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 [2,3,4,5,6]. 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 [7,8,9,10,11,12,13,14,15,16,17,18].

Recently, some new Opial’s inequalities for the conformable fractional integrals have been established (see [19,20,21,22]). 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 \([0,\infty)\).