For
\(0\leq a< b\)
and some fixed
\(m\in(0,1]\),
let
\(f: [a,\frac {b}{m} ]\rightarrow\mathbb{R}\)
be a continuous and
q-
differentiable function on
\((a,\frac{b}{m} )\),
and let
\({}_{a}D_{q}f\)
be integrable on
\([a,\frac{b}{m} ]\)
with
\(0< q <1\).
Then the inequality
$$\begin{aligned} & \biggl\vert \lambda \bigl[\mu f(b)+(1-\mu)f(a) \bigr]+(1-\lambda)f \bigl(\mu b+(1-\mu)a \bigr)-\frac{1}{b-a} \int_{a}^{b}f(x)\,{}_{a} \mathrm{d}_{q}x \biggr\vert \\ &\quad \leq\min \bigl\{ \mathcal{H}_{1}(\lambda,\mu,\alpha,m),\mathcal {H}_{2}(\lambda,\mu,\alpha,m) \bigr\} \end{aligned}$$
holds for all
\(\lambda,\mu\in[0,1]\)
if
\(|{}_{a}D_{q}f|\)
is
\((\alpha ,m)\)-
convex on
\([a,\frac{b}{m} ]\)
with
\(\alpha,m\in(0,1]^{2}\),
where
$$\begin{aligned}& \begin{aligned} \mathcal{H}_{1}(\lambda,\mu,\alpha,m) &=(b-a) \biggl\{ \bigl[\Phi_{1}(\lambda,\mu,\alpha)+\Phi_{2}( \lambda,\mu,\alpha )-\Phi_{3}(\lambda,\mu,\alpha) \bigr] \bigl\vert {}_{a}D_{q}f(b) \bigr\vert \\ &\quad {} +m \bigl[\Phi_{4}(\lambda,\mu)+\Phi_{5}(\lambda, \mu)-\Phi_{6}(\lambda,\mu )-\Phi_{1}(\lambda,\mu,\alpha) \\ &\quad {} -\Phi_{2}(\lambda,\mu,\alpha)+\Phi_{3}(\lambda, \mu,\alpha) \bigr] \biggl\vert {}_{a}D_{q}f \biggl( \frac{a}{m} \biggr) \biggr\vert \biggr\} , \end{aligned} \\& \begin{aligned} \mathcal{H}_{2}(\lambda,\mu,\alpha,m) &=(b-a) \biggl\{ \bigl[\Phi_{7}(\lambda,\mu,\alpha)+\Phi_{8}(\lambda,\mu, \alpha )-\Phi_{9}(\lambda,\mu,\alpha)\bigr] \bigl\vert {}_{a}D_{q}f(a) \bigr\vert \\ &\quad {} +m\bigl[\Phi_{4}(\lambda,\mu)+\Phi_{5}(\lambda, \mu)-\Phi_{6}(\lambda,\mu )-\Phi_{7}(\lambda,\mu,\alpha) \\ &\quad {} -\Phi_{8}(\lambda,\mu,\alpha)+\Phi_{9}(\lambda, \mu,\alpha)\bigr] \biggl\vert {}_{a}D_{q}f\biggl( \frac{b}{m}\biggr) \biggr\vert \biggr\} , \end{aligned} \\ & \begin{aligned}\Phi_{1}(\lambda,\mu,\alpha)&= \int_{0}^{\mu}t^{\alpha} \bigl\vert qt-( \lambda -\lambda\mu) \bigr\vert \,{}_{0}\mathrm{d}_{q}t \\ &= \textstyle\begin{cases} \frac{\mu^{\alpha+1}(1-q)(\lambda-\lambda\mu)}{1-q^{\alpha+1}}-\frac {q\mu^{\alpha+2}(1-q)}{1-q^{\alpha+2}}, &(\lambda+q)\mu\leq\lambda, \\ \frac{2(1-q)^{2}(\lambda-\lambda\mu)^{\alpha+2}}{(1-q^{\alpha +1})(1-q^{\alpha+2})}+\frac{q\mu^{\alpha+2}(1-q)}{1-q^{\alpha+2}}-\frac {\mu^{\alpha+1}(1-q)(\lambda-\lambda\mu)}{1-q^{\alpha+1}}, &(\lambda+q)\mu>\lambda, \end{cases}\displaystyle \end{aligned} \\ & \begin{aligned}[b] \Phi_{2}(\lambda,\mu,\alpha)&= \int_{0}^{1} t^{\alpha} \bigl\vert qt-(1- \lambda\mu ) \bigr\vert \,{}_{0}\mathrm{d}_{q}t \\ &= \textstyle\begin{cases} \frac{(1-q)(1-\lambda\mu)}{1-q^{\alpha+1}}-\frac{q(1-q)}{1-q^{\alpha+2}}, &\lambda\mu+q\leq1, \\ \frac{2(1-q)^{2}(1-\lambda\mu)^{\alpha+2}}{(1-q^{\alpha+1})(1-q^{\alpha +2})}+\frac{q(1-q)}{1-q^{\alpha+2}}-\frac{(1-q)(1-\lambda\mu )}{1-q^{\alpha+1}}, &\lambda\mu+q>1, \end{cases}\displaystyle \end{aligned} \end{aligned}$$
(3.1)
$$\begin{aligned}& \begin{aligned} \Phi_{3}(\lambda,\mu,\alpha)&= \int_{0}^{\mu}t^{\alpha} \bigl\vert qt-(1- \lambda\mu ) \bigr\vert \,{}_{0}\mathrm{d}_{q}t \\ &= \textstyle\begin{cases} \frac{\mu^{\alpha+1}(1-\lambda\mu)(1-q)}{1-q^{\alpha+1}}-\frac{q\mu ^{\alpha+2}(1-q)}{1-q^{\alpha+2}}, &(\lambda+q)\mu\leq1, \\ \frac{2(1-q)^{2}(1-\lambda\mu)^{\alpha+2}}{(1-q^{\alpha+1})(1-q^{\alpha +2})}+\frac{q\mu^{\alpha+2}(1-q)}{1-q^{\alpha+2}}-\frac{\mu^{\alpha +1}(1-\lambda\mu)(1-q)}{1-q^{\alpha+1}}, &(\lambda+q)\mu> 1, \end{cases}\displaystyle \end{aligned} \\ & \begin{aligned} \Phi_{4}(\lambda,\mu)&= \int_{0}^{\mu}\bigl\vert qt-(\lambda-\lambda\mu) \bigr\vert \,{}_{0}\mathrm{d}_{q}t \\ &= \textstyle\begin{cases} \lambda\mu(1-\mu)-\frac{q\mu^{2}}{1+q}, &(\lambda+q)\mu\leq\lambda, \\ \frac{2(\lambda-\lambda\mu)^{2}}{1+q}+\frac{q\mu^{2}}{1+q}- \lambda\mu (1-\mu), &(\lambda+q)\mu>\lambda, \end{cases}\displaystyle \end{aligned} \\ & \begin{aligned}[b] \Phi_{5}(\lambda,\mu)&= \int_{0}^{1} \bigl\vert qt-(1-\lambda\mu) \bigr\vert \,{}_{0}\mathrm {d}_{q}t \\ &= \textstyle\begin{cases} \frac{1}{1+q}-\lambda\mu, &\lambda\mu+q\leq1, \\ \frac{2(1-\lambda\mu)^{2}}{1+q}+\lambda\mu-\frac{1}{1+q}, &\lambda\mu+q>1, \end{cases}\displaystyle \end{aligned} \end{aligned}$$
(3.2)
