Inequalities for Katugampola conformable partial derivatives
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- 01-12-2019
- Research
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Abstract
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
where this constant
\(a/4\)
is best possible.
$$ \int_{0}^{a} \bigl\vert f(x)f'(x) \bigr\vert \,dx\leq\frac{a}{4} \int _{0}^{a}\bigl(f'(x) \bigr)^{2}\,dx, $$
(1.1)
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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 [2‐6]. 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 [7‐18].
Recently, some new Opial’s inequalities for the conformable fractional integrals have been established (see [19‐22]). 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
and
provided the limits exist. If f is fully differentiable at t, then
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.
$$ D_{\alpha}(f) (t)=\lim_{\varepsilon\rightarrow0}\frac {f(te^{\varepsilon t^{-\alpha}})-f(t)}{\varepsilon}, $$
(2.1)
$$ D_{\alpha}(f) (0)=\lim_{t\rightarrow0}D_{\alpha}(f) (t), $$
(2.2)
$$D_{\alpha}(f) (t)=t^{1-\alpha}\frac{df}{dt}(t). $$
Definition 2.1
(α-conformable partial derivative)
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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
provided the limits exist, and is called α-conformable partially derivable.
$$ \frac{\partial}{\partial s}(f)_{\alpha}(s,t)=\lim_{\varepsilon \rightarrow0} \frac{f(se^{\varepsilon s^{-\alpha}},t)-f(s,t)}{ \varepsilon}, $$
(2.3)
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
provided the limits exist, and is called Katugampola conformable partially derivable.
$$ \frac{\partial^{2}}{\partial s\partial t}(f)_{\alpha^{2}}(s,t)=\lim_{\varepsilon\rightarrow 0} \frac{\frac{\partial}{\partial s}(f)_{\alpha}(s,te^{\varepsilon t^{-\alpha}})-\frac{\partial}{\partial s}(f)_{\alpha}(s,t)}{\varepsilon}, $$
(2.4)
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
exists and is finite.
$$ \int_{a}^{b} \int_{c}^{d}f(x,y)\, d_{\alpha}x\, d_{\alpha}y:= \int _{a}^{b} \int_{c}^{d} (xy)^{\alpha-1}f(x,y)\, dx\, dy $$
(2.5)
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
where
f
has derivative at
\(g(s,t)\).
$$ \frac{\partial^{2}}{\partial s\partial t}(f\circ g)_{\alpha^{2}}(s,t)=f'\bigl(g(s,t) \bigr)\cdot\frac{\partial^{2}}{\partial s\partial t}(g)_{\alpha^{2}}(s,t)+ \frac{\partial}{\partial t}(g)_{\alpha}(s,t)\cdot\frac{\partial}{\partial t}\bigl(f'\bigl(g(s,t)\bigr)\bigr)_{\alpha}(s,t), $$
(2.6)
Proof
From Definitions 2.1 and 2.2, we obtain
Hence
$$\begin{aligned} \frac{\partial}{\partial s}(f\circ g)_{\alpha}(s,t) =&\frac{\partial}{\partial s}\bigl(f \bigl(g(s,t)\bigr)\bigr)_{\alpha}(s,t) \\ =&s^{1-\alpha}\frac{\partial}{\partial s}\bigl(f\bigl(g(s,t)\bigr)\bigr) \\ =&s^{1-\alpha}f'\bigl(g(s,t)\bigr)\frac{\partial}{\partial s}\bigl(g(s,t) \bigr) \\ =&f'\bigl(g(s,t)\bigr)\frac{\partial}{\partial s}(g)_{\alpha}(s,t). \end{aligned}$$
