We start this section by our nice observation about Abel-Gontscharoff identity (5) and the related Green’s function for ‘two-point right focal problem’. Therefore, keeping in view the Abel-Gontscharoff Green’s function for ‘two-point right focal problem’, we would like to introduce some new types of Green’s functions \(G_{k}:[\delta_{1},\delta_{2}]\times[\delta_{1},\delta_{2}]\rightarrow \mathbb {R}\), (\(k=2,3,4\)) defined as
$$\begin{aligned}& G_{2}(z,w) = \textstyle\begin{cases} z-\delta_{2}, & \delta_{1}\le w \le z,\\ w-\delta_{2}, & z\le w \le\delta_{2}. \end{cases}\displaystyle \end{aligned}$$
(7)
$$\begin{aligned}& G_{3}(z,w) = \textstyle\begin{cases} z-\delta_{1}, & \delta_{1}\le w \le z,\\ w-\delta_{1}, & z\le w \le\delta_{2}. \end{cases}\displaystyle \end{aligned}$$
(8)
$$\begin{aligned}& G_{4}(z,w) = \textstyle\begin{cases} \delta_{2}-w, & \delta_{1}\le w \le z,\\ \delta_{2}-z, & z\le w \le\delta_{2}. \end{cases}\displaystyle \end{aligned}$$
(9)
The graphical representations of \(G_{k}\), \(k=1,2,3,4\), are depicted in Figure 1 which shows that all four Green’s functions are continuous and symmetric. Moreover, all functions are convex with respect to both the variables z and w. These new Green’s functions enable us to introduce some new identities, stated in the form of the following lemma.
Lemma 1
Let
\(\psi:[\delta_{1},\delta_{2}]\to \mathbb {R}\)
be a twice differentiable function and
\(G_{k}\) (\(k=1,2,3,4\)) be the new Green’s functions defined above. Then along with (5) the following identities hold:
$$\begin{aligned}& \psi(z)=\psi(\delta_{2})+(\delta_{2}-z) \psi'(\delta_{1})+ \int _{\delta_{1}}^{\delta_{2}}{G_{2}(z,w) \psi''(w)}\,dw, \end{aligned}$$
(10)
$$\begin{aligned}& \psi(z)=\psi(\delta_{2})-(\delta_{2}- \delta_{1})\psi'(\delta _{2})+(z- \delta_{1})\psi'(\delta_{1})+ \int_{\delta_{1}}^{\delta _{2}}{G_{3}(z,w) \psi''(w)}\,dw, \end{aligned}$$
(11)
$$\begin{aligned}& \psi(z)=\psi(\delta_{1})+(\delta_{2}- \delta_{1})\psi'(\delta _{1})-( \delta_{2}-z)\psi'(\delta_{2})+ \int_{\delta_{1}}^{\delta _{2}}{G_{4}(z,w) \psi''(w)}\,dw. \end{aligned}$$
(12)
Proof
We can give the proofs of the above identities by following the same integrating scheme. Therefore we would like to give the proof of (12) only.
As
$$\begin{aligned} &\int_{\delta_{1}}^{\delta_{2}}{G_{4}(z, w) \psi''(w)}\,dw\\ &\quad = \int_{\delta _{1}}^{z}G_{4}(z,w) \psi''(w)\,dw+ \int_{z}^{\delta_{2}}G_{4}(z,w) \psi''(w)\,dw \\ &\quad = \int_{\delta_{1}}^{z}(\delta_{2}-w) \psi''(w)\,dw+ \int_{z}^{\delta _{2}}(\delta_{2}-z) \psi''(w)\,dw \\ &\quad =(\delta_{2}-w)\psi'(w)\mid_{\delta_{1}}^{z}- \int_{\delta _{1}}^{z}-1.\psi'(w)\,dw+( \delta_{2}-z)\bigl[\psi'(\delta_{2})- \psi'(z)\bigr] \\ &\quad =(\delta_{2}-z)\psi'(z)-(\delta_{2}- \delta_{1})\psi'(\delta_{1})+\psi (z)-\psi( \delta_{1})+(\delta_{2}-z)\psi'( \delta_{2})-(\delta_{2}-z)\psi '(z) \\ &\quad =(\delta_{2}-z)\psi'(\delta_{2})-( \delta_{2}-\delta_{1})\psi'(\delta _{1})-\psi(\delta_{1})+\psi(z). \end{aligned}$$
Now by simplifying terms, we will get our identity (12). □
Remark 3
Lemma 1 gives another proof of the special case of Abel-Gontscharoff identity (5). \(G_{3}\) and \(G_{4}\) are new Green’s functions, but the results are not so simple as in other two cases.
