Based on Lemma 3, some (G. J. M)-type inequalities can be represented pertaining local fractional integral forms as follows.
Theorem 7
Assume that \(\Psi \in GHK_{\varrho }(I)\) on the interval \(I=[\mathfrak{m} , \mathfrak{M}] \subset (0, \infty )\), then the following inequality for local fractional integrals holds:
$$\begin{aligned} \Psi \biggl( \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }- \frac{ 1}{\bar{a}}} \biggr) & \leq \underset{\imath =1 }{\overset{s}{\sum }} \gamma _{ \imath } ^{\varrho } \Psi \biggl( \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }- ( \frac{1-\vartheta }{\bar{a}}+\frac{\vartheta }{a_{\imath } } )} \biggr) \\ & \leq \Psi (\mathfrak{M})+\Psi (\mathfrak{m} )- \underset{\imath =1 }{ \overset{s}{\sum }} \gamma _{\imath }^{\varrho }\Psi (a_{\imath }) \end{aligned}$$
(6)
for all \(a_{\imath } \in [ \mathfrak{m} ,\mathfrak{M} ]\) and \(\gamma _{\imath }\in [0,1 ]\) (\(\imath =1 ,2,\ldots ,s \)) with \(\underset{\imath =1 }{\overset{s}{\sum }} \gamma _{\imath }=1\), where \(\frac{1}{\bar{a}}=\underset{\imath =1 }{\overset{s}{\sum }} \gamma _{ \imath }\frac{1}{a_{\imath }}\).
Proof
Since \(\Psi \in GHK_{\varrho }(I)\), we have
$$\begin{aligned} \underset{\imath =1 }{\overset{s}{\sum }} \gamma _{\imath } ^{\varrho } \Psi \biggl( \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }- ( \frac{1-\vartheta }{\bar{a}}+\frac{\vartheta }{a_{\imath } } )} \biggr)& \geq \Psi \biggl( \frac{1}{ \underset{\imath =1 }{\overset{s}{\sum }} \gamma _{\imath } ( \frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }- ( \frac{1-\vartheta }{\bar{a}}+\frac{\vartheta }{a_{\imath } } ) )} \biggr) \\ &= \Psi \biggl( \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }- \frac{1}{\bar{a}}+\frac{\vartheta }{\bar{a}}-\underset{\imath =1 }{\overset{s}{\sum }} \gamma _{\imath } \frac{\vartheta }{a_{\imath } }} \biggr) \\ & = \Psi \biggl( \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }- \frac{ 1}{\bar{a}}} \biggr). \end{aligned}$$
(7)
On the other hand,
$$\begin{aligned} &\underset{\imath =1 }{\overset{s}{\sum }} \gamma _{\imath } ^{\varrho } \Psi \biggl( \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }- ( \frac{1-\vartheta }{\bar{a}}+\frac{\vartheta }{a_{\imath } } )} \biggr) \\ &\quad = \underset{\imath =1 }{\overset{s}{\sum }} \gamma _{\imath } ^{\varrho } \Psi \biggl( \frac{1}{(1-\vartheta ) ( \frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{ \bar{a}} )+ \vartheta ( \frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{ a_{\imath }} )} \biggr) \\ &\quad \leq \underset{\imath =1 }{\overset{s}{\sum }} \gamma _{\imath } ^{ \varrho } \biggl[ (1-\vartheta )^{\varrho }\Psi \biggl( \frac{1}{ \frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{\bar{a}} } \biggr) +\vartheta ^{\varrho } \Psi \biggl( \frac{1}{ \frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{a_{\imath } } } \biggr) \biggr] \\ &\quad \leq \underset{\imath =1 }{\overset{s}{\sum }} \gamma _{\imath } ^{ \varrho } \Bigl[ (1-\vartheta )^{\varrho } \Bigl(\Psi ( \mathfrak{M})+ \Psi (\mathfrak{m} )- \underset{j=1}{\overset{s}{\sum }} \gamma _{j}^{ \varrho }\Psi (a_{j}) \Bigr) \\ &\qquad {} +\vartheta ^{\varrho } \bigl( \Psi (\mathfrak{M})+ \Psi (\mathfrak{m} )- \Psi (a_{\imath } ) \bigr) \Bigr] \\ &\quad = \Psi (\mathfrak{M})+\Psi (\mathfrak{m} )- \underset{\imath =1 }{\overset{s}{\sum }} \gamma _{\imath }^{\varrho } \Psi (a_{\imath }) . \end{aligned}$$
(8)
Combining inequalities (7) and (8), we get (6). □
Remark 3
If we take \(\varrho =1\) in Theorem 7, then it gives [[33], Theorem 2.3].
