Biomedical diagnosis perspective of epigenetic detections using alpha-hemolysin nanopore

Yong Wang; Li-qun Gu; Yong Wang; Li-qun Gu

doi:10.3934/matersci.2015.4.448

AIMS Materials Science

2015, Volume 2, Issue 4: 448-472. doi: 10.3934/matersci.2015.4.448

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Review Topical Sections

Biomedical diagnosis perspective of epigenetic detections using alpha-hemolysin nanopore

Yong Wang ^,,
Li-qun Gu ^,

Department of Biological Engineering, Dalton Cardiovascular Research Center, University of Missouri-Columbia, Columbia, MO 65211, USA

Received: 28 September 2015 Accepted: 02 November 2015 Published: 16 December 2015

The α-hemolysin nanopore has been studied for applications in DNA sequencing, various single-molecule detections, biomolecular interactions, and biochips. The detection of single molecules in a clinical setting could dramatically improve cancer detection and diagnosis as well as develop personalized medicine practices for patients. This brief review shortly presents the current solid state and protein nanopore platforms and their applications like biosensing and sequencing. We then elaborate on various epigenetic detections (like microRNA, G-quadruplex, DNA damages, DNA modifications) with the most widely used alpha-hemolysin pore from a biomedical diagnosis perspective. In these detections, a nanopore electrical current signature was generated by the interaction of a target with the pore. The signature often was evidenced by the difference in the event duration, current level, or both of them. An ideal signature would provide obvious differences in the nanopore signals between the target and the background molecules. The development of cancer biomarker detection techniques and nanopore devices have the potential to advance clinical research and resolve health problems. However, several challenges arise in applying nanopore devices to clinical studies, including super low physiological concentrations of biomarkers resulting in low sensitivity, complex biological sample contents resulting in false signals, and fast translocating speed through the pore resulting in poor detections. These issues and possible solutions are discussed.

Keywords:

Citation: Yong Wang, Li-qun Gu. Biomedical diagnosis perspective of epigenetic detections using alpha-hemolysin nanopore[J]. AIMS Materials Science, 2015, 2(4): 448-472. doi: 10.3934/matersci.2015.4.448

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Abstract

1. Introduction

In 1960, Opial ^[12] established the following inequality:

Theorem A Suppose $ f\in C^{1}[0, h] $ satisfies $ f(0) = f(h) = 0 $ and $ f(x) > 0 $ for all $ x\in (0, h). $ Then the inequality holds

$\begin{align} \int_{0}^h\left|f(x)f'(x)\right|dx\leq\frac{h}{4}\int_{0}^{h}(f'(x))^{2}dx, \end{align}$ $

(1.1)

where this constant $ h/4 $ is best possible.

Many generalizations and extensions of Opial's inequality were established ^{[2,4,5,6,7,8,9,10,11,15,16,17,18,19]}. For an extensive survey on these inequalities, see ^[13]. Opial's inequality and its generalizations and extensions play a fundamental role in the ordinary and partial differential equations as well as difference equation ^{[2,3,4,6,7,9,10,11,17]}. In particular, Agarwal and Pang ^[3] proved the following Opial-Wirtinger's type inequalities.

Theorem B Let $ \lambda\geq 1 $ be a given real number, and let $ p(t) $ be a nonnegative and continuous function on $ [0, a] $. Further, let $ x(t) $ be an absolutely continuous function on $ [0, a] $, with $ x(0) = x(a) = 0 $. Then

$\begin{align} \int_{0}^{a}p(t)|x(t)|^{\lambda}dt\leq\frac{1}{2}\int_{0}^{a}[t(a-t)]^{(\lambda-1)/2}p(t)dt\int_{0}^{a}\left|x'(t) \right|^{\lambda}dt. \end{align}$ $

(1.2)

The first aim of the present paper is to establish Opial-Wirtinger's type inequalities involving Katugampola conformable partial derivatives and $ \alpha $-conformable integrals (see Section 2). Our result is given in the following theorem, which is a generalization of (1.2).

