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# Inequalities for Katugampola conformable partial derivatives

Journal of Inequalities and Applications20192019:51

https://doi.org/10.1186/s13660-019-2000-3

• Accepted: 18 February 2019
• Published:

## Abstract

In the paper, we introduce two concepts of Katugampola conformable partial derivatives and α-conformable integrals. As applications, we establish Opial type inequalities for Katugampola conformable partial derivatives and α-conformable integrals. The new inequalities in special cases yield some of the recent results on inequality of this type.

## Keywords

• Convex function
• Conformable fractional integrals
• Katugampola conformable derivative
• Opial’s inequality
• Jensen’s inequality

• 26D15
• 26A51

## 1 Introduction

In 1960, Opial [1] established the following interesting and important inequality.

### Theorem A

Suppose that $$f\in C^{1}[0,a]$$ satisfies $$f(0)=f(a)=0$$ and $$f(x)>0$$ for all $$x\in(0,a)$$. Then the inequality holds
$$\int_{0}^{a} \bigl\vert f(x)f'(x) \bigr\vert \,dx\leq\frac{a}{4} \int _{0}^{a}\bigl(f'(x) \bigr)^{2}\,dx,$$
(1.1)
where this constant $$a/4$$ is best possible.

Opial’s inequality and its generalizations, extensions, and discretizations play a fundamental role in establishing the existence and uniqueness of initial and boundary value problems for ordinary and partial differential equations as well as difference equations [26]. Inequality (1.1) has received considerable attention, and a large number of papers dealing with new proofs, extensions, generalizations, variants, and discrete analogues of Opial’s inequality have appeared in the literature [718].

Recently, some new Opial’s inequalities for the conformable fractional integrals have been established (see [1922]). In the paper, we introduce two new concepts of Katugampola conformable partial derivatives and α-conformable integrals. As applications, we establish some Opial type inequalities for Katugampola conformable partial derivatives and α-conformable integrals.

## 2 Inequalities for Katugampola conformable partial derivatives

We recall the well-known Katugampola derivative formulation of conformable derivative of order for $$\alpha\in(0,1]$$ and $$t\in[0,\infty)$$, given by
$$D_{\alpha}(f) (t)=\lim_{\varepsilon\rightarrow0}\frac {f(te^{\varepsilon t^{-\alpha}})-f(t)}{\varepsilon},$$
(2.1)
and
$$D_{\alpha}(f) (0)=\lim_{t\rightarrow0}D_{\alpha}(f) (t),$$
(2.2)
provided the limits exist. If f is fully differentiable at t, then
$$D_{\alpha}(f) (t)=t^{1-\alpha}\frac{df}{dt}(t).$$
A function f is α-differentiable at a point $$t\geq0$$ if the limits in (2.1) and (2.2) exist and are finite. Inspired by this, we propose a new concept of α-conformable partial derivative. In the way of (2.1), we define α-conformable partial derivative.

### Definition 2.1

(α-conformable partial derivative)

Let $$\alpha\in(0,1]$$ and $$s,t\in[0,\infty)$$. Suppose that $$f(s,t)$$ is a continuous function and partially derivable, the α-conformable partial derivative at a point $$s\geq0$$, denoted by $$\frac{\partial}{\partial s}(f)_{\alpha}(s,t)$$, is defined by
$$\frac{\partial}{\partial s}(f)_{\alpha}(s,t)=\lim_{\varepsilon \rightarrow0} \frac{f(se^{\varepsilon s^{-\alpha}},t)-f(s,t)}{ \varepsilon},$$
(2.3)
provided the limits exist, and is called α-conformable partially derivable.

To generalize Definition 2.1, we give the following definition.

### Definition 2.2

(Katugampola conformable partial derivative)

Let $$\alpha\in(0,1]$$ and $$s,t\in[0,\infty)$$. Suppose that $$f(s,t)$$ and $$\frac{\partial}{\partial s}(f)_{\alpha}(s,t)$$ are continuous functions and partially derivable, the Katugampola conformable partial derivative, denoted by $$\frac{\partial^{2}}{\partial s\partial t}(f)_{\alpha^{2}}(s,t)$$, is defined by
$$\frac{\partial^{2}}{\partial s\partial t}(f)_{\alpha^{2}}(s,t)=\lim_{\varepsilon\rightarrow 0} \frac{\frac{\partial}{\partial s}(f)_{\alpha}(s,te^{\varepsilon t^{-\alpha}})-\frac{\partial}{\partial s}(f)_{\alpha}(s,t)}{\varepsilon},$$
(2.4)
provided the limits exist, and is called Katugampola conformable partially derivable.

