For our convenience, we use the following notations and assumptions:
$$\begin{aligned}& \mathbb{S}_{1}(\psi ,f,c,d)=\psi \bigl(f(c)\bigr)+ \int _{c}^{d}\psi ^{ \prime }(z)f^{\prime }(z) \,dz-\psi \bigl(f(d)\bigr), \\& \mathbb{S}_{2}(\psi ,f,d)= \int _{0}^{d}\psi ^{\prime }(z)f(z) \,dz- \psi \biggl( \int _{0}^{d}f(z) \,dz \biggr) , \\& \mathbb{S}_{3}(\psi ,f,\lambda ,d)= \int _{0}^{d}\varLambda (z) \psi ^{\prime }(z)f(z) \,dz- \int _{0}^{d}\lambda (t)\psi \biggl( \int _{0}^{t}f(z) \,dz \biggr) \,dt. \end{aligned}$$
- (\(A_{1}\)):
-
For \(n\in \mathbb{N}\), \(n\geq 3\), let \(\psi :[c,d]\rightarrow \mathbb{R}\) be an n times differentiable function with \(\psi ^{(n-1)}\) absolutely continuous on \([c,d]\).
- (\(A_{2}\)):
-
For \(n\in \mathbb{N}\), \(n\geq 3\), let \(\psi :[0,d]\rightarrow \mathbb{R}\) be an n times differentiable function with \(\psi (0)=0\) and \(\psi ^{(n-1)}\) absolutely continuous on \([0,d]\).
The first part of this section is the generalization of (5). For this, we start with the following theorem.
Theorem 4
Consider (\(A_{1}\)) with
f
be as in Corollary 1\((i)\)
and
\(R_{n}\)
be defined by (26), then:
-
\((a)\)
:
-
For
\(j=1,2,4,5\), we have
$$ \begin{aligned}[b] &\mathbb{S}_{1}(\psi ,f,c,d)\\ &\quad = \frac{\psi {^{\prime }(c)-\psi ^{\prime }(d)}}{ {d-c}} \int _{c}^{d}\mathbb{S}_{1} \bigl(G_{\ast ,j}(\cdot,s),f,c,d\bigr)\,ds \\ &\qquad {}+\frac{1}{{d-c}} \int _{c}^{d}\mathbb{S}_{1} \bigl(G_{\ast ,j}(\cdot,s),f,c,d\bigr)\\ &\qquad {}\times { \Biggl( {\sum _{l=2}^{n-1}{\frac{l}{{(l-1)!}} \bigl( \psi {{^{ ( l ) }} ( c ) {{ ( {s-c} ) } ^{l-1}}-\psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) }^{l-1}}} \bigr) }} \Biggr) } \,ds \\ &\qquad {}+\frac{1}{{ ( {n-3} ) !}} \int _{c}^{d}\psi {{^{(n)}}}(v) \biggl( \int _{c}^{d}\mathbb{S}_{1} \bigl(G_{\ast ,j}(\cdot,s),f,c,d\bigr){{\tilde{R}} _{n-2}} ( {s,v} ) \,ds \biggr) \,dv. \end{aligned} $$
(27)
-
\((b)\)
:
-
If
\(\psi ^{\prime }(c)=0\), then
$$ \begin{aligned}[b] &\mathbb{S}_{1}(\psi ,f,c,d)\\ &\quad = \frac{\psi {^{\prime }(c)-\psi ^{\prime }(d)}}{ {d-c}} \int _{c}^{d}\mathbb{S}_{1} \bigl(G_{\ast ,3}(\cdot,s),f,c,d\bigr)\,ds \\ &\qquad {}+\frac{1}{{d-c}} \int _{c}^{d}\mathbb{S}_{1} \bigl(G_{\ast ,3}(\cdot,s),f,c,d\bigr)\\ &\qquad {}\times { \Biggl( {\sum _{l=2}^{n-1}{\frac{l}{{(l-1)!}} \bigl( \psi {{^{ ( l ) }} ( c ) {{ ( {s-c} ) } ^{l-1}}-\psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) }^{l-1}}} \bigr) }} \Biggr) } \,ds \\ &\qquad {}+\frac{1}{{ ( {n-3} ) !}} \int _{c}^{d}\psi {{^{(n)}}}(v) \biggl( \int _{c}^{d}\mathbb{S}_{1} \bigl(G_{\ast ,3}(\cdot,s),f,c,d\bigr){{\tilde{R}} _{n-2}} ( {s,v} ) \,ds \biggr) \,dv, \end{aligned} $$
(28)
where
$$ \tilde{R}_{n-2} ( {s,v} ) = \textstyle\begin{cases} {\frac{1}{{d-c}} [ {\frac{{{{ ( {s-v} ) }^{n-2}}}}{ {(n-2)}}+ ( {s-c} ) {{ ( {s-v} ) }^{n-3}}} ] ,} & {c\leq v\leq s,} \\ {\frac{1}{{d-c}} [ {\frac{{{{ ( {s-v} ) }^{n-2}}}}{ {(n-2)}}+ ( {s-d} ) {{ ( {s-v} ) }^{n-3}}} ] ,} & {s< v\leq d,} \end{cases} $$
(29)
and
-
\((a^{\prime })\)
:
-
For
\(j=1,2,4,5\), we have
$$ \begin{aligned}[b] &\mathbb{S}_{1}(\psi ,f,c,d)\\ &\quad = \biggl( { \frac{\psi {^{\prime }(d)- \psi ^{\prime }(c)}}{{d-c}}} \biggr) \int _{c}^{d}\mathbb{S}_{1} \bigl(G_{ \ast ,j}(\cdot,s),f,c,d\bigr)\,ds \\ &\qquad {} +\frac{1}{{d-c}} \int _{c}^{d}\mathbb{S}_{1} \bigl(G_{\ast ,j}(\cdot,s),f,c,d\bigr)\\ &\qquad {}\times { \Biggl( {\sum _{l=3}^{n-1}{\frac{\psi {{^{ ( l ) }} ( c ) {{ ( {s-c} ) }^{l-1}}-\psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) }^{l-1}}}}{ {(l-3)!(l-1)}}}} \Biggr) }\,ds \\ &\qquad {}+\frac{1}{{ ( {n-3} ) !}} \int _{c}^{d}\psi {{^{ ( n ) }} ( v ) \biggl( { \int _{c}^{d}\mathbb{S}_{1}\bigl(G _{\ast ,j}(\cdot,s),f,c,d\bigr){R_{n-2}}(s,v)\,ds} \biggr) \,dv}. \end{aligned} $$
