In 2018, Alp et al. established the q-Hermite–Hadamard inequality in [2]. Here we give a new proof, which is more concise.
Theorem 3.1
Let
\(f:I\rightarrow\mathbb{R}\)
be a convex function on
\([a,b]\)
with
\(0< q<1\). Then we have
$$ f \biggl(\frac{qa+b}{1+q} \biggr)\leq\frac{1}{b-a} \int_{a}^{b}f(x)\,{}_{a}\mathrm {d}_{q}x\leq\frac{qf(a)+f(b)}{1+q}. $$
Proof
It is obvious that \(\sum_{n=0}^{\infty}(1-q)q^{n}=1\), \(0< q<1\). Since Jensen’s inequality defined on convex sets for infinite sums still remains true, utilizing this fact and Definition 1.2, we have
$$\begin{aligned} f \biggl(\frac{qa+b}{1+q} \biggr) &=f \Biggl(\sum _{n=0}^{\infty}(1-q)q^{n} \bigl(q^{n}b+ \bigl(1-q^{n}\bigr)a \bigr) \Biggr) \\ &\leq\sum_{n=0}^{\infty}(1-q)q^{n}f \bigl(q^{n}b+\bigl(1-q^{n}\bigr)a \bigr) \\ &=\frac{1}{b-a} \int_{a}^{b}f(x)\,{}_{a} \mathrm{d}_{q}x. \end{aligned}$$
Using Definition 1.2 and the convexity of f, we get
$$\begin{aligned} \frac{1}{b-a} \int_{a}^{b}f(x)\,{}_{a} \mathrm{d}_{q}x &= \sum_{n=0}^{\infty}(1-q)q^{n}f \bigl(q^{n}b+\bigl(1-q^{n}\bigr)a \bigr) \\ &\leq\sum_{n=0}^{\infty}(1-q)q^{n} \bigl[q^{n}f(b)+\bigl(1-q^{n}\bigr)f(a) \bigr] \\ &=\frac{qf(a)+f(b)}{1+q}. \end{aligned}$$
The proof is completed. □
Using Lemma 2.1, we can obtain the following theorem.
Theorem 3.2
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
$$ \begin{aligned} &\Phi_{9}(\lambda,\mu,\alpha) \\ &\quad = \int_{0}^{\mu}(1-t)^{\alpha} \bigl\vert qt-(1-\lambda\mu) \bigr\vert \,{}_{0}\mathrm {d}_{q}t \\ &\quad = \textstyle\begin{cases} (1-q)\mu\sum_{n=0}^{\infty}q^{n} (1-\lambda\mu-q^{n+1}\mu ) (1-q^{n}\mu )^{\alpha}, &(\lambda+q)\mu\leq1, \\ \left [ \textstyle\begin{array}{l} 2(1-q)(1-\lambda\mu)^{2}\sum_{n=0}^{\infty}q^{n-1} (1-q^{n} ) [1-q^{n-1}(1-\lambda\mu) ]^{\alpha}\\ \quad {}-(1-q)\mu\sum_{n=0}^{\infty}q^{n} (1-\lambda\mu-q^{n+1}\mu ) (1-q^{n}\mu )^{\alpha} \end{array}\displaystyle \right ], &(\lambda+q)\mu>1. \end{cases}\displaystyle \end{aligned} $$
Proof
From Lemma 2.1, utilizing the property of the modulus and the \((\alpha,m)\)-convexity of \(|{}_{a}D_{q}f|\), we have
$$\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(b-a) \biggl\{ \int_{0}^{\mu} \vert qt+\lambda\mu-\lambda \vert \bigl\vert {}_{a}D_{q}f\bigl(tb+(1-t)a\bigr) \bigr\vert \,{}_{0}\mathrm{d}_{q}t \\ &\qquad {} + \int_{\mu}^{1} \vert qt+\lambda\mu-1 \vert \bigl\vert {}_{a}D_{q}f\bigl(tb+(1-t)a \bigr) \bigr\vert \,{}_{0}\mathrm{d}_{q}t \biggr\} \\ &\quad \leq(b-a) \biggl\{ \int_{0}^{\mu}\bigl\vert qt-(\lambda-\lambda\mu) \bigr\vert \biggl[t^{\alpha}\bigl\vert {}_{a}D_{q}f(b) \bigr\vert +m\bigl(1-t^{\alpha}\bigr) \biggl\vert {}_{a}D_{q}f \biggl(\frac {a}{m}\biggr) \biggr\vert \biggr]\,{}_{0} \mathrm{d}_{q}t \\ &\qquad {} + \int_{\mu}^{1} \bigl\vert qt-(1-\lambda\mu) \bigr\vert \biggl[t^{\alpha}\bigl\vert {}_{a}D_{q}f(b) \bigr\vert +m\bigl(1-t^{\alpha}\bigr) \biggl\vert {}_{a}D_{q}f \biggl(\frac{a}{m}\biggr) \biggr\vert \biggr]\,{}_{0} \mathrm{d}_{q}t \biggr\} \\ &\quad =(b-a) \biggl\{ \int_{0}^{\mu}\bigl\vert qt-(\lambda-\lambda\mu) \bigr\vert \biggl[t^{\alpha}\bigl\vert {}_{a}D_{q}f(b) \bigr\vert +m\bigl(1-t^{\alpha}\bigr) \biggl\vert {}_{a}D_{q}f \biggl(\frac {a}{m}\biggr) \biggr\vert \biggr]\,{}_{0} \mathrm{d}_{q}t \\ &\qquad {} + \int_{0}^{1} \bigl\vert qt-(1-\lambda\mu) \bigr\vert \biggl[t^{\alpha}\bigl\vert {}_{a}D_{q}f(b) \bigr\vert +m\bigl(1-t^{\alpha}\bigr) \biggl\vert {}_{a}D_{q}f \biggl(\frac{a}{m}\biggr) \biggr\vert \biggr]\,{}_{0} \mathrm{d}_{q}t \\ &\qquad {} - \int_{0}^{\mu}\bigl\vert qt-(1-\lambda\mu) \bigr\vert \biggl[t^{\alpha}\bigl\vert {}_{a}D_{q}f(b) \bigr\vert +m\bigl(1-t^{\alpha}\bigr) \biggl\vert {}_{a}D_{q}f \biggl(\frac{a}{m}\biggr) \biggr\vert \biggr]\,{}_{0} \mathrm{d}_{q}t \biggr\} \\ &\quad =(b-a) \biggl\{ \biggl[ \int_{0}^{\mu}t^{\alpha}\bigl\vert qt-( \lambda-\lambda\mu) \bigr\vert \,{}_{0}\mathrm{d}_{q}t + \int_{0}^{1} t^{\alpha}\bigl\vert qt-(1- \lambda\mu) \bigr\vert \,{}_{0}\mathrm{d}_{q}t \\ &\qquad {} - \int_{0}^{\mu}t^{\alpha}\bigl\vert qt-(1- \lambda\mu) \bigr\vert \,{}_{0}\mathrm{d}_{q}t \biggr] \bigl\vert {}_{a}D_{q}f(b) \bigr\vert \\ &\qquad {} +m \biggl[ \int_{0}^{\mu}\bigl\vert qt-(\lambda-\lambda\mu) \bigr\vert \,{}_{0}\mathrm{d}_{q}t + \int_{0}^{1} \bigl\vert qt-(1-\lambda\mu) \bigr\vert \,{}_{0}\mathrm{d}_{q}t \\ &\qquad {} - \int_{0}^{\mu}\bigl\vert qt-(1-\lambda\mu) \bigr\vert \,{}_{0}\mathrm{d}_{q}t - \int_{0}^{\mu}t^{\alpha}\bigl\vert qt-( \lambda-\lambda\mu) \bigr\vert \,{}_{0}\mathrm {d}_{q}t \\ &\qquad {} - \int_{0}^{1} t^{\alpha}\bigl\vert qt-(1- \lambda\mu) \bigr\vert \,{}_{0}\mathrm{d}_{q}t + \int_{0}^{\mu}t^{\alpha}\bigl\vert qt-(1- \lambda\mu) \bigr\vert \,{}_{0}\mathrm{d}_{q}t \biggr] \biggl\vert {}_{a}D_{q}f\biggl(\frac{a}{m}\biggr) \biggr\vert \biggr\} . \end{aligned}$$
