) ^{\gamma} \bigl( f ( s ) -z \bigr)\,d\mu ( s ) \geq0, $$

for \(z= \int_{\Omega}f ( s )\,d\mu ( s ) >0\) and \(f\geq 0\), \(f^{\gamma}\) when \(\gamma\geq0 \) are integrable functions on the probability measure space \(( \Omega,\mu ) \).

Remark 1

By using the results in Examples 1 and 2 it is possible to derive Hardy-type inequalities with other ‘breaking points’ (the point where the inequality reverses) than the usual breaking point \(p=1\). See [5, 7, 8] and [9].

Remark 2

In the recent paper [6] it was proved that the notion of γ-superconvexity has sense also for the case \(-1\leq\gamma \leq0\) and in fact this was used even to derive there some new two-sided Jensen type inequalities.

Example 3

(see [10])

In his paper Walker studied the Jensen gap for the special case \(f\equiv1\) i.e. for \(J ( \varphi,\mu) :=J ( \varphi,\mu,1 ) \) and found an estimate of the type

$$ J ( \varphi,\mu ) \geq\frac{1}{2}C ( \varphi,\mu ) \biggl( \int_{\Omega}s^{2}\,d\mu ( s ) - \biggl( \int_{\Omega }s\,d\mu ( s ) \biggr) ^{2} \biggr) , $$

where the positive constant \(C=C ( \varphi,\mu ) \) is easily computed.

In his paper it was assumed that φ admits a Taylor power series representation \(\varphi ( x ) =\sum_{n=1}^{\infty }a_{n}x^{n}\), \(a_{n}\geq0\), \(n=0,1,2,\ldots\) , \(0< x\leq A<\infty\). In another recent paper Dragomir [11] derived some other Jensen integral inequalities for this power series case. A comparison between these two results and our results is given in our concluding remarks.

Inspired by these results, we derive some new results of the same type. In Theorem 1 we get an estimate like that of Walker in [10] but for the general case of \(J ( \varphi,\mu,f ) \). In Theorem 2 we prove another complement of the Walker result by considering the Jensen functional

$$ J_{\alpha} \bigl( t^{\alpha},\mu \bigr) = \int_{\Omega}y^{\alpha}\,d\mu ( y ) - \biggl( \int_{\Omega}y\,d\mu ( y ) \biggr) ^{\alpha }, \quad\alpha \geq2, $$

and get an estimate for this Jensen gap which even reduces to equality for \(\alpha=N\), \(N=2,3,\ldots\) . By using this result it is possible to derive a similar equality for the Jensen gap whenever it can be represented by a Taylor power series (see Theorem 3).

In Section 3 we show that our lower bound of the Jensen gap is better than the lower bound in [11] when the function that we deal with has a Taylor series expansion with non-negative coefficients. Moreover, we prove that by our technique we can in such cases derive also upper bounds and not only lower bounds as in [10].

2 The main results

Our first main result reads as follows.

Theorem 1

Let \(\phi: [ 0,A ) \rightarrow \mathbb{R} \) have a Taylor power series representation on \([ 0,A )\), \(0< A\leq\infty:\phi ( x ) =\sum_{n=0}^{\infty }a_{n}x^{n}\).

Let φ be a convex increasing function on \([ 0,A ) \) that is related to ϕ by

$$ \varphi ( x ) =\frac{\phi ( x ) -\phi(0)}{x}=\sum_{n=0}^{\infty}a_{n+1}x^{n}. $$

(a) If \(f\geq0\) and f, \(f^{2}\), and \(\phi\circ f\) are integrable functions on Ω, \(z=\int_{\Omega}f\,d\mu>0\), where μ is a probability measure on Ω, then

$$ \int_{\Omega}\phi ( f )\,d\mu-\phi ( z ) \geq \biggl( \frac{\phi ( z ) -\phi(0)}{z} \biggr) ^{\prime} \biggl( \int_{\Omega}f^{2}\,d\mu-z^{2} \biggr) \geq0. $$