### Definition 2.3

(α-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 α-conformable integrable if the integral
$$\int_{a}^{b} \int_{c}^{d}f(x,y)\, d_{\alpha}x\, d_{\alpha}y:= \int _{a}^{b} \int_{c}^{d} (xy)^{\alpha-1}f(x,y)\, dx\, dy$$
(2.5)
exists and is finite.

### Lemma 2.1

Let $$\alpha\in(0,1]$$, $$s,t\in[0,\infty)$$, and $$f(s,t)$$, $$g(s,t)$$ be Katugampola conformable partially differentiable, then
$$\frac{\partial^{2}}{\partial s\partial t}(f\circ g)_{\alpha^{2}}(s,t)=f'\bigl(g(s,t) \bigr)\cdot\frac{\partial^{2}}{\partial s\partial t}(g)_{\alpha^{2}}(s,t)+ \frac{\partial}{\partial t}(g)_{\alpha}(s,t)\cdot\frac{\partial}{\partial t}\bigl(f'\bigl(g(s,t)\bigr)\bigr)_{\alpha}(s,t),$$
(2.6)
where f has derivative at $$g(s,t)$$.

### Proof

From Definitions 2.1 and 2.2, we obtain
\begin{aligned} \frac{\partial}{\partial s}(f\circ g)_{\alpha}(s,t) =&\frac{\partial}{\partial s}\bigl(f \bigl(g(s,t)\bigr)\bigr)_{\alpha}(s,t) \\ =&s^{1-\alpha}\frac{\partial}{\partial s}\bigl(f\bigl(g(s,t)\bigr)\bigr) \\ =&s^{1-\alpha}f'\bigl(g(s,t)\bigr)\frac{\partial}{\partial s}\bigl(g(s,t) \bigr) \\ =&f'\bigl(g(s,t)\bigr)\frac{\partial}{\partial s}(g)_{\alpha}(s,t). \end{aligned}
Hence
\begin{aligned} \frac{\partial^{2}}{\partial s\partial t}(f\circ g)_{\alpha^{2}}(s,t) =&\frac{\partial}{\partial t} \biggl( \frac{\partial}{\partial s}(f\circ g)_{\alpha}(s,t) \biggr)_{\alpha}(s,t) \\ =&\frac{\partial}{\partial t} \biggl(f'\bigl(g(s,t)\bigr) \frac{\partial }{\partial s}(g)_{\alpha}(s,t) \biggr)_{\alpha}(s,t) \\ =&t^{1-\alpha}\frac{\partial}{\partial t} \biggl(f'\bigl(g(s,t)\bigr) \cdot\frac{\partial}{\partial s}(g)_{\alpha}(s,t) \biggr) \\ =&t^{1-\alpha}\frac{\partial}{\partial t} \bigl(f'\bigl(g(s,t)\bigr) \bigr)\cdot\frac{\partial}{\partial t}(g)_{\alpha}(s,t) +t^{1-\alpha}f'\bigl(g(s,t)\bigr)\cdot\frac{\partial}{\partial t}\biggl(\frac{\partial}{\partial s}(g)_{\alpha}(s,t)\biggr) \\ =&\frac{\partial}{\partial t}(g)_{\alpha}(s,t)\cdot\frac{\partial}{\partial t}\bigl(f'\bigl(g(s,t)\bigr)\bigr)_{\alpha}(s,t) +f'\bigl(g(s,t)\bigr)\cdot\frac{\partial^{2}}{\partial s\partial t}(g)_{\alpha^{2}}(s,t). \end{aligned}

This completes the proof. □

This similar chain rule theorem is important, but it is also understood. In order for the reader to better understand this theorem, we give another proof below.