Similarly, we get
$$\begin{aligned} & \biggl\vert \lambda\bigl[\mu f(b)+(1-\mu)f(a)\bigr]+(1-\lambda)f\bigl(\mu b+(1-\mu)a\bigr)-\frac{1}{b-a} \int_{a}^{b}f(x)\,{}_{a} \mathrm{d}_{q}x \biggr\vert \\ &\quad \leq(b-a) \biggl\{ \biggl[ \int_{0}^{\mu}(1-t)^{\alpha}\bigl\vert qt-( \lambda-\lambda \mu) \bigr\vert \,{}_{0}\mathrm{d}_{q}t + \int_{0}^{1} (1-t)^{\alpha}\bigl\vert qt-(1-\lambda\mu) \bigr\vert \,{}_{0}\mathrm{d}_{q}t \\ &\qquad {} - \int_{0}^{\mu}(1-t)^{\alpha}\bigl\vert qt-(1-\lambda\mu) \bigr\vert \,{}_{0}\mathrm {d}_{q}t \biggr] \bigl\vert {}_{a}D_{q}f(a) \bigr\vert \\ &\qquad {} +m\biggl[ \int_{0}^{\mu}\bigl\vert qt-(\lambda-\lambda\mu) \bigr\vert \,{}_{0}\mathrm{d}_{q}t + \int_{0}^{1} \bigl\vert qt-(1-\lambda\mu) \bigr\vert \,{}_{0}\mathrm{d}_{q}t \\ &\qquad {} - \int_{0}^{\mu}\bigl\vert qt-(1-\lambda\mu) \bigr\vert \,{}_{0}\mathrm{d}_{q}t - \int_{0}^{\mu}(1-t)^{\alpha}\bigl\vert qt-( \lambda-\lambda\mu) \bigr\vert \,{}_{0}\mathrm {d}_{q}t \\ &\qquad {} - \int_{0}^{1} (1-t)^{\alpha}\bigl\vert qt-(1-\lambda\mu) \bigr\vert \,{}_{0}\mathrm{d}_{q}t + \int_{0}^{\mu}(1-t)^{\alpha}\bigl\vert qt-(1-\lambda\mu) \bigr\vert \,{}_{0}\mathrm {d}_{q}t \biggr] \biggl\vert {}_{a}D_{q}f\biggl(\frac{b}{m} \biggr) \biggr\vert \biggr\} . \end{aligned}$$
Using Lemma 2.3, Lemma 2.4 and Lemma 2.5, we get the desired result. This completes the proof. □
Corollary 3.1
In Theorem 3.2, putting
\(\mu=\frac{1}{1+q}\), we have
$$\begin{aligned} & \biggl\vert \lambda\frac{qf(a)+f(b)}{1+q}+(1-\lambda)f \biggl(\frac {qa+b}{1+q} \biggr)-\frac{1}{b-a} \int_{a}^{b}f(x)\,{}_{a} \mathrm{d}_{q}x \biggr\vert \\ &\quad \leq\min \biggl\{ \mathcal{H}_{1} \biggl(\lambda, \frac{1}{1+q},\alpha,m \biggr),\mathcal{H}_{2} \biggl(\lambda, \frac{1}{1+q},\alpha,m \biggr) \biggr\} . \end{aligned}$$
Remark 3.1
Consider Corollary 3.1.
(i) Putting \(\lambda=0\), we get the midpoint-like integral inequality
$$\begin{aligned} & \biggl\vert f \biggl(\frac{qa+b}{1+q} \biggr)-\frac{1}{b-a} \int _{a}^{b}f(x)\,{}_{a} \mathrm{d}_{q}x \biggr\vert \\ &\quad \leq\min \biggl\{ \mathcal{H}_{1} \biggl(0,\frac{1}{1+q}, \alpha,m \biggr),\mathcal{H}_{2} \biggl(0,\frac{1}{1+q},\alpha,m \biggr) \biggr\} , \end{aligned}$$
where
$$\begin{aligned} &\mathcal{H}_{1} \biggl(0,\frac{1}{1+q},\alpha,m \biggr) \\ &\quad =(b-a) \biggl\{ \frac{[(1+q)^{\alpha+2}-(1+q^{\alpha +2})](1-q)^{2}}{(1+q)^{\alpha+2}(1-q^{\alpha+1})(1-q^{\alpha+2})} \bigl\vert {}_{a}D_{q}f(b) \bigr\vert \\ &\qquad {} +m \biggl[\frac{2q}{(1+q)^{3}}-\frac{[(1+q)^{\alpha+2}-(1+q^{\alpha +2})](1-q)^{2}}{(1+q)^{\alpha+2}(1-q^{\alpha+1})(1-q^{\alpha+2})} \biggr] \biggl\vert {}_{a}D_{q}f \biggl(\frac{a}{m} \biggr) \biggr\vert \biggr\} \end{aligned}$$
and
$$\begin{aligned} &\mathcal{H}_{2}\biggl(0,\frac{1}{1+q},\alpha,m\biggr) \\ &\quad = (b-a)\biggl\{ \biggl[\Phi_{7}\biggl(0,\frac{1}{1+q}, \alpha\biggr)+\Phi _{8}\biggl(0,\frac{1}{1+q},\alpha\biggr)- \Phi_{9}\biggl(0,\frac{1}{1+q},\alpha \biggr)\biggr] \bigl\vert {}_{a}D_{q}f(a) \bigr\vert \\ &\qquad {} +m\biggl[\frac{2q}{(1+q)^{3}}-\Phi_{7}\biggl(0, \frac{1}{1+q},\alpha \biggr)-\Phi_{8}\biggl(0,\frac{1}{1+q}, \alpha\biggr) \\ &\qquad {} +\Phi_{9}\biggl(0,\frac{1}{1+q},\alpha\biggr)\biggr] \biggl\vert {}_{a}D_{q}f \biggl(\frac{b}{m}\biggr) \biggr\vert \biggr\} . \end{aligned}$$
Specially, taking \(\alpha=1=m\), we obtain
$$\begin{aligned} & \biggl\vert f \biggl(\frac{qa+b}{1+q} \biggr)-\frac{1}{b-a} \int _{a}^{b}f(x)\,{}_{a} \mathrm{d}_{q}x \biggr\vert \\ &\quad \leq(b-a) \biggl\{ \frac{3q}{(1+q)^{3}(1+q+q^{2})} \bigl\vert {}_{a}D_{q}f(b) \bigr\vert +\frac{-q+2q^{2}+2q^{3}}{(1+q)^{3}(1+q+q^{2})} \bigl\vert {}_{a}D_{q}f(a) \bigr\vert \biggr\} , \end{aligned}$$
which is established by Alp et al. in [2, Theorem 13].
(ii) Putting \(\lambda=\frac{1}{3}\) and \(\alpha=1=m\), we get the Simpson-like integral inequality
$$\begin{aligned} & \biggl\vert \frac{1}{3} \biggl[\frac{qf(a)+f(b)}{1+q}+2f \biggl( \frac {qa+b}{1+q} \biggr) \biggr]-\frac{1}{b-a} \int_{a}^{b}f(x)\,{}_{a} \mathrm{d}_{q}x \biggr\vert \\ &\quad \leq\min \biggl\{ \mathcal{H}_{1} \biggl(\frac{1}{3}, \frac{1}{1+q},1,1 \biggr),\mathcal{H}_{2} \biggl( \frac{1}{3},\frac{1}{1+q},1,1 \biggr) \biggr\} . \end{aligned}$$
Specially, if \(q\rightarrow1^{-}\), then we obtain
$$ \biggl\vert \frac{1}{3} \biggl[\frac{f(a)+f(b)}{2}+2f \biggl( \frac{a+b}{2} \biggr) \biggr]-\frac{1}{b-a} \int_{a}^{b}f(x)\, \mathrm{d}x \biggr\vert \leq \frac{5(b-a)}{72} \bigl[ \bigl\vert f'(b) \bigr\vert + \bigl\vert f'(a) \bigr\vert \bigr], $$
which is established by Alomari et al. in [1, Corollary 1].