Theorem 1.1 Let $ \lambda\geq 1 $ be a real number and $ \alpha\in (0, 1] $, and let $ p(s, t) $ be a nonnegative and continuous functions on $ [0, a]\times [0, b] $. Further, let $ x(s, t) $ be an absolutely continuous function and Katugampola partial derivable on $ [0, a]\times [0, b] $, with $ x(s, 0) = x(0, t) = x(0, 0) = 0 $ and $ x(a, b) = x(a, t) = x(s, b) = 0 $. If $ p > 1, $ $ \frac{1}{p}+\frac{1}{q} = 1 $ Then

$ \int_{0}^{a}\int_{0}^{b}p(s, t)|x(s, t)|^{\lambda}d_{\alpha}sd_{\alpha}t\leq\frac{p+q}{pq} \left(\frac{1}{\alpha^{2}}\right)^{\lambda-1}\left(\int_{0}^{a}\int_{0}^{b}\Gamma_{abpq\lambda\alpha}(s, t)\cdot p(s, t)d_{\alpha}sd_{\alpha}t\right) $

$\begin{align} \times\int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}(x)_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t, \end{align}$ $

(1.3)

where

$ \Gamma_{abpq\lambda\alpha}(s, t) = \left\{(st)^{1/p}[(a-s)(b-t)]^{1/q}\right\}^{\alpha(\lambda-1)}. $

Remark 1.1 Let $ x(s, t) $ reduce to $ s(t) $ and with suitable modifications, and $ p = q = 2 $ and $ \alpha = 1 $, (1.3) become (1.2).

Theorem C Let $ \lambda\geq 1 $ be a given real number, and let $ p(t) $ be a nonnegative and continuous function on $ [0, a] $. Further, let $ x(t) $ be an absolutely continuous function on $ [0, a] $, with $ x(0) = x(a) = 0 $. Then

$\begin{align} \int_{0}^{a}p(t)|x(t)|^{\lambda}dt\leq\frac{1}{2}\left(\frac{a}{2} \right)^{\lambda-1}\left(\int_{0}^{a}p(t)dt\right)\int_{0}^{a}\left|x'(t)\right|^{\lambda}dt. \end{align}$ $

(1.4)

Another aim of this paper is to establish the following inequality involving Katugampola conformable partial derivatives and $ \alpha $-conformable integrals. Our result is given in the following theorem.

Theorem 1.2 Let $ j = 1, 2 $ and $ \lambda\geq 1 $ be a real number, and let $ p_{j}(s, t) $ be a nonnegative and continuous functions on $ [0, a]\times [0, b] $. Further, let $ x_{j}(s, t) $ be an absolutely continuous function and Katugampola partial derivable on $ [0, a]\times [0, b] $, with $ x_{j}(s, 0) = x_{j}(0, t) = x_{j}(0, 0) = 0 $ and $ x_{j}(a, b) = x_{j}(a, t) = x_{j}(s, b) = 0 $. Then for $ \alpha\in(0, 1] $

$ \int_{0}^{a}\int_{0}^{b}\left(p_{1}(s, t)|x_{1}(s, t)|^{\lambda}+p_{2}(s, t)|x_{2}(s, t)|^{\lambda}\right)d_{\alpha}sd_{\alpha}t\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; $

$ \leq\frac{1}{2^{\lambda}}\left(\frac{1}{\alpha^{2}}\right)^{\lambda-1}\Bigg[\left(\int_{0}^{a}\int_{0}^{b}(st)^{\alpha(\lambda-1)}p_{1}(s, t) d_{\alpha}sd_{\alpha}t\right)\int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}(x_{1})_{\alpha^{2}}(s, t)\right|^{\lambda} d_{\alpha}sd_{\alpha}t $

$\begin{align} +\left(\int_{0}^{a}\int_{0}^{b}(st)^{\alpha(\lambda-1)}p_{2}(s, t)d_{\alpha}sd_{\alpha}t\right)\int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}(x_{2})_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t\Bigg]. \end{align}$ $

(1.5)

2. Katugampola conformable partial derivatives

Here, let's recall the well-known Katugampola derivative formulation of conformable derivative of order for $ \alpha\in(0, 1] $ and $ t\in[0, \infty) $, given by

$\begin{align} {\cal D}_{\alpha}(f)(t) = \lim\limits_{\varepsilon\rightarrow 0}\frac{f(te^{\varepsilon t^{-\alpha}})-f(t)}{\varepsilon}, \end{align}$ $

(2.1)

and

$\begin{align} {\cal D}_{\alpha}(f)(0) = \lim\limits_{t\rightarrow 0}D_{\alpha}(f)(t), \end{align}$ $

