Abstract
In this paper, some integral inequalities for uniformly convex functions are studied by using unordered submajorization for cumulative functions. Strongly convex functions and superquadratic functions are considered, too. A Levin–Stečkin like theorem is obtained for such functions. As applications, some bounds for the Fejér functional are derived. A result on the Schur-convexity of averages of convex functions is extended to uniformly convex functions. Some specifications for symmetric functions are also given. A corollary for symmetric probability density functions is established. A Levin–Stečkin type inequality for generalized \( \psi \)-uniformly convex functions is provided. Some interpretations for Simpson distributions are presented.
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1 Introduction and summary
Throughout \( I \subset \mathbb {R}\) is an interval. For \( {{\mathbf {z}}} = ( z_1 , z_2 , \ldots , z_n ) \in \mathbb {R}^{n}\) and \( i = 1,2,\ldots ,n \), the symbol \( z_{[i]} \) stands for the ith largest entry of \( {\mathbf z} \).
An n-tuple \( {{\mathbf {y}} } = ( y_1 , y_2 , \ldots , y_n ) \in I^n \) is said to be weakly majorized by an n-tuple \( {\mathbf x } = ( x_1 , x_2 , \ldots , x_n ) \in I^n \), written as \( {\mathbf y } \prec _w {{\mathbf {x}} } \), if
(see [9, p. 12]). If, in addition,
then \( {{\mathbf {y}} } \) is said to be majorized by \( {{\mathbf {x}} } \), written as \( {{\mathbf {y}} } \prec {{\mathbf {x}} } \) (see [9, p. 8]).
A function \( F : I^n \rightarrow \mathbb {R}\) is said to be Schur-convex (resp. Schur-concave) on \( I^n \) if
provided \( {{\mathbf {x}} } , {{\mathbf {y}} } \in I^n \) (see [9, p. 80]).
Let \( g_1 , g_2 : [ a , b ] \rightarrow \mathbb {R}\) be two integrable real functions. The function \( g_2 \) is said to be unordered submajorized by \( g_1 \), written as \( g_2 \prec _w^u g_1 \), if
If, moreover,
then \( g_2 \) is said to be (unordered) majorized by \( g_1 \), written as \( g_2 \prec ^u g_1 \) (see [4], cf. [9, p. 22]).
By a cumulative function induced by an integrable function \( g : [a,b] \rightarrow \mathbb {R}\), we mean the integral function
In what follows, we assume that there exist all integrals under consideration.
Elezović and Pečarić in [5] established the following result.
Theorem A
[5] Let f be a continuous function on an interval I. Then the function
is Schur-convex (Schur-concave) on \( I^2 \) iff f is convex (concave) on I.
It is well-known that if \( f : I \rightarrow \mathbb {R}\) is a convex function on an interval \( I \subset \mathbb {R}\), \( a , b \in I \) with \( a < b \), then the following Hermite–Hadamard inequality holds:
(see [3, p. 137]).
A more general result is incorporated in the following [1, 7, 11].
Theorem B
[1] Let \( f : I \rightarrow \mathbb {R}\) be a convex function on an interval \( I \subset \mathbb {R}\), \( a , b \in I \) with \( a < b \), and let \( p : [a,b] \rightarrow \mathbb {R}\) be a non-negative integrable weight on I. Assume that p is symmetric about \( \frac{a+b}{2} \). Then the following Fejér inequality holds:
Throughout, we denote by \( G_1 \) and \( G_2 \) the cumulative functions of \( g_1 \) and \( g_2 \) on [a, b] , respectively, in the sense that
Likewise, we denote by \( {{\mathcal {G}}}_1 \) and \( {{\mathcal {G}}}_2 \) the cumulative functions of \( G_1 \) and \( G_2 \) on [a, b] , respectively, that is
We now present the Levin–Stečkin theorem [8].
Theorem C
[8] Let \( g_1 , g_2 : [ a , b ] \rightarrow \mathbb {R}\) be integrable functions and \( G_1 , G_2 , {{\mathcal {G}}}_1 , {{\mathcal {G}}}_2 : [ a , b ] \rightarrow \mathbb {R}\) defined by (5), (6) be functions satisfying the condition
If
then
for all continuously twice differentiable convex functions \( f : [ a , b ] \rightarrow \mathbb {R}\).