(iii) Putting \(\lambda=\frac{1}{2}\) and \(\alpha=1=m\), we get the averaged midpoint-trapezoid-like integral inequality
$$\begin{aligned} & \biggl\vert \frac{1}{2} \biggl[\frac{qf(a)+f(b)}{1+q}+f \biggl( \frac {qa+b}{1+q} \biggr) \biggr]-\frac{1}{b-a} \int_{a}^{b}f(x)\,{}_{a} \mathrm{d}_{q}x \biggr\vert \\ &\quad \leq\min \biggl\{ \mathcal{H}_{1} \biggl(\frac{1}{2}, \frac{1}{1+q},1,1 \biggr),\mathcal{H}_{2} \biggl( \frac{1}{2},\frac{1}{1+q},1,1 \biggr) \biggr\} . \end{aligned}$$
Specially, if \(q\rightarrow1^{-}\), then we obtain
$$ \biggl\vert \frac{1}{2}\biggl[\frac{f(a)+f(b)}{2}+f\biggl( \frac{a+b}{2} \biggr)\biggr]-\frac{1}{b-a} \int_{a}^{b}f(x)\mathrm{d}x \biggr\vert \leq \frac{b-a}{16} \bigl[ \bigl\vert f'(b) \bigr\vert + \bigl\vert f'(a) \bigr\vert \bigr], $$
which is established by Xi and Qi in [30, Corollary 3.4].
(iv) Putting \(\lambda=1\), we get the trapezoid-like integral inequality
$$\begin{aligned} & \biggl\vert \frac{qf(a)+f(b)}{1+q}-\frac{1}{b-a} \int_{a}^{b}f(x)\,{}_{a}\mathrm {d}_{q}x \biggr\vert \\ &\quad \leq\min \biggl\{ \mathcal{H}_{1}\biggl(1,\frac{1}{1+q}, \alpha,m \biggr),\mathcal{H}_{2}\biggl(1,\frac{1}{1+q},\alpha,m \biggr) \biggr\} , \end{aligned}$$
where
$$\begin{aligned} &\mathcal{H}_{1}\biggl(1,\frac{1}{1+q},\alpha,m\biggr) \\ &\quad = (b-a) \biggl\{ \frac{2q^{\alpha+2}(1-q)^{2}+q^{2}(1+q)^{\alpha +1}(1-q)(1-q^{\alpha})}{(1+q)^{\alpha+2}(1-q^{\alpha+1})(1-q^{\alpha +2})} \bigl\vert {}_{a}D_{q}f(b) \bigr\vert \\ &\qquad {} +m\biggl[\frac{2q^{2}}{(1+q)^{3}}-\frac{2q^{\alpha +2}(1-q)^{2}+q^{2}(1+q)^{\alpha+1}(1-q)(1-q^{\alpha})}{(1+q)^{\alpha +2}(1-q^{\alpha+1})(1-q^{\alpha+2})}\biggr] \biggl\vert {}_{a}D_{q}f \biggl(\frac {a}{m} \biggr) \biggr\vert \biggr\} \end{aligned}$$
and
$$\begin{aligned} &\mathcal{H}_{2} \biggl(1,\frac{1}{1+q},\alpha,m \biggr) \\ &\quad = (b-a)\biggl\{ \biggl[\Phi_{7} \biggl(1,\frac{1}{1+q}, \alpha \biggr)+\Phi _{8} \biggl(1,\frac{1}{1+q},\alpha \biggr)- \Phi_{9} \biggl(1,\frac{1}{1+q},\alpha \biggr) \biggr] \bigl\vert {}_{a}D_{q}f(a) \bigr\vert \\ &\qquad {} +m \biggl[\frac{2q^{2}}{(1+q)^{3}}-\Phi_{7} \biggl(1, \frac{1}{1+q},\alpha \biggr)-\Phi_{8} \biggl(1,\frac{1}{1+q}, \alpha \biggr) \\ &\qquad {} +\Phi_{9} \biggl(1,\frac{1}{1+q},\alpha \biggr) \biggr] \biggl\vert {}_{a}D_{q}f \biggl(\frac{b}{m} \biggr) \biggr\vert \biggr\} . \end{aligned}$$
Specially, taking \(\alpha=1=m\), we obtain
$$\begin{aligned} & \biggl\vert \frac{qf(a)+f(b)}{1+q}-\frac{1}{b-a} \int_{a}^{b}f(x)\,{}_{a}\mathrm {d}_{q}x \biggr\vert \\ &\quad \leq(b-a) \biggl\{ \frac{q^{2}(1+4q+q^{2})}{(1+q)^{4}(1+q+q^{2})} \bigl\vert {}_{a}D_{q}f(b) \bigr\vert +\frac{q^{2}(1+3q^{2}+2q^{3})}{(1+q)^{4}(1+q+q^{2})} \bigl\vert {}_{a}D_{q}f(a) \bigr\vert \biggr\} , \end{aligned}$$
which is established by Sudsutad et al. in [26, Theorem 4.1].
If \(|{}_{a}D_{q}f|^{r}\) for \(r\geq1\) is \((\alpha,m)\)-convex, then the following theorem can be obtained.
Theorem 3.3
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
$$\begin{aligned} & \biggl\vert \lambda \bigl[\mu f(b)+(1-\mu)f(a) \bigr]+(1- \lambda)f \bigl(\mu b+(1-\mu)a \bigr)-\frac{1}{b-a} \int_{a}^{b}f(x)\,{}_{a} \mathrm{d}_{q}x \biggr\vert \\ &\quad \leq(b-a)\min \bigl\{ \mathcal{J}_{1}(\lambda,\mu,\alpha,m,r), \mathcal {J}_{2}(\lambda,\mu,\alpha,m,r) \bigr\} \end{aligned}$$
holds for all
\(\lambda,\mu\in[0,1]\)
if
\(|{}_{a}D_{q}f|^{r}\)
for
\(r\geq1\)
is
\((\alpha,m)\)-convex on
\([a,\frac{b}{m} ]\)
with
\(\alpha,m\in (0,1]^{2}\), where
$$\begin{aligned}& \begin{aligned} &\mathcal{J}_{1}( \lambda,\mu,\alpha,m,r) \\ &\quad =\Phi_{5}^{1-\frac{1}{r}}(\lambda,\mu) \biggl[ \Phi_{2}(\lambda,\mu,\alpha ) \bigl\vert {}_{a}D_{q}f(b) \bigr\vert ^{r} +m\bigl(\Phi_{5}(\lambda,\mu)- \Phi_{2}(\lambda,\mu,\alpha)\bigr) \biggl\vert {}_{a}D_{q}f \biggl(\frac{a}{m}\biggr) \biggr\vert ^{r} \biggr]^{\frac{1}{r}} \\ &\qquad {} +(1-\lambda)\mu^{1-\frac{1}{r}} \biggl[\Upsilon_{1}(\mu, \alpha) \bigl\vert {}_{a}D_{q}f(b) \bigr\vert ^{r}+m\Upsilon_{2}(\mu,\alpha) \biggl\vert {}_{a}D_{q}f\biggl(\frac {a}{m}\biggr) \biggr\vert ^{r} \biggr]^{\frac{1}{r}}, \end{aligned} \\& \begin{aligned} &\mathcal{J}_{2}(\lambda,\mu, \alpha,m,r) \\ &\quad =\Phi_{5}^{1-\frac{1}{r}}(\lambda,\mu) \biggl[ \Phi_{8}(\lambda,\mu,\alpha ) \bigl\vert {}_{a}D_{q}f(a) \bigr\vert ^{r} +m\bigl(\Phi_{5}(\lambda,\mu)- \Phi_{8}(\lambda,\mu,\alpha)\bigr) \biggl\vert {}_{a}D_{q}f \biggl(\frac{b}{m}\biggr) \biggr\vert ^{r} \biggr]^{\frac{1}{r}} \\ &\qquad {} +(1-\lambda)\mu^{1-\frac{1}{r}} \biggl[\Upsilon_{3}(\mu, \alpha) \bigl\vert {}_{a}D_{q}f(a) \bigr\vert ^{r}+m\Upsilon_{4}(\mu,\alpha) \biggl\vert {}_{a}D_{q}f\biggl(\frac {b}{m}\biggr) \biggr\vert ^{r} \biggr]^{\frac{1}{r}}, \end{aligned} \\& \Upsilon_{1}(\mu,\alpha)= \int_{0}^{\mu}t^{\alpha} \,{}_{0} \mathrm{d}_{q}t=\frac {\mu^{\alpha+1}(1-q)}{1-q^{\alpha+1}}, \end{aligned}$$
(3.4)
$$\begin{aligned}& \Upsilon_{2}(\mu,\alpha)= \int_{0}^{\mu} \bigl(1-t^{\alpha} \bigr) \,{}_{0}\mathrm {d}_{q}t=\mu-\frac{\mu^{\alpha+1}(1-q)}{1-q^{\alpha+1}}, \end{aligned}$$
(3.5)
$$\begin{aligned}& \Upsilon_{3}(\mu,\alpha)= \int_{0}^{\mu}(1-t)^{\alpha} \,{}_{0}\mathrm {d}_{q}t=(1-q)\mu\sum _{n=0}^{\infty}q^{n} \bigl(1-q^{n}\mu \bigr)^{\alpha}, \end{aligned}$$
(3.6)
$$\begin{aligned}& \Upsilon_{4}(\mu,\alpha)= \int_{0}^{\mu} \bigl(1-(1-t)^{\alpha} \bigr) \,{}_{0}\mathrm{d}_{q}t=\mu-(1-q)\mu\sum _{n=0}^{\infty}q^{n} \bigl(1-q^{n}\mu \bigr)^{\alpha}, \end{aligned}$$
(3.7)
and
\(\Phi_{2}(\lambda,\mu,\alpha)\), \(\Phi_{5}(\lambda,\mu)\), \(\Phi_{8}(\lambda ,\mu,\alpha)\)
are defined by (3.1), (3.2) and (3.3), respectively.