(2.2)

provided the limits exist. If $ f $ is fully differentiable at $ t $, then

$ {\cal D}_{\alpha}(f)(t) = t^{1-\alpha}\frac{df}{dt}(t). $

A function $ f $ is $ \alpha $-differentiable at a point $ t\geq 0 $, if the limits in (2.1) and (2.2) exist and are finite. Inspired by this, we propose a new concept of $ \alpha $-conformable partial derivative. In the way of (1.4), $ \alpha $-conformable partial derivative is defined in as follows:

Definition 2.1 ^[20] ($ \alpha $-conformable partial derivative) Let $ \alpha\in (0, 1] $ and $ s, t\in[0, \infty) $. Suppose $ f(s, t) $ is a continuous function and partial derivable, the $ \alpha $-conformable partial derivative at a point $ s\geq 0 $, denoted by $ \frac{\partial}{\partial s}(f)_{\alpha}(s, t) $, defined by

$\begin{align} \frac{\partial}{\partial s}(f)_{\alpha}(s, t) = \lim\limits_{\varepsilon\rightarrow 0}\frac{f(se^{\varepsilon s^{-\alpha}}, t)-f(s, t)} {\varepsilon}, \end{align}$ $

(2.3)

provided the limits exist, and call $ \alpha $-conformable partial derivable.

Recently, Katugampola conformable partial derivative is defined in as follows:

Definition 2.2 ^[20] (Katugampola conformable partial derivatives) Let $ \alpha\in (0, 1] $ and $ s, t\in[0, \infty) $. Suppose $ f(s, t) $ and $ \frac{\partial}{\partial s}(f)_{\alpha}(s, t) $ are continuous functions and partial derivable, the Katugampola conformable partial derivative, denoted by $ \frac{\partial^{2}}{\partial s\partial t}(f)_{\alpha^{2}}(s, t) $, defined by

$\begin{align} \frac{\partial^{2}}{\partial s\partial t}(f)_{\alpha^{2}}(s, t) = \lim\limits_{\varepsilon\rightarrow 0}\frac{\frac{\partial}{\partial s}(f)_{\alpha}(s, te^{\varepsilon t^{-\alpha}})-\frac{\partial}{\partial s}(f)_{\alpha}(s, t)}{\varepsilon}, \end{align}$ $

(2.4)

provided the limits exist, and call Katugampola conformable partial derivable.

Definition 2.3 ^[20] ($ \alpha $-conformable integral) Let $ \alpha\in (0, 1] $, $ 0\leq a < b $ and $ 0\leq c < d $. A function $ f(x, y):[a, b]\times[c, d]\rightarrow {\Bbb R} $ is $ \alpha $-conformable integrable, if the integral

$\begin{align} \int_{a}^{b}\int_{c}^{d}f(x, y)d_{\alpha}xd_{\alpha}y: = \int_{a}^{b}\int_{c}^{d} (xy)^{\alpha-1}f(x, y)dxdy \end{align}$ $

(2.5)

exists and is finite.

3. Main results

Theorem 3.1 Let $ \lambda\geq 1 $ be a real number and $ \alpha\in (0, 1] $, and let $ p(s, t) $ be a nonnegative and continuous functions on $ [0, a]\times [0, b] $. Further, let $ x(s, t) $ be an absolutely continuous function and Katugampola partial derivable on $ [0, a]\times [0, b] $, with $ x(s, 0) = x(0, t) = x(0, 0) = 0 $ and $ x(a, b) = x(a, t) = x(s, b) = 0 $. If $ p > 1, $ $ \frac{1}{p}+\frac{1}{q} = 1 $ Then

$\begin{align} \times\int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}(x)_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t, \end{align}$ $

(3.1)

where

$ \Gamma_{abpq\lambda\alpha}(s, t) = \left\{(st)^{1/p}[(a-s)(b-t)]^{1/q}\right\}^{\alpha(\lambda-1)}. $

Proof From (2.4) and (2.5), we have

$ x(s, t) = \int_{0}^{s}\int_{0}^{t}\frac{\partial^{2}}{\partial s\partial t}(x)_{\alpha^{2}}(s, t)d_{\alpha}sd_{\alpha}t. $

By using Hölder's inequality with indices $ \lambda $ and $ \lambda/(\lambda-1) $, we have