In this paper, we study integral inequalities of type (8) for uniformly convex functions, strongly convex functions and superquadratic functions. Our purpose is to establish some further results related to Theorems A, B and C. Similar problems for real convex functions f are well-known (see [8, 12,13,14, 16,17,18,19]).
The paper is arranged as follows. In Sect. 2, first we point out that for a given generalized uniformly convex function \( f : [a,b] \rightarrow \mathbb {R}\), the unordered submajorization of cumulative functions \( G_1 \) and \( G_2 \) induced by \( g_1 \) and \( g_2 \), respectively, implies a refinement of inequality (8) (see Theorem 1).
Next, we provide some sufficient conditions under which the cumulative functions are unordered submajorized (see Lemma 2). In consequence, we are able to demonstrate sufficient conditions on two given functions \( g_1 \) and \( g_2 \) so that the refinement of inequality (8) holds (see Theorem 2). As an application, for uniformly convex functions we refine a result due to Elezović and Pečarić [5] (see Theorem A). This corresponds to the case of Theorem 1 when \( g_1 \) and \( g_2 \) represent two pdf’s of uniform distribution.
In Sect. 3 we focus on symmetric functions. This leads to some simplifications of the results of Sect. 2. After giving some properties of cumulative functions (see Lemma 3), we interpret the previous results for symmetric functions (see Theorem 3). We establish a Levin–Stečkin type inequality with uniformly convex f. We also specify the obtained results for symmetric probability density functions (see Corollary 3). Finally, we show applications for Simpson distributions.
2 Results
Let \( I = [a,b] \) be an interval and \( \psi : [0,b-a] \rightarrow \mathbb {R}\) be a function. A function \( f : [a,b] \rightarrow \mathbb {R}\) is said to be generalized\( \psi \)-uniformly convex if
(cf. [2]). If in addition \( \psi \ge 0 \), then f is said to be \(\psi \)-uniformly convex (see [15, 20]).
Observe that the case \( \psi = 0 \) corresponds to usual convex functions. Moreover, a \( \psi \)-uniformly convex function f (so, \( \psi \ge 0 \)) is necessarily convex. Conversely, if \( \psi \le 0 \), then a (usual) convex function f is generalized \(\psi \)-uniformly convex.
In general, if \( \psi _1 \le \psi _2 \), then generalized \(\psi _2\)-uniform convexity implies generalized \(\psi _1\)-uniform convexity.
We are now in a position to prove a Levin–Stečkin type theorem for generalized \( \psi \)-uniformly convex functions. Some simplifications of conditions (10) and (11) will be discussed after the end of the proof of Theorem 1. A similar approach for convex or n-convex functions can be found in [17,18,19].
Theorem 1
Let \( I = [a,b] \) be an interval and \( \psi : [0,b-a] \rightarrow \mathbb {R}\) be a function. Let \( f : [ a , b ] \rightarrow \mathbb {R}\) be a continuously twice differentiable generalized \( \psi \)-uniformly convex function on [a, b] . Denote \( \varphi (t) = \frac{\psi (t)}{t^2} \) for \( t \in (0,b-a] \) and \( \varphi (0) = \lim \limits _{t \rightarrow 0^+} \varphi (t) \).
Let \( g_1 , g_2 : [ a , b ] \rightarrow \mathbb {R}\) be integrable functions and \( G_1 , G_2 , {{\mathcal {G}}}_1, {{\mathcal {G}}}_2 : [ a , b ] \rightarrow \mathbb {R}\) defined by (5), (6) be functions satisfying the condition
If
then
where \( R = 2 \varphi (0) \int \limits _a^b ( {{\mathcal {G}}}_1 (t) - {{\mathcal {G}}}_2 (t) ) \, d t \). In particular, \( R \ge 0 \) whenever f is a \( \psi \)-uniformly convex function on [a, b] .