In other words,

$$\begin{aligned} J ( \phi,\mu,f ) =& \int_{\Omega}\phi ( f )\,d\mu -\phi ( z ) \\ =&\sum_{n=0}^{\infty}a_{n+1} \int_{\Omega}f^{n+1}\,d\mu-\sum _{n=0}^{\infty }a_{n+1}z^{n+1} \\ \geq&\sum_{n=0}^{\infty} ( n+1 ) a_{n+2}z^{n} \biggl( \int _{\Omega }f^{2}\,d\mu-z^{2} \biggr) \geq0. \end{aligned}$$

(b) For \(\overline{x}=\sum_{i=1}^{m}\alpha_{i}x_{i}\), \(\ \sum_{i=1}^{m}\alpha_{i}=1\), \(0\leq\alpha_{i}\leq1\), \(0\leq x_{i}< A\), \(i=1,\ldots,m\), it yields

$$ \sum_{i=1}^{m}\alpha_{i}\phi ( x_{i} ) -\phi ( \overline {x} ) \geq \biggl( \frac{\phi ( \overline{x} ) -\phi(0)}{\overline{x}} \biggr) ^{\prime} \Biggl( \sum _{i=1}^{m}\alpha _{i}x_{i}^{2}-\overline{x}^{2} \Biggr) \geq0. $$

In other words,

$$ \sum_{i=1}^{m}\sum _{n=0}^{\infty}\alpha _{i}a_{n+1}x_{i}^{n+1}- \sum_{n=0}^{\infty}a_{n+1} \overline{x}^{n+1}\geq \sum_{n=0}^{\infty} ( n+1 ) a_{n+2}\overline{x}^{n} \Biggl( \sum _{i=1}^{m}\alpha_{i}x_{i}^{2}- \overline{x}^{2} \Biggr) \geq0. $$

Proof

For \(\phi ( x ) =\sum_{n=0}^{\infty}a_{n}x^{n}\), \(0\leq x< A\), by denoting the function \(\psi: [ 0,A ) \rightarrow \mathbb{R} _{+}\) \(\psi ( x ) =\phi ( x ) -\phi ( 0 ) =\sum_{n=0}^{\infty}a_{n+1}x^{n+1}\), \(0\leq x< A\), and \(\varphi ( x ) =\frac{\psi ( x ) }{x}\Leftrightarrow x\varphi ( x ) =\psi ( x )\), \(0\leq x< A\), we see that \(\psi ( x ) \) is 1-quasiconvex function (see [6]), \(\varphi ( x ) =\sum_{n=0}^{\infty}a_{n+1}x^{n}\), \(0\leq x< A\), and \(\varphi ^{\prime} ( x ) =\sum_{n=0}^{\infty} ( n+1 ) a_{n+2}x^{n}\).

The functions ϕ, ψ, φ, and \(\varphi^{\prime}\) are differentiable functions on \([ 0,A ) \). From the convexity of \(\varphi ( x ) \) we have

$$ \varphi ( y ) -\varphi ( x ) >\varphi^{\prime } ( x ) ( y-x ) ,\quad x,y\in [ 0,A ), $$

and, therefore,

$$ \psi ( y ) -\psi ( x ) =y\varphi ( y ) -x\varphi ( x ) \geq\varphi ( x ) ( y-x ) +\varphi^{\prime} ( x ) y ( y-x ) ,\quad x,y\geq0. $$

Since \(\psi ( x ) =\phi ( x ) -\phi ( 0 ) \) we get

$$ \phi ( y ) -\phi ( x ) =\psi ( y ) -\psi ( x ) \geq\varphi ( x ) ( y-x ) + \varphi ^{\prime} ( x ) y ( y-x ) . $$