$$\begin{aligned} \frac{\partial^{2}}{\partial s\partial t}(f\circ g)_{\alpha^{2}}(s,t) =&\frac{\partial}{\partial t} \biggl( \frac{\partial}{\partial s}(f\circ g)_{\alpha}(s,t) \biggr)_{\alpha}(s,t) \\ =&\frac{\partial}{\partial t} \biggl(f'\bigl(g(s,t)\bigr) \frac{\partial }{\partial s}(g)_{\alpha}(s,t) \biggr)_{\alpha}(s,t) \\ =&t^{1-\alpha}\frac{\partial}{\partial t} \biggl(f'\bigl(g(s,t)\bigr) \cdot\frac{\partial}{\partial s}(g)_{\alpha}(s,t) \biggr) \\ =&t^{1-\alpha}\frac{\partial}{\partial t} \bigl(f'\bigl(g(s,t)\bigr) \bigr)\cdot\frac{\partial}{\partial t}(g)_{\alpha}(s,t) +t^{1-\alpha}f'\bigl(g(s,t)\bigr)\cdot\frac{\partial}{\partial t}\biggl(\frac{\partial}{\partial s}(g)_{\alpha}(s,t)\biggr) \\ =&\frac{\partial}{\partial t}(g)_{\alpha}(s,t)\cdot\frac{\partial}{\partial t}\bigl(f'\bigl(g(s,t)\bigr)\bigr)_{\alpha}(s,t) +f'\bigl(g(s,t)\bigr)\cdot\frac{\partial^{2}}{\partial s\partial t}(g)_{\alpha^{2}}(s,t). \end{aligned}$$
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
Obviously, if \(\varepsilon\rightarrow0\), then \(\delta\rightarrow0\). From the hypotheses, we obtain
$$\delta=g\bigl(se^{\varepsilon s^{-\alpha}},t\bigr)-g(s,t) . $$
$$\begin{aligned} \frac{\partial^{2}}{\partial s\partial t}(f\circ g)_{\alpha^{2}}(s,t) =&\frac{\partial}{\partial t} \biggl( \frac{\partial}{\partial s}\bigl(f\bigl(g(s,t)\bigr)\bigr)_{\alpha}(s,t) \biggr)_{\alpha}(s,t) \\ =&\frac{\partial}{\partial t} \biggl(\lim_{\varepsilon\rightarrow 0}\frac{f(g(se^{\varepsilon s^{-\alpha}},t))-f(g(s,t))}{ \varepsilon} \biggr)_{\alpha}(s,t) \\ =&\frac{\partial}{\partial t} \biggl(\lim_{\delta\rightarrow 0}\frac{f(g(s,t)+\delta)-f(g(s,t))}{\delta}\cdot \lim_{\varepsilon\rightarrow 0}\frac{\delta}{\varepsilon} \biggr)_{\alpha}(s,t) \\ =&\frac{\partial}{\partial t} \biggl(f'\bigl(g(s,t)\bigr) \frac{\partial}{\partial s}(g)_{\alpha}(s,t) \biggr)_{\alpha}(s,t) \\ =&f'\bigl(g(s,t)\bigr)\cdot\frac{\partial^{2}}{\partial s\partial t}(g)_{\alpha^{2}}(s,t)+ \frac{\partial}{\partial t}(g)_{\alpha}(s,t)\cdot\frac{\partial}{\partial t}\bigl(f'\bigl(g(s,t)\bigr)\bigr)_{\alpha}(s,t). \end{aligned}$$
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
Then
where
$$ z(s,t)= \int_{a}^{s} \int_{c}^{t}\frac{\partial^{2}}{\partial \sigma\partial \tau}(p)_{\alpha^{2}}( \sigma,\tau)\cdot\varphi \biggl(\frac{ \vert \frac{\partial^{2}}{\partial \sigma\partial \tau}(u)_{\alpha^{2}}(\sigma,\tau) \vert }{\frac{\partial ^{2}}{\partial \sigma\partial \tau}(p)_{\alpha^{2}}(\sigma,\tau)} \biggr)\, d_{\alpha} \sigma \, d_{\alpha}\tau. $$
(2.7)
$$\begin{aligned}& \int_{a}^{b} \int_{c}^{d} \biggl\{ \frac{\partial^{2}}{\partial s\partial t}(z)_{\alpha^{2}}(s,t)\cdot F' \biggl(p(s,t)\cdot \varphi \biggl( \frac{|u(s,t)|}{p(s,t)} \biggr) \biggr) \\& \qquad {}+\frac{\partial }{\partial t}(z)_{\alpha}(s,t)\cdot \frac{\partial}{\partial t}\bigl(F'\bigl(z(s,t)\bigr)\bigr)_{\alpha}(s,t) \biggr\} \, d_{\alpha}s\, d_{\alpha }t \\& \quad \leq F \biggl( \int_{a}^{b} \int_{c}^{d}\frac{\partial^{2}}{\partial s\partial t}(p)_{\alpha^{2}}(s,t) \cdot\varphi \biggl(\frac{ \vert \frac {\partial^{2}}{\partial s\partial t}(u)_{\alpha^{2}}(s,t) \vert }{\frac{\partial^{2}}{\partial s\partial t}(p)_{\alpha^{2}}(s,t)} \biggr)\, d_{\alpha}s\, d_{\alpha}t \biggr), \end{aligned}$$
(2.8)
$$\frac{\partial}{\partial t}\bigl(F'\bigl(z(s,t)\bigr)\bigr)_{\alpha}(s,t) =t^{1-\alpha} \frac{\partial}{\partial t}F'\bigl(z(s,t)\bigr). $$