(30)
-
\((b^{\prime })\)
:
-
If
\(\psi ^{\prime }(c)=0\), then
$$ \begin{aligned} &\mathbb{S}_{1}(\psi ,f,c,d)\\ &\quad = \biggl( { \frac{\psi {^{\prime }(d)- \psi ^{\prime }(c)}}{{d-c}}} \biggr) \int _{c}^{d}\mathbb{S}_{1} \bigl(G_{ \ast ,3}(\cdot,s),f,c,d\bigr)\,ds \\ &\qquad{} +\frac{1}{{d-c}} \int _{c}^{d}\mathbb{S}_{1} \bigl(G_{\ast ,3}(\cdot,s),f,c,d\bigr) { \Biggl( {\sum _{l=3}^{n-1}{\frac{\psi {{^{ ( l ) }} ( c ) {{ ( {s-c} ) }^{l-1}}-\psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) }^{l-1}}}}{ {(l-3)!(l-1)}}}} \Biggr) }\,ds \\ &\qquad {} +\frac{1}{{ ( {n-3} ) !}} \int _{c}^{d}\psi {{^{ ( n ) }} ( v ) \biggl( { \int _{c}^{d}\mathbb{S}_{1}\bigl(G _{\ast ,3}(\cdot,s),f,c,d\bigr){R_{n-2}}(s,v)\,ds} \biggr) \,dv}. \end{aligned} $$
Proof
By using (8) and (18) for ψ and \(\psi ^{ \prime }\) respectively and then applying (24), we get
$$ \mathbb{S}_{1}{(\psi ,f,c,d)}={ \int _{c}^{d}\mathbb{S}_{1} \bigl(G_{\ast ,j}(\cdot,s),f,c,d\bigr)} \psi ^{{\prime \prime }}(s)\,ds. $$
(31)
Differentiating (25) twice with respect to first variable, we have
$$ \begin{aligned}[b] \psi ^{\prime \prime } ( s ) &= \frac{\psi {^{\prime } ( c ) -\psi ^{\prime } ( d ) }}{{d-c}}+\sum_{l=2} ^{n-1}{ \biggl( {\frac{l}{{(l-1)!}}} \biggr) } \biggl( {\frac{\psi {{^{ ( l ) }} ( c ) {{ ( {s-c} ) } ^{l-1}}-\psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) }^{l-1}}}}{{d-c}}} \biggr) \\ &\quad {}+\frac{1}{{ ( {n-3} ) !}} \int _{c}^{d}{{{\tilde{R}}_{n-2}}} ( {s,v} ) \psi {^{ ( n ) }} ( v ) \,dv. \end{aligned} $$
(32)
Using (32) in (31), we get
$$\begin{aligned} &\mathbb{S}_{1}(\psi {,f,c,d)} \\ &\quad =\frac{\psi {^{\prime } ( c ) -\psi ^{\prime } ( d ) }}{{d-c}} \int _{c}^{d}\mathbb{S}_{1} { \bigl(G_{\ast ,1}(\cdot,s),f,c,d\bigr)\,ds} \\ &\qquad {}+\sum_{l=2}^{n-1}{ \biggl( { \frac{l}{{(l-1)!}}} \biggr) } \int _{c}^{d}\mathbb{S}_{1}{ \bigl(G_{\ast ,1}(\cdot,s),f,c,d\bigr)} \biggl( {\frac{\psi {{^{ ( l ) }} ( c ) {{ ( {s-c} ) } ^{l-1}}-\psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) }^{l-1}}}}{{d-c}}} \biggr) \,ds \\ &\qquad {}+\frac{1}{{ ( {n-3} ) !}} \int _{c}^{d} \mathbb{S} _{1}{ \bigl(G_{\ast ,1}(\cdot,s),f,c,d\bigr)} \biggl( { \int _{a}^{b}{{\tilde{R}}_{n-2}}} ( {s,v} ) \psi {^{ ( n ) }} ( v ) \,dv \biggr) \,ds. \end{aligned}$$
By applying Fubini’s theorem in the last term, we have (27). Next, using formula (25) on the function \({\psi }^{\prime \prime }\), replacing n by \(n-2\)
\((n\geq 3)\), and rearranging the indices, we have
$$ \begin{aligned}[b] \psi ^{\prime \prime } ( s ) &= \biggl( { \frac{\psi {^{\prime }(d)-\psi ^{\prime }(c)}}{{d-c}}} \biggr)\\ &\quad {} +\sum_{l=3}^{n-1} { \biggl( {\frac{1}{{(l-3)!(l-1)}}} \biggr) } \biggl( {\frac{\psi {{^{ ( {l} ) }} ( c ) {{ ( {s-c} ) }^{l-1}}- \psi {^{ ( {l} ) }} ( b ) {{ ( {s-d} ) }^{l-1}}}}{{d-c}}} \biggr) \\ &\quad {} +\frac{1}{{ ( {n-3} ) !}} \int _{c}^{d}{{R_{n-2}}(s,v) \psi {^{ ( n ) }} ( v ) \,dv}. \end{aligned} $$
(33)
Similarly, using (33) in (31) and applying Fubini’s theorem, we get (30). The proof for \(j=2,3,4,5\) can be obtained in a similar way except for the use of (9)–(12) and (19)–(20).
