Putting
\(x=\frac{t}{2}b+\frac{(2-t)}{2}\frac{a}{m}\) and
\(y= \frac{t}{2}a+m\frac{(2-t)}{2}b\) in (
7), we get
$$\begin{aligned} f \biggl( \frac{a+bm}{2} \biggr) \leq h \biggl( \frac{1}{2} \biggr) \biggl[ mf \biggl( \frac{t}{2}b+ \frac{(2-t)}{2}\frac{a}{m} \biggr)+f \biggl( \frac{t}{2}a+m \frac{(2-t)}{2}b \biggr) \biggr]. \end{aligned}$$
(12)
Multiplying (
12) by
\(t^{\alpha -1}E^{\gamma ,\delta ,k,c}_{ \mu ,\alpha ,l}(\omega t^{\mu };p)\) on both sides, then integrating over
\([0,1]\), we have
$$\begin{aligned} &f \biggl( \frac{a+bm}{2} \biggr) \int _{0}^{1}t^{\alpha -1}E^{\gamma , \delta ,k,c}_{\mu ,\alpha ,l} \bigl(\omega t^{\mu };p\bigr)\,dt \\ &\quad \leq h \biggl( \frac{1}{2} \biggr) \biggl[ \int _{0}^{1}t^{\alpha -1}E ^{\gamma ,\delta ,k,c}_{\mu ,\alpha ,l} \bigl(\omega t^{\mu };p\bigr) mf \biggl( \frac{t}{2}a+m \frac{(2-t)}{2}b \biggr)\,dt \\ &\qquad {} + \int _{0}^{1}t^{\alpha -1}E^{\gamma ,\delta ,k,c}_{\mu , \alpha ,l} \bigl(\omega t^{\mu };p\bigr)f \biggl( \frac{t}{2}b+ \frac{(2-t)}{2} \frac{a}{m} \biggr) \,dt \biggr]. \end{aligned}$$
Putting
\(x=\frac{t}{2}b+\frac{(2-t)}{2}\frac{a}{m}\) and
\(y= \frac{t}{2}a+m\frac{(2-t)}{2}b\), then, by using (
4) and (
5), we get
$$\begin{aligned} &f \biggl( \frac{a+bm}{2} \biggr) \bigl(\epsilon ^{\gamma ,\delta ,k,c} _{\mu ,\alpha ,l,\omega ^{o}2^{\mu }, ( \frac{a+bm}{2} ) ^{+}}1 \bigr) (mb;p) \\ &\quad \leq h \biggl( \frac{1}{2} \biggr) \biggl[ \bigl(\epsilon ^{\gamma ,\delta ,k,c} _{\mu ,\alpha ,l,\omega ^{o}2^{\mu }, ( \frac{a+bm}{2} ) ^{+}}f \bigr) (mb;p)+m^{\alpha +1} \bigl(\epsilon ^{\gamma ,\delta ,k,c} _{\mu ,\alpha ,l,\omega ^{o}(2m)^{\mu }, ( \frac{a+bm}{2m} ) ^{-}}f \bigr) \biggl(\frac{a}{m};p \biggr) \biggr]. \end{aligned}$$
(13)
Again by using
\((h-m)\)-convexity of
f, we have
$$\begin{aligned} \begin{aligned}[b] &f \biggl( \frac{t}{2}a+m\frac{(2-t)}{2}b \biggr) +mf \biggl( \frac{t}{2}b+\frac{(2-t)}{2}\frac{a}{m} \biggr) \\ &\quad \leq h \biggl( \frac{t}{2} \biggr)f(a)+mh \biggl( \frac{2-t}{2} \biggr)f(b)+mh \biggl( \frac{t}{2} \biggr) f(b)+ m^{2}h \biggl( \frac{2-t}{2} \biggr) f \biggl( \frac{a}{m^{2}} \biggr) \\ &\quad =m \biggl[ mf \biggl( \frac{a}{m^{2}} \biggr) +f(b) \biggr]h \biggl( \frac{2-t}{2} \biggr)+ \bigl[ mf(b)+f(a) \bigr]h \biggl( \frac{t}{2} \biggr). \end{aligned} \end{aligned}$$
(14)
Multiplying (
14) by
\(h ( \frac{1}{2} )t^{\alpha -1}E ^{\gamma ,\delta ,k,c}_{\mu ,\alpha ,l}(\omega t^{\mu };p)\) on both sides, then integrating over
\([0,1]\), we have
$$\begin{aligned} &h \biggl( \frac{1}{2} \biggr) \biggl[ \int _{0}^{1}t^{\alpha -1}E^{ \gamma ,\delta ,k,c}_{\mu ,\alpha ,l} \bigl(\omega t^{\mu };p\bigr)f \biggl( \frac{t}{2}a+m \frac{(2-t)}{2}b \biggr)\,dt \\ &\qquad {}+ \int _{0}^{1}t^{\alpha -1}E^{\gamma ,\delta ,k,c}_{\mu , \alpha ,l} \bigl(\omega t^{\mu };p\bigr)mf \biggl( \frac{t}{2}b+ \frac{(2-t)}{2} \frac{a}{m} \biggr)\,dt \biggr] \\ &\quad \leq h \biggl(\frac{1}{2} \biggr) \biggl\lbrace m \biggl[ mf \biggl( \frac{a}{m ^{2}} \biggr) +f(b) \biggr] \int _{0}^{1}t^{\alpha -1}E^{\gamma , \delta ,k,c}_{\mu ,\alpha ,l} \bigl(\omega t^{\mu };p\bigr)h \biggl( \frac{2-t}{2} \biggr) \,dt \\ &\qquad {}+ \bigl[ mf(b)+f(a) \bigr] \int _{0}^{1}t^{\alpha -1}E^{ \gamma ,\delta ,k,c}_{\mu ,\alpha ,l} \bigl(\omega t^{\mu };p\bigr)h \biggl( \frac{t}{2} \biggr) \,dt \biggr\rbrace . \end{aligned}$$
By using (
4) and (
5), we get
$$\begin{aligned} &h \biggl( \frac{1}{2} \biggr) \biggl[ \bigl(\epsilon ^{\gamma ,\delta ,k,c}_{\mu ,\alpha ,l,\omega ^{o}2^{\mu }, ( \frac{a+bm}{2} ) ^{+}}f \bigr) (mb;p)+m^{\alpha +1} \bigl(\epsilon ^{\gamma ,\delta ,k,c} _{\mu ,\alpha ,l,\omega ^{o}(2m)^{\mu }, ( \frac{a+bm}{2m} ) ^{-}}f \bigr) \biggl(\frac{a}{m};p \biggr) \biggr] \\ &\quad \leq h \biggl(\frac{1}{2} \biggr) \frac{(bm-a)^{\alpha }}{2^{\alpha }} \biggl\lbrace m \biggl[ mf \biggl( \frac{a}{m ^{2}} \biggr) +f(b) \biggr] \int _{0}^{1}t^{\alpha -1}E^{\gamma , \delta ,k,c}_{\mu ,\alpha ,l} \bigl(\omega t^{\mu };p\bigr)h \biggl( \frac{2-t}{2} \biggr) \,dt \\ &\qquad {}+ \bigl[ mf(b)+f(a) \bigr] \int _{0}^{1}t^{\alpha -1}E^{ \gamma ,\delta ,k,c}_{\mu ,\alpha ,l} \bigl(\omega t^{\mu };p\bigr)h \biggl( \frac{t}{2} \biggr) \,dt \biggr\rbrace . \end{aligned}$$
From the above inequality and (
13), we get the required inequality (
11). □