The inequality of Popoviciu, which was improved by Vasić and Stanković [1], is generalized by using the above new Green’s functions. In Theorem 1 we have that \(q_{i}\) (\(i=1,\ldots,s\)) are positive real numbers. Now we give the generalization of that result for real values of \(q_{i}\) (\(i=1,\ldots,s\)) with \(\sum_{i=1}^{s} q_{i} =1\) using the new Green’s functions \(G_{k}\), \(k=1,2,3,4\), as defined in Lemma 1.
Theorem 3
Let
\([\delta_{1},\delta_{2}]\)
be an interval in
\(\mathbb {R}\), for integers
\(m\geq3\), \(2\leq s\leq m-1\), consider the tuples
\(\mathbf{z} \in [\delta_{1},\delta_{2}]^{m}\), q
be a real
m-tuple such that
\(\sum_{j = 1}^{s} {{q_{{i_{j}}}}} \ne0\)
for any
\({1 \le{i_{1}} < \cdots < {i_{s}} \le m}\)
and
\(\sum_{i = 1}^{m} {{q_{i}}} = 1\). Also, let
\(\frac{{\sum_{j = 1}^{s} {{q_{{i_{j}}}}{z_{{i_{j}}}}}} }{{\sum_{j = 1}^{s} {{q_{{i_{j}}}}}} } \in[\delta_{1},\delta_{2} ]\)
for any
\({1 \le{i_{1}} < \cdots < {i_{s}} \le m}\). Then the following statements are equivalent:
-
(i)
For every continuous convex function
\(\psi:[\delta _{1},\delta_{2}]\rightarrow \mathbb {R}\),
$$ {\psi_{s,m}}(\mathbf{z},\mathbf{q}) \le\frac{{m - s}}{{m - 1}} {\psi _{1,m}}(\mathbf{z},\mathbf{q}) + \frac{{s - 1}}{{m - 1}}{\psi _{m,m}}(\mathbf{z},\mathbf{q}) $$
(13)
holds, where
$$ {\psi_{s,m}}(\mathbf{z},\mathbf{q}): = \frac{1}{{C_{s - 1}^{m - 1}}}\sum _{1 \le{i_{1}} < \cdots < {i_{s}} \le m} { \Biggl( {\sum_{j = 1}^{s} {{q_{{i_{j}}}}}} \Biggr)\psi \biggl( {\frac{{\sum_{j = 1}^{s} {{q_{{i_{j}}}}{z_{{i_{j}}}}}} }{{\sum_{j = 1}^{s} {{q_{{i_{j}}}}}} }} \biggr)}; $$
-
(ii)
For all
\(w\in[\delta_{1},\delta_{2}]\)
and
\(k=1,2,3,4\),
$$ {G_{s,m}^{k}}(\mathbf{z},w;\mathbf{q}) \le \frac{{m - s}}{{m - 1}}{G_{1,m}^{k}}(\mathbf{z},w;\mathbf{q}) + \frac{{s - 1}}{{m - 1}}{G^{k}_{m,m}}(\mathbf{z},w;\mathbf{q}), $$
(14)
where
$$\begin{gathered} {G_{s,m}^{k}}(\mathbf{z},w; \mathbf{q}) \\ \quad: = \frac{1}{{ C_{s - 1}^{m - 1}}}\sum_{1 \le{i_{1}} < \cdots < {i_{s}} \le m} { \Biggl( { \sum_{j = 1}^{s} {{q_{{i_{j}}}}}} \Biggr)G_{k} \biggl( {\frac{{\sum_{j = 1}^{s} {{q_{{i_{j}}}}{z_{{i_{j}}}}}} }{{\sum_{j = 1}^{s} {{q_{{i_{j}}}}}} },w} \biggr)};\quad2 \le s \le m, \end{gathered} $$
for the functions
\(G_{k}:[\delta_{1},\delta_{2}]\times[\delta_{1},\delta _{2}]\rightarrow \mathbb {R}\).