From the next two theorems, we get generalization of Steffensen’s inequality and its reverse by generalizing (5) and its reverse. □
Theorem 5
Consider (\(A_{1}\)) with
f
be as in Corollary 1\((i)\), and let
\(R_{n}\), \({\tilde{R}}_{n-2}\)
be as in Theorem 4. If
ψ
is
n-convex and
$$ \int _{c}^{d}\mathbb{S}_{1}{ \bigl(G_{\ast ,j}(\cdot,s),f,c,d\bigr){{\tilde{R}}_{n-2}} ( {s,v} ) \,ds\geq 0}, $$
(34)
then
$$ \begin{aligned}[b] \mathbb{S}_{1}({\psi },{f,c,d})&\geq \frac{{\psi ^{\prime }(c)- \psi ^{\prime }(d)}}{{d-c}} \int _{c}^{d}\mathbb{S}_{1} \bigl(G_{\ast ,j}(\cdot,s), {f,c,d}\bigr)\,ds \\ &\quad {}+\frac{1}{{d-c}} \int _{c}^{d}\mathbb{S}_{1} \bigl(G_{\ast ,j}(\cdot,s),f,c,d\bigr)\\ &\quad {}\times { \Biggl( {\sum _{l=2}^{n-1}{\frac{l}{{(l-1)!}} \bigl( {\psi {^{ ( l ) }} ( c ) {{ ( {s-c} ) } ^{l-1}}-\psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) }^{l-1}}} \bigr) }} \Biggr) } \,ds \end{aligned} $$
(35)
and if
$$ { \int _{c}^{d}\mathbb{S}_{1} \bigl(G_{\ast ,j}(\cdot,s),f,c,d\bigr){R_{n-2}}(s,v)\,ds \geq 0,} $$
(36)
then
$$ \begin{aligned}[b] &\mathbb{S}_{1}({\psi },{f,c,d})\\ &\quad \geq \biggl( {\frac{{\psi ^{\prime }(d)- \psi ^{\prime }(c)}}{{d-c}}} \biggr) \int _{c}^{d}\mathbb{S}_{1} \bigl(G_{ \ast ,j}(\cdot,s),f,c,d\bigr)\,ds \\ &\qquad {} +\frac{1}{{d-c}} \int _{c}^{d}\mathbb{S}_{1} \bigl(G_{\ast ,j}(\cdot,s),f,c,d\bigr) { \Biggl( {\sum _{l=3}^{n-1}{\frac{{\psi {^{ ( l ) }} ( c ) {{ ( {s-c} ) }^{l-1}}-\psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) }^{l-1}}}}{ {(l-3)!(l-1)}}}} \Biggr) }\,ds \end{aligned} $$
(37)
for
\(j=1,2,\ldots ,5\), where
\(\psi ^{\prime }(c)=0\)
for
\(j=3\).
Proof
Since \({\psi }^{(n-1)}\) is absolutely continuous on \([c,d]\), \({\psi }^{(n)}\) exists almost everywhere. As ψ is n-convex, so \({\psi }^{(n)}(x)\geq 0\) for all \(x\in {}[ c,d]\) (see [16], p. 16). Hence we can apply Theorem 4 to obtain (35) and (37) respectively. □
Theorem 6
Consider (\(A_{1}\)) with
f
be as in Corollary 1\((i)\), and let
\(R_{n}\), \({\tilde{R}}_{n-2}\)
be as in Theorem 4. If
ψ
is
n-convex, then:
-
(i)
If
n
is even and
\(n\geq 4\), then (35) holds.
-
(ii)
Let inequality (35) be satisfied and
$$ {\sum_{l=1}^{n-1}{\frac{l}{{(l-1)!}} \bigl( {\psi {^{ ( l ) }} ( c ) {{ ( {s-c} ) }^{l-1}}-\psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) } ^{l-1}}} \bigr) \geq 0}}; \quad \forall s\in {}[ c,d], $$
(38)
OR
(37) be satisfied and
$$ {\psi }^{\prime }(d)-{\psi }^{\prime }(c)+{\sum _{l=3}^{n-1} {\frac{{\psi {^{ ( l ) }} ( c ) {{ ( {s-c} ) }^{l-1}}-\psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) }^{l-1}}}}{{(l-3)!(l-1)}}}}\geq 0\quad \forall s\in {}[ c,d]{,} $$
(39)
then we have
$$ \mathbb{S}_{1}({\psi },f,c,d)\geq 0. $$
(40)
Proof
Since Green’s function G is convex and f is as in Corollary 1\((a)\), therefore, \(S_{1}(G_{\ast ,j}(\cdot,s),f,c,d)\geq 0\) holds by virtue of Corollary 1\((a)\). Moreover, \(\tilde{R}_{n-2} ( {s,v} ) \geq 0\) for \(n=4,6,\ldots \) , so (34) holds. As ψ is n-convex, hence by Theorem 5, we obtain (35). Further, by using (38) in (35) or (39) in (37), we have (40). □
Remark 2
Inequalities (35) and (37) hold in reverse directions if either the inequalities in (34) and (36) are reversed or −ψ is n-convex. By using these reverse inequalities and applying a similar technique as in Theorem 6, one may prove \(\mathbb{S}_{1}({\psi },f,c,d)\leq 0\), which gives the reverse of (1) inequality.
In the next theorem, we prove a few identities which enable us to prove generalization of (6) and its reverse.
Theorem 7
Consider
\((A_{2})\)
and let
f
be as in Corollary 2\((i)\), then:
-
\((a)\)
:
-
$$\begin{aligned} &\mathbb{S}_{2}({\psi },f,d)\\ &\quad =\frac{{\psi ^{\prime }(0)-\psi ^{\prime }(d)}}{ {d}} \int _{0}^{d}\mathbb{S}_{2} \bigl(G_{\ast ,j}(\cdot,s),f,d\bigr)\,ds \\ &\qquad {}+\frac{1}{{d}} \int _{0}^{d}\mathbb{S}_{2} \bigl(G_{\ast ,j}(\cdot,s),f,d\bigr){ \Biggl( {\sum _{l=2}^{n-1}{\frac{l}{{(l-1)!}} \bigl( {\psi {^{ ( l ) }} ( 0 ) {{ ( {s} ) }^{l-1}}-\psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) } ^{l-1}}} \bigr) }} \Biggr) }\,ds \\ &\qquad {}+\frac{1}{{ ( {n-3} ) !}} \int _{0}^{d}\psi {{^{(n)}}}(v) \biggl( { \int _{0}^{d}}\mathbb{S}_{2} \bigl(G_{\ast ,j}(\cdot,s),f,d\bigr){{\tilde{R}} _{0,n-2}} ( {s,v} ) \,ds \biggr) \,dv \end{aligned}$$
for
\(j=1,2\).