Proof
Let
such that
and \(y(s,t)\geq|u(s,t)|\). Since φ is convex and increasing, by using Jensen’s inequality, we get
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
This completes the proof. □
$$y(s,t)= \int_{a}^{s} \int_{c}^{t} \biggl\vert \frac{\partial^{2}}{\partial s\partial t}(u)_{\alpha^{2}}( \sigma,\tau) \biggr\vert \, d_{\alpha}\sigma\, d_{\alpha}\tau $$
$$\frac{\partial^{2}}{\partial s\partial t}(y)_{\alpha^{2}}(s,t)= \biggl\vert \frac{\partial^{2}}{\partial s\partial t}(u)_{\alpha^{2}}(s,t) \biggr\vert $$
$$\begin{aligned} \varphi \biggl(\frac{ \vert u(s,t) \vert }{p(s,t)} \biggr) \leq& \varphi \biggl( \frac{y(s,t)}{p(s,t)} \biggr) \\ =& \varphi \biggl(\frac{\int_{a}^{s}\int_{c}^{t}\frac{\partial ^{2}}{\partial \sigma\partial \tau}(p)_{\alpha^{2}}(\sigma,\tau)\frac{ \vert \frac{\partial ^{2}}{\partial \sigma\partial \tau}(u)_{\alpha^{2}}(\sigma,\tau) \vert }{\frac{\partial ^{2}}{\partial \sigma\partial\tau}(p)_{\alpha^{2}}(\sigma,\tau)}\,d_{\alpha }\sigma \,d_{\alpha}\tau}{\int_{a}^{s}\int_{c}^{t}\frac{\partial ^{2}}{\partial \sigma\partial \tau}(p)_{\alpha^{2}}(\sigma,\tau)\,d_{\alpha}\sigma \,d_{\alpha}\tau } \biggr) \\ \leq&\frac{1}{p(s,t)} \int_{a}^{s} \int_{c}^{t}\frac{\partial ^{2}}{\partial \sigma\partial \tau}(p)_{\alpha^{2}}( \sigma,\tau)\cdot\varphi \biggl(\frac{ \vert \frac{\partial^{2}}{\partial \sigma\partial \tau}(u)_{\alpha^{2}}(\sigma,\tau) \vert }{\frac{\partial ^{2}}{\partial \sigma\partial \tau}(p)_{\alpha^{2}}(\sigma,\tau)} \biggr)\,d_{\alpha}\sigma \,d_{\alpha}\tau \\ =&\frac{1}{p(s,t)} \int_{a}^{s} \int_{c}^{t}\frac {\partial^{2}}{\partial \sigma\partial \tau}(p)_{\alpha^{2}}( \sigma,\tau)\cdot\varphi \biggl(\frac{\frac {\partial^{2}}{\partial \sigma\partial \tau}(y)_{\alpha^{2}}(\sigma,\tau)}{\frac{\partial^{2}}{\partial \sigma\partial \tau}(p)_{\alpha^{2}}(\sigma,\tau)} \biggr)\, d_{\alpha} \sigma \, d_{\alpha}\tau. \end{aligned}$$
(2.9)
$$\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}$$
Remark 2.1
Putting \(\varphi(x)=x\) in (2.7), we have
where
$$\begin{aligned}& \int_{a}^{b} \int_{c}^{d} \biggl\{ \biggl\vert \frac{\partial^{2}}{\partial s\partial t}(u)_{\alpha^{2}}(s,t) \biggr\vert \cdot F' \bigl(\bigl|u(s,t)\bigr|\bigr) \\& \qquad {}+\frac{\partial}{\partial t} (y)_{\alpha}(s,t)\cdot\frac{\partial}{\partial t}\bigl(F'\bigl(y(s,t)\bigr)\bigr)_{\alpha}(s,t)\biggr\} \, d_{\alpha}s\, d_{\alpha}t \\& \quad \leq F \biggl( \int_{a}^{b} \int_{c}^{d} \biggl\vert \frac{\partial ^{2}}{\partial s\partial t}(u)_{\alpha^{2}}(s,t) \biggr\vert \, d_{\alpha}s\, d_{\alpha}t \biggr), \end{aligned}$$
(2.10)
$$y(s,t)= \int_{a}^{s} \int_{c}^{t} \biggl\vert \frac{\partial^{2}}{\partial s\partial t}(u)_{\alpha^{2}}( \sigma,\tau) \biggr\vert \, d_{\alpha}\sigma \, d_{\alpha}\tau. $$
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
Then
where
and
$$ \frac{\partial^{2}}{\partial s\partial t}(F\circ z)_{\alpha ^{2}}(s,t)\cdot\phi \biggl( \frac{1}{\frac{\partial^{2}}{\partial s\partial t}(z)_{\alpha^{2}}(s,t)} \biggr)\leq \frac{F(z(a,b))}{z(a,b)}\cdot\phi' \biggl( \frac{t}{z(a,b)} \biggr). $$
(2.11)