Proof
Inequality (11) means that
By using (6) and (13) we obtain
By integrating by parts twice [6, p. 129], we have (see (5) and (6))
It is easily seen from (5), (6) that
In consequence, by (16) and (10),
It follows from (9) that
In fact, for \( x,y \in I \) and \( t \in [0,1] \), (9) gives
and further for \( t \in (0,1] \),
Hence for \( x,y \in I \), \( x \ne y \),
Therefore,
For \( x = y \) inequality (22) also holds, because \( \psi (0) \le 0 \) is satisfied by (9).
By replacing the roles of x and y in (22), we get
By multiplying both sides by \( -1 \), we obtain
Now, subtracting inequalities (24) and(22) by sides yields (18), as claimed.
It holds that
To see this, observe that (18) implies
because \( \psi (x-y) = (x-y)^2 \varphi (x-y) \).
Consequently,
which gives (25).
In conclusion, we get
Therefore, by (15), (17) and (28), we deduce that
In addition, \( R \ge 0 \) provided that f is \(\psi \)-uniformly convex, because \( {{\mathcal {G}}}_1 (t) - {{\mathcal {G}}}_2 (t) \ge 0 \) for \( t \in [a,b] \) by (14), and \( \psi \ge 0 \) implies \( \varphi \ge 0 \).
This completes the proof of (12). \(\square \)
Let \( m \ge 0 \) be a nonnegative number. A function \( f : I =[a,b] \rightarrow \mathbb {R}\) is said to be m-strongly convex if it is \( \psi \)-uniformly convex for \( \psi (t) = \frac{m}{2} t^2 \), i.e.,
Note that m-strongly convex functions with \( m = 0 \) are simply convex.
Corollary 1
Under the hypothesis of Theorem 1, let \( f : [ a , b ] \rightarrow \mathbb {R}\) be a continuously twice differentiable m-uniformly convex function on [a, b] with \( m \ge 0 \). If conditions (10), (11) are fulfilled, then inequality (12) holds with \( R = m \int \limits _a^b ( {{\mathcal {G}}}_1 (t) - {{\mathcal {G}}}_2 (t) ) \, d t \).
Proof
It is enough to use Theorem 1 with \( \psi (t) = \frac{m}{2} t^2 \) and \( \varphi (t) = \frac{m}{2} \) for \( t \in [0,b-a] \). \(\square \)
Let \( f : [ 0 , b ] \rightarrow \mathbb {R}\) be a differentiable function. The function f is said to be superquadratic on [0, b] if
Corollary 2
Under the hypothesis of Theorem 1, let \( f : [ 0 , b ] \rightarrow \mathbb {R}\) be a continuously twice differentiable superquadratic function on [0, b] . If conditions (10), (11) are fulfilled, then inequality (12) holds with \( R = 2 \varphi (0) \int \limits _0^b ( {{\mathcal {G}}}_1 (t) - {{\mathcal {G}}}_2 (t) ) \, d t \), and \( \varphi (t) = \frac{f (t)}{t^2} \) for \( t \in (0,b] \) and \( \varphi (0) = \lim \limits _{t \rightarrow 0^+} \varphi (t) \).
Proof
Proceeding as in the proof of Theorem 1 with \( a = 0 \), \( \psi (t) = f (t) \) for \( t \in [0,b] \), and \( \varphi (t) = \frac{f (t)}{t^2} \) for \( t \in (0,b] \), we can see that the superquadracity of f on [0, b] leads to the validity of inequality (12).
Indeed, property (30) guarantees that inequalities (22) and (24) are met with \( \psi = f \), which implies (18) and (25) with \( \psi = f \) and \( \varphi (t) = \frac{f (t)}{t^2} \) for \( t \in (0,b] \) and \( \varphi (0) = \lim \limits _{t \rightarrow 0^+} \varphi (t) \). Hence (28) is satisfied.
Finally, by compiling (15), (17) and (28) we get
This completes the proof of (12) for a superquadratic function f. \(\square \)
We now discuss sufficient conditions for majorization inequalities (11) and (13) to be valid.
The following lemma is based on a discrete result due to Marshall et al. (see [9, Proposition B.1., p. 186]). It is also inspired by Ohlin’s Lemma [13], see also [14, Lemma 1].