$$\begin{aligned}& \begin{aligned} \Phi_{6}(\lambda,\mu) & = \int_{0}^{\mu}\bigl\vert qt-(1-\lambda\mu) \bigr\vert \,{}_{0}\mathrm{d}_{q}t \\ & = \textstyle\begin{cases} \mu(1-\lambda\mu)-\frac{q\mu^{2}}{1+q}, &(\lambda+q)\mu\leq1, \\ \frac{2(1-\lambda\mu)^{2}}{1+q}+\frac{q\mu^{2}}{1+q}-\mu(1-\lambda\mu), &(\lambda+q)\mu> 1, \end{cases}\displaystyle \end{aligned} \\ & \begin{aligned} &\Phi_{7}(\lambda,\mu,\alpha) \\ &\quad = \int_{0}^{\mu}(1-t)^{\alpha} \bigl\vert qt-( \lambda-\lambda\mu) \bigr\vert \,{}_{0}\mathrm {d}_{q}t \\ &\quad = \textstyle\begin{cases} (1-q)\mu\sum_{n=0}^{\infty}q^{n} (\lambda-\lambda\mu-q^{n+1}\mu ) (1-q^{n}\mu )^{\alpha}, &(\lambda+q)\mu\leq\lambda, \\ \left [ \textstyle\begin{array}{l} 2(1-q)(\lambda-\lambda\mu)^{2}\sum_{n=0}^{\infty}q^{n-1} (1-q^{n} ) [1-q^{n-1}(\lambda-\lambda\mu) ]^{\alpha}\\ \quad {}-(1-q)\mu\sum_{n=0}^{\infty}q^{n} (\lambda-\lambda\mu-q^{n+1}\mu ) (1-q^{n}\mu )^{\alpha} \end{array}\displaystyle \right ], &(\lambda+q)\mu>\lambda, \end{cases}\displaystyle \end{aligned} \\ & \begin{aligned}[b] &\Phi_{8}(\lambda,\mu,\alpha) \\ &\quad = \int_{0}^{1} (1-t)^{\alpha} \bigl\vert qt-(1-\lambda\mu) \bigr\vert \,{}_{0}\mathrm {d}_{q}t \\ &\quad = \textstyle\begin{cases} (1-q)\sum_{n=0}^{\infty}q^{n} (1-\lambda\mu-q^{n+1} ) (1-q^{n} )^{\alpha}, &\lambda\mu+q\leq1, \\ \left [ \textstyle\begin{array}{l} 2(1-q)(1-\lambda\mu)^{2}\sum_{n=0}^{\infty}q^{n-1} (1-q^{n} ) [1-q^{n-1}(1-\lambda\mu) ]^{\alpha}\\ \quad {}-(1-q)\sum_{n=0}^{\infty}q^{n} (1-\lambda\mu-q^{n+1} ) (1-q^{n} )^{\alpha} \end{array}\displaystyle \right ], &\lambda\mu+q>1, \end{cases}\displaystyle \end{aligned} \end{aligned}$$
(3.3)
and
$$ \begin{aligned} &\Phi_{9}(\lambda,\mu,\alpha) \\ &\quad = \int_{0}^{\mu}(1-t)^{\alpha} \bigl\vert qt-(1-\lambda\mu) \bigr\vert \,{}_{0}\mathrm {d}_{q}t \\ &\quad = \textstyle\begin{cases} (1-q)\mu\sum_{n=0}^{\infty}q^{n} (1-\lambda\mu-q^{n+1}\mu ) (1-q^{n}\mu )^{\alpha}, &(\lambda+q)\mu\leq1, \\ \left [ \textstyle\begin{array}{l} 2(1-q)(1-\lambda\mu)^{2}\sum_{n=0}^{\infty}q^{n-1} (1-q^{n} ) [1-q^{n-1}(1-\lambda\mu) ]^{\alpha}\\ \quad {}-(1-q)\mu\sum_{n=0}^{\infty}q^{n} (1-\lambda\mu-q^{n+1}\mu ) (1-q^{n}\mu )^{\alpha} \end{array}\displaystyle \right ], &(\lambda+q)\mu>1. \end{cases}\displaystyle \end{aligned} $$