Corollary 1
Let all assumptions of Theorem 7hold and \(\Psi (x) \in I^{(\varrho )}_{x}[\mathfrak{m} ,\mathfrak{M}]\), then
$$\begin{aligned} \begin{aligned} \Psi \biggl( \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }- \frac{ 1}{\bar{a}}} \biggr) &\leq \underset{\imath =1 }{\overset{s}{\sum }} \frac{ \gamma _{\imath } ^{\varrho } \Gamma ( 1+\varrho )}{ (\frac{1}{a_{\imath }}-\frac{1}{\bar{a}} )^{\varrho }} {}_{ \frac{1}{ \frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{\bar{a}} } } I^{(\varrho )}_{ \frac{1}{ \frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{a_{\imath } } } }\frac{ \Psi (x)}{x^{2\varrho }} \\ & \leq \Psi (\mathfrak{M})+\Psi (\mathfrak{m} )- \underset{\imath =1 }{ \overset{s}{\sum }} \gamma _{\imath }^{\varrho } \Psi (a_{\imath }) . \end{aligned} \end{aligned}$$
(9)
Proof
Multiplying by \(\frac{1}{\Gamma ( 1+\varrho )}\) on both sides of (6) and then integrating w.r.t ϑ over \([0,1]\), we get (9). Here, we used the fact
$$\begin{aligned} & \frac{1}{\Gamma ( 1+\varrho )} \int _{0}^{1} \Psi \biggl( \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }- ( \frac{1-\vartheta }{\bar{a}}+\frac{\vartheta }{a_{\imath } } )} \biggr) (d\vartheta )^{\varrho } \\ &\quad = \frac{1}{\Gamma ( 1+\varrho )} \int _{0}^{1} \Psi \biggl( \frac{1}{ (1-\vartheta )\Psi (\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{\bar{a}} ) +\vartheta \Psi ( \frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{a_{\imath } } ) } \biggr) (d\vartheta )^{\varrho } \\ &\quad = \frac{1}{ (\frac{1}{a_{\imath }}-\frac{1}{\bar{a}} )^{\varrho }} \biggl[\frac{1}{\Gamma ( 1+\varrho )} \int _{ \frac{1}{ \frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{\bar{a}} }}^{ \frac{1}{ \frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{a_{\imath } } }} \frac{ \Psi (t)}{t^{2\varrho }}(dt)^{\varrho } \biggr] \\ &\quad = \frac{1}{ (\frac{1}{a_{\imath }}-\frac{1}{\bar{a}} )^{\varrho }} {}_{ \frac{1}{ \frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{\bar{a}} } } I^{(\varrho )}_{ \frac{1}{ \frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{a_{\imath } } } } \frac{ \Psi (x)}{x^{2\varrho }} . \end{aligned}$$
□
Remark 4
If we take \(\varrho =1\) in Corollary 1, then it gives [[33], Corollary 2.4] and if put \(\varrho =1\), \(s=2\), \(\gamma _{1}=\gamma _{2}=\frac{1}{2}\), \(a_{1}=a\), and \(a_{2}=b\) in Corollary 1, then it gives [[33], Theorem 2.1].
Now, we present a more precise estimate in the following theorem.