-
\((b)\)
:
-
If
\({\psi }^{\prime }(0)=0\), then
$$\begin{aligned} &{\mathbb{S}_{2}}({\psi },f,d)+{\psi }(d)\\ &\quad =\frac{{\psi ^{\prime }(0)- \psi ^{\prime }(d)}}{{d}} \int _{0}^{d}\mathbb{S}_{2} \bigl(G_{\ast ,3}(\cdot,s),f,d\bigr)\,ds \\ &\qquad {}+\frac{1}{{d}} \int _{0}^{d}\mathbb{S}_{2} \bigl(G_{\ast ,3}(\cdot,s),f,d\bigr){ \Biggl( {\sum _{l=2}^{n-1}{\frac{l}{{(l-1)!}} \bigl( {\psi {^{ ( l ) }} ( 0 ) {{ ( {s} ) }^{l-1}}-\psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) } ^{l-1}}} \bigr) }} \Biggr) }\,ds \\ &\qquad {}+\frac{1}{{ ( {n-3} ) !}} \int _{0}^{d}\psi {{^{(n)}}}(v) \biggl( \int _{0}^{d}{\mathbb{S}_{2}} \bigl(G_{\ast ,3}(\cdot,s),f,d\bigr){{\tilde{R}} _{0,n-2}} ( {s,v} ) \,ds \biggr) \,dv. \end{aligned}$$
-
\((c)\)
:
-
$$\begin{aligned} &{\mathbb{S}_{2}}({\psi },f,d)+{\psi }(d)-d{\psi }^{\prime }(d)\\ &\quad =\frac{ {\psi ^{\prime }(0)-\psi ^{\prime }(d)}}{{d}} \int _{0}^{d}\mathbb{S}_{2}\bigl(G _{\ast ,4}(\cdot,s),f,d\bigr)\,ds \\ &\qquad {}+\frac{1}{{d}} \int _{0}^{d}\mathbb{S}_{2} \bigl(G_{\ast ,4}(\cdot,s),f,d\bigr){ \Biggl( {\sum _{l=2}^{n-1}{\frac{l}{{(l-1)!}} \bigl( {\psi {^{ ( l ) }} ( 0 ) {{ ( {s} ) }^{l-1}}-\psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) } ^{l-1}}} \bigr) }} \Biggr) }\,ds \\ &\qquad {}+ \frac{1}{{ ( {n-3} ) !}} \int _{0}^{d}\psi {{^{(n)}}}(v) \biggl( \int _{0}^{d}{\mathbb{S}_{2}} \bigl(G_{\ast ,4}(\cdot,s),f,d\bigr){{\tilde{R}} _{0,n-2}} ( {s,v} ) \,ds \biggr) \,dv. \end{aligned}$$
-
\((d)\)
:
-
If
\({\psi }^{\prime }(0)=0\), then
$$\begin{aligned} &{\mathbb{S}_{2}}({\psi },f,d)-d{\psi }^{\prime }(d)\\ &\quad ={ \frac{{\psi ^{ \prime }(0)-\psi ^{\prime }(d)}}{{d}} \int _{0}^{d}\mathbb{S}_{2}} \bigl(G_{ \ast ,5}(\cdot,s),f,d\bigr)\,ds \\ &\qquad {}+\frac{1}{{d}} \int _{0}^{d}\mathbb{S}_{2} \bigl(G_{\ast ,5}(\cdot,s),f,d\bigr){ \Biggl( {\sum _{l=2}^{n-1}{\frac{l}{{(l-1)!}} \bigl( {\psi {^{ ( l ) }} ( 0 ) {{ ( {s} ) }^{l-1}}-\psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) } ^{l-1}}} \bigr) }} \Biggr) }\,ds \\ &\qquad {}+\frac{1}{{ ( {n-3} ) !}} \int _{0}^{d}\psi {{^{(n)}}}(v) \biggl( \int _{0}^{d}{\mathbb{S}_{2}} \bigl(G_{\ast ,5}(\cdot,s),f,d\bigr){{\tilde{R}} _{0,n-2}} ( {s,v} ) \,ds \biggr) \,dv, \end{aligned}$$
where
\(\tilde{R}_{0,n-2}\)
is obtained by taking
\(c=0\)
in (29), and
-
\((a^{\prime })\)
:
-
$$\begin{aligned} {\mathbb{S}_{2}}({\psi },f,d)&= \biggl( {\frac{{\psi ^{\prime }(d)- \psi ^{\prime }(0)}}{{d}}} \biggr) \int _{0}^{d}\mathbb{S}_{2} \bigl(G_{ \ast ,j}(\cdot,s),f,d\bigr)\,ds \\ &\quad {} +\frac{1}{{d}} \int _{0}^{d}\mathbb{S}_{2} \bigl(G_{\ast ,j}(\cdot,s),f,d\bigr) { \Biggl( {\sum _{l=3}^{n-1}{\frac{{\psi {^{ ( l ) }} ( 0 ) {{ ( {s} ) }^{l-1}}-\psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) }^{l-1}}}}{ {(l-3)!(l-1)}}}} \Biggr) }\,ds \\ &\quad {}+\frac{1}{{ ( {n-3} ) !}} \int _{0}^{d}\psi {^{ ( n ) }} ( v ) \biggl( { \int _{0}^{d}\mathbb{S}_{2}\bigl(G _{\ast ,j}(\cdot,s),f,d\bigr){R_{0,n-2}}(s,v)\,ds} \biggr) \,dv \end{aligned}$$
for
\(j=1,2\).