Now using this inequality with \(x=z\), \(y=f\), and integrating, we find that

$$\begin{aligned} &\int_{\Omega}\phi ( f )\,d\mu-\phi ( z )\\ &\quad\geq\varphi ( z ) \biggl( \int_{\Omega}f\,d\mu- \int_{\Omega }z\,d\mu \biggr) +\varphi^{\prime} ( z ) \biggl( \int_{\Omega }f^{2}\,d\mu-z^{2} \biggr)\\ &\quad=0+ \biggl( \frac{\phi ( z ) -\phi ( 0 ) }{z} \biggr) ^{\prime} \biggl( \int_{\Omega}f^{2}\,d\mu-z^{2} \biggr) \geq0. \end{aligned}$$

In the last inequality we have used \(z=\int_{\Omega}f\,d\mu>0\) and φbeing convex increasing, where \(\varphi ( z ) =\frac{\phi ( z ) -\phi ( 0 ) }{z}\).

Hence (a) is proved and since (b) is just a special case of (a), the proof is complete. □

For the proof of our next main result we need the following lemma, which is also of independent interest.

Lemma 1

Let φ be a differentiable function on \(I\subset \mathbb{R} \), and let \(x,y\subseteq I\). Then, for \(N=2,3,\ldots\) ,

$$\begin{aligned} &\varphi ( x ) \bigl( y^{N-1}-x^{N-1} \bigr) +\varphi ^{\prime } ( x ) y^{N-1} ( y-x ) \\ &\quad= \bigl( x^{N-1}\varphi ( x ) \bigr) ^{\prime} ( y-x ) + ( y-x ) ^{2}\sum_{k=1}^{N-1}y^{k-1} \bigl( x^{N-k-1}\varphi ( x ) \bigr) ^{\prime}. \end{aligned}$$
(2.1)

In particular, for \(N=2\) we have

$$ \varphi ( x ) ( y-x ) +\varphi^{\prime} ( x ) y ( y-x ) = \bigl( x\varphi ( x ) \bigr) ^{\prime } ( y-x ) +\varphi^{\prime} ( x ) ( y-x ) ^{2}. $$
(2.2)

Proof

A simple calculation shows that (2.2) holds, i.e., that (2.1) holds for \(N=2\). For \(N=3\) (2.1) reads

$$\begin{aligned} &\varphi ( x ) \bigl( y^{2}-x^{2} \bigr) + \varphi^{\prime } ( x ) y^{2} ( y-x ) = \bigl( x^{2}\varphi ( x ) \bigr) ^{\prime} ( y-x ) + ( y-x ) ^{2} \bigl( \bigl( x\varphi ( x ) \bigr) ^{\prime}+y \varphi^{\prime} ( x ) \bigr) . \end{aligned}$$
(2.3)

Moreover, it is easy to verify the identity

$$\begin{aligned} &\varphi ( x ) \bigl( y^{2}-x^{2} \bigr) + \varphi^{\prime } ( x ) y^{2} ( y-x ) =\varphi^{\prime} ( x ) y ( y-x ) ^{2}+x\varphi ( x ) ( y-x ) + \bigl( x\varphi ( x ) \bigr) ^{\prime }y ( y-x ). \end{aligned}$$
(2.4)

By using (2.4) together with (2.2) and making some straightforward calculations we obtain (2.3). The general proof follows in the same way using induction and the more general (than (2.4)) identity

$$\begin{aligned}[b] &\varphi ( x ) \bigl( y^{N-1}-x^{N-1} \bigr) +\varphi ^{\prime } ( x ) y^{N-1} ( y-x )\\ &\qquad{}- \bigl[ \bigl( x\varphi ( x ) \bigr) \bigl( y^{N-2}-x^{N-2} \bigr) + \bigl( x\varphi ( x ) \bigr) ^{\prime }y^{N-2} ( y-x ) \bigr]\\ &\quad=\varphi^{\prime} ( x ) y^{N-2} ( y-x ) ^{2},\quad N=2,3,4, \ldots. \end{aligned} $$

 □

Now we are ready to state our next main result.