Proof
Using Lemma 2.1 and the power mean inequality, we have
$$\begin{aligned} & \biggl\vert \lambda\bigl[\mu f(b)+(1-\mu)f(a)\bigr]+(1-\lambda)f\bigl(\mu b+(1-\mu)a\bigr)-\frac{1}{b-a} \int_{a}^{b}f(x)\,{}_{a} \mathrm{d}_{q}x \biggr\vert \\ &\quad \leq(b-a) \biggl\{ \biggl( \int_{0}^{1} \bigl\vert qt-(1-\lambda\mu) \bigr\vert \,{}_{0}\mathrm {d}_{q}t\biggr)^{1-\frac{1}{r}} \\ &\qquad {} \times\biggl( \int_{0}^{1} \bigl\vert qt-(1-\lambda\mu) \bigr\vert \bigl\vert {}_{a}D_{q}f \bigl(tb+(1-t)a\bigr) \bigr\vert ^{r}\,{}_{0}\mathrm{d}_{q}t \biggr)^{\frac{1}{r}} \\ &\qquad {} +(1-\lambda) \biggl( \int_{0}^{\mu}1 \,{}_{0} \mathrm{d}_{q}t\biggr)^{1-\frac{1}{r}} \biggl( \int_{0}^{\mu}\bigl\vert {}_{a}D_{q}f \bigl(tb+(1-t)a\bigr) \bigr\vert ^{r}\,{}_{0}\mathrm {d}_{q}t\biggr)^{\frac{1}{r}} \biggr\} . \end{aligned}$$
(3.8)
Utilizing the \((\alpha,m)\)-convexity of \(|{}_{a}D_{q}f|^{r}\), we get
$$\begin{aligned} & \int_{0}^{1} \bigl\vert qt-(1-\lambda\mu) \bigr\vert \bigl\vert {}_{a}D_{q}f\bigl(tb+(1-t)a \bigr) \bigr\vert ^{r}\,{}_{0}\mathrm{d}_{q}t \\ &\quad \leq \int_{0}^{1} \bigl\vert qt-(1-\lambda\mu) \bigr\vert \biggl[t^{\alpha} \bigl\vert {}_{a}D_{q}f(b) \bigr\vert ^{r}+m\bigl(1-t^{\alpha}\bigr) \biggl\vert {}_{a}D_{q}f\biggl(\frac{a}{m} \biggr) \biggr\vert ^{r} \biggr]\,{}_{0}\mathrm{d}_{q}t \\ & \quad = \biggl( \int_{0}^{1}t^{\alpha}\bigl\vert qt-(1- \lambda\mu) \bigr\vert \,{}_{0}\mathrm {d}_{q}t \biggr) \bigl\vert {}_{a}D_{q}f(b) \bigr\vert ^{r} \\ &\qquad {} +m \biggl( \int_{0}^{1} \bigl\vert qt-(1-\lambda\mu) \bigr\vert \,{}_{0}\mathrm{d}_{q}t- \int _{0}^{1}t^{\alpha} \bigl\vert qt-(1- \lambda\mu) \bigr\vert \,{}_{0}\mathrm{d}_{q}t \biggr) \biggl\vert {}_{a}D_{q}f\biggl(\frac{a}{m}\biggr) \biggr\vert ^{r} \end{aligned}$$
(3.9)
and
$$\begin{aligned} & \int_{0}^{\mu}\bigl\vert {}_{a}D_{q}f \bigl(tb+(1-t)a\bigr) \bigr\vert ^{r}\,{}_{0} \mathrm{d}_{q}t \\ &\quad \leq \int_{0}^{\mu}\biggl[t^{\alpha} \bigl\vert {}_{a}D_{q}f(b) \bigr\vert ^{r}+m \bigl(1-t^{\alpha }\bigr) \biggl\vert {}_{a}D_{q}f \biggl(\frac{a}{m}\biggr) \biggr\vert ^{r} \biggr] \,{}_{0}\mathrm{d}_{q}t \\ &\quad = \biggl( \int_{0}^{\mu}t^{\alpha}\,{}_{0} \mathrm{d}_{q}t \biggr) \bigl\vert {}_{a}D_{q}f(b) \bigr\vert ^{r} +m \biggl( \int_{0}^{\mu}\bigl(1-t^{\alpha}\bigr) \,{}_{0}\mathrm{d}_{q}t \biggr) \biggl\vert {}_{a}D_{q}f\biggl(\frac{a}{m}\biggr) \biggr\vert ^{r}. \end{aligned}$$
(3.10)
Using (3.9) and (3.10) in (3.8), we get
$$\begin{aligned} & \biggl\vert \lambda\bigl[\mu f(b)+(1-\mu)f(a)\bigr]+(1-\lambda)f\bigl(\mu b+(1-\mu)a\bigr)-\frac{1}{b-a} \int_{a}^{b}f(x)\,{}_{a} \mathrm{d}_{q}x \biggr\vert \\ &\quad \leq(b-a)\biggl\{ \biggl[ \int_{0}^{1} \bigl\vert qt-(1-\lambda\mu) \bigr\vert \,{}_{0}\mathrm {d}_{q}t\biggr]^{1-\frac{1}{r}} \\ &\qquad {} \times\biggl[\biggl( \int_{0}^{1}t^{\alpha}\bigl\vert qt-(1- \lambda\mu) \bigr\vert \,{}_{0}\mathrm{d}_{q}t\biggr) \bigl\vert {}_{a}D_{q}f(b) \bigr\vert ^{r} \\ &\qquad {} +m\biggl( \int_{0}^{1} \bigl\vert qt-(1-\lambda\mu) \bigr\vert \,{}_{0}\mathrm{d}_{q}t- \int _{0}^{1}t^{\alpha} \bigl\vert qt-(1- \lambda\mu) \bigr\vert \,{}_{0}\mathrm{d}_{q}t\biggr) \biggl\vert {}_{a}D_{q}f\biggl(\frac{a}{m}\biggr) \biggr\vert ^{r}\biggr]^{\frac{1}{r}} \\ &\qquad {} +(1-\lambda)\mu^{1-\frac{1}{r}}\biggl[\biggl( \int_{0}^{\mu}t^{\alpha }\,{}_{0} \mathrm{d}_{q}t\biggr) \bigl\vert {}_{a}D_{q}f(b) \bigr\vert ^{r} \\ &\qquad {} +m\biggl( \int_{0}^{\mu}\bigl(1-t^{\alpha}\bigr) \,{}_{0}\mathrm{d}_{q}t\biggr) \biggl\vert {}_{a}D_{q}f\biggl(\frac{a}{m}\biggr) \biggr\vert ^{r}\biggr]^{\frac{1}{r}}\biggr\} . \end{aligned}$$
(3.11)
Similarly, we get
$$\begin{aligned} & \int_{0}^{1} \bigl\vert qt-(1-\lambda\mu) \bigr\vert \bigl\vert {}_{a}D_{q}f\bigl(tb+(1-t)a \bigr) \bigr\vert ^{r}\,{}_{0}\mathrm{d}_{q}t \\ &\quad \leq \int_{0}^{1} \bigl\vert qt-(1-\lambda\mu) \bigr\vert \biggl[(1-t)^{\alpha} \bigl\vert {}_{a}D_{q}f(a) \bigr\vert ^{r}+m\bigl(1-(1-t)^{\alpha}\bigr) \biggl\vert {}_{a}D_{q}f\biggl(\frac {b}{m}\biggr) \biggr\vert ^{r} \biggr]\,{}_{0}\mathrm{d}_{q}t \\ &\quad = \biggl( \int_{0}^{1}(1-t)^{\alpha}\bigl\vert qt-(1- \lambda\mu) \bigr\vert \,{}_{0}\mathrm {d}_{q}t \biggr) \bigl\vert {}_{a}D_{q}f(a) \bigr\vert ^{r} \\ &\qquad {} +m \biggl( \int_{0}^{1} \bigl\vert qt-(1-\lambda\mu) \bigr\vert \,{}_{0}\mathrm{d}_{q}t- \int _{0}^{1}(1-t)^{\alpha} \bigl\vert qt-(1-\lambda\mu) \bigr\vert \,{}_{0}\mathrm{d}_{q}t \biggr) \\ &\qquad {}\times\biggl\vert {}_{a}D_{q}f\biggl(\frac{b}{m} \biggr) \biggr\vert ^{r} \end{aligned}$$
(3.12)
and