Lemma 1
Let \( g_1 , g_2 : [ a , b ] \rightarrow \mathbb {R}\) be integrable functions such that
and, in addition, there exists \( c \in [a,b] \) satisfying
Then
for \( s \in [a,b] \).
Proof
It follows from the first inequality in (32) that (33) holds for \( s \in [a,c) \).
Assume that \( s \in [c,b] \). Due to (31) we can see that
the last inequality being a consequence of the second inequality in (32).
Summarizing all of this, inequality (33) holds true for all \( s \in [a,b] \). \(\square \)
In the next lemma we utilize interlaced functions \( g_1 \) and \( g_2 \) (see (35), (36)). In consequence we obtain the required inequalities (11) and (13) for the corresponding cumulative functions \( G_1 \) and \( G_2 \) (see (38)).
Lemma 2
Let \( g_1 , g_2 : [ a , b ] \rightarrow \mathbb {R}\) be integrable functions and \( G_1 , G_2 : [ a , b ] \rightarrow \mathbb {R}\) be functions defined by (5). Assume that there exists \( c \in [a,b] \) satisfying
and, in addition, there exist \( d_1 \in [a,c) \) and \( d_2 \in [c,b] \) satisfying (a.e.)
If
then
Proof
We consider the restrictions of \( g_1 \) and \( g_2 \) to the interval [a, c] . In light of Lemma 1 applied to the interval [a, c] , by using (35) and the first part of (34), we find that
with equality for \( t = c \) (see (34)).
Likewise, consider the restrictions of \( g_1 \) and \( g_2 \) to the interval [c, b] . Denote
Hence
By making use of Lemma 1, applied to the interval [c, b] via (36) and the second part of (34), we derive
with equality for \( t = b \) (see (34)).
By combining (40) and (41), with \( G_1 (c) = G_2 (c) \) (see (34)), we obtain
According to Lemma 1 applied to the functions \( G_1 \) and \( G_2 \) on the interval [a, b] , properties (39), (42) and (37) imply (38), as desired. \(\square \)
Remark 1
The conditions (35), (36) say that the pair \( (g_2 , g_1 ) \) crosses two times (see [14, Definition 1]).
Remark 2
In Lemma 2, conditions (34), (35), (36) and (37) ensure that
Theorem 2
Let \( I = [a,b] \) be an interval and \( \psi : [0,b-a] \rightarrow \mathbb {R}\) be a function. Let \( f : [ a , b ] \rightarrow \mathbb {R}\) be a continuously twice differentiable generalized \( \psi \)-uniformly convex function on [a, b] . Denote \( \varphi (t) = \frac{\psi (t)}{t^2} \) for \( t \in (0,b-a] \) and \( \varphi (0) = \lim \limits _{t \rightarrow 0^+} \varphi (t) \).
Let \( g_1 , g_2 : [ a , b ] \rightarrow \mathbb {R}\) be integrable functions and \( G_1 , G_2 , {{\mathcal {G}}}_1 , {{\mathcal {G}}}_2 : [ a , b ] \rightarrow \mathbb {R}\) be functions defined by (5), (6). Assume that there exist \( c \in [a,b] \), \( d_1 \in [a,c) \) and \( d_2 \in [c,b] \) satisfying conditions (34), (35), (36) and (37).
If
then
where \( R = 2 \varphi (0) \int \limits _a^b ( {{\mathcal {G}}}_1 (t) - {{\mathcal {G}}}_2 (t) ) \, d t \). In particular, \( R \ge 0 \) whenever f is a \( \psi \)-uniformly convex function on [a, b] .
Proof
In light of (34) one has \( G_1 (b) = G_2 (b) \), so \( f (b) [ G_1 (b) - G_2 (b) ] = 0 \). Therefore (10) reduces to (43).
Simultaneously, conditions (34), (35), (36) and (37) of Lemma 2 ensure that (38) is satisfied. Therefore (11) is fulfilled. Now, it is sufficient to apply Theorem 1 to get (44). \(\square \)
2.1 Uniform distributions
In order to illustrate the above results, we now show how to use Theorem 2 to extend the sufficiency part of Theorem A [5] to uniformly convex functions.