### Second proof

Let
$$\delta=g\bigl(se^{\varepsilon s^{-\alpha}},t\bigr)-g(s,t) .$$
Obviously, if $$\varepsilon\rightarrow0$$, then $$\delta\rightarrow0$$. From the hypotheses, we obtain
\begin{aligned} \frac{\partial^{2}}{\partial s\partial t}(f\circ g)_{\alpha^{2}}(s,t) =&\frac{\partial}{\partial t} \biggl( \frac{\partial}{\partial s}\bigl(f\bigl(g(s,t)\bigr)\bigr)_{\alpha}(s,t) \biggr)_{\alpha}(s,t) \\ =&\frac{\partial}{\partial t} \biggl(\lim_{\varepsilon\rightarrow 0}\frac{f(g(se^{\varepsilon s^{-\alpha}},t))-f(g(s,t))}{ \varepsilon} \biggr)_{\alpha}(s,t) \\ =&\frac{\partial}{\partial t} \biggl(\lim_{\delta\rightarrow 0}\frac{f(g(s,t)+\delta)-f(g(s,t))}{\delta}\cdot \lim_{\varepsilon\rightarrow 0}\frac{\delta}{\varepsilon} \biggr)_{\alpha}(s,t) \\ =&\frac{\partial}{\partial t} \biggl(f'\bigl(g(s,t)\bigr) \frac{\partial}{\partial s}(g)_{\alpha}(s,t) \biggr)_{\alpha}(s,t) \\ =&f'\bigl(g(s,t)\bigr)\cdot\frac{\partial^{2}}{\partial s\partial t}(g)_{\alpha^{2}}(s,t)+ \frac{\partial}{\partial t}(g)_{\alpha}(s,t)\cdot\frac{\partial}{\partial t}\bigl(f'\bigl(g(s,t)\bigr)\bigr)_{\alpha}(s,t). \end{aligned}

This completes the proof. □

### Theorem 2.1

Let $$p(s,t), u(s,t):[a,b]\times[c,d]\rightarrow {\Bbb {R}}$$ with $$a,c\geq 0$$ be Katugampola conformable partially derivable such that $$\frac{\partial^{2}}{\partial s\partial t}(p)_{\alpha^{2}}(s,t)>0$$, $$\alpha\in(0,1]$$ and $$p(a,c)=p(a,d)=p(b,c)=p(b,d)=0$$, and F be derivable on $$[0,\infty)$$ and $$F'$$ be increasing. Let φ be a convex and increasing function on $$[0,\infty)$$, and define
$$z(s,t)= \int_{a}^{s} \int_{c}^{t}\frac{\partial^{2}}{\partial \sigma\partial \tau}(p)_{\alpha^{2}}( \sigma,\tau)\cdot\varphi \biggl(\frac{ \vert \frac{\partial^{2}}{\partial \sigma\partial \tau}(u)_{\alpha^{2}}(\sigma,\tau) \vert }{\frac{\partial ^{2}}{\partial \sigma\partial \tau}(p)_{\alpha^{2}}(\sigma,\tau)} \biggr)\, d_{\alpha} \sigma \, d_{\alpha}\tau.$$
(2.7)
Then
\begin{aligned}& \int_{a}^{b} \int_{c}^{d} \biggl\{ \frac{\partial^{2}}{\partial s\partial t}(z)_{\alpha^{2}}(s,t)\cdot F' \biggl(p(s,t)\cdot \varphi \biggl( \frac{|u(s,t)|}{p(s,t)} \biggr) \biggr) \\& \qquad {}+\frac{\partial }{\partial t}(z)_{\alpha}(s,t)\cdot \frac{\partial}{\partial t}\bigl(F'\bigl(z(s,t)\bigr)\bigr)_{\alpha}(s,t) \biggr\} \, d_{\alpha}s\, d_{\alpha }t \\& \quad \leq F \biggl( \int_{a}^{b} \int_{c}^{d}\frac{\partial^{2}}{\partial s\partial t}(p)_{\alpha^{2}}(s,t) \cdot\varphi \biggl(\frac{ \vert \frac {\partial^{2}}{\partial s\partial t}(u)_{\alpha^{2}}(s,t) \vert }{\frac{\partial^{2}}{\partial s\partial t}(p)_{\alpha^{2}}(s,t)} \biggr)\, d_{\alpha}s\, d_{\alpha}t \biggr), \end{aligned}
(2.8)
where
$$\frac{\partial}{\partial t}\bigl(F'\bigl(z(s,t)\bigr)\bigr)_{\alpha}(s,t) =t^{1-\alpha} \frac{\partial}{\partial t}F'\bigl(z(s,t)\bigr).$$