Theorem 2

Let μ be a probability measure on \(\Omega= (0,\infty)\), \(z=\int_{\Omega}y\,d\mu ( y ) >0\), and \(N=2,3,\ldots\) . Then the refined Jensen-type inequality

$$ \int_{\Omega}y^{\alpha}\,d\mu ( y ) -z^{\alpha}\geq \int _{\Omega } ( y-z ) ^{2}\sum _{k=1}^{N-1} ( \alpha-k ) x^{k-1}z^{\alpha-k-1}\,d\mu,\quad y\geq0, $$
(2.5)

holds for any \(\alpha\geq N\). Moreover, for \(N-1<\alpha\leq N\) (2.5) holds in the reversed direction. In particular, for \(\alpha=N\) we have equality in (2.5).

Proof

A convex differentiable function on \(\varphi ( x ) \) is characterized by

$$ \varphi ( y ) -\varphi ( x ) \geq\varphi^{\prime } ( x ) ( y-x ) $$

and this inequality holds in the reversed direction if \(\varphi ( x ) \) is concave. For \(\varphi ( x ) =x\) we have equality. Therefore, when \(\varphi ( x ) \) is convex it yields

$$ \varphi ( y ) y^{N-1}-\varphi ( x ) x^{N-1}\geq \varphi ( x ) \bigl( y^{N-1}-x^{N-1} \bigr) +\varphi^{\prime} ( x ) y^{N-1} ( y-x ) ,\quad x,y\geq0. $$

Hence in view of Lemma 1 we find that

$$ \varphi ( y ) y^{N-1}-\varphi ( x ) x^{N-1}\geq \bigl( x^{N-1}\varphi ( x ) \bigr) ^{\prime} ( y-x ) + ( y-x ) ^{2}\sum_{k=1}^{N-1}y^{k-1} \bigl( x^{N-k-1}\varphi ( x ) \bigr) ^{\prime}. $$

By using this inequality with the convex function \(\varphi ( x ) =x^{\alpha-N+1}\), \(x\geq0\), \(\alpha\geq N\), we obtain

$$ y^{\alpha}-x^{\alpha}\geq\alpha x^{\alpha-1} ( y-x ) + ( y-x ) ^{2}\sum_{k=1}^{N-1} ( \alpha-k ) y^{k-1}x^{\alpha -k-1}. $$

By now choosing \(x=z\), integrating over Ω, and using the fact that \(\int_{\Omega} ( y-z )\,d\mu ( y ) =0\) we obtain (2.5). For the reversed inequality we use the concave function \(\varphi ( x ) =x^{\alpha-N+1}\), \(( N-1 ) <\alpha\leq N\), and all inequalities above reverse. For \(\alpha=N\) we get an equality, so the proof is complete. □

Corollary 1

Let \(x_{i}\geq0\), \(\alpha_{i}\geq0\), \(i=1,2,\ldots,m\), \(\sum_{i=1}^{m}\alpha_{i}=1\), and \(\overline{x}=\sum_{i=1}^{m}\alpha _{i}x_{i}\). Then, for \(N=2,3,\ldots\) ,

$$ \sum_{i=1}^{m}\alpha_{i}x_{i}^{\alpha}- \overline{x}^{\alpha}\geq \sum_{i=1}^{m} \alpha_{i} ( x_{i}-\overline{x} ) ^{2}\sum _{k=1}^{N-1} ( \alpha-k ) x_{i}^{k-1} \overline {x}^{\alpha -k-1} $$
(2.6)

holds for any \(\alpha\geq N\). Moreover, for \(N-1<\alpha\leq1\) (2.6) holds in the reversed direction. In particular, for \(\alpha =N\), (2.6) reduces to an equality.

Our final main result reads as follows.