Let \( I = [a,b] \) be an interval, \( \psi : [0,b-a] \rightarrow \mathbb {R}\) be a function, \( \varphi (t) = \frac{\psi (t)}{t^2} \) for \( t \in (0,b-a] \) and \( \varphi (0) = \lim \limits _{t \rightarrow 0^+} \varphi (t) \). Take \( f : [a,b] \rightarrow \mathbb {R}\) to be a continuously twice differentiable generalized \( \psi \)-uniformly convex function on [a, b] .
Assume that \( x_1 , x_2 , y_1 , y_2 \in [a,b] \) such that \( (x_2,y_2) \prec (x_1,y_1) \) and \( a \le x_1 \le x_2< \frac{a+b}{2} < y_2 \le y_1 \le b \), with \( c = \frac{a+b}{2} = \frac{x_1 + y_1}{2} = \frac{x_2 + y_2}{2} \). Set
By putting \( d_1 = x_2 \) and \( d_2 = y_2 \), we see that conditions (35), (36) are satisfied. Furthermore, (34) holds in the form
In this way, we have \( G_1 (b) = G_2 (b) \). We also find by a straightforward calculation that
So, we infer that (37) is valid.
Since \( {{\mathcal {G}}}_1 (b) = {{\mathcal {G}}}_2 (b) \), condition (43) is satisfied trivially. Taking Theorem 2 into consideration, we obtain (44) with the above \( g_1 \) and \( g_2 \), as follows:
where \( R = 2 \varphi (0) \int \limits _a^b ( {{\mathcal {G}}}_1 (t) - {{\mathcal {G}}}_2 (t) ) \, d t \) (see (46)).
By direct computations, we find that
Hence we derive
Therefore we have
Because \( x_1 + y_1 = x_2 + y_2 \), a bit of algebra gives
So, we deduce from (45) that
In particular, for an m-strongly convex function f we obtain the inequality
Also, for a superquadratic function f inequality (46) holds valid with \( \varphi (0) = \lim \limits _{t \rightarrow 0^+} \frac{f (t)}{t^2} \). If, moreover, f is positive, then f must be convex, and in this case (46) refines the original inequality of Theorem A due to [5].
3 Applications for symmetric functions
We are interested in simplifying the assumptions of the results in the previous section. To this end we employ symmetric functions.
A function \( g : [a,b] \rightarrow \mathbb {R}\) is said to symmetric about \( c = \frac{a+b}{2} \) if
Lemma 3
Let \( g : [a,b] \rightarrow \mathbb {R}\) be an integrable symmetric function about \( c = \frac{a+b}{2} \), and \( G : [a,b] \rightarrow \mathbb {R}\) be the cumulative function of g defined by (2).
Then
- (i)
G is rotational symmetric around the point (c, G(c)) , i.e.,
$$\begin{aligned} G (c) - G (c-u) = G (c+u) - G (c) \;\;\; \text{ for } u \in \left[ 0, \frac{b-a}{2} \right] , \end{aligned}$$(48) - (ii)
the following equality holds:
$$\begin{aligned} \int \limits _a^b G (t) \, d t = (b-a) G (c) . \end{aligned}$$(49)
Proof
-
(i)
Fix any \( u \in [ 0, \frac{b-a}{2} ] \). It is not hard to check that
$$\begin{aligned} G (c-u)= & {} \int \limits _a^{c-u} g (t) \, d t = \int \limits _a^c g (t) \, d t + \int \limits _c^{c-u} g (t) \, d t = G (c) - \int \limits _0^u g (c-v) \, d v ,\\ G (c+u)= & {} \int \limits _a^{c+u} g (t) \, d t = \int \limits _a^c g (t) \, d t + \int \limits _c^{c+u} g (t) \, d t = G (c) + \int \limits _0^u g (c+v) \, d v . \end{aligned}$$Therefore, by (47), we derive
$$\begin{aligned} G (c) - G (c-u) = \int \limits _0^u g (c-v) \, d v = \int \limits _0^u g (c+v) \, d v = G (c+u) - G (c) , \end{aligned}$$which proves (48).