### Proof

Let
$$y(s,t)= \int_{a}^{s} \int_{c}^{t} \biggl\vert \frac{\partial^{2}}{\partial s\partial t}(u)_{\alpha^{2}}( \sigma,\tau) \biggr\vert \, d_{\alpha}\sigma\, d_{\alpha}\tau$$
such that
$$\frac{\partial^{2}}{\partial s\partial t}(y)_{\alpha^{2}}(s,t)= \biggl\vert \frac{\partial^{2}}{\partial s\partial t}(u)_{\alpha^{2}}(s,t) \biggr\vert$$
and $$y(s,t)\geq|u(s,t)|$$. Since φ is convex and increasing, by using Jensen’s inequality, we get
\begin{aligned} \varphi \biggl(\frac{ \vert u(s,t) \vert }{p(s,t)} \biggr) \leq& \varphi \biggl( \frac{y(s,t)}{p(s,t)} \biggr) \\ =& \varphi \biggl(\frac{\int_{a}^{s}\int_{c}^{t}\frac{\partial ^{2}}{\partial \sigma\partial \tau}(p)_{\alpha^{2}}(\sigma,\tau)\frac{ \vert \frac{\partial ^{2}}{\partial \sigma\partial \tau}(u)_{\alpha^{2}}(\sigma,\tau) \vert }{\frac{\partial ^{2}}{\partial \sigma\partial\tau}(p)_{\alpha^{2}}(\sigma,\tau)}\,d_{\alpha }\sigma \,d_{\alpha}\tau}{\int_{a}^{s}\int_{c}^{t}\frac{\partial ^{2}}{\partial \sigma\partial \tau}(p)_{\alpha^{2}}(\sigma,\tau)\,d_{\alpha}\sigma \,d_{\alpha}\tau } \biggr) \\ \leq&\frac{1}{p(s,t)} \int_{a}^{s} \int_{c}^{t}\frac{\partial ^{2}}{\partial \sigma\partial \tau}(p)_{\alpha^{2}}( \sigma,\tau)\cdot\varphi \biggl(\frac{ \vert \frac{\partial^{2}}{\partial \sigma\partial \tau}(u)_{\alpha^{2}}(\sigma,\tau) \vert }{\frac{\partial ^{2}}{\partial \sigma\partial \tau}(p)_{\alpha^{2}}(\sigma,\tau)} \biggr)\,d_{\alpha}\sigma \,d_{\alpha}\tau \\ =&\frac{1}{p(s,t)} \int_{a}^{s} \int_{c}^{t}\frac {\partial^{2}}{\partial \sigma\partial \tau}(p)_{\alpha^{2}}( \sigma,\tau)\cdot\varphi \biggl(\frac{\frac {\partial^{2}}{\partial \sigma\partial \tau}(y)_{\alpha^{2}}(\sigma,\tau)}{\frac{\partial^{2}}{\partial \sigma\partial \tau}(p)_{\alpha^{2}}(\sigma,\tau)} \biggr)\, d_{\alpha} \sigma \, d_{\alpha}\tau. \end{aligned}
(2.9)
From (2.9) and noting that $$F'$$ is increasing, and Lemma 2.1, (2.7) and in view of that F is derivable on $$[0,\infty)$$, we obtain
\begin{aligned}& \int_{a}^{b} \int_{c}^{d} \biggl\{ \frac{\partial^{2}}{\partial s\partial t}(z)_{\alpha^{2}}(s,t)\cdot F' \biggl(p(s,t)\cdot \varphi \biggl( \frac{|u(s,t)|}{p(s,t)} \biggr) \biggr) \\& \qquad {}+\frac{\partial }{\partial t}(z)_{\alpha}(s,t)\cdot\frac{\partial}{\partial t}\bigl(F'\bigl(z(s,t)\bigr)\bigr)_{\alpha}(s,t) \biggr\} \,d_{\alpha}s\,d_{\alpha }t \\& \quad \leq \int_{a}^{b} \int_{c}^{d} \biggl\{ \frac {\partial^{2}}{\partial s\partial t}(z)_{\alpha^{2}}(s,t) \cdot F' \bigl(z(s,t) \bigr) \\& \qquad {}+ \frac{\partial}{\partial t}(z)_{\alpha}(s,t)\cdot\frac{\partial}{\partial t}\bigl(F'\bigl(z(s,t)\bigr)\bigr)_{\alpha}(s,t) \biggr\} \,d_{\alpha}s\,d_{\alpha}t \\& \quad = \int_{a}^{b} \int_{c}^{d}\frac{\partial^{2}}{\partial s\partial t}(F\circ z)_{\alpha^{2}}(s,t)\,d_{\alpha}s\,d_{\alpha}t \\& \quad = \int_{a}^{b} \int_{c}^{d}\frac{\partial^{2}}{\partial s\partial t} \biggl(F \biggl( \int_{a}^{s} \int_{b}^{t}\frac{\partial ^{2}}{\partial \sigma\partial \tau}(p)_{\alpha^{2}}( \sigma,\tau) \\& \qquad {}\cdot\varphi \biggl(\frac{\frac {\partial^{2}}{\partial \sigma\partial \tau}(y)_{\alpha^{2}}(\sigma,\tau)}{\frac{\partial^{2}}{\partial \sigma\partial \tau}(p)_{\alpha^{2}}(\sigma,\tau)} \biggr)\,d_{\alpha}\sigma \,d_{\alpha}\tau \biggr) \biggr)_{\alpha^{2}}(s,t)\,d_{\alpha}s \,d_{\alpha}t \\& \quad = F \biggl( \int_{a}^{b} \int_{c}^{d}\frac{\partial^{2}}{\partial \sigma\partial \tau}(p)_{\alpha^{2}}( \sigma,\tau)\cdot\varphi \biggl(\frac{\frac {\partial^{2}}{\partial \sigma\partial \tau}(y)_{\alpha^{2}}(\sigma,\tau)}{\frac{\partial^{2}}{\partial \sigma\partial \tau}(p)_{\alpha^{2}}(\sigma,\tau)} \biggr)\,d_{\alpha}\sigma \,d_{\alpha}\tau \biggr) \\& \quad = F \biggl( \int_{a}^{b} \int_{c}^{d}\frac{\partial^{2}}{\partial s\partial t}(p)_{\alpha^{2}}(s,t) \cdot\varphi \biggl(\frac{ \vert \frac {\partial^{2}}{\partial s\partial t}(u)_{\alpha^{2}}(s,t) \vert }{\frac{\partial^{2}}{\partial s\partial t}(p)_{\alpha^{2}}(s,t)} \biggr)\,d_{\alpha}s \,d_{\alpha}t \biggr). \end{aligned}
This completes the proof. □