-
\((b^{\prime })\)
:
-
If
\({\psi }^{\prime }(0)=0\), then
$$\begin{aligned} &{\mathbb{S}_{2}}({\psi },f,d)+{\psi }(d)\\ &\quad = \biggl( { \frac{{\psi ^{ \prime }(d)-\psi ^{\prime }(0)}}{{d}}} \biggr) \int _{0}^{d}\mathbb{S} _{2} \bigl(G_{\ast ,3}(\cdot,s),f,d\bigr)\,ds \\ &\qquad {} +\frac{1}{{d}} \int _{0}^{d}\mathbb{S}_{2} \bigl(G_{\ast ,3}(\cdot,s),f,d\bigr) { \Biggl( {\sum _{l=3}^{n-1}{\frac{{\psi {^{ ( l ) }} ( 0 ) {{ ( {s} ) }^{l-1}}-\psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) }^{l-1}}}}{ {(l-3)!(l-1)}}}} \Biggr) }\,ds \\ &\qquad {}+\frac{1}{{ ( {n-3} ) !}} \int _{0}^{d}\psi {^{ ( n ) }} ( v ) \biggl( { \int _{0}^{d}\mathbb{S}_{2}\bigl(G _{\ast ,3}(\cdot,s),f,d\bigr){R_{0,n-2}}(s,v)\,ds} \biggr) \,dv. \end{aligned}$$
-
\((c^{\prime })\)
:
-
$$\begin{aligned} &{\mathbb{S}_{2}}({\psi },f,d)+{\psi }(d)-d{\psi }^{\prime }(d)\\ &\quad = \biggl( {\frac{{\psi ^{\prime }(d)-\psi ^{\prime }(0)}}{{d}}} \biggr) \int _{0} ^{d}\mathbb{S}_{2} \bigl(G_{\ast ,4}(\cdot,s),f,d\bigr)\,ds \\ &\qquad {} +\frac{1}{{d}} \int _{0}^{d}\mathbb{S}_{2} \bigl(G_{\ast ,4}(\cdot,s),f,d\bigr) { \Biggl( {\sum _{l=3}^{n-1}{\frac{{\psi {^{ ( l ) }} ( 0 ) {{ ( {s} ) }^{l-1}}-\psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) }^{l-1}}}}{ {(l-3)!(l-1)}}}} \Biggr) }\,ds \\ &\qquad {}+\frac{1}{{ ( {n-3} ) !}} \int _{0}^{d}\psi {^{ ( n ) }} ( v ) \biggl( { \int _{0}^{d}\mathbb{S}_{2}\bigl(G _{\ast ,4}(\cdot,s),f,d\bigr){R_{0,n-2}}(s,v)\,ds} \biggr) \,dv. \end{aligned}$$
-
\((d^{\prime })\)
:
-
If
\({\psi }^{\prime }(0)=0\), then
$$\begin{aligned} &{\mathbb{S}_{2}}({\psi },f,d)-d{\psi }^{\prime }(d)\\ &\quad = \biggl( {\frac{ {\psi ^{\prime }(d)-\psi ^{\prime }(0)}}{{d}}} \biggr) \int _{0}^{d} \mathbb{S}_{2} \bigl(G_{\ast ,5}(\cdot,s),f,d\bigr)\,ds \\ &\qquad {} +\frac{1}{{d}} \int _{0}^{d}\mathbb{S}_{2} \bigl(G_{\ast ,5}(\cdot,s),f,d\bigr) { \Biggl( {\sum _{l=3}^{n-1}{\frac{{\psi {^{ ( l ) }} ( 0 ) {{ ( {s} ) }^{l-1}}-\psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) }^{l-1}}}}{ {(l-3)!(l-1)}}}} \Biggr) }\,ds \\ &\qquad {}+\frac{1}{{ ( {n-3} ) !}} \int _{0}^{d}{\psi {^{ ( n ) }} ( v ) \biggl( { \int _{0}^{d}\mathbb{S}_{2}\bigl(G _{\ast ,5}(\cdot,s),f,d\bigr){R_{0,n-2}}(s,v)\,ds} \biggr)\,dv}, \end{aligned}$$
where
\({R_{0,n-2}}\)
is obtained by taking
\(c=0\)
in (26).
Proof
We give the proof of our results by fixing \(j=1\), other cases can be followed in a similar pattern. By using (8) and (18) for ψ and \(\psi ^{\prime }\) respectively and applying assumption \(\psi (0)=0\), we get
$$ {\mathbb{S}_{2}}({\psi },f,d)= \int _{0}^{d}{\mathbb{S}_{2}} \bigl(G_{\ast ,1}(\cdot,s),f,d\bigr) {\psi }^{\prime \prime }(s) \,ds. $$
The rest is a similar application of (25) as in the proof of Theorem 4. □
Similar to Theorem 5 (from Theorem 4), we may get the following theorem (from Theorem 7).
Theorem 8
Consider (\(A_{2}\)) and let
f
be as in Corollary 2\((i)\). If
ψ
is
n-convex and
$$ { \int _{0}^{d}{\mathbb{S}_{2}} \bigl(G_{\ast ,j}(\cdot,s),f,d\bigr){{\tilde{R}}_{0,n-2}} ( {s,v} ) \,ds\geq 0{,}} $$
(41)
then
$$ \begin{aligned}[b]& {{\mathbb{S}_{2}}}({\psi },f,d)\\ &\quad \geq \frac{{\psi ^{\prime }(0)- \psi ^{\prime }(d)}}{{d}} \int _{0}^{d}{\mathbb{S}_{2}} \bigl(G_{\ast ,j}(\cdot,s),f,d\bigr)\,ds \\ &\qquad {}+\frac{1}{{d}} \int _{0}^{d}{\mathbb{S}_{2}} \bigl(G_{\ast ,j}(\cdot,s),f,d\bigr) { \Biggl( {\sum _{l=2}^{n-1}{\frac{l}{{(l-1)!}} \bigl( {\psi {^{ ( l ) }} ( 0 ) {{ ( {s} ) }^{l-1}}- \psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) }^{l-1}}} \bigr) }} \Biggr) } \,ds \end{aligned} $$
(42)
and if
$$ \int _{0}^{d}{\mathbb{S}_{2}} \bigl(G_{\ast ,j}(\cdot,s),f,d\bigr){R_{0,n-2}}(s,v)\,ds \geq 0, $$
(43)
then
$$ \begin{aligned}[b] {{\mathbb{S}_{2}}}({\psi },f,d)&\geq \biggl( {\frac{{\psi ^{\prime }(d)- \psi ^{\prime }(0)}}{{d}}} \biggr) \int _{0}^{d}{\mathbb{S}_{2}} \bigl(G_{ \ast ,j}(\cdot,s),f,d\bigr)\,ds \\ &\quad {} +\frac{1}{{d}} \int _{0}^{d}{\mathbb{S}_{2}} \bigl(G_{\ast ,j}(\cdot,s),f,d\bigr) { \Biggl( {\sum _{l=3}^{n-1}{\frac{{{f^{ ( l ) }} ( 0 ) {{ ( {s} ) }^{l-1}}-f{^{ ( l ) }} ( d ) {{ ( {s-d} ) }^{l-1}}}}{{(l-3)!(l-1)}}}} \Biggr) }\,ds \end{aligned} $$
(44)
for
\(j=1,2\), where
\({\tilde{R}}_{0,n-2}\)
and
\({R_{0,n-2}}\)
are as described in Theorem 7.
The following theorem yields generalization of (6).
Theorem 9
Consider (\(A_{2}\)) and let
f
be as in Corollary 2\((i)\). If
ψ
and
-
(i)
If
n
is even and
\(n\geq 4\), then (42) holds.