-
(ii)
It follows that
$$\begin{aligned} \int \limits _a^c G (t) \, d t= & {} \int \limits _a^c G (c) \, d t - \left( \int \limits _a^c ( G (c) - G (t) ) \, d t \right) \nonumber \\= & {} \int \limits _a^c G (c) \, d t - P_1 = (c-a) G (c) - P_1 , \end{aligned}$$(50)and
$$\begin{aligned} \int \limits _c^b G (t) \, d t= & {} \int \limits _c^b G (c) \, d t + \left( \int \limits _c^b ( G (t) - G (c) ) \, d t \right) \nonumber \\= & {} \int \limits _c^b G (c) \, d t + P_2 = (b-c) G (c) + P_2 , \end{aligned}$$(51)where
$$\begin{aligned} P_1 = \int \limits _a^c ( G (c) - G (t) ) \, d t = \int \limits _0^{b-c} ( G (c) - G (c-v) ) \, d v \end{aligned}$$and
$$\begin{aligned} P_2 = \int \limits _c^b ( G (t) - G (c) ) \, d t = \int \limits _0^{b-c} ( G (c+v) - G (c) ) \, d v . \end{aligned}$$In view of (48) we find that \( P_1 = P_2 \). Hence, by (50) and (51),
$$\begin{aligned} \int \limits _a^b G (t) \, d t {=} \int \limits _a^c G (t) \, d t {+} \int \limits _c^b G (t) \, d t {=} (c {-} a {+} b {-} c) G (c) {-} P_1 {+} P_2 {=} (b - a) G (c) . \end{aligned}$$Thus we see that (49) holds valid.
\(\square \)
Theorem 3
(Symmetric functions.) Let \( I = [a,b] \) be an interval and \( \psi : [0,b-a] \rightarrow \mathbb {R}\) be a function. Let \( f : [ a , b ] \rightarrow \mathbb {R}\) be a continuously twice differentiable generalized \( \psi \)-uniformly convex function on [a, b] . Denote \( \varphi (t) = \frac{\psi (t)}{t^2} \) for \( t \in (0,b-a] \) and \( \varphi (0) = \lim \limits _{t \rightarrow 0^+} \varphi (t) \).
Let \( g_1 , g_2 : [a,b] \rightarrow \mathbb {R}\) be integrable symmetric functions about \( c = \frac{a+b}{2} \), and \( G_1 , G_2 : [a,b] \rightarrow \mathbb {R}\) be the cumulative functions of \( g_1 \) and \( g_2 \) defined by (5), respectively, and \( {{\mathcal {G}}}_1 , {{\mathcal {G}}}_2 : [a,b] \rightarrow \mathbb {R}\) be the cumulative functions of \( G_1 \) and \( G_2 \) defined by (6), respectively.
Assume that
and, in addition, there exists \( d_2 \in [c,b] \) satisfying (a.e.)
Then
where \( R = 2 \varphi (0) \int \limits _a^b ( {{\mathcal {G}}}_1 (t) - {{\mathcal {G}}}_2 (t) ) \, d t \).
Proof
Because of (52), we have \( G_1 (b) = 2 G_1 (c) = 2 G_2 (c) = G_2 (b) \). For symmetric functions conditions (34), (35), (36) are reduced to (52) and (53). To see (37), we apply \( G_2 (c) = G_1 (c) \) via Lemma 3, part (ii), and we derive
(see (6)). Moreover, condition (43) is fulfilled, too. We appeal now to Theorem 2 to get the desired result. \(\square \)
A result for symmetric probability density functions is given as follows.
Corollary 3
(Symmetric p.d.f.) Under the assumptions of Theorem 3 with deleted condition (52), let \( g_1 , g_2 : [a,b] \rightarrow \mathbb {R}\) be probability density functions symmetric about \( c = \frac{a+b}{2} \).
Then inequality (54) holds.
Proof
For symmetric p.d. functions \( g_1 \) and \( g_2 \), condition (52) holds, because
So, the result is true according to Theorem 3. \(\square \)
3.1 Levin–Stečkin type inequalities for uniformly convex functions
We now demonstrate the use of Theorem 3 to derive a Levin–Stečkin type inequality with uniformly convex f.