### Remark 2.1

Putting $$\varphi(x)=x$$ in (2.7), we have
\begin{aligned}& \int_{a}^{b} \int_{c}^{d} \biggl\{ \biggl\vert \frac{\partial^{2}}{\partial s\partial t}(u)_{\alpha^{2}}(s,t) \biggr\vert \cdot F' \bigl(\bigl|u(s,t)\bigr|\bigr) \\& \qquad {}+\frac{\partial}{\partial t} (y)_{\alpha}(s,t)\cdot\frac{\partial}{\partial t}\bigl(F'\bigl(y(s,t)\bigr)\bigr)_{\alpha}(s,t)\biggr\} \, d_{\alpha}s\, d_{\alpha}t \\& \quad \leq F \biggl( \int_{a}^{b} \int_{c}^{d} \biggl\vert \frac{\partial ^{2}}{\partial s\partial t}(u)_{\alpha^{2}}(s,t) \biggr\vert \, d_{\alpha}s\, d_{\alpha}t \biggr), \end{aligned}
(2.10)
where
$$y(s,t)= \int_{a}^{s} \int_{c}^{t} \biggl\vert \frac{\partial^{2}}{\partial s\partial t}(u)_{\alpha^{2}}( \sigma,\tau) \biggr\vert \, d_{\alpha}\sigma \, d_{\alpha}\tau.$$
This inequality (2.10) is just a two-dimensional generalization of the following inequality which was established in [20] and [21]:
$$\int_{a}^{b} \bigl\vert D_{\alpha}u(t) \bigr\vert \cdot F' \bigl( \bigl\vert u(t) \bigr\vert \bigr)\, d_{\alpha}t\leq F \biggl( \int_{a}^{b} \bigl\vert D_{\alpha}u(t) \bigr\vert \, d_{\alpha}t \biggr).$$