$ |x(s, t)|^{\lambda/p}\leq \left[\left(\int_{0}^{s}\int_{0}^{t}\left|\frac{\partial^{2}}{\partial s\partial t} (x)_{\alpha^{2}}(s, t)\right|d_{\alpha}sd_{\alpha}t\right)^{\lambda}\right]^{1/p}\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; $

$\begin{align} \leq\left(\frac{1}{\alpha^{2}}(st)^{\alpha}\right)^{(\lambda-1)/p}\left(\int_{0}^{s}\int_{0}^{t}\left|\frac{\partial^{2}}{\partial s\partial t}(x)_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t\right)^{1/p}. \end{align}$ $

(3.2)

Similarly, from

$ x(s, t) = \int_{s}^{a}\int_{t}^{b}\frac{\partial^{2}}{\partial s\partial t}(x)_{\alpha^{2}}(s, t)d_{\alpha}sd_{\alpha}t, $

we obtain

$\begin{align} |x(s, t)|^{\lambda/q}\leq \left(\frac{1}{\alpha^{2}}[(a-s)(b-t)]^{\alpha}\right)^{(\lambda-1)/q} \left(\int_{s}^{a}\int_{t}^{b}\left|\frac{\partial^{q}}{\partial s\partial t}(x)_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t\right)^{1/q}. \end{align}$ $

(3.3)

Now a multiplication of (3.2) and (3.3), and by using the well-known Young inequality gives

$\begin{eqnarray*} |x(s, t)|^{\lambda}&\leq &\left(\frac{1}{\alpha^{2}}\right)^{\lambda-1}\cdot\Gamma_{abpq\lambda\alpha}(s, t)\cdot\left(\int_{0}^{s}\int_{0}^{t} \left|\frac{\partial^{2}}{\partial s\partial t}(x)_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t\right)^{1/p}\\ &\times&\left(\int_{s}^{a}\int_{t}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}(x)_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t\right)^{1/q}\\ &\leq &\left(\frac{1}{\alpha^{2}}\right)^{\lambda-1}\cdot\Gamma_{abpq\lambda\alpha}(s, t)\cdot\Bigg(\frac{1}{p}\int_{0}^{s}\int_{0}^{t}\left|\frac{\partial^{2}}{\partial s\partial t}(x)_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t\\ &+&\frac{1}{q}\int_{s}^{a}\int_{t}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}(x)_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t\Bigg) \end{eqnarray*}$ $

$\begin{align} \; \; \; \; \; \; \; \; \; \; = \frac{p+q}{pq}\left(\frac{1}{\alpha^{2}}\right)^{\lambda-1}\cdot\Gamma_{abpq\lambda\alpha}(s, t)\int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}(x)_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t, \end{align}$ $

(3.4)

where

$ \Gamma_{abpq\lambda\alpha}(s, t) = \Big\{(st)^{1/p}[(a-s)(b-t)]^{1/q}\Big\} ^{\alpha(\lambda-1)}. $

Multiplying the both sides of (3.4) by $ p(s, t) $ and $ \alpha $–conformable integrating both sides over $ t $ from $ 0 $ to $ b $ first and then integrating the resulting inequality over $ s $ from $ 0 $ to $ a $, we obtain

$ \int_{0}^{a}\int_{0}^{b}p(s, t)|x(s, t)|^{\lambda}d_{\alpha}sd_{\alpha}t\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; $

$ \leq\frac{p+q}{pq}\left(\frac{1}{\alpha^{2}}\right)^{\lambda-1}\cdot\int_{0}^{a}\int_{0}^{b}\Gamma_{abpq\lambda\alpha}(s, t)\cdot p(s, t)\left(\int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}(x)_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t\right)d_{\alpha}sd_{\alpha}t $

$ = \frac{p+q}{pq}\left(\frac{1}{\alpha^{2}}\right)^{\lambda-1}\cdot\left(\int_{0}^{a}\int_{0}^{b}\Gamma_{abpq\lambda\alpha}(s, t)\cdot p(s, t)d_{\alpha}sd_{\alpha}t\right)\int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}(x)_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t. $

This completes the proof.

Remark 3.1 Let $ x(s, t) $ reduce to $ s(t) $ and with suitable modifications, (3.1) becomes the following result.