$$\begin{aligned} & \int_{0}^{\mu}\bigl\vert {}_{a}D_{q}f \bigl(tb+(1-t)a\bigr) \bigr\vert ^{r}\,{}_{0} \mathrm{d}_{q}t \\ &\quad \leq \int_{0}^{\mu}\biggl[(1-t)^{\alpha} \bigl\vert {}_{a}D_{q}f(a) \bigr\vert ^{r}+m \bigl(1-(1-t)^{\alpha}\bigr) \biggl\vert {}_{a}D_{q}f \biggl(\frac{b}{m}\biggr) \biggr\vert ^{r} \biggr] \,{}_{0}\mathrm{d}_{q}t \\ &\quad = \biggl( \int_{0}^{\mu}(1-t)^{\alpha}\,{}_{0} \mathrm{d}_{q}t \biggr) \bigl\vert {}_{a}D_{q}f(a) \bigr\vert ^{r} \\ &\qquad {} +m \biggl( \int_{0}^{\mu}\bigl(1-(1-t)^{\alpha}\bigr) \,{}_{0}\mathrm{d}_{q}t \biggr) \biggl\vert {}_{a}D_{q}f\biggl(\frac{b}{m}\biggr) \biggr\vert ^{r}. \end{aligned}$$
(3.13)
Using (3.12) and (3.13) in (3.8), we get
$$\begin{aligned} & \biggl\vert \lambda\bigl[\mu f(b)+(1-\mu)f(a)\bigr]+(1-\lambda)f\bigl(\mu b+(1-\mu)a\bigr)-\frac{1}{b-a} \int_{a}^{b}f(x)\,{}_{a} \mathrm{d}_{q}x \biggr\vert \\ & \quad \leq(b-a) \biggl\{ \biggl[ \int_{0}^{1} \bigl\vert qt-(1-\lambda\mu) \bigr\vert \,{}_{0}\mathrm {d}_{q}t\biggr]^{1-\frac{1}{r}} \\ &\qquad {} \times\biggl[\biggl( \int_{0}^{1}(1-t)^{\alpha}\bigl\vert qt-(1- \lambda\mu) \bigr\vert \,{}_{0}\mathrm{d}_{q}t\biggr) \bigl\vert {}_{a}D_{q}f(a) \bigr\vert ^{r} \\ &\qquad {} +m\biggl( \int_{0}^{1} \bigl\vert qt-(1-\lambda\mu) \bigr\vert \,{}_{0}\mathrm{d}_{q}t- \int _{0}^{1}(1-t)^{\alpha} \bigl\vert qt-(1-\lambda\mu) \bigr\vert \,{}_{0}\mathrm{d}_{q}t \biggr) \biggl\vert {}_{a}D_{q}f\biggl(\frac{b}{m} \biggr) \biggr\vert ^{r}\biggr]^{\frac{1}{r}} \\ &\qquad {} +(1-\lambda)\mu^{1-\frac{1}{r}}\biggl[\biggl( \int_{0}^{\mu}(1-t)^{\alpha }\,{}_{0} \mathrm{d}_{q}t\biggr) \bigl\vert {}_{a}D_{q}f(a) \bigr\vert ^{r} \\ &\qquad {} +m\biggl( \int_{0}^{\mu}\bigl(1-(1-t)^{\alpha}\bigr) \,{}_{0}\mathrm{d}_{q}t \biggr) \biggl\vert {}_{a}D_{q}f\biggl(\frac{b}{m}\biggr) \biggr\vert ^{r}\biggr]^{\frac{1}{r}} \biggr\} . \end{aligned}$$
(3.14)
From (3.11) and (3.14), utilizing (3.1), (3.2), (3.3) and Lemma 2.2, we can deduce the desired result. The proof is complete. □
If \(|{}_{a}D_{q}f|^{r}\) for \(r>1\) is \((\alpha,m)\)-convex, then the following theorem can be obtained.
Theorem 3.4
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
$$\begin{aligned} & \biggl\vert \lambda \bigl[\mu f(b)+(1-\mu)f(a) \bigr]+(1-\lambda)f \bigl(\mu b+(1-\mu)a \bigr)-\frac{1}{b-a} \int_{a}^{b}f(x)\,{}_{a} \mathrm{d}_{q}x \biggr\vert \\ &\quad \leq(b-a)\min \bigl\{ \mathcal{K}_{1}(\lambda,\mu,\alpha,m), \mathcal {K}_{2}(\lambda,\mu,\alpha,m) \bigr\} \end{aligned}$$
holds for all
\(\lambda,\mu\in[0,1]\)
if
\(|{}_{a}D_{q}f|^{r}\)
for
\(r> 1\)
with
\(r^{-1}+s^{-1}=1\)
is
\((\alpha,m)\)-convex on
\([a,\frac{b}{m} ]\)
with
\(\alpha,m\in(0,1]^{2}\), where
$$\begin{aligned}& \begin{aligned} &\mathcal{K}_{1}(\lambda,\mu,\alpha,m) \\ &\quad =\Psi_{1}^{\frac{1}{s}}(\lambda,\mu) \biggl[ \Psi_{2}(\alpha) \bigl\vert {}_{a}D_{q}f(b) \bigr\vert ^{r}+m\bigl(1-\Psi_{2}(\alpha )\bigr) \biggl\vert {}_{a}D_{q}f\biggl(\frac{a}{m}\biggr) \biggr\vert ^{r}\biggr]^{\frac{1}{r}} \\ &\qquad {} +(1-\lambda)\mu^{\frac{1}{s}} \biggl[\Upsilon_{1}(\mu, \alpha) \bigl\vert {}_{a}D_{q}f(b) \bigr\vert ^{r}+m\Upsilon_{2}(\mu ,\alpha) \biggl\vert {}_{a}D_{q}f\biggl(\frac{a}{m}\biggr) \biggr\vert ^{r}\biggr]^{\frac{1}{r}}, \end{aligned} \\& \begin{aligned} &\mathcal{K}_{2}(\lambda,\mu,\alpha,m) \\ &\quad =\Psi_{1}^{\frac{1}{s}}(\lambda,\mu) \biggl[ \Psi_{3}(\alpha) \bigl\vert {}_{a}D_{q}f(a) \bigr\vert ^{r}+m\bigl(1-\Psi_{3}(\alpha )\bigr) \biggl\vert {}_{a}D_{q}f\biggl(\frac{b}{m}\biggr) \biggr\vert ^{r}\biggr]^{\frac{1}{r}} \\ &\qquad {} +(1-\lambda)\mu^{\frac{1}{s}} \biggl[\Upsilon_{3}(\mu, \alpha) \bigl\vert {}_{a}D_{q}f(a) \bigr\vert ^{r}+m\Upsilon_{4}(\mu ,\alpha) \biggl\vert {}_{a}D_{q}f\biggl(\frac{b}{m}\biggr) \biggr\vert ^{r}\biggr]^{\frac{1}{r}}, \end{aligned} \\& \begin{aligned} \Psi_{1}(\lambda,\mu)&= \int_{0}^{1} \bigl\vert qt-(1-\lambda\mu) \bigr\vert ^{s}\,{}_{0}\mathrm{d}_{q}t \\ &= \textstyle\begin{cases} (1-q)\sum_{n=0}^{\infty}q^{n}(1-\lambda\mu-q^{n+1})^{s}, &0\leq\lambda\mu\leq1-q, \\ \left [ \textstyle\begin{array}{l} (1-q)(1-\lambda\mu)^{s+1}\sum_{n=0}^{\infty}q^{n-1}(1-q^{n})^{s}\\ \quad {}+(1-q)\sum_{n=0}^{\infty}q^{n}(q^{n+1}-1+\lambda\mu)^{s}\\ \quad {}-(1-q)(1-\lambda\mu)^{s+1}\sum_{n=0}^{\infty}q^{n-1}(q^{n}-1)^{s} \end{array}\displaystyle \right ], &1-q< \lambda\mu\leq1, \end{cases}\displaystyle \end{aligned} \\& \Psi_{2}(\alpha)= \int_{0}^{1} t^{\alpha} \,{}_{0} \mathrm{d}_{q}t=\frac {1-q}{1-q^{\alpha+1}}, \\& \Psi_{3}(\alpha)= \int_{0}^{1} (1-t)^{\alpha} \,{}_{0}\mathrm{d}_{q}t=(1-q)\sum _{n=0}^{\infty}q^{n}\bigl(1-q^{n} \bigr)^{\alpha}, \end{aligned}$$
and
\(\Upsilon_{1}(\mu,\alpha)\), \(\Upsilon_{2}(\mu,\alpha)\), \(\Upsilon_{3}(\mu ,\alpha)\), \(\Upsilon_{4}(\mu,\alpha)\)
are defined by (3.4), (3.5), (3.6) and (3.7), respectively.