### Theorem 2.2

Let α, $$p(s,t)$$, $$u(s,t)$$, $$z(s,t)$$, φ, F be as in Theorem 2.1 and replace $$[a,b]\times[c,d]$$ by $$[0,a]\times[0,b]$$. Let h be a concave and increasing function on $$[0,\infty)$$, and ϕ be a continuous and positive function on $$[0,\infty)$$ and such that
$$\frac{\partial^{2}}{\partial s\partial t}(F\circ z)_{\alpha ^{2}}(s,t)\cdot\phi \biggl( \frac{1}{\frac{\partial^{2}}{\partial s\partial t}(z)_{\alpha^{2}}(s,t)} \biggr)\leq \frac{F(z(a,b))}{z(a,b)}\cdot\phi' \biggl( \frac{t}{z(a,b)} \biggr).$$
(2.11)
Then
\begin{aligned}& \int_{0}^{a} \int_{0}^{b} \biggl\{ \psi\biggl(\frac{\partial^{2}}{\partial s\partial t}(p)_{\alpha^{2}}(s,t) \cdot\varphi \biggl(\frac{ \vert \frac{\partial^{2}}{\partial s\partial t}(u)_{\alpha^{2}}(s,t) \vert }{\frac{\partial^{2}}{\partial s\partial t}(p)_{\alpha^{2}}(s,t)} \biggr) \biggr)\cdot F' \biggl(p(s,t)\cdot\varphi \biggl( \frac{|u(s,t)|}{p(s,t)} \biggr) \biggr) \\& \qquad {} +\psi \biggl(\frac{\partial^{2}}{\partial s\partial t}(z)_{\alpha^{2}}(s,t) \biggr)\cdot \frac{\partial}{\partial t}\bigl(F'\bigl(z(s,t)\bigr)\bigr)_{\alpha}(s,t) \cdot\frac{\frac{\partial}{\partial t}(z(s,t))_{\alpha}(s,t)}{\frac{\partial^{2}}{\partial s\partial t}(z)_{\alpha^{2}}(s,t)} \biggr\} \, d_{\alpha}s\, d_{\alpha}t \\& \quad \leq\varPhi \biggl( \int_{0}^{a} \int_{0}^{b}\frac{\partial ^{2}}{\partial s\partial t}(p)_{\alpha^{2}}(s,t) \cdot\varphi \biggl(\frac{ \vert \frac {\partial^{2}}{\partial s\partial t}(u)_{\alpha^{2}}(s,t) \vert }{\frac{\partial^{2}}{\partial s\partial t}(p)_{\alpha^{2}}(s,t)} \biggr)\, d_{\alpha}s\, d_{\alpha}t \biggr), \end{aligned}
(2.12)
where
$$\psi(r)=rh \biggl(\phi \biggl(\frac{1}{r} \biggr) \biggr),$$
(2.13)
and
$$\varPhi(r)=F(r)\cdot h \biggl(\frac{1}{r} \int_{0}^{a} \int_{0}^{b}\phi ' \biggl( \frac{t}{r} \biggr)\, d_{\alpha}s\, d_{\alpha}t \biggr).$$
(2.14)