Moreover, the statements (i) and (ii) are also equivalent if we change the sign of the inequality in both (13) and (14).
Proof
\(\mbox{(i)}\Rightarrow\mbox{(ii)}\): Let (i) be valid. Fix \(k=1,2,3,4\). Then as the functions for all k
\(G_{k}(\cdot, w)\) (\(w \in[\delta_{1},\delta_{2}]\)) are also continuous and convex, it follows that for these functions (13) also holds for each fix k, i.e., (14) is valid.
\(\mbox{(ii)}\Rightarrow\mbox{(i)}\): Let \(\psi:[\delta_{1},\delta_{2}]\rightarrow \mathbb {R}\) be a convex function, \(\psi\in C^{2}([\delta_{1},\delta_{2}])\) and (ii) holds. Then, by Lemma 1, we can represent a function ψ in the form (5), (10), (11) and (12) for their respective fixed k. Now, by means of some simple calculations, we can write
$$ \begin{gathered}[b] \frac{{m - s}}{{m - 1}} { \psi_{1,m}}(\mathbf{z},\mathbf{q}) + \frac{{s - 1}}{{m - 1}}{ \psi_{m,m}}(\mathbf{z},\mathbf{q}) - {\psi_{s,m}}(\mathbf{z}, \mathbf{q}) \\ \quad = \int_{\delta_{1}} ^{\delta_{2}} { \biggl( \frac{{m - s}}{{m - 1}}{{G_{1,m}^{k}}(\mathbf{z},w;\mathbf{q}) + \frac{{s - 1}}{{m - 1}}{G^{k}_{m,m}}(\mathbf{z},w;\mathbf{q}) - {G_{s,m}^{k}}(\mathbf{z},w;\mathbf{q})} \biggr)} \psi''(w)\,dw. \end{gathered} $$
(15)
By the convexity of ψ, we have \(\psi''(w)\ge0\) for all \(w\in [\delta_{1},\delta_{2}]\). Hence, if for every \(w\in[\delta_{1},\delta _{2}]\), (14) is valid for each \(k=1,2,3,4\), then it follows that for every convex function \(\psi:[\delta_{1},\delta_{2}]\rightarrow \mathbb {R}\), with \(\psi\in C^{2}([\delta_{1},\delta_{2}])\), (13) is valid.
Here we can eliminate the differentiability condition due to the fact that it is possible to approximate uniformly a continuous convex function by convex polynomials (see [2], p.172).
Analogous to the above proof, we can give the proof of the last part of our theorem. □
Next we formulate generalized identities with the help of identities defined in Lemma 1 and Fink’s identity.