Theorem 8
Assume that \(\Psi \in GHK_{\varrho }(I)\) on the interval \(I=[\mathfrak{m} , \mathfrak{M}] \subset (0, \infty )\), then the following inequality for local fractional integrals holds:
$$\begin{aligned} \begin{aligned} \Psi \biggl( \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }- \frac{ 1}{\bar{a}}} \biggr) & \leq \underset{\imath =1 }{\overset{s}{\sum }} \gamma _{ \imath } ^{\varrho } \Psi \biggl( \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{2} ( \frac{1}{\bar{a}}+\frac{1}{a_{\imath } } )} \biggr) \\ & \leq \underset{\imath =1 }{\overset{s}{\sum }} \frac{ \gamma _{\imath } ^{\varrho } \Gamma ( 1+\varrho )}{ (\frac{1}{a_{\imath }}-\frac{1}{\bar{a}} )^{\varrho }} {}_{ \frac{1}{ \frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{\bar{a}} } } I^{(\varrho )}_{ \frac{1}{ \frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{a_{\imath } } } }\frac{ \Psi (x)}{x^{2\varrho }} \\ & \leq \Psi (\mathfrak{M})+\Psi (\mathfrak{m} )- \underset{\imath =1 }{ \overset{s}{\sum }} \gamma _{\imath }^{\varrho }\Psi (a_{\imath }) \end{aligned} \end{aligned}$$
(10)
for all \(a_{\imath } \in [ \mathfrak{m} ,\mathfrak{M} ]\) and \(\gamma _{\imath }\in [0,1 ]\) (\(\imath =1 ,2,\ldots ,s \)) with \(\underset{\imath =1 }{\overset{s}{\sum }} \gamma _{\imath }=1\), where \(\frac{1}{\bar{a}}=\underset{\imath =1 }{\overset{s}{\sum }} \gamma _{ \imath }\frac{1}{a_{\imath }}\).
Proof
As \(\Psi \in GHK_{\varrho }(I)\), then for any \(a_{1},b_{1} \in [\mathfrak{m} ,\mathfrak{M}]\) and \(\vartheta \in [0,1]\), we have
$$\begin{aligned} \Psi \biggl( \frac{2a_{1}b_{1}}{a_{1}+b_{1}} \biggr)&=\Psi \biggl( \frac{1}{\frac{1}{2}(\frac{\vartheta }{a_{1}}+\frac{1-\vartheta }{b_{1}}+\frac{1-\vartheta }{a_{1}}+\frac{\vartheta }{b_{1}})} \biggr) \\ & \leq \frac{1}{2^{\varrho }} \biggl[ \Psi \biggl( \frac{1}{\frac{\vartheta }{a_{1}}+\frac{1-\vartheta }{b_{1}}} \biggr) + \Psi \biggl( \frac{1}{\frac{1-\vartheta }{a_{1}}+\frac{\vartheta }{b_{1}}} \biggr) \biggr] \\ & \leq \frac{\Psi (a_{1})+\Psi (b_{1})}{2^{\varrho }}. \end{aligned}$$
Writing \(a_{1}= \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }- \frac{ 1}{a}}\) and \(b_{1}= \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }- \frac{ 1}{b}}\) for \(a,b \in [\mathfrak{m} ,\mathfrak{M}]\), we get
$$\begin{aligned} \Psi \biggl( \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{a+b}{2ab}} \biggr)& \leq \frac{1}{2^{\varrho }} \biggl[ \Psi \biggl( \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-(\frac{\vartheta }{a}+\frac{1-\vartheta }{b}) } \biggr)+ \Psi \biggl( \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-(\frac{1-\vartheta }{a}+\frac{\vartheta }{b})} \biggr) \biggr] \\ & \leq \frac{1}{2^{\varrho }} \biggl[ \Psi \biggl( \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{a}} \biggr)+ \Psi \biggl( \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{b}} \biggr) \biggr]. \end{aligned}$$
Multiplying by \(\frac{1}{\Gamma ( 1+\varrho )}\) on both sides of the above equation, then integrating w.r.t ϑ over \([0,1]\) and using fact that