Proof
Using Lemma 2.1 and the Hölder inequality, we have
$$\begin{aligned} & \biggl\vert \lambda\bigl[\mu f(b)+(1-\mu)f(a)\bigr]+(1-\lambda)f\bigl(\mu b+(1-\mu)a\bigr)-\frac{1}{b-a} \int_{a}^{b}f(x)\,{}_{a} \mathrm{d}_{q}x \biggr\vert \\ &\quad \leq(b-a) \biggl\{ \biggl( \int_{0}^{1} \bigl\vert qt-(1-\lambda\mu) \bigr\vert ^{s}\,{}_{0}\mathrm{d}_{q}t \biggr)^{\frac{1}{s}} \\ &\qquad {} \times\biggl( \int_{0}^{1} \bigl\vert {}_{a}D_{q}f \bigl(tb+(1-t)a\bigr) \bigr\vert ^{r}\,{}_{0} \mathrm{d}_{q}t\biggr)^{\frac{1}{r}} \\ &\qquad {} +(1-\lambda) \biggl( \int_{0}^{\mu} 1^{s} \,{}_{0} \mathrm{d}_{q}t\biggr)^{\frac{1}{s}} \biggl( \int_{0}^{\mu} \bigl\vert {}_{a}D_{q}f \bigl(tb+(1-t)a\bigr) \bigr\vert ^{r}\,{}_{0}\mathrm {d}_{q}t\biggr)^{\frac{1}{r}} \biggr\} . \end{aligned}$$
(3.15)
Utilizing the \((\alpha,m)\)-convexity of \(|{}_{a}D_{q}f|^{r}\), we get
$$\begin{aligned} & \int_{0}^{1} \bigl\vert {}_{a}D_{q}f \bigl(tb+(1-t)a\bigr) \bigr\vert ^{r}\,{}_{0} \mathrm{d}_{q}t \\ &\quad \leq \int_{0}^{1} \biggl[t^{\alpha} \bigl\vert {}_{a}D_{q}f(b) \bigr\vert ^{r}+m \bigl(1-t^{\alpha }\bigr) \biggl\vert {}_{a}D_{q}f \biggl(\frac{a}{m}\biggr) \biggr\vert ^{r} \biggr] \,{}_{0}\mathrm{d}_{q}t \\ &\quad = \biggl( \int_{0}^{1} t^{\alpha} \,{}_{0} \mathrm{d}_{q}t \biggr) \bigl\vert {}_{a}D_{q}f(b) \bigr\vert ^{r}+m \biggl( \int_{0}^{1} \bigl(1-t^{\alpha}\bigr) \,{}_{0}\mathrm {d}_{q}t \biggr) \biggl\vert {}_{a}D_{q}f\biggl(\frac{a}{m}\biggr) \biggr\vert ^{r} \end{aligned}$$
(3.16)
and
$$\begin{aligned} & \int_{0}^{\mu} \bigl\vert {}_{a}D_{q}f \bigl(tb+(1-t)a\bigr) \bigr\vert ^{r}\,{}_{0}\mathrm {d}_{q}t \\ &\quad \leq \int_{0}^{\mu} \biggl[t^{\alpha} \bigl\vert {}_{a}D_{q}f(b) \bigr\vert ^{r}+m \bigl(1-t^{\alpha}\bigr) \biggl\vert {}_{a}D_{q}f \biggl(\frac{a}{m}\biggr) \biggr\vert ^{r} \biggr] \,{}_{0}\mathrm{d}_{q}t \\ &\quad = \biggl( \int_{0}^{\mu}t^{\alpha} \,{}_{0} \mathrm{d}_{q}t \biggr) \bigl\vert {}_{a}D_{q}f(b) \bigr\vert ^{r}+m \biggl( \int_{0}^{\mu}\bigl(1-t^{\alpha} \bigr) \,{}_{0}\mathrm{d}_{q}t \biggr) \biggl\vert {}_{a}D_{q}f\biggl(\frac{a}{m}\biggr) \biggr\vert ^{r}. \end{aligned}$$
(3.17)
Using (3.16) and (3.17) in (3.15), we get
$$\begin{aligned} & \biggl\vert \lambda\bigl[\mu f(b)+(1-\mu)f(a)\bigr]+(1-\lambda)f\bigl(\mu b+(1-\mu)a\bigr)-\frac{1}{b-a} \int_{a}^{b}f(x)\,{}_{a} \mathrm{d}_{q}x \biggr\vert \\ &\quad \leq(b-a) \biggl\{ \biggl( \int_{0}^{1} \bigl\vert qt-(1-\lambda\mu) \bigr\vert ^{s}\,{}_{0}\mathrm{d}_{q}t \biggr)^{\frac{1}{s}} \\ &\qquad {} \times\biggl[\biggl( \int_{0}^{1} t^{\alpha} \,{}_{0} \mathrm{d}_{q}t\biggr) \bigl\vert {}_{a}D_{q}f(b) \bigr\vert ^{r}+m\biggl( \int_{0}^{1} \bigl(1-t^{\alpha}\bigr) \,{}_{0}\mathrm {d}_{q}t\biggr) \biggl\vert {}_{a}D_{q}f\biggl(\frac{a}{m}\biggr) \biggr\vert ^{r}\biggr]^{\frac {1}{r}} \\ &\qquad {} +(1-\lambda)\mu^{\frac{1}{s}} \biggl[\biggl( \int_{0}^{\mu}t^{\alpha} \,{}_{0} \mathrm{d}_{q}t\biggr) \bigl\vert {}_{a}D_{q}f(b) \bigr\vert ^{r} \\ &\qquad {} +m\biggl( \int_{0}^{\mu}\bigl(1-t^{\alpha}\bigr) \,{}_{0}\mathrm{d}_{q}t\biggr) \biggl\vert {}_{a}D_{q}f\biggl(\frac{a}{m}\biggr) \biggr\vert ^{r}\biggr]^{\frac{1}{r}} \biggr\} . \end{aligned}$$
(3.18)
Similarly, we get
$$\begin{aligned} & \int_{0}^{1} \bigl\vert {}_{a}D_{q}f \bigl(tb+(1-t)a\bigr) \bigr\vert ^{r}\,{}_{0} \mathrm{d}_{q}t \\ &\quad \leq \int_{0}^{1} \biggl[(1-t)^{\alpha} \bigl\vert {}_{a}D_{q}f(a) \bigr\vert ^{r}+m \bigl(1-(1-t)^{\alpha}\bigr) \biggl\vert {}_{a}D_{q}f \biggl(\frac{b}{m}\biggr) \biggr\vert ^{r} \biggr] \,{}_{0}\mathrm{d}_{q}t \\ &\quad = \biggl( \int_{0}^{1} (1-t)^{\alpha} \,{}_{0}\mathrm{d}_{q}t \biggr) \bigl\vert {}_{a}D_{q}f(a) \bigr\vert ^{r} \\ &\qquad {} +m \biggl( \int_{0}^{1} \bigl(1-(1-t)^{\alpha}\bigr) \,{}_{0}\mathrm{d}_{q}t \biggr) \biggl\vert {}_{a}D_{q}f\biggl(\frac{b}{m}\biggr) \biggr\vert ^{r} \end{aligned}$$
(3.19)
and
$$\begin{aligned} & \int_{0}^{\mu} \bigl\vert {}_{a}D_{q}f \bigl(tb+(1-t)a\bigr) \bigr\vert ^{r}\,{}_{0}\mathrm {d}_{q}t \\ &\quad \leq \int_{0}^{\mu} \biggl[(1-t)^{\alpha} \bigl\vert {}_{a}D_{q}f(a) \bigr\vert ^{r}+m \bigl(1-(1-t)^{\alpha}\bigr) \biggl\vert {}_{a}D_{q}f \biggl(\frac{b}{m}\biggr) \biggr\vert ^{r} \biggr] \,{}_{0}\mathrm{d}_{q}t \\ &\quad = \biggl( \int_{0}^{\mu}(1-t)^{\alpha} \,{}_{0}\mathrm{d}_{q}t \biggr) \bigl\vert {}_{a}D_{q}f(a) \bigr\vert ^{r} \\ &\qquad {} +m \biggl( \int_{0}^{\mu}\bigl(1-(1-t)^{\alpha}\bigr) \,{}_{0}\mathrm{d}_{q}t \biggr) \biggl\vert {}_{a}D_{q}f\biggl(\frac{b}{m}\biggr) \biggr\vert ^{r}. \end{aligned}$$