Theorem 3

Let \(0< A\leq\infty\) and let \(\phi: ( 0,A ] \rightarrow \mathbb{R} \) have a Taylor expansion \(\phi ( x ) =\sum_{n=0}^{\infty }a_{n}x^{n}\), on \(( 0,A ] \). If μ is a probability measure on \(( 0,A ] \) and \(z=\int_{0}^{A}x\,d\mu ( x ) >0\), then

$$ \int_{\Omega}\phi ( x )\,d\mu-\phi ( z ) =\sum _{n=2}^{\infty}a_{n} \int_{0}^{A} ( x-z ) ^{2}\sum _{k=1}^{n-1} ( n-k ) x^{k-1}z^{n-k-1}\,d\mu. $$
(2.7)

Proof

We note that

$$ \int_{0}^{A}\phi ( x )\,d\mu-\phi ( z ) = \int_{0}^{A}\sum_{n=0}^{\infty}a_{n} \bigl( x^{n}-z^{n} \bigr)\,d\mu =\sum _{n=0}^{\infty}a_{n} \int_{0}^{A} \bigl( x^{n}-z^{n} \bigr)\,d\mu. $$

Obviously, \(\int_{0}^{A} ( x^{n}-z^{n} )\,d\mu=0\), for \(n=0,1\), and hence (2.7) follows from the equality cases in (2.5) in Theorem 2, i.e. when \(\alpha=N=2,3,\ldots\) .

The proof is complete. □

Corollary 2

Let \(0< A\leq\infty\) and let \(\phi: [ 0,A ) \) have a Taylor expansion \(\phi ( x ) =\sum_{n=0}^{\infty}a_{n}x^{n}\), on \([ 0,A ) \). If \(\overline{x}=\sum_{i=1}^{m}\alpha_{i}x_{i}\), \(\sum_{i=1}^{m}\alpha_{i}=1\), \(0\leq \alpha_{i}\leq1\), \(0\leq x_{i}\leq A\), \(i=1,2,\ldots,m\), then

$$ J=\sum_{i=1}^{m}\alpha_{i}\phi ( x_{i} ) -\phi ( \overline{x} ) =\sum _{n=2}^{\infty}a_{n} \Biggl( \sum _{i=1}^{m}\alpha _{i}x_{i}^{2}-\overline{x}^{2} \Biggr) \sum_{k=1}^{n-1} ( n-k ) x^{k-1}\overline{x}^{n-k-1}. $$

Corollary 3

Let \(0< a< b<\infty\), and μ be a probability measure on \(( a,b ) \). Then we have the following estimate of the Jensen gap \(J_{N}:=\int_{a}^{b}x^{N}\,d\mu- ( \int_{a}^{b}x\,d\mu ) ^{N}\), \(N=2,3,\ldots\) :

$$ \frac{N ( N-1 ) }{2}a^{N-2}J_{2}\leq J_{N}\leq \frac{N ( N-1 ) }{2}b^{N-2}J_{2}. $$
(2.8)

Proof

We use Theorem 2 with \(\alpha=N\) and find that

$$ J_{N}= \int_{a}^{b} ( x-z ) ^{2}\sum _{k=1}^{N-1} ( N-k ) x^{k-1}z^{N-k-1}\,d\mu. $$

We note that if \(a< x< b\), then \(a< z< b \) so that \(a^{N-2}\leq x^{k-1}z^{N-k-1}\leq b^{N-2}\). Moreover, \(\sum_{k=1}^{N-1} ( N-k ) =\frac{N ( N-1 ) }{2}\) and

$$ \int_{a}^{b} ( x-z ) ^{2}\,d\mu= \int_{a}^{b}x^{2}\,d\mu- \biggl( \int_{a}^{b}x\,d\mu \biggr) ^{2}=J_{2}, $$

so (2.8) is proved. □

Remark 3

For the case \(N=2\) both inequalities in (2.8) reduce to equalities. Moreover, for the discrete case we have: If \(0< a< x_{i}< b\), \(\alpha_{i}\geq0\), \(i=1,2,\ldots,m\), \(\sum_{i=1}^{m}\alpha_{i}=1\), \(\overline{x}=\sum_{i=1}^{m}\alpha_{i}x_{i}\), then, for \(N=2,3,\ldots\) ,

$$\begin{aligned} &\frac{N ( N-1 ) }{2}a^{N-2} \Biggl( \sum_{i=1}^{m}a_{i}x_{i}^{2}-\overline{x}^{2} \Biggr) \\ &\quad\leq\sum_{i=1}^{m}a_{i}x_{i}^{N}- \overline{x}^{N}\leq\frac{N ( N-1 ) }{2}b^{N-2} \Biggl( \sum _{i=1}^{m}a_{i}x_{i}^{2}- \overline{x}^{2} \Biggr) . \end{aligned}$$
(2.9)