$$\begin{aligned} &\frac{1}{\Gamma ( 1+\varrho )} \int _{0}^{1} \Psi \biggl( \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-(\frac{\vartheta }{a}+\frac{1-\vartheta }{b}) } \biggr) (d\vartheta )^{\varrho } \\ &\quad = \frac{1}{\Gamma ( 1+\varrho )} \int _{0}^{1} \Psi \biggl( \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-(\frac{1-\vartheta }{a}+\frac{\vartheta }{b})} \biggr) (d\vartheta )^{\varrho }, \end{aligned}$$
we infer that
$$\begin{aligned} \frac{1}{\Gamma ( 1+\varrho )} \Psi \biggl( \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{a+b}{2ab}} \biggr)& \leq \frac{1}{\Gamma ( 1+\varrho )} \int _{0}^{1} \Psi \biggl( \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-(\frac{\vartheta }{a}+\frac{1-\vartheta }{b}) } \biggr) (d\vartheta )^{\varrho } \\ & \leq \frac{1}{2^{\varrho }\Gamma ( 1+\varrho )} \biggl[ \Psi \biggl( \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{a}} \biggr)+ \Psi \biggl( \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{b}} \biggr) \biggr]. \end{aligned}$$
Since \(\frac{1}{a_{\imath }}\), \(\frac{1}{\bar{a}} \in [\mathfrak{m} ,\mathfrak{M}]\), we can write
$$\begin{aligned} \Psi \biggl( \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{2} ( \frac{1}{\bar{a}}+\frac{1}{a_{\imath } } )} \biggr) & \leq \frac{ \Gamma ( 1+\varrho )}{ (\frac{1}{a_{\imath }}-\frac{1}{\bar{a}} )^{\varrho }} {}_{ \frac{1}{ \frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{\bar{a}} } } I^{(\varrho )}_{ \frac{1}{ \frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{a_{\imath } } } }\frac{ \Psi (x)}{x^{2\varrho }} \\ & \leq \frac{1}{2^{\varrho }} \biggl[ \Psi \biggl( \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{\bar{a}}} \biggr)+ \Psi \biggl( \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{a_{\imath }}} \biggr) \biggr], \end{aligned}$$
due to
$$\begin{aligned} \frac{1}{\Gamma ( 1+\varrho )} \int _{0}^{1} \Psi \biggl( \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-(\frac{\vartheta }{\bar{a}}+\frac{1-\vartheta }{a_{\imath }}) } \biggr) (d\vartheta )^{\varrho }= \frac{ 1}{ (\frac{1}{a_{\imath }}-\frac{1}{\bar{a}} )^{\varrho }} {}_{ \frac{1}{ \frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{\bar{a}} } } I^{(\varrho )}_{ \frac{1}{ \frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{a_{\imath } } } }\frac{ \Psi (x)}{x^{2\varrho }}. \end{aligned}$$
Multiplying by \(\gamma _{\imath }^{\varrho }>0\) (\(\imath =1 ,\ldots, s\)) and summing over ı from 1 to s, we may deduce
$$\begin{aligned} \begin{aligned} &\underset{\imath =1 }{\overset{s}{ \sum }} \gamma _{\imath } ^{\varrho } \Psi \biggl( \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{2} ( \frac{1}{\bar{a}}+\frac{1}{a_{\imath } } )} \biggr) \\ &\quad \leq \underset{\imath =1 }{\overset{s}{\sum }} \frac{\gamma _{\imath } ^{\varrho } \Gamma ( 1+\varrho )}{ (\frac{1}{a_{\imath }}-\frac{1}{\bar{a}} )^{\varrho }} {}_{ \frac{1}{ \frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{\bar{a}} } } I^{(\varrho )}_{ \frac{1}{ \frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{a_{\imath } } } }\frac{ \Psi (x)}{x^{2\varrho }} \\ &\quad \leq \frac{1}{2^{\varrho }} \biggl[ \Psi \biggl( \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{\bar{a}}} \biggr)+ \underset{\imath =1 }{\overset{s}{\sum }} \gamma _{\imath } ^{ \varrho } \Psi \biggl( \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{a_{\imath }}} \biggr) \biggr]. \end{aligned} \end{aligned}$$
(11)
On the other hand, by (G. J. I) for \(\Psi \in GHK_{\varrho }(I)\)
$$\begin{aligned} \begin{aligned} \Psi \biggl( \frac{1}{ \frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{\bar{a}} } \biggr)&= \Psi \biggl( \frac{1}{\underset{\imath =1 }{\overset{s}{\sum }} \gamma _{\imath } (\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{2} ( \frac{1}{\bar{a}}+\frac{1}{a_{\imath } } ) )} \biggr) \\ & \leq \underset{\imath =1 }{\overset{s}{\sum }} \gamma _{\imath } ^{ \varrho } \Psi \biggl( \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{2} ( \frac{1}{\bar{a}}+\frac{1}{a_{\imath } } )} \biggr) , \end{aligned} \end{aligned}$$
(12)
and by Lemma 3 and (G. J. M. I) for \(\Psi \in GHK_{\varrho }(I)\)
$$\begin{aligned} \begin{aligned} & \frac{1}{2^{\varrho }} \biggl[ \Psi \biggl( \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{\bar{a}}} \biggr)+\underset{\imath =1 }{\overset{s}{\sum }} \gamma _{\imath } ^{ \varrho } \Psi \biggl( \frac{1}{\frac{1}{\mathfrak{M}}+\frac{1}{\mathfrak{m} }-\frac{1}{a_{\imath }}} \biggr) \biggr] \\ &\quad \leq \frac{1}{2^{\varrho }} \Bigl[ \Psi (\mathfrak{M})+\Psi ( \mathfrak{m} )- \underset{j=1}{\overset{s}{\sum }} \gamma _{j}^{ \varrho } \Psi (a_{j})+\Psi (\mathfrak{M})+\Psi (\mathfrak{m} )- \underset{\imath =1 }{\overset{s}{\sum }} \gamma _{\imath } ^{\varrho } \Psi (a_{\imath }) \Bigr] \\ &\quad = \Psi (\mathfrak{M})+\Psi (\mathfrak{m} )- \underset{\imath =1 }{\overset{s}{\sum }} \gamma _{\imath }^{\varrho }\Psi (a_{\imath }) . \end{aligned} \end{aligned}$$
(13)
Combining (11), (12), and (13), we get (10). □
Remark 5
If we take \(\varrho =1\) in Theorem 8, then it gives [[33], Theorem 2.6].
Remark 6
Let all the assumptions of Theorem 8 hold, then
$$\begin{aligned} \Psi \biggl( \frac{1}{\underset{\imath =1 }{\overset{s}{\sum }} \frac{\gamma _{\imath }}{a_{\imath }}} \biggr) & \leq \underset{\imath =1 }{ \overset{s}{\sum }} \gamma _{ \imath } ^{\varrho } \Psi \biggl( \frac{2}{ \underset{\imath =1 }{\overset{s}{\sum }} \frac{\gamma _{\imath }}{a_{\imath }}+\frac{1}{a_{\imath } }} \biggr) \\ & \leq \underset{\imath =1 }{\overset{s}{\sum }} \frac{ \gamma _{\imath } ^{\varrho } \Gamma ( 1+\varrho )}{ (\underset{\imath =1 }{\overset{s}{\sum }} \frac{\gamma _{\imath }}{a_{\imath }}-\frac{1}{a_{\imath }} )^{\varrho }} {}_{ \frac{1}{\underset{\imath =1 }{\overset{s}{\sum }} \frac{\gamma _{\imath }}{a_{\imath }}} } I^{(\varrho )}_{a_{\imath } }\frac{ \Psi (x)}{x^{2\varrho }} \\ & \leq \underset{\imath =1 }{\overset{s}{\sum }} \gamma _{\imath }^{\varrho }\Psi (a_{\imath }) . \end{aligned}$$