### Proof

From (2.9), we have
$$\varphi \biggl(\frac{ \vert u(s,t) \vert }{p(s,t)} \biggr)\leq\frac {z(s,t)}{p(s,t)}.$$
(2.15)
From (2.7), (2.15), (2.13) (2 times), Lemma 2.1, and noting that h is a concave, increasing and using reverse Jensen’s inequality, and (2.11) and (2.14), we obtain
\begin{aligned}& \int_{0}^{a} \int_{0}^{b} \biggl\{ \psi \biggl( \frac{\partial ^{2}}{\partial s\partial t}(p)_{\alpha^{2}}(s,t) \cdot\varphi \biggl(\frac{ \vert \frac{\partial^{2}}{\partial s\partial t}(u)_{\alpha^{2}}(s,t) \vert }{\frac{\partial^{2}}{\partial s\partial t}(p)_{\alpha^{2}}(s,t)} \biggr) \biggr)\cdot F' \biggl(p(s,t)\cdot\varphi \biggl( \frac{|u(s,t)|}{p(s,t)} \biggr) \biggr) \\& \qquad {} + \psi \biggl(\frac{\partial^{2}}{\partial s\partial t}(z)_{\alpha^{2}}(s,t) \biggr)\cdot \frac{\partial}{\partial t}\bigl(F'\bigl(z(s,t)\bigr)\bigr)_{\alpha}(s,t) \cdot\frac{\frac{\partial}{\partial t}(z)_{\alpha}(s,t)}{\frac{\partial^{2}}{\partial s\partial t}(z)_{\alpha^{2}}(s,t)} \biggr\} \, d_{\alpha}s\, d_{\alpha}t \\& \quad \leq \int_{0}^{a} \int_{0}^{b} \biggl\{ \psi \biggl( \frac{\partial ^{2}}{\partial s\partial t}(z)_{\alpha^{2}}(s,t) \biggr)\cdot F'\bigl(z(s,t) \bigr) \\& \qquad {} + h \biggl(\phi \biggl(\frac{1}{\frac{\partial^{2}}{\partial s\partial t}(z)_{\alpha^{2}}(s,t)} \biggr) \biggr) \frac{\partial}{\partial t}(z)_{\alpha}(s,t) \frac{\partial}{\partial t}\bigl(F'\bigl(z(s,t)\bigr)\bigr)_{\alpha}(s,t) \biggr\} \, d_{\alpha}s\, d_{\alpha}t \\& \quad = \int_{0}^{a} \int_{0}^{b}h \biggl(\phi \biggl(\frac{1}{\frac {\partial^{2}}{\partial s\partial t}(z)_{\alpha^{2}}(s,t)} \biggr) \biggr)\cdot \biggl(\frac{\partial ^{2}}{\partial s\partial t}(z)_{\alpha^{2}}(s,t)\cdot F'\bigl(z(s,t)\bigr) \\& \qquad {}+\frac{\partial}{\partial t}(z)_{\alpha}(s,t)\cdot \frac{\partial}{\partial t}\bigl(F'\bigl(z(s,t)\bigr)\bigr)_{\alpha}(s,t)\biggr)\, d_{\alpha}s \, d_{\alpha}t \\& \quad = \frac{\int_{0}^{a}\int_{0}^{b}\frac{\partial^{2}}{\partial s\partial t}(F\circ z )_{\alpha^{2}}(s,t)\cdot h (\phi (\frac{1}{\frac{\partial^{2}}{\partial s\partial t}(z)_{\alpha^{2}}(s,t)} ) )\, d_{\alpha}s\, d_{\alpha}t}{\int_{0}^{a}\int_{0}^{b}\frac{\partial^{2}}{\partial s\partial t} (F\circ z )_{\alpha^{2}}(s,t)\, d_{\alpha}s\, d_{\alpha}t} \int_{0}^{a} \int _{0}^{b}\frac{\partial^{2}}{\partial s\partial t}(F\circ z)_{\alpha^{2}}(s,t)\, d_{\alpha}s\, d_{\alpha}t \\& \quad \leq h \biggl(\frac{\int_{0}^{a}\int_{0}^{b}\frac{\partial^{2}}{\partial s\partial t}(F\circ z)_{\alpha^{2}}(s,t)\cdot \phi (\frac{1}{\frac{\partial^{2}}{\partial s\partial t}(z)_{\alpha^{2}}(s,t)} ) \, d_{\alpha}s\, d_{\alpha}t}{\int_{0}^{a}\int_{0}^{b}\frac{\partial ^{2}}{\partial s\partial t}(F\circ z )_{\alpha^{2}}(s,t)\, d_{\alpha}s\, d_{\alpha}t} \biggr)F\bigl(z(a,b)\bigr) \\& \quad \leq h \biggl(\frac{\int_{0}^{a}\int_{0}^{b}\frac{F(z(a,b))}{z(a,b)}\phi ' (\frac{t}{z(a,b)} )\, d_{\alpha}s\, d_{\alpha }t}{F(z(a,b))} \biggr)F\bigl(z(a,b)\bigr) \\& \quad = \varPhi\bigl(z(a,b)\bigr) \\& \quad = \varPhi \biggl( \int_{0}^{a} \int_{0}^{b}\frac{\partial^{2}}{\partial s\partial t}(p)_{\alpha^{2}}(s,t) \cdot\varphi \biggl(\frac{ \vert \frac {\partial^{2}}{\partial s\partial t}(u)_{\alpha^{2}}(s,t) \vert }{\frac{\partial^{2}}{\partial s\partial t}(p)_{\alpha^{2}}(s,t)} \biggr)\, d_{\alpha}s\, d_{\alpha}t \biggr). \end{aligned}
This completes the proof. □