$$\begin{aligned}& \int_{0}^{a} \int_{0}^{b} \biggl\{ \psi\biggl(\frac{\partial^{2}}{\partial s\partial t}(p)_{\alpha^{2}}(s,t) \cdot\varphi \biggl(\frac{ \vert \frac{\partial^{2}}{\partial s\partial t}(u)_{\alpha^{2}}(s,t) \vert }{\frac{\partial^{2}}{\partial s\partial t}(p)_{\alpha^{2}}(s,t)} \biggr) \biggr)\cdot F' \biggl(p(s,t)\cdot\varphi \biggl( \frac{|u(s,t)|}{p(s,t)} \biggr) \biggr) \\& \qquad {} +\psi \biggl(\frac{\partial^{2}}{\partial s\partial t}(z)_{\alpha^{2}}(s,t) \biggr)\cdot \frac{\partial}{\partial t}\bigl(F'\bigl(z(s,t)\bigr)\bigr)_{\alpha}(s,t) \cdot\frac{\frac{\partial}{\partial t}(z(s,t))_{\alpha}(s,t)}{\frac{\partial^{2}}{\partial s\partial t}(z)_{\alpha^{2}}(s,t)} \biggr\} \, d_{\alpha}s\, d_{\alpha}t \\& \quad \leq\varPhi \biggl( \int_{0}^{a} \int_{0}^{b}\frac{\partial ^{2}}{\partial s\partial t}(p)_{\alpha^{2}}(s,t) \cdot\varphi \biggl(\frac{ \vert \frac {\partial^{2}}{\partial s\partial t}(u)_{\alpha^{2}}(s,t) \vert }{\frac{\partial^{2}}{\partial s\partial t}(p)_{\alpha^{2}}(s,t)} \biggr)\, d_{\alpha}s\, d_{\alpha}t \biggr), \end{aligned}$$
(2.12)
$$ \psi(r)=rh \biggl(\phi \biggl(\frac{1}{r} \biggr) \biggr), $$
(2.13)
$$ \varPhi(r)=F(r)\cdot h \biggl(\frac{1}{r} \int_{0}^{a} \int_{0}^{b}\phi ' \biggl( \frac{t}{r} \biggr)\, d_{\alpha}s\, d_{\alpha}t \biggr). $$
(2.14)
Proof
From (2.9), we have
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
This completes the proof. □
$$ \varphi \biggl(\frac{ \vert u(s,t) \vert }{p(s,t)} \biggr)\leq\frac {z(s,t)}{p(s,t)}. $$
(2.15)
$$\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}$$
Remark 2.2
Putting \(\varphi(x)=x\) in (2.12), we have
where
$$\begin{aligned}& \int_{0}^{b}\psi \biggl( \biggl\vert \frac{\partial^{2}}{\partial s\partial t}(u)_{\alpha^{2}}(s,t) \biggr\vert \biggr)\cdot F' \bigl(\bigl|u(s,t)\bigr| \bigr)\, d_{\alpha}s\, d_{\alpha}t \\& \quad \leq\varPhi \biggl( \int_{0}^{a} \int_{0}^{b} \biggl\vert \frac{\partial ^{2}}{\partial s\partial t}(u)_{\alpha^{2}}(s,t) \biggr\vert \, d_{\alpha}s\, d_{\alpha}t \biggr)-N_{\alpha}(a,b), \end{aligned}$$
(2.16)
$$N_{\alpha}(a,b)=\int_{0}^{a}\int_{0}^{b}\psi \biggl(\frac{\partial^{2}}{\partial s\partial t}(z)_{\alpha^{2}}(s,t)\biggr)\cdot\frac{\partial}{\partial t}\bigl(F'\bigl(z(s,t)\bigr)\bigr)_{\alpha}(s,t)\cdot\frac{\frac{\partial}{\partial t}(z)_{\alpha}(s,t)}{\frac{\partial^{2}}{\partial s\partial t}(z)_{\alpha^{2}}(s,t)}\,d_{\alpha}s\,d_{\alpha}t. $$
This inequality (2.16) is just a two-dimensional generalization of the following inequality which was established in [21]:
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 \([0,\infty)\).
$$\begin{aligned}& \int_{0}^{b}\psi \biggl(D_{\alpha}p(t) \cdot\varphi \biggl(\frac { \vert D_{\alpha}u(t) \vert }{D_{\alpha}p(t)} \biggr) \biggr)\cdot F' \biggl(p(t)\cdot\varphi \biggl(\frac{|u(t)|}{p(t)} \biggr) \biggr)\, d_{\alpha}t \\& \quad \leq\varPhi \biggl( \int_{0}^{b}D_{\alpha}p(t)\cdot\varphi \biggl(\frac { \vert D_{\alpha}u(t) \vert }{ D_{\alpha}p (t)} \biggr)\, d_{\alpha}t \biggr), \end{aligned}$$
Acknowledgements
The first author expresses his gratitude to professor G. Leng and W. Li for their valuable helps.
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