Theorem 4
Let all the assumptions of Theorem
2
be valid with
\(n>2\), and let
\(m, s\in{\mathbb{N}}\), \(m\geq3\), \(2\leq s\leq m-1\), \(\mathbf{z} \in[\delta_{1},\delta_{2}]^{m}\), q
be a real
m-tuple such that
\(\sum_{j = 1}^{s} {{q_{{i_{j}}}}} \ne0\)
for any
\({1 \le{i_{1}} < \cdots < {i_{s}} \le m}\)
and
\(\sum_{i = 1}^{m} {{q_{i}}} = 1\). Also let
\(\frac{{\sum_{j = 1}^{s} {{q_{{i_{j}}}}{z_{{i_{j}}}}}} }{{\sum_{j = 1}^{s} {{q_{{i_{j}}}}}} } \in[\delta_{1},\delta_{2} ]\)
for any
\({1 \le{i_{1}} < \cdots < {i_{s}} \le m}\)
with
\(R_{n}( \cdot,v)\)
and
\(G_{k}(\cdot, w)\), (\(k=1,2,3,4\)) be the same as defined in (4) and Lemma
1, respectively. Then we have the following new identities for
\(k=1,2,3,4\):
$$ \begin{aligned}[b] &\operatorname{POP}\bigl[\mathbf{z}, \mathbf{q};\psi(z)\bigr]\\ &\quad = \biggl(\frac {{\psi'(\delta_{1} ) - \psi'(\delta_{2} )}}{{\delta_{2} - \delta_{1} }} \biggr) \int_{\delta_{1}}^{\delta_{2}} {\operatorname{POP}\bigl[\mathbf{z}, \mathbf{q};G_{k}(z,w)\bigr]}\,dw \\ &\qquad{} + \frac{1}{{\delta_{2} - \delta_{1}} } \int_{\delta_{1}} ^{\delta_{2}} \operatorname{POP}\bigl[\mathbf{z}, \mathbf{q};G_{k}(z,w)\bigr] \\ &\qquad {}\times\Biggl(\sum_{l = 2}^{n - 1} \frac{l}{{(l - 1)!}} \bigl( {\psi^{ ( l )}} ( \delta_{1} ){{ ( {w - \delta_{1}} )}^{l - 1}}- {\psi^{ ( l )}} ( \delta_{2} ){{ ( {w - \delta_{2}} )}^{l - 1}} \bigr) \Biggr) \,dw \\ &\qquad{} + \frac{1}{{ ( {n - 3} )!}} \int_{\delta_{1}} ^{\delta_{2}} {{\psi^{(n)}}} (v) \biggl( \int_{\delta_{1}} ^{\delta_{2}} {\operatorname{POP}\bigl[\mathbf{z}, \mathbf{q};G_{k}(z,w)\bigr]} {{\tilde{R}}_{n - 2}} ( {w,v} )\,dw \biggr)\,dv, \end{aligned} $$
(16)
where
$$\tilde{R}_{n-2} ( {w,v} )= \textstyle\begin{cases} {\frac{1}{{\delta_{2} - \delta_{1}} } [ {\frac{{{{ ( {w - v} )}^{n - 2}}}}{{(n - 2)}} + ( {w - \delta_{1}} ){{ ( {w - v} )}^{n - 3}}} ],}&{\delta_{1} \le v \le w,}\\ {\frac{1}{{\delta_{2} - \delta_{1}} } [ {\frac{{{{ ( {w - v} )}^{n - 2}}}}{{(n - 2)}} + ( {w - \delta_{2}} ){{ ( {w - v} )}^{n - 3}}} ],}&{w < v \le\delta_{2},} \end{cases} $$
and
$$ \begin{aligned}[b] &\operatorname{POP}\bigl[\mathbf{z}, \mathbf{q};\psi(z)\bigr] \\ &\quad = \biggl( {\frac {{\psi'(\delta_{2} ) - \psi'(\delta_{1} )}}{{\delta_{2} - \delta_{1}} }} \biggr) \int_{\delta_{1}} ^{\delta_{2}} {\operatorname{POP}\bigl[\mathbf{z}, \mathbf{q};G_{k}(z,w)\bigr]} \,dw \\ &\qquad{} + \frac{1}{{\delta_{2} - \delta_{1}} } \int_{\delta_{1}} ^{\delta_{2}} \operatorname{POP}\bigl[\mathbf{z}, \mathbf{q};G_{k}(z,w)\bigr]\\ &\qquad{}\times \Biggl({\sum_{l = 3}^{n - 1} {\frac{{{\psi^{ ( l )}} (\delta_{1} ){{ ( {w - \delta_{1}} )}^{l - 1}} - {\psi ^{ ( l )}} ( \delta_{2} ){{ ( {w - \delta _{2}} )}^{l - 1}}}}{{(l - 3)!