$\begin{align} \int_{0}^{a}p(t)|x(t)|^{\lambda}d_{\alpha}t\leq\frac{p+q}{pq}\left(\frac{1}{\alpha^{2}}\right)^{\lambda-1}\cdot\int_{0}^{a} \Gamma_{apq\lambda\alpha}(t)p(t)d_{\alpha}t\int_{0}^{a}\left|{\cal D}_{\alpha}(x)(t) \right|^{\lambda}d_{\alpha}t, \end{align}$ $

(3.5)

where $ {\cal D}_{\alpha}(x)(t) $ is Katugampola derivative (2.1) stated in the introduction, and

$ \Gamma_{apq\lambda\alpha}(t) = \Big\{t^{1/p}(a-t)^{1/q}\Big\} ^{\alpha(\lambda-1)}. $

Putting $ p = q = 2 $ and $ \alpha = 1 $ in (3.5), (3.5) becomes inequality (1.2) established by Agarwal and Pang ^[3] stated in the introduction.

Taking for $ \alpha = 1 $, $ p = q = 2 $ and $ p(s, t) = {\rm constant} $ in (3.1), we have the following interesting result.

$ \int_{0}^{a}\int_{0}^{b}|x(s, t)|^{\lambda}dsdt\leq \frac{1}{2}(ab)^{\lambda}\left[B\left(\frac{\lambda+1}{2}, \frac{\lambda+1}{2}\right)\right] ^{2}\int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}x(s, t)\right|^{\lambda}dsdt, $

where $ B $ is the Beta function.

Theorem 3.2 Let $ j = 1, 2 $ and $ \lambda\geq 1 $ be a real number, and let $ p_{j}(s, t) $ be a nonnegative and continuous functions on $ [0, a]\times [0, b] $. Further, let $ x_{j}(s, t) $ be an absolutely continuous function and Katugampola partial derivable on $ [0, a]\times [0, b] $, with $ x_{j}(s, 0) = x_{j}(0, t) = x_{j}(0, 0) = 0 $ and $ x_{j}(a, b) = x_{j}(a, t) = x_{j}(s, b) = 0 $. Then for $ \alpha\in(0, 1] $

(3.6)

Proof Because

$ x_{1}(s, t) = \int_{0}^{s}\int_{0}^{t}\frac{\partial^{2}}{\partial s\partial t}(x_{1})_{\alpha^{2}}(s, t)d_{\alpha}sd_{\alpha}t = \int_{s}^{a}\int_{t}^{b}\frac{\partial^{2}}{\partial s\partial t}(x_{1})_{\alpha^{2}}(s, t)d_{\alpha}sd_{\alpha}t. $

Hence

$ |x_{1}(s, t)|\leq\frac{1}{2}\int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}(x_{1})_{\alpha^{2}}(s, t)\right|d_{\alpha}sd_{\alpha}t. $

By Hölder's inequality with indices $ \lambda $ and $ \lambda/(\lambda-1) $, it follows that

$ p_{1}(s, t)|x_{1}(s, t)|^{\lambda}\leq\frac{1}{2^{\lambda}}p_{1}(s, t)\left(\int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t} (x_{1})_{\alpha^{2}}(s, t)\right|d_{\alpha}sd_{\alpha}t \right)^{\lambda}\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; $

$\begin{align} \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \leq\frac{1}{2^{\lambda}}\left(\frac{1}{\alpha^{2}}\right)^{\lambda-1}(st)^{\alpha(\lambda-1)}p_{1}(s, t)\int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}(x_{1})_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t, \end{align}$ $

(3.7)

Similarly

$\begin{align} p_{2}(s, t)|x_{2}(s, t)|^{\lambda}\leq\frac{1}{2^{\lambda}}\left(\frac{1}{\alpha^{2}}\right)^{\lambda-1}(st)^{\alpha(\lambda-1)}p_{2}(s, t)\int_{0}^{a}\int_{0}^{b} \left|\frac{\partial^{2}}{\partial s\partial t}(x_{2})_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t, \end{align}$ $

(3.8)

Taking the sum of (3.7) and (3.8) and $ \alpha $-integrating the resulting inequalities over $ t $ from $ 0 $ to $ b $ first and then over $ s $ from $ 0 $ to $ a $, we obtain

