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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

  • Received: 21 December 2018
  • 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

MSC

  • 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.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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

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