Let \( I = [a,b] \) be an interval and \( \psi : [0,b-a] \rightarrow \mathbb {R}\) be a function. We denote \( \varphi (t) = \frac{\psi (t)}{t^2} \) for \( t \in (0,b-a] \) with \( \varphi (0) = \lim \limits _{t \rightarrow 0^+} \varphi (t) \).
Let \( f : I \rightarrow \mathbb {R}\) be a continuously twice differentiable generalized \( \psi \)-uniformly convex function on I. Let \( p : [a,b] \rightarrow \mathbb {R}\) be a non-negative integrable weight on I. Suppose that p is symmetric about \( c = \frac{a+b}{2} \).
We also introduce
In the case when there exists \( d_2 \in [c,b] \) satisfying (a.e.)
we set
Thus (53) is fulfilled.
By referring to the symmetry of p about \( c = \frac{a+b}{2} \) we can write \( b-c = \frac{1}{2} (b-a) \) and
which easily leads to (52) as follows
To sum up, inequality (54) in Theorem 3 quarantees that
which is a Levin–Stečkin type inequality for a generalized \( \psi \)-uniformly convex function f (cf. [10]). Here \( R = 2 \varphi (0) \int \limits _a^b ( {{\mathcal {G}}}_1 (t) - {{\mathcal {G}}}_2 (t) ) \, d t \) (see below).
Additionally, we have
Hence
So, we infer that
On the other hand, in the case when there exists \( d_2 \in [c,b] \) satisfying (a.e.)
we put
For this reason (53) is satisfied.
As previously, by the symmetry of p about \( c = \frac{a+b}{2} \), and thanks to (55) and (60) we can write
This forces (52), because
Finally, we deduce from inequality (54) in Theorem 3 that
with \( R = 2 \varphi (0) \int \limits _a^b ( {{\mathcal {G}}}_1 (t) - {{\mathcal {G}}}_2 (t) ) \, d t \) (see below). This is a Levin–Stečkin type inequality for a generalized \( \psi \)-uniformly convex function f (cf. [10]).
Furthermore,
and
Therefore, we conclude that
3.2 Simpson distributions
Recall that Theorem A corresponds to uniform distribution on an interval [a, b] . We shall establish a similar result corresponding to the Simpson (triangle) distribution on an interval [a, b] .
As usual, \( f : [a,b] \rightarrow \mathbb {R}\) is a continuously twice differentiable generalized \( \psi \)-uniformly convex function, where \( \psi : [0,b-a] \rightarrow \mathbb {R}\) is a function. Also, \( \varphi (t) = \frac{\psi (t)}{t^2} \) for \( t \in (0,b-a] \) with \( \varphi (0) = \lim \limits _{t \rightarrow 0^+} \varphi (t) \).
We put \( c = \frac{a+b}{2} \) and take \( x_1 , x_2 , y_1 , y_2 \in [a,b] \) with \( (x_2,y_2) \prec (x_1,y_1) \) and \( a \le x_1< x_2< c< y_2 < y_1 \le b \).
We define \( g_1 \) and \( g_2 \) to be probability density functions of Simpson distributions on [a, b] with triangles based on intervals \( [x_1,y_1] \) and \( [x_2,y_2] \), respectively. That is,
By setting \( d_2 = \frac{\xi y_2 - y_1}{\xi -1 } \) with \( \xi = \left( \frac{y_1 - x_1}{y_2 - x_2} \right) ^2 \), we see that condition (53) is satisfied. Taking Corollary 3 into account, we can rewrite (54) as
where \( R = 2 \varphi (0) \int \limits _a^b ( {{\mathcal {G}}}_1 (t) - {{\mathcal {G}}}_2 (t) ) \, d t \).
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Niezgoda, M. An extension of Levin–Stečkin’s theorem to uniformly convex and superquadratic functions. Aequat. Math. 94, 303–321 (2020). https://doi.org/10.1007/s00010-019-00675-4
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DOI: https://doi.org/10.1007/s00010-019-00675-4