-
(ii)
Let inequality (42) be satisfied and
$$ {\sum_{l=1}^{n-1}{\frac{l}{{(l-1)!}} \bigl( {\psi {^{ ( l ) }} ( 0 ) {{ ( {s} ) }^{l-1}}-\psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) } ^{l-1}}} \bigr) }};\quad \forall s\in {}[ 0,d], $$
(45)
OR
(44) be satisfied and
$$ {\psi }^{\prime }(d)-{\psi }^{\prime }(0)+{\sum _{l=3}^{n-1} {\frac{{{\psi ^{ ( l ) }} ( 0 ) {{ ( {s} ) }^{l-1}}-\psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) }^{l-1}}}}{{(l-3)!(l-1)}}}}\geq 0 \quad \forall s\in {}[ 0,d]{,} $$
(46)
then we have
$$ {{\mathbb{S}_{2}}}({\psi },f,d)\geq 0. $$
(47)
Proof
The proof is similar to that of Theorem 6 except for the use of Theorem 8 and Corollary 2\((a)\). □
Remark 3
Inequalities (42) and (44) hold in reverse directions if either the inequalities in (41) and (43) are reversed or −ψ is n-convex. By using these reverse inequalities and applying a similar technique as in Theorem 9, one may prove \({{\mathbb{S}_{2}}}({\psi },f,d)\leq 0\), which gives the reverse of (6).
For the generalization of (7), we construct the following identities.
Theorem 10
Consider (\(A_{2}\)) and let
f
be as in Corollary 2\((i)\), then
-
\((a)\)
:
-
$$\begin{aligned} {{\mathbb{S}_{3}}}({\psi },f,\lambda ,d)&=\frac{{\psi ^{\prime }(0)- \psi ^{\prime }(d)}}{{d}} \int _{0}^{d}{\mathbb{S}_{3}} \bigl(G_{\ast ,j}(\cdot,s),f, \lambda ,d\bigr)\,ds \\ &\quad {}+\frac{1}{{d}} \int _{0}^{d}{\mathbb{S}_{3}} \bigl(G_{\ast ,j}(\cdot,s),f, \lambda ,d\bigr)\\ &\quad {}\times{ \Biggl( {\sum _{l=2}^{n-1}{\frac{l}{{(l-1)!}} \bigl( {\psi {^{ ( l ) }} ( 0 ) {{ ( {s} ) } ^{l-1}}-\psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) }^{l-1}}} \bigr) }} \Biggr) } \,ds \\ &\quad {}+\frac{1}{{ ( {n-3} ) !}} \int _{0}^{d}\psi {{^{(n)}}}(v) \biggl( \int _{0}^{d}{\mathbb{S}_{3}} \bigl(G_{\ast ,j}(\cdot,s),f,\lambda ,d\bigr){{\tilde{R}} _{0,n-2}} ( {s,v} ) \,ds\biggr)\,dv \end{aligned}$$
for
\(j=1,2\).
-
\((b)\)
:
-
If
\({\psi }^{\prime }(0)=0\), then
$$\begin{aligned} &{{\mathbb{S}_{3}}}({\psi },f,\lambda ,d)+{\psi }(d) \int _{0}^{d}\lambda (x) \,dx\\ &\quad ={ \frac{{\psi ^{\prime }(0)-\psi ^{\prime }(d)}}{{d}} \int _{0} ^{d}{\mathbb{S}_{3}}} \bigl(G_{\ast ,3}(\cdot,s),f,\lambda ,d\bigr)\,ds \\ &\qquad {}+\frac{1}{{d}} \int _{0}^{d}{\mathbb{S}_{3}} \bigl(G_{\ast ,3}(\cdot,s),f,\lambda ,d\bigr){ \Biggl( {\sum _{l=2}^{n-1}{\frac{l}{{(l-1)!}} \bigl( {\psi {^{ ( l ) }} ( 0 ) {{ ( {s} ) }^{l-1}}- \psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) }^{l-1}}} \bigr) }} \Biggr) } \\ &\qquad {}+\frac{1}{{ ( {n-3} ) !}} \int _{0}^{d}\psi {{^{(n)}}}(v) \biggl( \int _{0}^{d}{\mathbb{S}_{3}} \bigl(G_{\ast ,3}(\cdot,s),f,\lambda ,d\bigr){{\tilde{R}} _{0,n-2}} ( {s,v} ) \,ds\biggr)\,dv. \end{aligned}$$
-
\((c)\)
:
-
$$\begin{aligned} &{{\mathbb{S}_{3}}}({\psi },f,\lambda ,d)+\bigl({\psi }(d)-d{\psi }^{\prime }(d)\bigr) \int _{0}^{d}\lambda (x) \,dx\\ &\quad ={ \frac{{\psi ^{\prime }(0)-\psi ^{ \prime }(d)}}{{d}} \int _{0}^{d}{\mathbb{S}_{3}}} \bigl(G_{\ast ,4}(\cdot,s),f, \lambda ,d\bigr)\,ds \\ &\qquad {}+\frac{1}{{d}} \int _{0}^{d}{\mathbb{S}_{3}} \bigl(G_{\ast ,4}(\cdot,s),f,\lambda ,d\bigr)\\ &\qquad {}\times{ \Biggl( {\sum _{l=2}^{n-1}{\frac{l}{{(l-1)!}} \bigl( {\psi {^{ ( l ) }} ( 0 ) {{ ( {s} ) }^{l-1}}- \psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) }^{l-1}}} \bigr) }} \Biggr) } \,ds \\ &\qquad {}+\frac{1}{{ ( {n-3} ) !}} \int _{0}^{d}\psi {{^{(n)}}}(v) \biggl( \int _{0}^{d}{\mathbb{S}_{3}} \bigl(G_{\ast ,4}(\cdot,s),h,k,b\bigr){{\tilde{R}}_{0,n-2}} ( {s,v} ) \,ds\biggr)\,dv. \end{aligned}$$
-
\((d)\)
:
-
If
\({\psi }^{\prime }(0)=0\), then