3 Final remarks and examples

In this section we present some recent interesting results of Dragomir [11] and Walker [10]. Moreover, we point out the corresponding special cases of our results and compare these results with those of [11] and [10].

Example 4

In Dragomir’s paper [11], Theorem 2, it was proved that for

$$ \phi ( x ) =\sum_{n=0}^{\infty}a_{n}x^{n},\quad a_{n}\geq0, $$
(3.1)

which converges on \(0< x< R\leq\infty\), the following lower bound of the Jensen gap holds:

$$\begin{aligned} &\int_{\Omega}\phi\circ f\,d\mu-\phi \biggl( \int_{\Omega}f\,d\mu \biggr) \\ &\quad\geq\frac{1}{2} \biggl[ \int_{\Omega}f^{2}\,d\mu- \biggl( \int_{\Omega }f\,d\mu \biggr) ^{2} \biggr] \frac{\phi^{\prime} ( \int_{\Omega}f\,d\mu ) -\phi^{\prime} ( 0 ) }{\int_{\Omega}f\,d\mu}, \end{aligned}$$
(3.2)

when \(( \Omega,\mu ) \) is a probability measure space, \(f\geq 0\), and f, \(f^{2}\), and \(\phi\circ f\) are integrable on Ω and \(\int_{\Omega}f\,d\mu>0\).

Example 5

In Theorem 1 we proved that for convex increasing functions we get the inequalities

$$\begin{aligned} &\int_{\Omega}\phi\circ f\,d\mu-\phi \biggl( \int_{\Omega}f\,d\mu \biggr) \\ &\quad\geq \biggl[ \int_{\Omega}f^{2}\,d\mu- \biggl( \int_{\Omega}f\,d\mu \biggr) ^{2} \biggr] \biggl( \frac{\phi ( \int_{\Omega}f\,d\mu ) -\phi ( 0 ) }{\int_{\Omega}f\,d\mu} \biggr) ^{\prime}\geq0. \end{aligned}$$
(3.3)

A function that satisfies (3.1) is convex increasing and therefore Theorem 1 holds, which means that we get the inequalities in (3.3).

Remark 4

It is easily computed that when ϕ is of the form (3.1), then

$$ \frac{1}{2}\frac{\phi^{\prime} ( \int_{\Omega}f\,d\mu ) -\phi ^{\prime} ( 0 ) }{\int_{\Omega}f\,d\mu}\leq \biggl( \frac{\phi ( \int_{\Omega}f\,d\mu ) -\phi ( 0 ) }{\int_{\Omega }f\,d\mu} \biggr) ^{\prime} $$
(3.4)

holds, and from this we conclude that our bound in (3.3), when (3.1) is satisfied, is stronger than Dragomir’s (3.2). Indeed,

$$ \frac{1}{2}\frac{\phi^{\prime} ( z ) -\phi^{\prime} ( 0 ) }{z}=\sum_{n=0}^{\infty} \frac{1}{2} ( n+2 ) a_{n+2}z^{n} $$

and

$$ \biggl( \frac{\phi ( \int_{\Omega}f\,d\mu ) -\phi ( 0 ) }{\int_{\Omega}f\,d\mu} \biggr) ^{\prime}=\sum _{n=0}^{\infty} ( n+1 ) a_{n+2}z^{n}, $$

and our claim is obvious.