(l - 1)}}}} \Biggr) \,dw \\ &\qquad{} + \frac{1}{{ ( {n - 3} )!}} \int_{\delta_{1}} ^{\delta_{2}} {{\psi^{ ( n )}} ( v ) \biggl({ \int_{\delta_{1}} ^{\delta_{2}} {\operatorname{POP}\bigl[\mathbf{z}, \mathbf{q};G_{k}(z,w)\bigr]} {R_{n - 2}}(w,v)\,dw} \biggr)\,dv}. \end{aligned} $$
(17)
Proof
Fix \(k=1,2,3,4\). Applying Popoviciu’s functional (2) to identities (5), (10), (11), (12) along with their respective new Green’s functions and following the properties of \(\operatorname{POP}[\mathbf{z},\mathbf{q};\cdot]\), we get
$$ \operatorname{POP}\bigl[\mathbf{z},\mathbf{q};\psi(z)\bigr]= \int_{\delta _{1}} ^{\delta_{2}} {\operatorname{POP}\bigl[\mathbf{z}, \mathbf{q};G_{k}(z, w)\bigr]} \psi''(w)\,dw. $$
(18)
Differentiating (3) twice with respect to the first variable, we have
$$ \begin{aligned}[b] \psi'' ( w ) &= \frac{{\psi' ( \delta_{1} ) - \psi' ( \delta_{2} )}}{{\delta_{2} - \delta_{1}} } \\ &\quad{} + \sum_{l = 2}^{n - 1} { \biggl( { \frac{l}{{(l - 1)!}}} \biggr)} \biggl( {\frac{{{\psi^{ ( l )}} ( \delta _{1} ){{ ( {w - \delta_{1}} )}^{l - 1}} - {\psi ^{ ( l )}} ( \delta_{2} ){{ ( {w - \delta _{2}} )}^{l - 1}}}}{{\delta_{2} - \delta_{1}} }} \biggr) \\ &\quad{} + \frac{1}{{ ( {n - 3} )!}} \int_{\delta_{1}} ^{\delta_{2}} {{{\tilde{R}}_{n - 2}}} ( {w,v} ){\psi^{ ( n )}} ( v )\,dv. \end{aligned} $$
(19)
Using (19) in (18), we get
$$\begin{aligned} \operatorname{POP}\bigl[\mathbf{z},\mathbf{q};\psi(z) \bigr] &= \biggl(\frac {{\psi'(\delta_{1} ) - \psi'(\delta_{2} )}}{{\delta_{2} - \delta_{1} }} \biggr) \int_{\delta_{1}}^{\delta_{2}} {\operatorname{POP}\bigl[\mathbf{z}, \mathbf{q};G_{k}(z,w)\bigr]}\,dw \\ &\quad{} + \sum_{l = 2}^{n - 1} { \frac{l}{{(l - 1)!}}} \int_{\delta _{1}} ^{\delta_{2}} \operatorname{POP}\bigl[\mathbf{z}, \mathbf{q};G_{k}(z,w)\bigr]\\ &\quad {}\times \biggl( {\frac{{{\psi^{ ( l )}} (\delta_{1} ){{ ( {w - \delta_{1}} )}^{l - 1}} - {\psi ^{ ( l )}} ( \delta_{2} ){{ ( {w - \delta _{2}} )}^{l - 1}}}}{{\delta_{2} - \delta_{1}} }} \biggr) \,dw \\ &\quad{} + \frac{1}{{ ( {n - 3} )!}} \int_{\delta_{1}} ^{\delta_{2}} {\mathbf{P}\bigl(\mathbf{x}, \mathbf{p};G_{k}(z,w)\bigr)} \biggl( \int_{\delta _{1}} ^{\delta_{2}} {{{\tilde{R}}_{n - 2}}} ( {w,v} ){\psi ^{ ( n )}} ( v )\,dv\biggr)\,dw. \end{aligned} $$
By executing Fubini’s theorem in the last term, we have (16) respectively for \(k=1,2,3,4\).