### Remark 2.2

Putting $$\varphi(x)=x$$ in (2.12), we have
\begin{aligned}& \int_{0}^{b}\psi \biggl( \biggl\vert \frac{\partial^{2}}{\partial s\partial t}(u)_{\alpha^{2}}(s,t) \biggr\vert \biggr)\cdot F' \bigl(\bigl|u(s,t)\bigr| \bigr)\, d_{\alpha}s\, d_{\alpha}t \\& \quad \leq\varPhi \biggl( \int_{0}^{a} \int_{0}^{b} \biggl\vert \frac{\partial ^{2}}{\partial s\partial t}(u)_{\alpha^{2}}(s,t) \biggr\vert \, d_{\alpha}s\, d_{\alpha}t \biggr)-N_{\alpha}(a,b), \end{aligned}
(2.16)
where
$$N_{\alpha}(a,b)=\int_{0}^{a}\int_{0}^{b}\psi \biggl(\frac{\partial^{2}}{\partial s\partial t}(z)_{\alpha^{2}}(s,t)\biggr)\cdot\frac{\partial}{\partial t}\bigl(F'\bigl(z(s,t)\bigr)\bigr)_{\alpha}(s,t)\cdot\frac{\frac{\partial}{\partial t}(z)_{\alpha}(s,t)}{\frac{\partial^{2}}{\partial s\partial t}(z)_{\alpha^{2}}(s,t)}\,d_{\alpha}s\,d_{\alpha}t.$$
This inequality (2.16) is just a two-dimensional generalization of the following inequality which was established in [21]:
\begin{aligned}& \int_{0}^{b}\psi \biggl(D_{\alpha}p(t) \cdot\varphi \biggl(\frac { \vert D_{\alpha}u(t) \vert }{D_{\alpha}p(t)} \biggr) \biggr)\cdot F' \biggl(p(t)\cdot\varphi \biggl(\frac{|u(t)|}{p(t)} \biggr) \biggr)\, d_{\alpha}t \\& \quad \leq\varPhi \biggl( \int_{0}^{b}D_{\alpha}p(t)\cdot\varphi \biggl(\frac { \vert D_{\alpha}u(t) \vert }{ D_{\alpha}p (t)} \biggr)\, d_{\alpha}t \biggr), \end{aligned}
where $$D_{\alpha}p(t)=D_{\alpha}(p)(t)$$, $$\psi(r)=rh (\phi (\frac{1}{r} ) )$$ and $$\varPhi(r)=F(r)h (\phi (\frac{b}{r} ) )$$, and h is a concave and increasing function on $$[0,\infty)$$.

## Declarations

### Acknowledgements

The first author expresses his gratitude to professor G. Leng and W. Li for their valuable helps.

### Availability of data and materials

All data generated or analysed during this study are included in this published article.

### Funding

The first author’s research is supported by the Natural Science Foundation of China (11371334, 10971205). The second author’s research is partially supported by a HKU Seed Grant for Basic Research.

### Authors’ contributions

C-JZ and W-SC jointly contributed to the main results. All authors read and approved the final manuscript.

### Competing interests

The authors declare that they have no competing interests.

## Authors’ Affiliations

(1)
Department of Mathematics, China Jiliang University, Hangzhou, P.R. China
(2)
Department of Mathematics, The University of Hong Kong, Hong Kong, P.R. China

## References

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