$ \int_{0}^{a}\int_{0}^{b}\left(p_{1}(s, t)|x_{1}(s, t)|^{\lambda}+p_{2}(s, t)|x_{2}(s, t)|^{\lambda}\right)d_{\alpha}s d_{\alpha}t\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; $

$\begin{eqnarray*} &\leq&\frac{1}{2^{\lambda}}\left(\frac{1}{\alpha^{2}}\right)^{\lambda-1}\Bigg\{\int_{0}^{a}\int_{0}^{b}\left((st)^{\alpha(\lambda-1)}p_{1}(s, t) \int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}(x_{1})_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t\right)d_{\alpha}sd_{\alpha}t\\ &+&\int_{0}^{a}\int_{0}^{b}\left((st)^{\alpha(\lambda-1)}p_{2}(s, t)\int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}(x_{2})_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t\right)d_{\alpha}sd_{\alpha}t\Bigg\}\\ & = &\frac{1}{2^{\lambda}}\left(\frac{1}{\alpha^{2}}\right)^{\lambda-1}\Bigg[\left(\int_{0}^{a}\int_{0}^{b}(st)^{\alpha(\lambda-1)}p_{1}(s, t) d_{\alpha}sd_{\alpha}t\right)\int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}(x_{1})_{\alpha^{2}}(s, t)\right|^{\lambda} d_{\alpha}sd_{\alpha}t\\ &+&\left(\int_{0}^{a}\int_{0}^{b}(st)^{\alpha(\lambda-1)}p_{2}(s, t)d_{\alpha}sd_{\alpha}t\right)\int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t}(x_{2})_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t\Bigg]. \end{eqnarray*}$ $

Remark 3.2 Taking for $ x_{1}(s, t) = x_{2}(s, t) = x(s, t) $ and $ p_{1}(s, t) = p_{2}(s, t) = p(s, t) $ in (3.6), (3.6) changes to the following inequality.

$ \int_{0}^{a}\int_{0}^{b}p(s, t)|x(s, t)|^{\lambda}d_{\alpha}sd_{\alpha}t\leq\frac{1}{2^{\lambda}}\left(\frac{1} {\alpha^{2}}\right)^{\lambda-1}\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; $

$\begin{align} \times\left(\int_{0}^{a}\int_{0}^{b}(st)^{\alpha(\lambda-1)}p(s, t) d_{\alpha}sd_{\alpha}t\right) \int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t} (x)_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t. \end{align}$ $

(3.9)

Putting $ \alpha = 1 $ in (3.9), we have

$\begin{align} \int_{0}^{a}\int_{0}^{b}p(s, t)|x(s, t)|^{\lambda}dsdt\leq\frac{1}{2^{\lambda}}\left(\int_{0}^{a}\int_{0}^{b}(st)^{\lambda-1}p(s, t) dsdt\right)\int_{0}^{a}\int_{0}^{b}\left|\frac{\partial^{2}}{\partial s\partial t} x(s, t)\right|^{\lambda}dsdt. \end{align}$ $

(3.10)

Let $ x(s, t) $ reduce to $ s(t) $ and with suitable modifications, and $ \lambda = 1 $, (2.10) becomes the following result.

$\begin{align} \int_{0}^{a}p(t)|x(t)|dt\leq\frac{1}{2}\left(\int_{0}^{a}p(t)dt\right)\int_{0}^{a}\left|x'(t)\right|dt. \end{align}$ $

(3.11)

This is just a new inequality established by Agarwal and Pang ^[4]. For $ \lambda = 2 $ the inequality (3.11) has appear in the work of Traple ^[14], Pachpatte ^[13] proved it for $ \lambda = 2m $ ($ m\geq 1 $ an integer).

Remark 3.3 Let $ x_{j}(s, t) $ reduce to $ x_{j}(t) $ ($ j = 1, 2 $) and $ p_{j}(s, t) $ reduce to $ p_{j}(t) $ ($ j = 1, 2 $) with suitable modifications, (3.6) becomes the following interesting result.