Next, using formula (3) on the function \(\psi''\), replacing n by \(n-2\) (\(n\geq3\)) and rearranging the indices, we have
$$ \begin{aligned}[b] \psi'' ( w ) &= \biggl( {\frac{{\psi'(\delta_{2} ) - \psi '(\delta_{1} )}}{{\delta_{2} - \delta_{1}} }} \biggr) \\ &\quad{} + \sum_{l = 3}^{n - 1} { \biggl( { \frac{1}{{(l-3)!(l -1)}}} \biggr)} \biggl( {\frac{{{\psi^{ ( {l} )}} ( \delta _{1} ){{ ( {w - \delta_{1}} )}^{l-1}} - {\psi^{ ( {l} )}} ( \delta_{2} ){{ ( {w - \delta_{2}} )}^{l-1}}}}{{\delta_{2} - \delta_{1}} }} \biggr) \\ &\quad{} + \frac{1}{{ ( {n - 3} )!}} \int_{\delta_{1}} ^{\delta_{2}} {{R_{n - 2}}(w,v){ \psi^{ ( n )}} ( v )\,dv}. \end{aligned} $$
(20)
Similarly, using (20) in (18) and employing Fubini’s theorem, we get (17) respectively for \(k=1,2,3,4\). □
As an application of the above obtained identities, the next theorem gives artistic generalization of Popoviciu-type inequalities for n-convex functions involving new Green’s functions.
Theorem 5
Let all the assumptions of Theorem
4
be satisfied and
\(n\ge 3\). Also let
ψ
be an
n-convex function such that
\(\psi ^{(n-1)}\)
is absolutely continuous. Then we have the following two results:
If
$$ \int_{\delta_{1}} ^{\delta_{2}} {\operatorname{POP}\bigl[\mathbf{z}, \mathbf{q};G_{k}(z,w)\bigr]} {{\tilde{R}}_{n - 2}} ( {w,v} )\,dw \ge0, \quad v\in[\delta_{1},\delta_{2}] $$
(21)
for
\(k=1,2,3,4\), then
$$ \begin{aligned}[b] \operatorname{POP}\bigl[\mathbf{z}, \mathbf{q};\psi(z)\bigr] &\ge \biggl(\frac {{\psi'(\delta_{1} ) - \psi'(\delta_{2} )}}{{\delta_{2} - \delta_{1} }} \biggr) \int_{\delta_{1}}^{\delta_{2}} {\operatorname{POP}\bigl[\mathbf{z}, \mathbf{q};G_{k}(z,w)\bigr]}\,dw \\ &\quad{} + \frac{1}{{\delta_{2} - \delta_{1}} } \int_{\delta_{1}} ^{\delta_{2}} \operatorname{POP}\bigl[\mathbf{z}, \mathbf{q};G_{k}(z,w)\bigr] \\ &\quad{}\times\Biggl({\sum_{l = 2}^{n - 1} {\frac{l}{{(l - 1)!}} \bigl( {{\psi^{ ( l )}} ( \delta_{1} ){{ ( {w - \delta_{1}} )}^{l - 1}} - {\psi^{ ( l )}} ( \delta_{2} ){{ ( {w - \delta_{2}} )}^{l - 1}}} \bigr)}} \Biggr) \,dw; \end{aligned} $$
(22)
and if
$$ \int_{\delta_{1}} ^{\delta_{2}} {\operatorname{POP}\bigl[\mathbf{z}, \mathbf{q};G_{k}(z,w)\bigr]} {R_{n - 2}} ( {w,v} )\,dw\ge0, \quad v\in [\delta_{1},\delta_{2}] $$
(23)
for
\(k=1,2,3,4\), then
