From Lemma
2.1 and by using Hölder’s inequality, we have
$$\begin{aligned} & \biggl\vert \biggl( \int_{a}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha}\bigl(f(a)+f(mb) \bigr) \\ &\qquad{} -\alpha \int_{a}^{mb} \biggl( \int_{a}^{t}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha-1} g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k, c}\bigl(\omega t^{\mu}; p\bigr)f(t)\,dt \\ &\qquad{} -\alpha \int_{a}^{mb} \biggl( \int_{t}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha-1} g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k, c}\bigl(\omega t^{\mu}; p\bigr)f(t)\,dt \biggr\vert \\ & \quad \leq \biggl( \int_{a}^{mb} \biggl\vert \int_{a}^{t}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr\vert ^{\alpha p}\,dt \biggr)^{\frac{1}{p}} \biggl( \int_{a}^{mb} \bigl\vert f'(t) \bigr\vert ^{q}\,dt \biggr)^{\frac{1}{q}} \\ &\qquad{} + \biggl( \int_{a}^{mb} \biggl\vert \int_{t}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr\vert ^{\alpha p}\,dt \biggr)^{\frac{1}{p}} \biggl( \int_{a}^{mb} \bigl\vert f'(t) \bigr\vert ^{q}\,dt \biggr)^{\frac{1}{q}}. \end{aligned}$$
(18)
Using absolute convergence of the Mittag-Leffler function and
\(\Vert g \Vert_{\infty}= \sup_{t\in[a,b]}|g(t)|\), we have
$$\begin{aligned} & \biggl\vert \biggl( \int_{a}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha}\bigl(f(a)+f(mb) \bigr) \\ &\qquad{} -\alpha \int_{a}^{mb} \biggl( \int_{a}^{t}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha-1} g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k, c}\bigl(\omega t^{\mu}; p\bigr)f(t)\,dt \\ &\qquad{} -\alpha \int_{a}^{mb} \biggl( \int _{t}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha-1} g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k, c}\bigl(\omega t^{\mu}; p\bigr)f(t)\,dt \biggr\vert \\ & \quad \leq \Vert g \Vert _{\infty}^{\alpha}S^{\alpha} \biggl( \biggl( \int _{a}^{mb} \vert t-a \vert ^{\alpha p}\,dt \biggr)^{\frac{1}{p}} \\ &\qquad {}+ \biggl( \int _{a}^{mb} \vert mb-t \vert ^{\alpha p}\,dt \biggr)^{\frac{1}{p}} \biggr) \biggl( \int _{a}^{mb} \bigl\vert f'(t) \bigr\vert ^{q}\,dt \biggr)^{\frac{1}{q}}. \end{aligned}$$
(19)
Since
\(|f'(t)|^{q}\) is an
m-convex function, we have
$$ \bigl\vert f'(t) \bigr\vert ^{q}\leq \frac{mb-t}{mb-a} \bigl\vert f'(a) \bigr\vert ^{q}+m \frac{t-a}{mb-a} \bigl\vert f'(b) \bigr\vert ^{q}. $$
(20)
Using (
20) in (
19), we have
$$\begin{aligned} & \biggl\vert \biggl( \int_{a}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha}\bigl(f(a)+f(mb) \bigr) \\ &\qquad{} -\alpha \int_{a}^{mb} \biggl( \int_{a}^{t}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha-1} g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k, c}\bigl(\omega t^{\mu}; p\bigr)f(t)\,dt \\ &\qquad{} -\alpha \int_{a}^{mb} \biggl( \int_{t}^{mb}g(s)E_{\mu,\alpha,l}^{\gamma,\delta,k, c} \bigl(\omega s^{\mu}; p\bigr)\,ds \biggr)^{\alpha-1} g(t)E_{\mu,\alpha,l}^{\gamma,\delta,k, c}\bigl(\omega t^{\mu}; p\bigr)f(t)\,dt \biggr\vert \\ & \quad \leq \Vert g \Vert _{\infty}^{\alpha}S^{\alpha} \biggl( \biggl( \int _{a}^{mb} \vert t-a \vert ^{\alpha p}\,dt \biggr)^{\frac{1}{p}}+ \biggl( \int _{a}^{mb} \vert mb-t \vert ^{\alpha p}\,dt \biggr)^{\frac{1}{p}} \biggr) \\ & \qquad {}\times \biggl( \int_{a}^{mb}\frac{mb-t}{mb-a} \bigl\vert f'(a) \bigr\vert ^{q}+m\frac {t-a}{mb-a} \bigl\vert f'(b) \bigr\vert ^{q} \biggr)^{\frac{1}{q}} . \end{aligned}$$
(21)
After simple calculation of the above inequality, we get (
17) which is required. □