$$\begin{aligned} &{{\mathbb{S}_{3}}}({\psi },f,\lambda ,d)-d{\psi }^{\prime }(d) \int _{0}^{d}\lambda (x) \,dx\\ &\quad ={ \frac{{\psi ^{\prime }(0)-\psi ^{\prime }(d)}}{ {d}} \int _{0}^{d}{\mathbb{S}_{3}}} \bigl(G_{\ast ,5}(\cdot,s),f,\lambda ,d\bigr)\,ds \\ &\qquad {}+\frac{1}{{d}} \int _{0}^{d}{\mathbb{S}_{3}} \bigl(G_{\ast ,5}(\cdot,s),f,\lambda ,d\bigr)\\ &\qquad {}\times{ \Biggl( {\sum _{l=2}^{n-1}{\frac{l}{{(l-1)!}} \bigl( {\psi {^{ ( l ) }} ( 0 ) {{ ( {s} ) }^{l-1}}- \psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) }^{l-1}}} \bigr) }} \Biggr) } \,ds \\ &\qquad {}+\frac{1}{{ ( {n-3} ) !}} \int _{0}^{d}\psi {{^{(n)}}}(v) \biggl( \int _{0}^{d}{\mathbb{S}_{3}} \bigl(G_{\ast ,5}(\cdot,s),f,\lambda ,d\bigr){{\tilde{R}} _{0,n-2}} ( {s,v} ) \,ds\biggr)\,dv \end{aligned}$$
and
-
\((a^{\prime })\)
:
-
$$\begin{aligned} &{{\mathbb{S}_{3}}}({\psi },f,\lambda ,d)\\ &\quad = \biggl( { \frac{{\psi ^{ \prime }(d)-\psi ^{\prime }(0)}}{{d}}} \biggr) \int _{0}^{d}{\mathbb{S} _{3}} \bigl(G_{\ast ,j}(\cdot,s),f,\lambda ,d\bigr)\,ds \\ &\qquad {} +\frac{1}{{d}} \int _{0}^{d}{\mathbb{S}_{3}} \bigl(G_{\ast ,j}(\cdot,s),f, \lambda ,d\bigr){ \Biggl( {\sum _{l=3}^{n-1}{\frac{{\psi {^{ ( l ) }} ( 0 ) {{ ( {s} ) }^{l-1}}-\psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) } ^{l-1}}}}{{(l-3)!(l-1)}}}} \Biggr) }\,ds \\ &\qquad {}+\frac{1}{{ ( {n-3} ) !}} \int _{0}^{d}{\psi {^{ ( n ) }} ( v ) \biggl( { \int _{0}^{d}{\mathbb{S}_{3}} \bigl(G _{\ast ,j}(\cdot,s),f,\lambda ,d\bigr){R_{0,n-2}}(s,v) \,ds} \biggr) \,dv} \end{aligned}$$
for
\(j=1,2\).
-
\((b^{\prime })\)
:
-
If
\({\psi }^{\prime }(0)=0\), then
$$\begin{aligned} &{{\mathbb{S}_{3}}}({\psi },f,\lambda ,d)+{\psi }(d) \int _{0}^{d}\lambda (x) \,dx\\ &\quad = \biggl( { \frac{{\psi ^{\prime }(d)-\psi ^{\prime }(0)}}{{d}}} \biggr) \int _{0} ^{d}{\mathbb{S}_{3}} \bigl(G_{\ast ,3}(\cdot,s),f,\lambda ,d\bigr)\,ds \\ &\qquad {} +\frac{1}{{d}} \int _{0}^{d}{\mathbb{S}_{3}} \bigl(G_{\ast ,3}(\cdot,s),f, \lambda ,d\bigr){ \Biggl( {\sum _{l=3}^{n-1}{\frac{{\psi {^{ ( l ) }} ( 0 ) {{ ( {s} ) }^{l-1}}-\psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) } ^{l-1}}}}{{(l-3)!(l-1)}}}} \Biggr) }\,ds \\ &\qquad {}+\frac{1}{{ ( {n-3} ) !}} \int _{0}^{d}{\psi {^{ ( n ) }} ( v ) \biggl( { \int _{0}^{d}{\mathbb{S}_{3}} \bigl(G _{\ast ,3}(\cdot,s),f,\lambda ,d\bigr){R_{0,n-2}}(s,v) \,ds} \biggr) \,dv}. \end{aligned}$$
-
\((c^{\prime })\)
:
-
$$\begin{aligned} &{{\mathbb{S}_{3}}}({\psi },f,\lambda ,d)+\bigl({\psi }(d)-d{\psi }^{\prime }(d)\bigr) \int _{0}^{d}\lambda (x) \,dx\\ &\quad = \biggl( { \frac{{\psi ^{\prime }(d)- \psi ^{\prime }(0)}}{{d}}} \biggr) \int _{0}^{d}{\mathbb{S}_{3}} \bigl(G_{ \ast ,4}(\cdot,s),f,\lambda ,d\bigr)\,ds \\ &\qquad {} +\frac{1}{{d}} \int _{0}^{d}{\mathbb{S}_{3}} \bigl(G_{\ast ,4}(\cdot,s),f, \lambda ,d\bigr){ \Biggl( {\sum _{l=3}^{n-1}{\frac{{\psi {^{ ( l ) }} ( 0 ) {{ ( {s} ) }^{l-1}}-\psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) } ^{l-1}}}}{{(l-3)!(l-1)}}}} \Biggr) }\,ds \\ &\qquad {}+\frac{1}{{ ( {n-3} ) !}} \int _{0}^{d}{\psi {^{ ( n ) }} ( v ) \biggl( { \int _{0}^{d}{\mathbb{S}_{3}} \bigl(G _{\ast ,4}(\cdot,s),f,\lambda ,d\bigr){R_{0,n-2}}(s,v) \,ds} \biggr) \,dv}. \end{aligned}$$
-
\((d^{\prime })\)
:
-
If
\({\psi }^{\prime }(0)=0\), then
$$\begin{aligned} &{{\mathbb{S}_{3}}}({\psi },f,\lambda ,d)-d{\psi }^{\prime }(d) \int _{0}^{d}\lambda (x) \,dx\\ &\quad = \biggl( { \frac{{\psi ^{\prime }(d)-\psi ^{ \prime }(0)}}{{d}}} \biggr) \int _{0}^{d}{\mathbb{S}_{3}} \bigl(G_{\ast ,5}(\cdot,s),f, \lambda ,d\bigr)\,ds \\ &\qquad {} +\frac{1}{{d}} \int _{0}^{d}{\mathbb{S}_{3}} \bigl(G_{\ast ,5}(\cdot,s),f, \lambda ,d\bigr){ \Biggl( {\sum _{l=3}^{n-1}{\frac{{\psi {^{ ( l ) }} ( 0 ) {{ ( {s} ) }^{l-1}}-\psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) } ^{l-1}}}}{{(l-3)!(l-1)}}}} \Biggr) }\,ds \\ &\qquad {}+\frac{1}{{ ( {n-3} ) !}} \int _{0}^{d}{\psi {^{ ( n ) }} ( v ) \biggl( { \int _{0}^{d}{\mathbb{S}_{3}} \bigl(G _{\ast ,5}(\cdot,s),f,\lambda ,d\bigr){R_{0,n-2}}(s,v) \,ds} \biggr) \,dv}, \end{aligned}$$
where
\(\tilde{R}_{0,n-2}\)
and
\({R_{0,n-2}}\)
are as described in Theorem 7.