(3.20)
Using (3.19) and (3.20) in (3.15), we get
$$\begin{aligned} & \biggl\vert \lambda\bigl[\mu f(b)+(1-\mu)f(a)\bigr]+(1-\lambda)f\bigl(\mu b+(1-\mu)a\bigr)-\frac{1}{b-a} \int_{a}^{b}f(x)\,{}_{a} \mathrm{d}_{q}x \biggr\vert \\ &\quad \leq(b-a) \biggl\{ \biggl( \int_{0}^{1} \bigl\vert qt-(1-\lambda\mu) \bigr\vert ^{s}\,{}_{0}\mathrm{d}_{q}t \biggr)^{\frac{1}{s}} \\ &\qquad {} \times\biggl[\biggl( \int_{0}^{1} (1-t)^{\alpha} \,{}_{0}\mathrm{d}_{q}t\biggr) \bigl\vert {}_{a}D_{q}f(a) \bigr\vert ^{r} \\ &\qquad {} +m\biggl( \int_{0}^{1} \bigl(1-(1-t)^{\alpha}\bigr) \,{}_{0}\mathrm{d}_{q}t\biggr) \biggl\vert {}_{a}D_{q}f\biggl(\frac{b}{m}\biggr) \biggr\vert ^{r}\biggr]^{\frac{1}{r}} \\ &\qquad {} +(1-\lambda)\mu^{\frac{1}{s}} \biggl[\biggl( \int_{0}^{\mu}(1-t)^{\alpha} \,{}_{0}\mathrm{d}_{q}t\biggr) \bigl\vert {}_{a}D_{q}f(a) \bigr\vert ^{r} \\ &\qquad {} +m\biggl( \int_{0}^{\mu}\bigl(1-(1-t)^{\alpha}\bigr) \,{}_{0}\mathrm{d}_{q}t \biggr) \biggl\vert {}_{a}D_{q}f\biggl(\frac{b}{m}\biggr) \biggr\vert ^{r}\biggr]^{\frac{1}{r}} \biggr\} . \end{aligned}$$
(3.21)
From (3.18) and (3.21), utilizing (3.4), (3.5), (3.6), (3.7), Lemma 2.2 and Lemma 2.6, we can deduce the desired result. The proof is completed. □
Remark 3.2
For \(\mu=\frac{1}{1+q}\), if we put \(\lambda=0\), \(\lambda=\frac{1}{3}\), \(\lambda=\frac{1}{2}\) and \(\lambda=1\) in Theorem 3.3 and Theorem 3.4, then we can get the midpoint-like integral inequality, the Simpson-like integral inequality, the averaged midpoint-trapezoid-like integral inequality and the trapezoid-like integral inequality, respectively.
Next we establish the q-integral inequalities involving the product of two \((\alpha,m)\)-convex functions.
Theorem 3.5
For
\(0\leq a< b\)
and some fixed
\(m\in(0,1]\), let
\(f,g: [a,\frac {b}{m} ]\rightarrow\mathbb{R}\)
be continuous and nonnegative functions. Then the inequality
$$ \frac{1}{b-a} \int_{a}^{b} f(x)g(x)\,{}_{a} \mathrm{d}_{q}x\leq\min \bigl\{ \mathcal {L}_{1}( \alpha_{1},\alpha_{2},m),\mathcal{L}_{2}( \alpha_{1},\alpha_{2},m) \bigr\} $$
holds if
f
and
g
are
\((\alpha_{1},m)\)-convex and
\((\alpha _{2},m)\)-convex on
\([a,\frac{b}{m} ]\)
with
\(\alpha_{1},\alpha_{2}\in (0,1]^{2}\), respectively, where
$$\begin{aligned} &\mathcal{L}_{1}(\alpha_{1},\alpha_{2},m) \\ &\quad = \biggl[\frac{1-q}{1-q^{\alpha_{1}+\alpha_{2}+1}}-\frac{1-q}{1-q^{\alpha _{1}+1}}-\frac{1-q}{1-q^{\alpha_{2}+1}}+1 \biggr]m^{2}f \biggl(\frac{a}{m} \biggr)g \biggl( \frac{a}{m} \biggr) \\ &\qquad {} +\frac{1-q}{1-q^{\alpha_{1}+\alpha_{2}+1}}f(b)g(b)+ \biggl[\frac {1-q}{1-q^{\alpha_{2}+1}}- \frac{1-q}{1-q^{\alpha_{1}+\alpha_{2}+1}} \biggr]mf \biggl(\frac{a}{m} \biggr)g(b) \\ &\qquad {} + \biggl[\frac{1-q}{1-q^{\alpha_{1}+1}}-\frac{1-q}{1-q^{\alpha_{1}+\alpha _{2}+1}} \biggr]mf(b)g \biggl( \frac{a}{m} \biggr), \\ &\mathcal{L}_{2}(\alpha_{1},\alpha_{2},m) \\ &\quad = \bigl[\Theta(\alpha_{1},\alpha_{2})-\Theta( \alpha_{1})-\Theta(\alpha _{2})+1 \bigr]m^{2}f \biggl(\frac{b}{m} \biggr)g \biggl(\frac{b}{m} \biggr) \\ &\qquad {} +\Theta(\alpha_{1},\alpha_{2})f(a)g(a)+ \bigl[ \Theta(\alpha_{1})-\Theta(\alpha _{1},\alpha_{2}) \bigr]mf(a)g \biggl(\frac{b}{m} \biggr) \\ &\qquad {} + \bigl[\Theta(\alpha_{2})-\Theta(\alpha_{1}, \alpha_{2}) \bigr]mf \biggl(\frac {b}{m} \biggr)g(a), \\ &\Theta(\alpha_{1},\alpha_{2})= \int_{0}^{1} (1-t)^{\alpha_{1}+\alpha_{2}} \,{}_{0}\mathrm{d}_{q}t=(1-q)\sum _{n=0}^{\infty}q^{n} \bigl(1-q^{n} \bigr)^{\alpha _{1}+\alpha_{2}}, \end{aligned}$$
and
$$ \Theta(\alpha_{i})= \int_{0}^{1} (1-t)^{\alpha_{i}} \,{}_{0}\mathrm{d}_{q}t=(1-q)\sum _{n=0}^{\infty}q^{n} \bigl(1-q^{n} \bigr)^{\alpha_{i}},\quad i=1,2. $$
Proof
Using the \((\alpha_{1},m)\)-convexity of f and the \((\alpha _{2},m)\)-convexity of g, respectively, for all \(t\in[0,1]\), we have
$$ f \bigl(tb+(1-t)a \bigr)\leq t^{\alpha_{1}}f(b)+m \bigl(1-t^{\alpha_{1}}\bigr)f \biggl(\frac {a}{m} \biggr) $$