$ \int_{0}^{a}\left(p_{1}(t)|x_{1}(t)|^{\lambda}+p_{2}(t)|x_{2}(t)|^{\lambda}\right)d_{\alpha}t\leq\frac{1}{2^{\lambda}}\left(\frac{1} {\alpha^{2}}\right)^{\lambda-1}\Bigg[\left(\int_{0}^{a}t^{\alpha(\lambda-1)}p_{1}(t)d_{\alpha}t\right)\int_{0}^{a}\left|{\cal D }_{\alpha}(x'_{1})(t)\right|^{\lambda} d_{\alpha}t $

$\begin{align} \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; +\left(\int_{0}^{a}t^{\alpha(\lambda-1)}p_{2}(t)d_{\alpha}t\right)\int_{0}^{a}\left|{\cal D}_{\alpha}(x'_{2})(t)\right|^{\lambda}d_{\alpha}t\Bigg]. \end{align}$ $

(3.12)

Putting $ \lambda = 1 $ and $ \alpha = 1 $ in (3.12), we have the following interesting result.

$ \int_{0}^{a}\left(p_{1}(t)|x_{1}(t)|+p_{2}(t)|x_{2}(t)|\right)dt\leq\frac{1}{2}\Bigg(\int_{0}^{a}p_{1}(t)dt\int_{0}^{a}\left|x'_{1}(t)\right| dt+\int_{0}^{a}p_{2}(t)dt\int_{0}^{a}\left|x'_{2}(t)\right|dt\Bigg). $

Finally, we give an example to verify the effectiveness of the new inequalities. Estimate the following double integrals:

$ \int_{0}^{1}\int_{0}^{1}\Big[st(s-1)(t-1)\Big]^{\lambda}dsdt, $

where $ \lambda\geq 1 $.

Let $ x_{1}(s,t)=x_{2}(s,t)= x(s, t) = st(s-1)(t-1) $, $ p_{1}(s,t)=p_{2}(s,t)= p(s, t) = (st)^{1-\alpha} $, $ a = b = 1 $ and $ 0 < \alpha\leq 1 $, and by using Theorem 3.2, we obtain

$ \int_{0}^{1}\int_{0}^{1}\Big[st(s-1)(t-1)\Big]^{\lambda}dsdt\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; $

$\begin{eqnarray*} & = &\int_{0}^{1}\int_{0}^{1}p(s, t) \left|x(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t\\ &\leq&\frac{1}{2^{\lambda}}\left(\frac{1} {\alpha^{2}}\right)^{\lambda-1}\left(\int_{0}^{1}\int_{0}^{1}(st)^{\alpha(\lambda-1)}p(s, t) d_{\alpha}sd_{\alpha}t\right) \int_{0}^{1}\int_{0}^{1}\left|\frac{\partial^{2}}{\partial s\partial t} (x)_{\alpha^{2}}(s, t)\right|^{\lambda}d_{\alpha}sd_{\alpha}t\\ & = &\frac{1}{2^{\lambda}}\left(\frac{1} {\alpha^{2}}\right)^{\lambda-1}\left(\frac{1}{\alpha(\lambda-1)+1}\right)^{2} \int_{0}^{1}\int_{0}^{1}\Big[(2s-1)(2t-1)\Big]^{\lambda}(st)^{\alpha-1}dsdt\\ & = &\frac{1}{2^{\lambda}}\left(\frac{1} {\alpha^{2}}\right)^{\lambda-1}\left(\frac{1}{\alpha(\lambda-1)+1}\right)^{2} \left(\frac{1}{2^{\alpha-1}}\int_{-1}^{1}t^{\lambda}\frac{1}{(t+1)^{1-\alpha}}dt\right)^{2}\\ &\leq&\frac{1}{2^{\lambda}}\left(\frac{1} {\alpha^{2}}\right)^{\lambda-1}\left(\frac{1}{\alpha(\lambda-1)+1}\right)^{2} \left(\frac{1}{2^{\alpha-1}}\frac{2^{\alpha}}{\alpha}\right)^{2}\\ & = &\frac{2^{2-\lambda}}{\alpha^{2\lambda}\left(\alpha(\lambda-1)+1\right)^{2}}. \end{eqnarray*}$ $

4. Conclusions

We have introduced a general version of Opial-Wirtinger's type integral inequality for the Katugampola partial derivatives. The established results are generalization of some existing Opial type integral inequalities in the previous published studies. For further investigations we propose to consider the Opial-Wirtinger's type inequalities for other partial derivatives.

Acknowledgments

I would like to thank that research is supported by National Natural Science Foundation of China(11471334, 10971205).

Conflict of interest

The author declares no conflicts of interest.

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