Example 6

In Theorem 3.1 in Walker’s paper [10], a lower bound for the Jensen gap is given for a function ϕ that satisfies (3.1):

$$ \int_{\Omega}\phi ( s )\,d\mu ( s ) -\phi \biggl( \int_{\Omega}\,d\mu ( s ) \biggr) \geq\mu ( 1,R ) \tau \frac{1}{2}\sum_{n=2}^{\infty}a_{n}n ( n-1 ) $$

where

$$ \tau= \int_{\Omega}s^{2}\,d\mu_{2} ( s ) - \biggl( \int_{\Omega }s\,d\mu_{2} ( s ) \biggr) ^{2} $$

when μ is a probability measure defined on \(\Omega= ( 0,R ) \) and \(\mu_{2}\) is μ restricted and normalized to \(( 1,R ) \).

More generally, in Section 4 in [10], \(\mu ( 1,R ) \) was replaced by \(\mu ( a,R ) \) and we have

$$ \int_{\Omega}\phi ( s )\,d\mu ( s ) -\phi \biggl( \int_{\Omega}\,d\mu ( s ) \biggr) \geq\mu ( a,R ) \tau \frac{1}{2}\sum_{n=2}^{\infty}a^{n}a_{n}n ( n-1 ) , $$
(3.5)

where

$$ \tau= \int_{\Omega}s^{2}\,d\mu_{a} ( s ) - \biggl( \int_{\Omega }s\,d\mu_{a} ( s ) \biggr) ^{2}, $$

when \(\mu_{a}\) is μ restricted and normalized to \(\Omega= ( a,R ) \).

From Corollary 3 and Remark 3 we easily get the following.

Example 7

Let \(0< A\leq\infty\) and let \(\phi: ( 0,A ] \rightarrow \mathbb{R} \) have Taylor expansion \(\phi ( x ) =\sum_{n=0}^{\infty }a_{n}x^{n}\), \(a_{n}\geq0\), \(n=2,3,\ldots\) , on \(( 0,A ] \). If μ is a probability measure on \(( 0,A ] \), \(0\leq a< b\leq A \), and \(z=\int_{0}^{A}x\,d\mu ( x ) >0\), then

$$ \sum_{n=2}^{\infty}a_{n} \frac{n ( n-1 ) }{2}a^{n-2}J_{2}\leq J ( \phi,\mu ) \leq\sum _{n=2}^{\infty}a_{n} \frac{n ( n-1 ) }{2}b^{n-2}J_{2}. $$
(3.6)

Moreover, for the discrete case we have: If \(0< a< x_{i}< b\), \(\alpha _{i}\geq 0\), \(i=1,2,\ldots,m\), \(\sum_{i=1}^{m}a_{i}=1\), \(\overline{x}=\sum_{i=1}^{m}\alpha_{i}x_{i}\), then, for \(n=2,3,\ldots\) ,

$$\begin{aligned} &\sum_{n=2}^{\infty}a_{n} \frac{n ( n-1 ) }{2}a^{n-2} \Biggl( \sum_{i=1}^{m} \alpha_{i}x_{i}^{2}-\overline{x}^{2} \Biggr) \\ &\quad\leq\sum_{i=1}^{m}\alpha_{i} \bigl( \phi ( x_{i} ) -\phi ( \overline{x} ) \bigr) \leq\sum _{n=2}^{\infty}a_{n}\frac{n ( n-1 ) }{2}b^{n-2} \Biggl( \sum_{i=1}^{m}\alpha _{i}x_{i}^{2}-\overline{x}^{2} \Biggr) . \end{aligned}$$

Remark 5

The lower bound in (3.5) coincides with that in (3.6) when \(a=1\). The lower bound in (3.6) is better than that in (3.5) when \(a<1\), but Walker’s bound (3.5) is better than (3.6) for \(a>1\). It seems not to be possible to derive an upper bound like that in (3.5) by using the method in [10].

References

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Abramovich, S., Persson, LE. Some new estimates of the ‘Jensen gap’. J Inequal Appl 2016, 39 (2016). https://doi.org/10.1186/s13660-016-0985-4

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