Proof
We give the proof of our results by fixing \(j=1\), other cases can be followed in a similar pattern. By using (8) and (18) for ψ and \({\psi }^{\prime }\), \({\psi (0)=0}\) and \(\int _{0} ^{d}\lambda (x)\int _{0}^{x}f(t) \,dt \,dx=\int _{0}^{d}f(t) ( \int _{t}^{b}\lambda (x) \,dx ) \,dt=\int _{0}^{d}\varLambda (t)f(t)\,dt\), we get
$$ {{\mathbb{S}_{3}}}({\psi },f,\lambda ,d)= \int _{0}^{d}{{\mathbb{S}_{3}}} \bigl(G _{\ast ,1}(\cdot,s),f,\lambda ,d\bigr){\psi }^{\prime \prime }(s) \,ds. $$
The rest is a similar application of (25). □
Similar to Theorem 5 (from Theorem 4), we may get the following theorem (from Theorem 10).
Theorem 11
Consider (\(A_{2}\)) and let
f
be as in Corollary 2\((i)\). If
ψ
is
n-convex and
$$ { \int _{0}^{d}{\mathbb{S}_{3}} \bigl(G_{\ast ,j}(\cdot,s),f,\lambda ,d\bigr){{\tilde{R}} _{0,n-2}} ( {s,v} ) \,ds\geq 0}, $$
(48)
then
$$ \begin{aligned}[b] {{\mathbb{S}_{3}}}({\psi },f,\lambda ,d)&\geq \frac{{\psi ^{\prime }(0)- \psi ^{\prime }(d)}}{{d}} \int _{0}^{d}{\mathbb{S}_{3}} \bigl(G_{\ast ,j}(\cdot,s),f, \lambda ,d\bigr)\,ds \\ &\quad {}+\frac{1}{{d}} \int _{0}^{d}{\mathbb{S}_{3}} \bigl(G_{\ast ,j}(\cdot,s),f,\lambda ,d\bigr)\\ &\quad {}\times{ \Biggl( {\sum _{l=2}^{n-1}{\frac{l}{{(l-1)!}} \bigl( {\psi {^{ ( l ) }} ( 0 ) {{ ( {s} ) }^{l-1}}- \psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) }^{l-1}}} \bigr) }} \Biggr) } \,ds; \end{aligned} $$
(49)
and if
$$ { \int _{0}^{d}{\mathbb{S}_{3}} \bigl(G_{\ast ,j}(\cdot,s),f,\lambda ,d\bigr){R_{0,n-2}}(s,v)\,ds \geq 0}, $$
(50)
then
$$ \begin{aligned}[b] {{\mathbb{S}_{3}}}({\psi },f,\lambda ,d)&\geq \biggl( {\frac{{\psi ^{ \prime }(d)-\psi ^{\prime }(0)}}{{d}}} \biggr) \int _{0}^{d}{\mathbb{S} _{3}} \bigl(G_{\ast ,j}(\cdot,s),f,\lambda ,d\bigr)\,ds \\ &\quad {} +\frac{1}{{d}} \int _{0}^{d}{\mathbb{S}_{3}} \bigl(G_{\ast ,j}(\cdot,s),f, \lambda ,d\bigr)\\ &\quad {}\times{ \Biggl( {\sum _{l=3}^{n-1}{\frac{{\psi {^{ ( l ) }} ( 0 ) {{ ( {s} ) }^{l-1}}-\psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) } ^{l-1}}}}{{(l-3)!(l-1)}}}} \Biggr) }\,ds \end{aligned} $$
(51)
for
\(j=1,2\), where
\(\tilde{R}_{0,n-2}\)
and
\({R_{0,n-2}}\)
are as described in Theorem 7.
In the next theorem, we prove generalization of (7).
Theorem 12
Consider (\(A_{2}\)) and let
f, λ, Λ
be as in Corollary 3\((i)\). If
ψ
is
n-convex and
-
(i)
If
n
is even and
\(n\geq 4\), then (49) holds.
-
(ii)
Let inequality (49) be satisfied and
$$ {\sum_{l=1}^{n-1}{\frac{l}{{(l-1)!}} \bigl( {\psi {^{ ( l ) }} ( 0 ) {{ ( {s} ) }^{l-1}}-\psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) } ^{l-1}}} \bigr) }};\quad \forall s\in {}[ 0,d], $$
(52)
OR
(51) be satisfied and
$$ {\psi }^{\prime }(d)-{\psi }^{\prime }(0)+{\sum _{l=3}^{n-1} {\frac{{\psi {^{ ( l ) }} ( 0 ) {{ ( {s} ) }^{l-1}}-\psi {^{ ( l ) }} ( d ) {{ ( {s-d} ) }^{l-1}}}}{{(l-3)!(l-1)}}}}\geq 0;\quad \forall s\in {}[ 0,d]. $$
(53)
Then we have
$$ {{\mathbb{S}_{3}}}({\psi },f,\lambda ,d)\geq 0. $$
(54)
Proof
The proof is an application of Theorem 10, Theorem 11, and Corollary 3\((a)\). □
Remark 4
Inequalities (49) and (51) hold in reverse directions if either the inequalities in (48) and (50) are reversed or −ψ is n-convex. By using these reverse inequalities and applying similar technique as in Theorem 12, one may prove \({{\mathbb{S}_{3}}}({\psi },f,\lambda ,d) \leq 0\), which gives the reverse of (7).
Remark 5
Theorem 8 and Theorem 11 have been proved for \(j=1,2\). Both theorems can be extended to \(j=1,2,\ldots , 5\) according to each case in Theorem 7 and Theorem 10, respectively, which ultimately produce inequalities (as given in Theorem 9 and Theorem 12) related to the generalizations of (6) and (7) (and their reverse).