$$ \begin{aligned}[b] \operatorname{POP}\bigl[\mathbf{z}, \mathbf{q};\psi(z)\bigr] &\ge \biggl( {\frac {{\psi'(\delta_{2} ) - \psi'(\delta_{1} )}}{{\delta_{2} - \delta_{1}} }} \biggr) \int_{\delta_{1}} ^{\delta_{2}} {\operatorname{POP}\bigl[\mathbf{z}, \mathbf{q};G_{k}(z,w)\bigr]} \,dw \\ &\quad{} + \frac{1}{{\delta_{2} - \delta_{1}} } \int_{\delta_{1}} ^{\delta_{2}} \operatorname{POP}\bigl[\mathbf{z}, \mathbf{q};G_{k}(z,w)\bigr]\\ &\quad{}\times \Biggl({\sum_{l = 3}^{n - 1} {\frac{{{\psi^{ ( l )}} (\delta_{1} ){{ ( {w - \delta_{1}} )}^{l - 1}} - {\psi ^{ ( l )}} ( \delta_{2} ){{ ( {w - \delta _{2}} )}^{l - 1}}}}{{(l - 3)!(l - 1)}}}} \Biggr) \,dw. \end{aligned} $$
(24)
Proof
Fix \(k=1,2,3,4\). Since \(\psi^{(n-1)}\) is absolutely continuous on \([\delta_{1},\delta_{2}]\), \(\psi^{(n)}\) exists almost everywhere. As ψ is n-convex, so \(\psi^{(n)}(z)\ge0\) for all \(z\in[\delta _{1},\delta_{2}]\) (see [2], p.16). Hence we can apply Theorem 4 to obtain (22) and (24) respectively. □
Remark 4
Inequalities (22) and (23) hold in reverse directions if the inequalities in (21) and (23) are reversed.
Now we state the final result of this section in the form of the following theorem.
Theorem 6
In addition to the assumptions of Theorem
4, let
\(\mathbf {q}=(q_{1},\ldots,q_{m})\)
be a positive
m-tuple such that
\(\sum_{i = 1}^{m} {{q_{i}}} = 1\), and
\(\psi:[\delta_{1},\delta_{2}] \to \mathbb {R}\)
be an
n-convex function.
-
(i)
For fixed
\(k=1,2,3,4\), inequalities (22) and (24) hold provided that
n
is even and (\(n\ge4\)).
-
(ii)
For fixed
\(k=1,2,3,4\), let inequality (22) be satisfied and
$$ \sum_{l = 1}^{n - 1} { \frac{l}{{(l - 1)!}} \bigl( {{\psi^{ ( l )}} ( \delta_{1} ){{ ( {w - \delta_{1}} )}^{l - 1}} - {\psi^{ ( l )}} ( \delta_{2} ){{ ( {w - \delta_{2}} )}^{l - 1}}} \bigr)} \ge0;\quad \forall w \in[\delta_{1}, \delta_{2}], $$
(25)
OR (24) be satisfied and
$$ \begin{aligned}[b] &\psi'(\delta_{2} ) - \psi'( \delta_{1} )\\ &\quad {} + \sum_{l = 3}^{n - 1} { \frac {{{\psi^{ ( l )}} ( \delta_{1} ){{ ( {w - \delta_{1}} )}^{l - 1}} - {\psi^{ ( l )}} (\delta_{2} ){{ ( {w - \delta_{2}} )}^{l - 1}}}}{{(l - 3)!(l - 1)}}} \ge0;\quad\forall w \in[\delta_{1}, \delta_{2}]. \end{aligned} $$
(26)
Then we have
$$ \operatorname{POP}\bigl[\mathbf{z},\mathbf{q};\psi(z)\bigr]\ge0. $$
(27)
Proof
It is clear from Figure 1 that Green’s function \(G_{k}(z, w)\) is convex for all \(k=1,2,3,4\), and the weights are assumed to be positive. Therefore, applying Theorem 1 and taking into account Remark 1, we can obtain \(\operatorname{POP}[\mathbf{z},\mathbf{q};G_{k}(z, w)]\ge0\) for all \(k=1,2,3,4\).
-
(i)
\(\tilde{R}_{n-2} ( {w,v} )\ge0\) and \(R_{n-2} ( {w,v} )\ge0\) for \(n=4,6,\ldots\) , so (21) and (23) hold. As ψ is n-convex, hence by following Theorem 5, we obtain (22) and (24).
-
(ii)
Using (25) in (22) and (26) in (24), (27) is established for all \(k=1,2,3,4\). □