(3.22)
and
$$ g \bigl(tb+(1-t)a \bigr)\leq t^{\alpha_{2}}g(b)+m \bigl(1-t^{\alpha_{2}}\bigr)g \biggl(\frac {a}{m} \biggr). $$
(3.23)
Multiplying (3.22) with (3.23), we get
$$\begin{aligned} &f \bigl(tb+(1-t)a \bigr)g \bigl(tb+(1-t)a \bigr) \\ &\quad \leq t^{\alpha_{1}+\alpha_{2}}f(b)g(b)+\bigl(1-t^{\alpha_{1}}\bigr) \bigl(1-t^{\alpha _{2}}\bigr)m^{2}f \biggl(\frac{a}{m} \biggr)g \biggl(\frac{a}{m} \biggr) \\ &\qquad {} +t^{\alpha_{2}}\bigl(1-t^{\alpha_{1}}\bigr)mf \biggl( \frac{a}{m} \biggr)g(b)+t^{\alpha _{1}}\bigl(1-t^{\alpha_{2}} \bigr)mf(b)g \biggl(\frac{a}{m} \biggr). \end{aligned}$$
(3.24)
Taking the q-integral for (3.24) with respect to t on \((0,1)\) and using Lemma 2.2, we obtain
$$\begin{aligned} & \int_{0}^{1} f \bigl(tb+(1-t)a \bigr)g \bigl(tb+(1-t)a \bigr)\,{}_{0}\mathrm{d}_{q}t \\ &\quad \leq \biggl[\frac{1-q}{1-q^{\alpha_{1}+\alpha_{2}+1}}-\frac{1-q}{1-q^{\alpha _{1}+1}}-\frac{1-q}{1-q^{\alpha_{2}+1}}+1 \biggr]m^{2}f \biggl(\frac{a}{m} \biggr)g \biggl( \frac{a}{m} \biggr) \\ &\qquad {} +\frac{1-q}{1-q^{\alpha_{1}+\alpha_{2}+1}}f(b)g(b)+ \biggl[\frac {1-q}{1-q^{\alpha_{2}+1}}- \frac{1-q}{1-q^{\alpha_{1}+\alpha_{2}+1}} \biggr]mf \biggl(\frac{a}{m} \biggr)g(b) \\ &\qquad {} + \biggl[\frac{1-q}{1-q^{\alpha_{1}+1}}-\frac{1-q}{1-q^{\alpha_{1}+\alpha _{2}+1}} \biggr]mf(b)g \biggl( \frac{a}{m} \biggr). \end{aligned}$$
(3.25)
Similarly, we get
$$\begin{aligned} & \int_{0}^{1} f \bigl(tb+(1-t)a \bigr)g \bigl(tb+(1-t)a \bigr)\,{}_{0}\mathrm{d}_{q}t \\ &\quad \leq \biggl( \int_{0}^{1}(1-t)^{\alpha_{1}+\alpha_{2}}\,{}_{0} \mathrm{d}_{q}t- \int _{0}^{1}(1-t)^{\alpha_{1}}\,{}_{0} \mathrm{d}_{q}t \\ &\qquad {} - \int_{0}^{1}(1-t)^{\alpha_{2}}\,{}_{0} \mathrm{d}_{q}t+1 \biggr)m^{2}f \biggl(\frac {b}{m} \biggr)g \biggl(\frac{b}{m} \biggr) \\ &\qquad {} + \biggl( \int_{0}^{1}(1-t)^{\alpha_{1}+\alpha_{2}}\,{}_{0} \mathrm{d}_{q}t \biggr)f(a)g(a) \\ &\qquad {} + \biggl( \int_{0}^{1}(1-t)^{\alpha_{1}}\,{}_{0} \mathrm{d}_{q}t- \int_{0}^{1}(1-t)^{\alpha _{1}+\alpha_{2}}\,{}_{0} \mathrm{d}_{q}t \biggr)mf(a)g \biggl(\frac{b}{m} \biggr) \\ &\qquad {} + \biggl( \int_{0}^{1}(1-t)^{\alpha_{2}}\,{}_{0} \mathrm{d}_{q}t- \int _{0}^{1}(1-t)^{\alpha_{1}+\alpha_{2}}\,{}_{0} \mathrm{d}_{q}t \biggr)mf \biggl(\frac {b}{m} \biggr)g(a). \end{aligned}$$
(3.26)
A simple calculation shows that
$$ \int_{0}^{1} f \bigl(tb+(1-t)a \bigr)g \bigl(tb+(1-t)a \bigr)\,{}_{0}\mathrm{d}_{q}t = \frac{1}{b-a} \int_{a}^{b} f(x)g(x)\,{}_{a} \mathrm{d}_{q}x. $$
(3.27)
From (3.25), (3.26) and (3.27), we obtain the desired result. This ends the proof. □
Corollary 3.2
In Theorem 3.5, choosing
\(\alpha_{1}=\alpha_{2}=\alpha\), we obtain
$$ \frac{1}{b-a} \int_{a}^{b} f(x)g(x)\,{}_{a} \mathrm{d}_{q}x\leq\min \bigl\{ \mathcal {T}_{1}(\alpha,m), \mathcal{T}_{2}(\alpha,m) \bigr\} , $$
where
$$\begin{aligned} \mathcal{T}_{1}(\alpha,m) &= \frac{1-q}{1-q^{2\alpha+1}}f(b)g(b)+ \biggl[ \frac{1-q}{1-q^{2\alpha +1}}-\frac{2(1-q)}{1-q^{\alpha+1}}+1 \biggr]m^{2}f \biggl( \frac{a}{m} \biggr)g \biggl(\frac{a}{m} \biggr) \\ &\quad {} +\frac{q^{\alpha+1}(1-q)(1-q^{\alpha})}{(1-q^{\alpha+1})(1-q^{2\alpha +1})}m \biggl[f \biggl(\frac{a}{m} \biggr)g(b)+f(b)g \biggl(\frac{a}{m} \biggr) \biggr] \end{aligned}$$
and
$$\begin{aligned} \mathcal{T}_{2}(\alpha,m) &= \Biggl[(1-q)\sum _{n=0}^{\infty}q^{n} \bigl(1-q^{n} \bigr)^{2\alpha}-2(1-q)\sum_{n=0}^{\infty}q^{n} \bigl(1-q^{n} \bigr)^{\alpha}+1 \Biggr]m^{2}f \biggl( \frac {b}{m} \biggr)g \biggl(\frac{b}{m} \biggr) \\ &\quad {} +(1-q)\sum_{n=0}^{\infty}q^{n} \bigl(1-q^{n} \bigr)^{2\alpha}f(a)g(a) \\ &\quad {} + \Biggl[(1-q)\sum_{n=0}^{\infty}q^{n} \bigl(1-q^{n} \bigr)^{\alpha}-(1-q)\sum _{n=0}^{\infty}q^{n} \bigl(1-q^{n} \bigr)^{2\alpha} \Biggr] \\ &\quad {} \times \biggl[mf(a)g \biggl(\frac{b}{m} \biggr)+mf \biggl( \frac{b}{m} \biggr)g(a) \biggr]. \end{aligned}$$
Further, taking
\(\alpha=1=m\), we get
$$\begin{aligned} &\frac{1}{b-a} \int_{a}^{b} f(x)g(x)\,{}_{a} \mathrm{d}_{q}x \\ &\quad \leq \frac{1}{1+q+q^{2}}f(b)g(b)+\frac{q(1+q^{2})}{(1+q)(1+q+q^{2})}f(a)g(a) \\ &\qquad {} +\frac{q^{2}}{(1+q)(1+q+q^{2})} \bigl[f(a)g(b)+f(b)g(a) \bigr], \end{aligned}$$
which is established by Sudsutad et al. in [26, Theorem 4.3].
